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Comprehensive High Frequency Electron Paramagnetic Resonance Studies of Single Molecule Magnets

Permanent Link: http://ufdc.ufl.edu/UFE0021532/00001

Material Information

Title: Comprehensive High Frequency Electron Paramagnetic Resonance Studies of Single Molecule Magnets
Physical Description: 1 online resource (202 p.)
Language: english
Creator: Lawrence, Jonathan D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: epr, magnetic, molecules, nanoparticles, spectroscopy
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation presents research on a number of single molecule magnet (SMM) compounds conducted using high frequency, low temperature magnetic resonance spectroscopy of single crystals. By developing a new technique that incorporated other devices such as a piezoelectric transducer or Hall magnetometer with our high frequency microwaves, we were able to collect unique measurements on SMMs. This class of materials, which possess a negative, axial anisotropy barrier, exhibit unique magnetic properties such as quantum tunneling of a large magnetic moment vector. There are a number of spin Hamiltonians used to model these systems, the most common one being the giant spin approximation. Work done on two nickel systems with identical symmetry and microenvironments indicates that this model can contain terms that lack any physical significance. In this case, one must turn to a coupled single ion approach to model the system. This provides information on the nature of the exchange interactions between the constituent ions of the molecule. Additional studies on two similar cobalt systems show that, for these compounds, one must use a coupled single ion approach since the assumptions of the giant spin model are no longer valid. Finally, we conducted a collection of studies on the most famous SMM, Mn12Ac. Three different techniques were used to study magnetization dynamics in this system: stand-alone HFEPR in two different magnetization relaxation regimes, HFEPR combined with magnetometry, and HFEPR combined with surface acoustic waves. All of this research gives insight into the relaxation mechanisms in Mn12Ac.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan D Lawrence.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hill, Stephen O.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021532:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021532/00001

Material Information

Title: Comprehensive High Frequency Electron Paramagnetic Resonance Studies of Single Molecule Magnets
Physical Description: 1 online resource (202 p.)
Language: english
Creator: Lawrence, Jonathan D
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: epr, magnetic, molecules, nanoparticles, spectroscopy
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation presents research on a number of single molecule magnet (SMM) compounds conducted using high frequency, low temperature magnetic resonance spectroscopy of single crystals. By developing a new technique that incorporated other devices such as a piezoelectric transducer or Hall magnetometer with our high frequency microwaves, we were able to collect unique measurements on SMMs. This class of materials, which possess a negative, axial anisotropy barrier, exhibit unique magnetic properties such as quantum tunneling of a large magnetic moment vector. There are a number of spin Hamiltonians used to model these systems, the most common one being the giant spin approximation. Work done on two nickel systems with identical symmetry and microenvironments indicates that this model can contain terms that lack any physical significance. In this case, one must turn to a coupled single ion approach to model the system. This provides information on the nature of the exchange interactions between the constituent ions of the molecule. Additional studies on two similar cobalt systems show that, for these compounds, one must use a coupled single ion approach since the assumptions of the giant spin model are no longer valid. Finally, we conducted a collection of studies on the most famous SMM, Mn12Ac. Three different techniques were used to study magnetization dynamics in this system: stand-alone HFEPR in two different magnetization relaxation regimes, HFEPR combined with magnetometry, and HFEPR combined with surface acoustic waves. All of this research gives insight into the relaxation mechanisms in Mn12Ac.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jonathan D Lawrence.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Hill, Stephen O.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021532:00001


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918e34cc16fa0035e9f70eaa4fbb980d
a0cd2b294c23796274cce2821a0f06f718935157







COMPREHENSIVE HIGH FREQUENCY ELECTRON PARAMAGNETIC RESONANCE
STUDIES OF SINGLE MOLECULE MAGNETS




















By

JONATHAN D. LAWRENCE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007







































O 2007 Jonathan D. Lawrence

































To my family









ACKNOWLEDGMENTS

This thesis would not have been possible without the help and guidance of a number of

people. First I need to express my thanks to my advisor, Dr. Stephen Hill. For the past 4.5 years

Steve has provided a world class environment for cutting edge research in physics. He has

always been a strong mentor, guiding me toward challenging but rewarding proj ects. I have also

had the privilege to travel to numerous conferences to present my work and interact with my

peers in the same Hield of research. Steve has Einancially supported all of these endeavors, and

for that, I am quite grateful. I am truly indebted to Steve for teaching me how to be a

professional scientist, and giving me the opportunity to learn what being an independent

researcher is all about.

I would also like to thank my other committee members, Dr. George Christou, Dr. Mark

Meisel, Dr. Yasumasa Takano, and Dr. Selman Hershfield, for overseeing the completion of my

research work.

I wish to express my deepest gratitude to everyone in the machine shop here in the physics

department: Marc Link, Bill Malphurs, Ed Storch, Skip Frommeyer, Mike Herlevich, and John

Van Leer. Every experiment that I conducted used equipment built or modified by the machine

shop. They always provided helpful design suggestions and world class craftsmanship on every

project. Without the help of these kind, talented gentlemen, none of the research presented in

this dissertation would have been possible.

I am grateful to the technical staff in the electronics shop (Larry Phelps, Pete Axson, and

Rob Hamersma) for helpful discussions, advice, and assistance with regards to any electronic

equipment problems or design issues. They were always willing to help, and contributed

significantly to a number of my proj ects.










I want to thank the cryogenic engineers, Greg Labbe and John Graham, who are

responsible for the recovery and re-liquification of helium, as well as all the aspects of cryogenic

equipment implemented in our lab. Throughout my research there were many technical

challenges dealing with low temperature equipment. They were vastly knowledgeable and

always willing to help with any matter. Their generous assistance is much appreciated.

I want to thank Dr. Enrique del Barco and Dr. Chris Ramsey for their generous help in

fabrication of the Hall magnetometers used for certain experiments. They allowed the use of

their facilities and were patient enough to go through the fabrication process twice so that my

experiment could be successful. For this, I am deeply indebted.

For a great collaborative effort with a unique experiment, I want to thank Ferran Macia

from the Tej ada group at the University of Barcelona. This work was some of the best I have

done as a graduate student, and he was a large part of the success.

I wish to thank Christos Lampropoulos from the Christou group in the chemistry

department here at the University of Florida. He supplied the Mnl2Ac samples used in numerous

experiments.

I wish to thank En-Che Yang and Chris Beedle of the Hendrickson group in the chemistry

department at the University of California at San Diego. They supplied Ni4, NiZn, Co4, and

CoZn samples used in many studies.

I want to recognize the UF Alumni Association for providing me with a $500 scholarship

in order to travel to Denver, CO for the APS 2007 meeting where I was able to network with

scientists working in similar areas of research and present some of my work to a diverse

audience.









Finally, I want to thank all of the wonderful friends I have made during my time here. The

few hours each week not spent doing physics were made all the more enj oyable by you.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....

LIST OF FIGURES .............. ...............9.....

AB S TRAC T ........._. ............ ..............._ 12...

CHAPTER

1 INTRODUCTION TO SINGLE MOLECULE MAGNETS .............. ....................1


1.1 Basic Properties of Single Molecule Magnets ......._............. ._ ........._._.....14
1.2 The Magnetic Anisotropy Barrier in Single Molecule Magnets............... ................15
1.3 The Role of Magnetic Anisotropy .....___ ................ .. .. .. ......... ...... 1
1.4 Magnetic Hysteresis and Magnetic Quantum Tunneling............... ...............1
1.4.1 Zero Field Tunneling ............... ...............20....
1.4.2 Magnetic Field Induced Tunneling ........__............._. ............... 21.....
1.5 Applications ....._.. ................ ......._.. ..........2
1.6 Summary ................. ...............29........ ......

2 EXPERIMENTAL INS TRUMENTATION AND TECHNIQUE S ................ ................. .3 6

2.1 M easurement Techniques ............. ........... ... ... ....... ....................3
2.2 Electron Paramagnetic Resonance in the Context of Single Molecule Magnets.........36
2.3 Cavity Perturbation ................ ...............37........... ....
2.3.1 Technical Challenges ................ ...............37........... ....
2.3.2 Equipment ................. ...............38.................
2.4 Quasi Optical Setup .............. .... ...............45..
2.4.1 Piezoelectric Transducer Device ................. ...............51................
2.4.2 Hall Magnetometer Device ................. ...............55................
2.5 Hall Magnetometer Fabrication ................. ....__. ...............57. ....
2.6 Summary ........._.__............ ...............62.....

3 THEORETICAL BASIS OF THE SPIN HAMILTONIAN ................. .................7


3.1 Two Versions of the Spin Hamiltonian ................ ........ ......... ................75
3.1.1 The Giant Spin Hamiltonian ................ ...............76........... ...
3.1.2 Coupled Single lon Hamiltonian .............. ...............81....
3.2 Summary ................. ...............85.......... .....

4 CHARACTERIZATION OF DISORDER AND EXCHANGE INTERACTIONS IN
[ Ni(hmp) (d mb) Cl]4 ............ ...... ._ __ ...............86.....

4.1 The Tetranuclear Single Molecule Magnet [Ni(hmp)(dmb)Cl]4 .............. .................86
4.2 HFEPR Measurements of [Ni(hmp)(dmb)Cl]4 ................. ............... ......... ...88











4.2.1 Characterization of Easy Axis Data ................. ...............89...............
4.2.2 Peak Splittings Arising from Disorder............... .. ...............9
4.2.3 Peak Splittings Arising from Intermolecular Exchange .................. ...............92
4.3 Physical Origin of the Fast QTM in [Ni(hmp)(dmb)Cl]4 ............_.._ .....................95
4.4 Measuring the Exchange Interaction with HFEPR ....._____ .........__ ................99
4.5 Summary ................. ...............103......... ......

5 HFEPR CHARACTERIZATION OF SINGLE Co (II) IONS IN A TETRANUCLEAR
COM PLEX ................. ...............116......... ......

5.1 Introduction to the [Co(hmp)(dmb)Cl]4 and [Zn3CO(hmp)4(dmb)4C 4] COmplexes..116
5.2 HFEPR Measurements of [Zn3CO(hmp)4(dmb)414]............__.. ......___........._117
5.2.1 The [Zn3CO(hmp)4(dmb)4C 4] COmplex ....___ .............. ............... 117
5.2.2 Frequency and Temperature Dependent Measurements ..............._ ...............117
5.2.3 Angle Dependent Measurements .......__ ......... ___ ........_.. ........11
5.3 HFEPR Measurements of [Co(hmp)(dmb)Cl]4 .............. ...............122....
5.3.1 Measurements Along the Crystallographic c Axis .............. .....................2
5.3.2 Discussion of Spectra Within the ab Plane ......___ ........... ..............125
5.4 Spin Hamiltonian for the Tetranuclear system .............. ...............125....

6 HFEPR STUDIES OF MAGNETIC RELAXATION PROCESSES IN Mnl2Ac ...............139

6.1 Introduction to M nl2Ac ............... ... .......... ... ...............139...
6.1.1 Theory and Effects of Disorder in Mnl2Ac ......... ................. ............... 140
6. 1.2 Dipolar and Hyperfine Field Relaxation Mechanisms ................. .................1 43
6.1.3 High and Low Temperature Relaxation Regimes............... .................4
6.2 QTM Studied by HFEPR ................. ..... .......... .. ... .. ........ .......14
6.2.1 Experiment in Stretched Exponential Relaxation Regime. .............. ..... ..........147
6.2.2 Experiment in the t'/ Relaxation Regime ................. ........_ ................1 50
6.2.2a Low Temperature MQT .............. ...............151....
6.2.2b Magnetic Avalanches ................._.._.._ ......... ............15
6.2.3 Characterization of Minority Species ................. ...... .. ............15
6.3 Microwave Induced Tunneling Measured with Hall Magnetometry ................... ......156
6.4 Relaxation in Mnl2Ac Measured with HFEPR ....._____ .........__ .............. .161
6.4.1 Triggered Avalanches in Mnl2Ac ....__ ......_____ .......___ ............6
6.4.2 Pulsed Heating and Spin Lattice Relaxation............... ..............16
6.5 Summary ............ _...... ._ ...............170...

7 SUMM ARY ............ ..... .._ ...............190...

REFERENCE S .............. ...............194....

BIOGRAPHICAL SKETCH .............. ...............202....










LIST OF FIGURES


Figure page

1-1 M olecule of M nl2Ac. ............. ...............31.....

1-2 Energy barrier for a molecule of Mnl2Ac in zero field. ............. .......... ................32

1-3 Energy barrier for a molecule of Mnl2Ac with an external magnetic field applied
parallel to the easy axis. ............. ...............33.....

1-4 Low temperature hysteresis loop for a single crystal of Mnl2Ac with an external
magnetic field applied parallel to the easy axis. ............ ...............34.....

1-5 Two energy levels in a system of Mnl2Ac as they pass through the first non zero
resonance field. ............. ...............35.....

2-1 Energy levels in the Ni4 SMM with its easy axis aligned along the external field. .........64

2-2 Normal EPR spectrum for the Ni4 SMM. ............. ...............65.....

2-3 Typical experimental setup. ............. ...............66.....

2-4 Rotational capabilities of each magnet system. ................ ...............67........... ..

2-5 Free space Gaussian beam. ............. ...............68.....

2-6 Quasi optics equipment. ............ ...............69.....

2-7 Signal polarization as it changes from interactions with the respective components of
the quasi optical setup. ................ ...............70.......... .....

2-8 TE11 and HE11 modes in a circular, corrugated waveguide. ................ ........___.........._71

2-9 Piezoelectric device used in our experiments. ................ ...............72........... ..

2-10 Electronic equipment used in our avalanche experiments. ..........._... ......... ............73

2-11 Hall device used for our magnetometry measurement. ............ ...............14.....

4-1 Molecule of [Ni(hmp)(dmb)Cl]4.. ........... ...............105.....

4-2 172.2 GHz HFEPR spectra of [Ni(hmp)(dmb)Cl]4.fOr different temperatures. .................106

4-3 Peak positions of [Ni(hmp)(dmb)Cl]4.in magnetic field for different frequencies. ...........107

4-4 Influence of the spacing between the resonance branches due to a negative, axial,
fourth order anisotropy term. ............ ...............108.....











4-5 Peak splitting at 172.2 GHz in [Ni(hmp)(dmb)Cl]4. ................ ................. ....__. 109

4-6 Heat capacity measurements of [Ni(hmp)(dmb)Cl]4. ................ ......__ ................. 110

4-7 Comparison of the structure of [Ni(hmp)(dmb)Cl]4 at 173 K and 12 K. ........._._...............11 1

4-8 Temperature dependence of the peak splitting in Ni(hmp)(dmb)Cl]4.at a given
magnetic field for three frequencies. ........._._. ......__. ....__ ............1

4-9 Simulation of four coupled s = 1 spins. ................ ......... ...............113 ..

4-10 Expanded view of data obtained at a frequency of 198 GHz in the range from 4 T to
9 T. .......... ..... __ ...............114...

4-11 Temperature dependence of the intensity of transition B observed at 172 GHz. ...............1 15

5-1 Peak position of [Zn0.995COO.005(hmp)(dmb)Cl]4.aS a function of frequency. ............_.._. ...13 1

5-2 Temperature dependence of [Zn0.995COO.005(hmp)(dmb)Cl]4. ............ .....................13

5-3 Peak position as a function of angle in [Zn0.995COO.005(hmp)(dmb)Cl]4for two planes of
rotation. ............ ...............133.....

5-4 Magnetic core of a CoZn molecule. ............ ...............134.....

5-5 Temperature dependence of the peaks in [Co(hmp)(dmb)Cl]4 for two different
frequencies. ........... ...............135.....

5-6 Easy axis frequency dependence of the resonance peaks in [Co(hmp)(dmb)Cl]4 at 2 K. ..136

5-7 Data taken from [Co(hmp)(dmb)Cl]4 with the field aligned within the ab plane of the
crystal. ........._ ...... ...............137....

5-8 Simulation of the Co4 System for a frequency of 501 GHz at 30 K with the field aligned
along the c axis. ............. ...............138....

6-1 Six different Mnl2 isomers. ........... ...............172.....

6-2 Energy levels in Mnl2Ac as a function of magnetic field. ................ ....._.. .............173

6-3 2 K EPR spectra for different waiting times at 0.9 T. ............ ...............174.....

6-4 Area of the positive field peak as a function of wait time. ........... .... .._. ............175

6-5 Emerging resonance peak for different wait times at 1.8 T and sweeping back to -6 T. ..176

6-6 Peak area in Fig. 6-5 as a function of wait time. ............ ...............177.....










6-7 Spectrum for a wait time of 2400 s after fitting it to a simulation that combined two
different Gaussian peaks. ............ ...............178.....

6-8 Average D value and peak width vs. time. ............. ...............179....

6-9 Spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to -6 T from
waiting for 600 s at different magnetic fields. ........... ...............180.....

6-10 Minority species molecules in Mnl2Ac. ........... ...............181.....

6-1 1 Low temperature frequency dependence of the minority species peak. ........._...._ ............182

6-12 Hysteresis loops taken under the influence of a number of different microwave
frequencies. ............ ...............183.....

6-13 Percentage of magnetization reversal for the step at 0.5 T under the influence of 286
GHz microwave radiation for different duty cycles. ........... ...............184.....

6-14 Difference in the magnetization reversal for the data sets taken with and without
microwaves as a function of wait time at 0.5 T. ........... ...............185.....

6-15 Energy barrier diagram illustrating how spins move during an avalanche. ......................186

6-16 Transitions within the metastable well during an avalanche. ................ .........._. .......187

6-17 Transitions within the stable well during an avalanche. ............. ...............188....

6-18 EPR signal as a function of time for the ms = -9 to -8 transition. ............_ ..........._..... 189









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPREHENSIVE HIGH FREQUENCY ELECTRON PARAMAGNETIC RESONANCE
STUDIES OF SINGLE MOLECULE MAGNETS

By

Jonathan D. Lawrence

December 2007

Chair: Stephen O. Hill
Major: Physics

This dissertation presents research on a number of single molecule magnet (SMM)

compounds conducted using high frequency, low temperature magnetic resonance spectroscopy

of single crystals. By developing a new technique that incorporated other devices such as a

piezoelectric transducer or Hall magnetometer with our high frequency microwaves, we were

able to collect unique measurements on SMMs. This class of materials, which possess a

negative, axial anisotropy barrier, exhibit unique magnetic properties such as quantum tunneling

of a large magnetic moment vector.

There are a number of spin Hamiltonians used to model these systems, the most common

one being the giant spin approximation. Work done on two nickel systems with identical

symmetry and microenvironments indicates that this model can contain terms that lack any

physical significance. In this case, one must turn to a coupled single ion approach to model the

system. This provides information on the nature of the exchange interactions between the

constituent ions of the molecule. Additional studies on two similar cobalt systems show that, for

these compounds, one must use a coupled single ion approach since the assumptions of the giant

spin model are no longer valid.









Finally, we conducted a collection of studies on the most famous SMM, Mnl2Ac. Three

different techniques were used to study magnetization dynamics in this system: stand-alone

HFEPR in two different magnetization relaxation regimes, HFEPR combined with

magnetometry, and HFEPR combined with surface acoustic waves. All of this research gives

insight into the relaxation mechanisms in Mnl2Ac.









CHAPTER 1
INTRODUCTION TO SINGLE MOLECULE MAGNETS

1.1 Basic Properties of Single Molecule Magnets

Progressive research over the last decade in the inorganic chemistry community has

allowed the synthesis of an exciting class of materials with unique magnetic properties [1].

These new materials were given the name "single molecule magnets" (SMMs) for reasons that

will soon be clear. All SMMs have transition metal ions such as Fe, Mn, Ni and Co as the source

of their magnetic properties. The magnetic core of each complex comprises multiple ions with

unpaired electrons, which are strongly coupled to each other through intramolecular exchange

interactions isotropicc being the most dominant). This isotropic Heisenberg exchange interaction

is expressed as


Hs,, =C JLS2 Sj (1-1)


In Eq. 1-1, J,, is the magnitude of the isotropic Heisenberg interaction (positive for

antiferromagnetic coupling and negative for ferromagnetic coupling) between spin i and spin j. S

represents the spin operator for an individual ion within the molecule. This coupling leads to a

large magnetic moment for each molecule, and separates the energy spectrum into respective

spin multiplets. For traditional SMM systems the isotropic exchange interaction is sufficient to

isolate the ground state spin multiple from higher lying multiplets. The ground state multiple is

then modeled as a state with a perfectly rigid magnetic moment vector. Intramolecular exchange

is the dominant interaction between ions within a molecule, and the bulky organic ligands that

surround the magnetic core serve to isolate each molecule from surrounding neighbors. Thus,

intermolecular exchange interactions are rather weak and there is no long range ordering. Each

molecule can be considered as an independent magnetic nanocluster possessing a large magnetic









moment. Additionally, to a first approximation, all molecules within a crystalline sample are

identical. Since each molecule behaves like an isolated magnetic moment, the term single

molecule magnet was appropriately coined for these compounds [2]. Fig. 1-1 illustrates a

molecule of the most famous SMM, [Mnl2012(CH3COO) 16(H20)4]- 2CH3COOH-4H20, hereafter

Mnl2Ac. Eight Mn+3 ions and four Mn+4 ions are antiferromagnetically coupled through the

oxygen atoms, giving rise to a large S = 10 ground state for the molecule. The ligands that

surround the magnetic core minimize the interactions that each molecule experiences from its

neighbors by increasing the distance between effective dipole centers (~ 14 A+ for this particular

compound [3]). We present studies done on this SMM in Ch. 6.

1.2 The Magnetic Anisotropy Barrier in Single Molecule Magnets

The essential feature of all SMM systems is their significant negative axial anisotropy

that creates a barrier to reversal of the magnetization vector. From a quantum mechanical

perspective, the Hamiltonian in its simplest form can be expressed [1] as


H = DSz (1-2)

In Eq. 1-2, D represents the dominant, axial anisotropy of the molecule which must be negative

for any SMM. Sz is the spin operator for the magnetic moment of the molecule. For simplicity

we will consider each state to have no orbital angular momentum contribution and can therefore

be expressed as a pure spin multiple. The wavefunction for each energy state can be expressed

as the spin proj section, m, ). The number of spin projections (energy states) for any molecule is

given by 2S + 1, where S is the spin ground state of the molecule. In the absence of any

transverse anisotropies or transverse fields, energy states with equal but opposite spin proj sections

are degenerate in zero magnetic field. From the Hamiltonian in Eq. 1-2 we obtain the energy

eigenvalues for each eigenstate:










Ei~ms)= Dmzs2 (1-3)

where ms is the spin proj section along the z axis. The ground state for the system is the largest

magnitude of the spin proj section (ms = AS), since D is negative. The height of the energy barrier

in Fig. 1-2, and given by Eq. 1-4, is governed by both the magnitude of D and the magnitude of

the spin ground state multiple, S.

AE= D S2 (1-4)

This can be pictured as an inverse parabola potential, as shown in Fig. 1-2, which represents the

energy barrier for a S = 10 system like Mnl2Ac or the Fes SMM [1]. This dominant axial

anisotropy allows one to define a quantization axis, traditionally labeled as the z axis, for the

energy levels of the system, which are quantized. The energy barrier makes the magnetic

moment bi-stable due to the fact that energetically it prefers to point in one direction (+z) or a

direction that is antiparallel to the first (-z).

The above mentioned source of anisotropy occurs in the absence of an external magnetic

field. Any phenomenon such as this is known as zero field splitting, referring to the fact that the

energy degeneracy is lifted in the absence of any external field [4]. However, anisotropies can

manifest themselves in the presence of an external field as well, in terms of the Zeeman

interaction. In an external magnetic field, the Zeeman energy of an electron now depends on the

orientation of the field with respect to its spin projection along the field. The energy of the

Zeeman interaction is related to the strength of the external magnetic field by a proportionality

factor [5], g. Eq. 1-5a gives the general expression for the Zeeman energy


Ezeeman = IUBB J -S (1-5a)

EZeeman = -E u,BBms (1-5b)









where pBg is the Bohr magneton, B is the external field strength, g is the Lande tensor and S is the

spin operator for the molecule. For a free electron g is a scalar (= 2 ), but in a molecule g

becomes a tensor due to spin orbit induced anisotropies. In equation Eq. 1-5b we give the

expression for the case of the magnetic field aligned with the spin vector of the molecule, where

g, is the component of the Lande tensor along the external field direction and ms is the proj section

of the spin vector along the external field direction.

The zero field mechanism mentioned above contributes to anisotropic electron

distributions among electronic orbitals [6]. This manifests itself in the lifting of degeneracies of

energy levels through energy splitting. Measuring the energy splitting with high frequency

electron paramagnetic resonance (HFEPR) spectroscopy is a way to probe the anisotropies

present in the material of interest.

1.3 The Role of Magnetic Anisotropy

Anisotropy plays a fundamental role in the magnetic properties of all SMM systems. Since

they comprise multiple transition metal ions, SMMs constitute an exchange coupled system. The

anisotropy can come from many sources, including spin-spin dipolar and exchange coupling of

electrons, hyperfine interactions of electrons with the nuclei of the constituent atoms of the

molecule, and most notably spin orbit coupling of the electrons. All of these sources can lead to

anisotropic electron distributions on the molecule. This coupling is the dominant mechanism

responsible for the essential feature of SMMs, which is their negative axial anisotropy that

creates a barrier to magnetization reversal. Spin orbit coupling results from the interaction of

one electron's orbital angular momentum with its own spin angular momentum [7]. The orbital

momentum of an electron creates a magnetic field, which will couple to the spin magnetic

moment. This type of coupling is also present between the spin of the electron and the magnetic

field of a proton, which, from the electron's perspective, constitutes an orbital momentum. In









this scenario, the individual moment are not separately conserved. The sum of the two

moment, however, is conserved.

One source of anisotropy is the Jahn-Teller elongations of electronic orbitals. Such a

distortion of molecular orbitals of the Mn+3 ions is the significant contribution to the axial

anisotropy in the Mnl2Ac SMM system. The Jahn-Teller theorem states that "for a non-linear

molecule in an electronically degenerate state, a distortion must occur to lower the symmetry, to

remove the degeneracy, and lower the energy [8]." An atom in free space with a symmetric

distribution of electronic orbitals will have no anisotropy and, thus, the energy levels of a certain

orbital will be degenerate. But atoms in a molecule will have their molecular orbitals distorted.

This distortion can be a geometric compression or elongation and it causes degeneracies to be

lifted. In the case of a compression, the sign of the axial anisotropy parameter, D, is positive and

conversely, D is negative in the case of an elongation [9]. If the distortion occurs along one axis

only, then only axial anisotropies will develop. If the distortion occurs along multiple axes, then

both axial and transverse anisotropies will develop.

Spin-spin dipolar coupling between electrons arises from the interaction of one electron's

spin in another electron's dipolar field. Similar interactions between electrons and nuclei can

give rise to hyperfine couplings as well. The nuclei in most materials have magnetic moments

which couple to the orbital and spin angular momentum of the electrons. However, in exchange

coupled systems, like SMMs, the delocalization of electrons throughout the molecule usually

makes this a weak effect [3]. Although, we will show in Ch. 6 that the interactions between a

molecule and the fields from nuclear moments as well as other molecules play an important role

in the tunneling process.









Exchange interactions are classified by many different types, such as direct exchange,

indirect exchange, superexchange, itinerant exchange, and double exchange [10]. The common

feature for all of these is a weak bond between magnetic moment centers within the molecule

[l l, 12] which can be spread over distances on the order of a few A+. These exchange

interactions can manifest as both isotropic and anisotropic quantities. For the systems discussed

in this dissertation the dominant type of exchange is superexchange, which is the coupling of

localized magnetic moments in insulating materials through diamagnetic groups. In many SMM

systems the isotropic Heisenberg interaction, given by Eq. 1-1, is of a much larger magnitude

than the anisotropic interactions or any anisotropic or antisymmetric exchange. Under these

circumstances dipolar interactions, anisotropic exchange, and antisymmetric exchange are

usually ignored. However, in Ch. 5 we will discuss a system where both anisotropic and

antisymmetric exchange are considered.

1.4 Magnetic Hysteresis and Magnetic Quantum Tunneling

In the absence of any transverse anisotropies or transverse fields, energy states with equal

but opposite spin proj sections are degenerate in zero magnetic field. Application of an external

magnetic field shifts the energy levels of the potential well with respect to one another. At low

temperatures (kBT<< DS ) where only the ground state of a molecule is significantly populated,

one can observe magnetic hysteresis. Many studies have been done reporting magnetic

hysteresis loops in various systems [13, 14]. In most systems, magnetic hysteresis occurs due to

the formation of domain boundaries within the lattice. The exchange interaction between spins

is shorter ranged than the dipolar interactions, and the magnitude of the dipolar interactions can

become significant in bulk systems where large numbers of spins are involved [15]. The dipolar

energy can be reduced by dividing the system into uniformly magnetized domains, with each

domain having a different direction. This costs energy in terms of the exchange interaction due









to the fact that the individual spins within a domain will have their energy increased by spins in

neighboring domains with different orientations. However, this energy cost is minimal since

only those spins at the domain walls have this increase in energy and hence this effect is short

ranged. Conversely, the dipolar interaction is longer, and consequently the dipolar energy of

every spin in the system is lowered. Thus it is energetically favorable for the system to adopt

this configuration.

However, in SMM systems it is the anisotropy barrier of each individual molecule that is

responsible for the observed hysteresis, in contrast to the collective effect of the dipolar

interaction in domain wall formation. At low temperatures, the anisotropy barrier makes it

energetically favorable for all the spins to populate the ground state of the system. The barrier

prevents spins from directly reversing their spin state proj section from parallel to antiparallel with

respect to the quantization axis.

Perhaps most interesting is the appearance of steps in the hysteresis loops, which is an

indication of quantum tunneling of the magnetic moment through the anisotropy barrier [16].

After biasing the system with a magnetic Hield such that all the spins are in one of the wells, it is

possible for the spins to tunnel through the barrier at resonance Hields. By measuring the

magnetization of a single crystal, it is possible to observe drastic changes in the magnetization as

the external Hield is swept from a large biasing Hield through resonance fields. At resonance

Shields the magnetization is seen to have steps where it moves toward the opposite saturation

value. This indicates that spins are reversing their proj section state by tunneling through the

energy barrier at these resonance fields.

1.4.1 Zero Field Tunneling

For magnetic quantum tunneling to occur, there need to exist terms that break the axial

symmetry and mix the energy states on opposite sides of the energy barrier, such as transverse









anisotropies and transverse magnetic fields, both internal and external [17]. Transverse

anisotropies arise due to symmetries of the molecules, which is discussed in detail in Ch. 3.

Internal transverse fields arise due to dipolar interactions between neighboring molecules and

fields due to nuclear magnetic moments. External transverse fields arise due to misalignment of

the sample' s easy axis with respect to the magnetic field. All of these can be expressed in the

Hamiltonian as terms that do not commute with S,, and these off diagonal terms cause mixing

between states. As an example, a second order transverse anisotropy can be written

as~ E(-) nytrm such~ as1 this can causel the wavefunctio+;r nsl for the states~ on each sie~l of


the energy barrier to become mixed and extend to the opposite side of the energy barrier.

Without a transverse anisotropy, states with equal but opposite spin proj sections are

completely degenerate, but with such a term there is an energy difference between the new

states, which are symmetric and antisymmetric combinations of the unmixed states. This energy

difference is known as the tunnel splitting (A), and since the wavefunctions become mixed into

linear superposition states there is a probability for the proj section of the spin vector to be

measured in either state.

1.4.2 Magnetic Field Induced Tunneling

An external magnetic field will bias the energy levels with respect to each other, and at

certain magnetic field values, energy states with different and opposite spin proj sections can

become degenerate due to the Zeeman interaction. These are known as resonance fields and as

shown in Fig. 1-3, these are the non zero fields where magnetic quantum tunneling occurs [18].

With the magnetic field applied parallel to the easy magnetization axis of the molecule (B

parallel to z) and considering only the dominant second order term, D, and the Zeeman term, the

spin Hamiltonian will be










Hs D; +g B B- (1-6)

The energies given by Eq. 1-6 are

E = Dms" + g pBmS (1-7)


We can solve for the values of resonance Hields, with the expression for a resonance field where

quantum tunneling of the magnetization (QTM) occurs given by


Bre =k (1-8)


In Eq. 1-8, k = na + na' and can take on integer values, while na and na' represent the states

with opposite spin proj sections along the Hield quantization axis, with the lowest two states being

na = & S (k = 0 resonance). At resonance Hields, which occur in integer steps of approximately


,states with opposite spin proj sections become mixed by terms in the spin Hamiltonian that


do not commute with the Sz operator. From Eq. 1-8 we can calculate the values of the

longitudinal field where tunneling can occur for a given system. For Mnl2Ac, resonance Hields

appear approximately every 0.45 T. In this respect, the magnetic field can be used to switch

tunneling "on" and "off'. When the field does not correspond to a resonance value, tunneling of

the magnetic moment is forbidden. However, when the field matches a resonance value,

tunneling is allowed. Fig. 1-4 shows a typical hysteresis loop for the Mnl2Ac SMM. The flat

plateaus in the figure correspond to fields where no tunneling is allowed, but the sharp steps seen

at resonance fields are where the tunneling is switched on. The steps correspond to spins

changing their magnetization state by tunneling through the energy barrier, which changes the

value of the sample magnetization being measured. Thus the steps are the relaxation of the

magnetization of the spins toward the opposite saturated magnetization value.









For simplicity we will consider only the previously mentioned second order transverse


anisotropy. By rewriting this in terms of the raising and lowering operators (S, S = ,
S2

we see that this term will mix states that differ in na by two, and hence for an integer spin system

this will cause mixing of the pure spin multiple states, Im However, there is a symmetry

imposed by this term such that tunneling is possible only between certain states. To illustrate

this we start with the general expression for the eigenvectors [1].


Im = E p(Mm')m'(19


In Eq. 1-9,
all possible na '. The overlap, or amount of mixing, of two states is quantified by the matrix

elements between the respective states.


(n2 T, n')=(n2 ( S ) 2')(1-10)


Eq. 1-10 is zero unless the values of the states na and na differ by a multiple of two.

The eigenvectors of each state can take on one of two forms [1]


mi)= Ep(S-2-2p S-p) (1-11a)
p= U





p=U

where p is an integer. Eq. 1-11a applies to even integer states and Eq. 1-11b applies to odd

integer states. Since the states na and na' are mixed, the wavefunction describing each state will

have a component of both m and m '. From Eq. 1-11la it follows that m = S-2p and na = S-2p '

or from Eq. 1-11b it follows that m = S-2p 1 and m = S-2p' '-. Now we introduce a









longitudinal external Hield and formulate the selection rule for tunneling between states at

resonance Hields. By taking the difference between na and na we get a value of 2(p-p '). Since p

is an integer we write the selection rule as na n = 2n. Therefore, tunneling can only occur

between states that satisfy this condition and this occurs at Ilth order in perturbation theory. For

example, for a system with a given S, tunneling can occur between states nas = -S and n,=

-(S+2) in first order, but tunneling between states nas = -S and na, = S occurs in Sth order.

Consequently, tunneling between the lowest states is less probable than tunneling between higher

energy states. The above mentioned selection rule was obtained for a second order transverse

ani sotropy. Similar selection rules are applicable for higher order transverse anisotropies as


well. Another quite common one is the fourth order term 4~( 4) The selection rule for


this symmetry would be na n = 4n, otherwise no tunneling is allowed.

At resonance Hields, the non-commuting terms create an energy difference between the

symmetric and antisymmetric combinations of the spin projection states (tunnel splitting).

As the magnetic Hield is swept through a resonance value there is a probability for a spin to

change its proj section state, which is given by the Landau-Zener formula for tunneling [19]. Fig.

1-5 illustrates the energy levels of Mnl2Ac close to the first (k = 1) resonance field. The effect of

transverse anisotropies manifests itself as the repulsion of energy levels (red lines in Fig. 1-5)

close to the resonance field. This is known as an "anti-crossing", since without any transverse

anisotropies the energy levels cross (grey lines in Fig. 1-5) at the resonance field. While on

resonance the spins have a nonzero tunneling frequency that is given by Eq. 1-12 [1].

(m ~I m')
my) = (1-12)










Thus, the tunneling frequency depends on the matrix elements between the respective tunneling

states. The relation between the tunnel splitting and tunneling frequency is given by


wf""' = """ (1-13)
2h

Hence, small tunnel splitting generate low tunneling frequencies and minimal tunneling. In the

absence of any decoherence effects the spin would oscillate between states na and na at the

tunneling frequency given by Eq. 1-13, with a probability to find the spin in either state.

In Fig. 1-3, we illustrate a situation for the k = 2 resonance field, where the spins could tunnel

back and forth through the anisotropy barrier. However, finite lifetimes of excited state energy

levels arise from the possibility of phonon emission, and the oscillations between two states (nz

and na ', where one or both are excited states) in resonance are damped since the spin can relax

from the excited state to the ground state by emitting phonons after tunneling. Consequently the

spins that tunnel through the energy barrier quickly decay back to the ground state and do not

tunnel back again to the other side.

The solution to Eq. 1-12, for the case of a second order transverse anisotropy (E), as

given by Eq. 1-10 is presented in Eq. 1-14.


gk hE 2-


my = (1-14a)


Kk22S-k-2 [(s k /2 1) !]2 k!(14b

In Eq. 1-14, k must be an even integer, which is imposed by the second order transverse

anisotropy term. It is easily seen that for smaller values of k (tunneling between the lowest lying

levels), the tunneling frequency is small. However, as k increases, the tunneling frequency can

quickly increase by many orders of magnitude. From a physical standpoint, this demonstrates









that while tunneling between lower lying energy states is weak or even negligible, tunneling

between higher lying levels can be quite significant.

For the studies on the incoherent tunneling processes presented in section 6.2 we can

separate the tunneling into two regimes. As we sweep the magnetic Hield back and forth from >

+3 T, we pass through resonance Hields where there is a probability for QTM. Once the system is

fully biased and we sweep the Hield back through zero toward the reverse saturation Hield, the

spins have a chance to tunnel as we pass through each resonance field. The amount of spins that

tunnel increases as the Hield increases due to two effects. First, the tunnel splitting increases as

the difference between na and na' decreases and k increases, as can be seen Eq. 1-14. Second, the

effective energy barrier is lowered as the Hield increases and it is more probable for spins to

tunnel through the lowered effective barrier due to the finite lattice temperature. Consequently,

spins can be excited to higher states, which increases the amount of tunneling as we sweep

through a resonance field since tunneling between higher lying states is more probable (as shown

in Eq. 1-14). Eventually as the biasing field becomes large enough (g-pBBnas > |D|S ) the energy

barrier becomes non existent and all spins have reversed their proj section state. This corresponds

to a field of about 5 T for Mnl2Ac.

On the other hand, for much of the data collected, we would wait at a resonance field for

a fixed amount of time as opposed to sweeping through the resonance at a given rate. In this

case, there exists a tunneling probability per unit time for each spin. Approximate formulas for

the tunneling probability per unit time from the ground state into an excited state have been

derived [20, 21, 22] and take into account the lifetime of each state (ground and excited)

calculated without tunneling and the tunneling frequency calculated for an isolated spin. These

are directly applicable to our experiments, since they relate to conditions where two states (nz and










m ') are in resonance. We Eind it necessary to only mention the qualitative aspect, which shows

that the total amount of spins that tunnel will increase for longer wait times. Of course, within a

distribution, different molecules will have different probabilities and consequently will tunnel on

different time scales. This can be seen through Eq. 1-14, since there are distributions of the D

and E values among different molecules. The fact that we can observe which molecules are

tunneling on certain time scales is the main point of the HFEPR studies done to monitor the

QTM presented in Ch. 6.

Interesting phenomena such as quantum phase interference of spin tunneling traj ectories

have been observed in single molecule magnets [23, 24]. This manifests itself in oscillations of

the value of the tunnel splitting at a longitudinal resonance Hield while the value of the transverse

Hield is varied. While a significant amount of research has been done to characterize this

phenomenon, in this dissertation we will focus on quantum tunneling in the absence of an

externally applied transverse Hield. For our interests, there are two tunneling regimes that can be

considered:

* Thermally assisted regime. In this regime, spins can populate excited energy levels above
the ground state. The probability for spin tunneling increases for resonant levels higher up
the barrier. Thus, more tunneling takes place between higher lying states in resonance.
This can take place with or without an external magnetic Hield. An application of an
external magnetic Hield shifts the zero Hield energy levels with respect to each other. At
certain values of the field (given by Eq. 1-8) energy levels with opposite spin proj sections
along the quantization axis residing on opposite sides of the energy well become nearly
degenerate. At these resonance Hields, the spins can tunnel through the energy barrier and
relax back to the ground state through phonon emission. This process is illustrated in Fig.
1-3. Any spins on the left side of the barrier (ground state or excited states) that tunnel
through the barrier to an excited state (right side of the barrier) relax to the other ground
state by the process of phonon emission.

* Pure quantum tunneling regime. In the absence of an external magnetic Hield and at
extremely low temperatures (kBT<< |D|S ) spins will only populate the ground state energy
level. For an integer spin system the tunnel splitting between the symmetric and
antisymmetric linear combinations of the two ground state wavefunctions allows for spins
to tunnel through the energy barrier and reverse their spin proj section. No phonon









absorption or emission is involved in this process. The rate of this process depends on the
system, and can vary over many orders of magnitude [25]. For Mnl2Ac, the rate is
immeasurably small (< 10- s )~, while for the Ni4 System discussed in Ch. 4, the rate is
quite fast (2 x 10- s )~.

For all of the QTM presented in this dissertation the observed tunneling is a resonant quantum

tunneling process in the thermally assisted regime.

In addition to tunneling, spins can also change their energy state by thermal activation over

the anisotropy barrier. At elevated temperatures where the thermal energy is comparable to the

anisotropy barrier (kBT > |D|S2) the spins can relax by essentially going over the barrier through

a thermal activation process. Under these conditions the reversal of the magnetization can be an

ongoing process as spins move back and forth over the top of the barrier, which is pure thermal

relaxation. The activation energy is expressed as |D|S2 by considering a relaxation time that

follows an Arrhenius law [1].

1.5 Applications

SMMs offer opportunities for exciting research in both the physical chemistry and physics

communities. From a physics perspective, they allow investigation of the quantum mechanical

properties of individual nanoparticles and how these properties are influenced by the surrounding

environment. Additionally, they are unique in that they lie on the border of classical and

quantum mechanical physics [26]. Even though the magnetic moment of each molecule exhibits

quantum mechanical behavior by tunneling through a magnetic anisotropy barrier, the magnetic

moment is larger than a normal quantum mechanical system [16, 27]. Quantum phase

interference effects have also been predicted and reported in certain systems [24, 28, 29]. In the

sense that they are macroscopic systems exhibiting quantum mechanical behavior, SMMs lie in

both the classical and quantum regimes.









The fact that each molecule is magnetically bi-stable due to the negative axial anisotropy

barrier has created excitement for using these systems in possible technological applications.

Each molecule could potentially act as an individual computing cluster, with the entangled states

of the molecule acting as a qubit. In this way they could be used as a quantum computing

device, with the speed for a single computation on the order of 10-10 s in a system such as

Mnl2Ac [30, 31]. Another promising application would be as a magnetic memory storage device

[32]. The isolated magnetic moment of each molecule would represent one bit of information,

and an incredibly high density (30 x 109 mOleCUleS / cm2 [30 terabits]) of data storage would be

possible. Regardless of whether or not the realization of these applications is ever achieved,

SMMs are interesting to study from a purely scientific aspect.

1.6 Summary

In this chapter we gave an introduction to the SMM systems which are the focus of the

research presented in this dissertation. We explained the sources and importance of anisotropy

to the magnetic behavior and also described a unique feature of SMMs: quantum tunneling of the

large magnetic moment vector of a single molecule. The main goal of studying these materials is

to gain a deeper understanding of the magnetic behavior. There are various parameters that

control this behavior and in this dissertation we will present experiments and analysis done to

determine such parameters and use this to explain the observed phenomena. In Ch. 4 we outline

studies done on two nickel based compounds in order to determine the disorder and isotropic

exchange terms in the system that is a SMM. It is shown that the disorder and intermolecular

exchange interactions in this system have a dramatic effect on the low temperature magnetic

spectrum. In Ch. 5 we outline studies done on two cobalt based compounds in order to

determine the anisotropy and exchange terms in the system that is reported to be a SMM [33]. It

is shown that anisotropic exchange is a major component of the low temperature magnetic










properties. Finally, in Ch. 6 we outline studies done on Mnl2Ac that combine high frequency

electron paramagnetic resonance (HFEPR), Hall magnetometry, and surface acoustic waves

(SAWs) in order to characterize the effects of disorder on the QTM in addition to the relaxation

processes in this SMM.












nln~~s = 3/2


Oxygen i)

Carbon e




S =10

Figure 1-1. An illustration of a molecule of Mnl2Ac. Eight Mn+3 ions and four Mn+4 ions are
antiferromagnetically coupled through the oxygen atoms, giving rise to a large S= 10
ground state for the molecule. The central blue circles represent the giant
magnetization vector pointing out of the page, along the S4 Symmetry axis of the
molecule.










Spin projection m,
-10 -8 -6 -4 -2 0 2 4


6 8 10


55 te '


Figure 1-2. The energy barrier for a molecule ofMnl2Ac in zero field. The axial anisotropy
forces the magnetic moment to point either parallel ("up") or anitparallel ("down") to
the quantization (easy) axis. The energy barrier to magnetization reversal is given by
|D|IS.


"down "










-ID p -8 -7 -6 -5 -4 -3 2 -1 0 1 2 3 4 5 6 7 8 9 10


Figure 1-3. The energy barrier for a molecule of Mnl2Ac with an external magnetic field
applied parallel to the quantization (easy) axis. At values of the resonance fields the
magnetic moment (blue arrows) can change its proj section state by tunneling through
the energy barrier. As an example we show a spin tunneling from the ms = -10 to ms
= 8 state. The spin then relaxes back to the ground state (ms = 10) by emitting
phonons.


Phonon










1.0


O
-C)
Cd
C**l
-C,
d)




zt
a,

~


O


0.5














-1.0


2K

B 11z


-3 -2 -1 0 3

Magnetic Field (Tesla)
Figure 1-4. A hysteresis loop for a single crystal of Mnl2Ac at a temperature of 2 Kelvin and
with an external magnetic Hield applied parallel to the quantization (easy) axis. The
flat plateaus in the Eigure correspond to Hields where tunneling is switched off, but the
sharp steps seen at Hield values that correspond to multiples of approximately 0.45
Tesla are resonance fields where the tunneling is switched on.










mt -- *=


m =10


'--
S


0.40 01.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0E.50
Magnetic Field (Tesla)
Figure 1-5. A diagram of two energy levels in a system of Mnl2Ac as they pass through the first
non zero resonance field. The dashed lines show how the energy levels would behave
in the absence of any tunnel splitting term. The red lines show how the energy levels
are repelled due to the tunnel splitting term. A spin in the state nas = 10 state has a
probability, dependent upon A, to tunnel to the nas = -9 state as the field is swept
through the resonance field.









CHAPTER 2
EXPERIMENTAL INSTRUMENTATION AND TECHNIQUES

2.1 Measurement Techniques

There exist many different experimental techniques used to study SMMs. Some of these

include x-ray crystallography to measure the molecular structure [34, 35], susceptibility

measurements for basic magnetic characterization [17], neutron scattering to measure exchange

couplings and spin multiple excitations [36, 37, 38, 39], high frequency electron paramagnetic

resonance (HFEPR) to measure spin multiple energy spectra [40, 41, 42], and Hall or SQUID

magnetometry to measure magnetic hysteresis and observe quantum tunneling steps in the

magnetization [43, 44]. All of the studies presented in this dissertation were done using HFEPR

as the main experimental technique. Some work, presented in Ch. 6, combined HFEPR with

surface acoustic waves or magnetometry to study the influence of microwaves on non

equilibrium processes in single crystals of Mnl2Ac.

2.2 Electron Paramagnetic Resonance in the Context of Single Molecule Magnets

In Ch. 3 where we formulate the spin Hamiltonian for SMM systems, the Zeeman

interaction is expressed as


Hie,,,, = #gBB gf -S (2-1)


For a magnetic field given by B = Bo3 the energy of this interaction is


Exeenms = 8,PBBom (2-2)

where ms is the proj section of the electron' s magnetic moment along the z axis.


In order to induce a transition between states one must supply an energy of hwo. Therefore, the

resonance condition is such that this energy must match the energy difference between respective

states, or equivalently, hwo = AE. The energy comes in the form of microwave radiation at a










frequency coo. This is the basic principal behind which all of our EPR measurements lie. As an

example of this, in Fig. 2-1, we plot the energy levels in the Ni4 SMM (discussed in Ch. 4) with

its easy axis aligned along the external field. For a fixed frequency, there is a strong absorption

of the signal each time the resonance condition (energy difference between adj acent levels

matches the microwave radiation) is met as the magnetic field is swept. The black arrows

represent the magnetic fields where absorption of the microwaves causes transitions between

states.

In a typical experiment, fixed temperature, fixed frequency magnetic field sweeps are

performed at a number of different temperatures and frequencies. As the magnetic field is swept,

sharp inverted peaks appear in the spectrum when the resonance condition is met, which

corresponds to the transmission of the microwave signal through the experimental setup. Each

peak corresponds to a transition between spin proj section states, and the decrease in transmission

signal is due to the absorption of microwave radiation by the spins within the sample. A normal

EPR spectrum from a SMM with its easy axis aligned along the external field consists of a series

of absorption peaks corresponding to transitions between states within the ground state spin

multiple. At sufficient temperatures where all states within the multiple are populated, one

should see 2S peaks, corresponding to 2S transitions between 2S+1 states. This is illustrated in

Fig. 2-2 for the same system in Fig. 2-1. The larger intensity peaks correspond to transitions

within the ground state. The additional, smaller intensity peaks, which are attributed to

transitions within a higher lying spin multiple, are also discussed in Ch. 4.

2.3 Cavity Perturbation

2.3.1 Technical Challenges

The basis for any EPR measurement is to monitor the response of the sample to the

application of external microwave radiation and a magnetic field. This presents many obstacles









to overcome in terms of experimental techniques and instrumentation. First, there are space

restrictions in terms of positioning the sample in the center of the magnetic field. High field,

high homogeneity magnets possess rather narrow bores which leave little room for a sample

stage. Even more problematic is the propagation of microwaves through a probe and ensuring

proper coupling of the microwaves to the sample. Unwanted standing waves arise due to

multiple reflections from impedance mismatched components [45]. Another obstacle is

attenuation of the signal due to the finite conductivity of the waveguides, especially at high

frequencies and/or where the propagation is along considerable distances. The need for high

frequencies stems from the large zero field splitting energies (discussed in section 1.2) present in

SMMs. These materials would be EPR silent with typical X band (~ 9 GHz) spectroscopy.

Detecting a signal from a tiny single crystal, where the typical sample dimension used for our

experiments is on the order of <1 mm, is not trivial and requires extremely precise methods.

2.3.2 Equipment

An ideal method suited for fixed frequency magnetic resonance spectroscopy is known as

the cavity perturbation technique [46]. This technique relies on the resonant modes of a high

quality factor cavity to ensure good coupling of the radiation to the sample and compensate for

the small filling factor of the sample. The drawback to this method is that one is limited to

working at frequencies that correspond to the modes of the cavity. We use cylindrical copper

cavities that have a fundamental mode, TEonl, at close to 50 GHz, but can operate at frequencies

up to 400+ GHz. We have two types of cylindrical cavities: one is a standard vertical cavity

where the z axis of the cylinder is oriented vertically, and the other is a rotating cavity that allows

rotation of the sample with respect to the external magnetic field, where the z axis of the cylinder

is oriented horizontally [47, 48].









All experiments were done using a single crystal of the respective compound. If the mass

of the crystal is measured, and the molar ratio of each constituent compound is known, then the

number of spins participating in a given transition can be estimated from the area of the

respective resonance peak. For the purposes of our experiments, this was not a crucial number,

and therefore the mass of a given crystal was never measured. Additionally, the data was

recorded as a relative voltage signal, making an absolute measurement of the number of spins

contributing to a resonance peak impractical. Consequently, the transmission signal is quoted in

terms of arbitrary (arb.) units.

We now describe the orientation of a sample in the cavities in the interest of subsequent

chapters when we discuss experiments using these cavities. For experiments conducted using the

standard vertical cavity, the sample was placed on a circular endplate of a copper cavity at a

position halfway between the center and the edge of the endplate. For the fundamental mode of

the cavity, the magnetic component of the microwaves has a maximum at this position. Thus,

we maximize coupling of the microwaves to the sample. This endplate then fastens onto the end

of the cylindrical cavity. This does not allow for sample rotation within the cavity itself, but is

convenient when the magnetic axes of the sample are known from x ray crystallography

measurements and can be deduced by merely looking at the crystal. The sample can then be

placed in the proper orientation before insertion into the cryostat. For experiments conducted

using the rotating cavity, the sample was mounted in a similar manner, except the endplate

mounts in a horizontal fashion. The initial sample orientation depends upon the nature of the

experiment, but when used in our magnet system possessing a horizontal field, the rotating cavity

allows for two axis rotation, and thus any sample alignment. The cavities and the rotational

capabilities of each magnet system are illustrated in Fig. 2-3 and Fig. 2-4. A detailed discussion









of the rotating cavity is given elsewhere [47]. The capability of the rotating cavity is critical for

studies of anisotropic systems such as SMMs and must be used for systems where the

orientations of the magnetic axes with respect to the crystal are not known beforehand. The

sample temperature is always determined from a Cernox thermometer that is securely fastened to

the outside of the respective copper cavity. The waveguide probe used for measurements with a

cavity is also described in detail elsewhere [45].

Now we discuss a simple model of the resonant cavity where the sample is placed. An

ideal resonator is made from a continuous, perfectly conducting material and is filled with a

lossless dielectric. In this case the cavity modes are perfectly discrete, have the shape of a delta

function peak, and can be calculated considering the correct boundary conditions for the

appropriate geometry. Our cavities are cylindrical resonators, with the resonant modes of an

ideal resonator being given by [49]



m =("") +( ) (2-3)


In Eq. 2-3, pu is the magnetic permeability of the cavity dielectric, e is the electric permittivity of

the cavity dielectric, m, n, and p are integers, r is the radius of the cavity, d is the height of the

cavity, and x,,,, is the nth TOOt of the first derivative of the Bessel function. Of course our cavities

are not ideal resonators since they are made of a Einite conducting material (copper), and filled

with a Einite loss dielectric (helium thermal exchange gas and the sample). Additionally, we

must allow the microwaves from the waveguides to couple to the cavity, which is done through

two small coupling holes with a diameter of 0.97 mm (31/6 for a frequency of approximately 50

GHz). The dimensions (diameter and thickness) of the apertures are what determine the amount

of coupling to the cavity. There is an important trade off between: (i) strong coupling (large










apertures), which ensures good power throughput from the source to the detector and, hence, a

large dynamic range, and (ii) weak coupling (small apertures), which limits radiation losses from

the cavity, resulting in higher cavity Q values and increased sensitivity, at the expense of some

dynamic range. It is found empirically that the optimum coupling apertures should be small for

our setup, hence the choice of a diameter of3l /6. Additionally, it is necessary for the coupling

plate to be sufficiently thin (~1~ /20), since the signal is obviously attenuated as it passes through

the apertures, which are way below cutoff.

The critical instrument for all of our EPR measurements is a Millimeter Vector Network

Analyzer (MVNA), which acts as both a microwave source and detector [50]. The instrument

contains two YIG (Yttrium Iron Garnet) oscillators, which are continuously tunable in the range

of 8 to 18.5 GHz. These sources have an intrinsic stability that drifts by only a few MHz per

hour when working around 100 GHz. Although the cm sources are phase locked to each other,

their absolute frequencies must also be stabilized, which can be achieved through phase locking

the MVNA sources to an external frequency counter with an internal quartz reference. The first

oscillator produces a signal Fl and is locked to an external frequency counter to stabilize the

frequency with a feedback loop. This becomes useful for improving frequency resolution and

stability when measuring high Q factors (> 104) Of modes of our resonance cavities. To this end,

a Phase Matrix 575 source locking frequency counter [51] is used, which provides both the

stability and precision necessary for our measurements.

The second oscillator produces a signal F2 Fbeat and is phase locked to the first oscillator,

and consequently the frequency counter as well. The beat frequency is derived from an internal

reference oscillator operating at 50 MHz and is programmed by the software controlling the

MVNA hardware. The MVNA receiver can operate at one of two exceptionally precise










frequencies (either 9.01048828125 MHz or 34.01048828125 MHz), which corresponds to the

harmonic multiplication of the frequency difference between Fl and F2 (N[F1 F2 -.

The signal from the first oscillator is sent to the harmonic generator (HG) along low loss,

flexible coax cable. The signal from the second oscillator is sent to the harmonic mixer (HM),

also along low loss, flexible coax cable. These cables introduce a combined insertion loss of 4-5

dB [52]. The two frequencies are beat together, and the amplitude and phase of this signal are

what is processed by the MVNA receiver. The beat frequency, Fbeat, iS given by NF1 -N' F2,

where N and N' are the harmonic numbers that multiply the respective fundamental frequencies.

These values will be 1 for frequencies up to 18.5 GHz; for frequencies above 18.5 GHz we

employ external Schottky diodes to work on harmonics of the fundamental frequencies. The

Schottky diodes are passive, non-linear devices that can generate harmonics from N= 3 (V band)

up to N= 15+ (D band). We use pairs of Schottky diodes that will produce the same harmonic

(N = N') for the HG and HM. In this respect, Fbeat = N(F1 F2). Additionally the phase

difference, Obeat, between the two frequencies is given by N 01 N'0s. With the above

mentioned configuration (N = N'), the two phases are locked to each other (01= 02), which

cancels any phase noise associated with the beat signal and allows for a incredibly low noise

level. However, any phase difference between the two frequencies is upheld in the beat signal.

Thus, we can measure both the amplitude and phase of the beat frequency which constitutes a

vector measurement. The choice of the beat frequency (receiver frequency) is determined by the

software, and is made depending on the harmonic number. For harmonics up to three, the

9.01048828125 MHz receiver is chosen, and for harmonics greater than this the 34.01048828125

MHz receiver is chosen. For signal detection, these frequencies are down converted by beating

intermediate frequencies with the receiver frequency. If using the 34.01048828125 MHz









receiver, the first stage consists of beating this signal with a 25 MHz signal from the internal 50

MHz oscillator. This signal (9.01048828125 MHz) is then beat with a 9 MHz signal from the 50

MHz oscillator which leaves a 10.488 k
9.01048828125 MHz receiver, except the first step (beating with 25 MHz) is obviously not

needed. Since all signals used in the down conversion process are correlated to the same

reference oscillator, the phase information is maintained during the process. Finally, the 10.488

k
amplitude and phase information.

As the harmonic number increases, the power output from the HG decreases. For

harmonics greater than nine (or frequencies greater than 160 GHz) we use a Gunn diode as an

external microwave source that works in conjunction with the MVNA and Schottky diodes that

can produce higher frequencies with slightly increased power outputs. The non-linear effects

from the HG and HM create a comb of frequencies (NF2), which mix with the frequencies from

the Gunn source, creating an intermediate frequency such that FIF = M~Fcunn -NF2, where M~and

N are integers. The frequency of the Gunn source is locked to the first YIG oscillator (and by

default the second, as well), with an offset equal to the 50 MHz reference oscillator, such that

FGunn = kF1 50 MHz, where k is the harmonic number of the first oscillator. The relation

between the harmonics of the YIG oscillators and Gunn source is given by N = kM~ The

intermediate frequency can now be expressed as FIF = N(F1 F2) -M(50 MHz). As mentioned

previously, the MVNA receiver operates at one of two precise frequencies, FR. Therefore, in

order to properly tune the hardware, the software programs the MVNA electronics such that

difference frequency between the YIG oscillators is F; F2 = FR/N + 50 MHz / k. Since the

frequency of the Gunn source is phase locked to the internal oscillators of the MVNA, we









achieve the same low noise level as when we are solely using the MVNA oscillators. We have

two Gunn diodes that cover fundamental frequencies from 69 GHz to 102.5 GHz with power

outputs of ~40 mW at 92 GHz. The Gunn diodes have two attachment options depending on the

type of measurement desired. If we are doing a continuous wave (CW) EPR experiment, we use

a multi harmonic multiplier to achieve frequencies up to 500+ GHz. The multi harmonic

multiplier has four mechanical tuning knobs that allow precise optimization of the signal for a

given frequency. Two of the knobs are for optimization of the incoming power from the Gunn at

the frequency FGunn. The other two knobs are for optimization of the output power at a

frequency M~FGunn. Various filters are used at different frequencies (138 GHz, 235 GHz, 345

GHz, 460 GHz, 560 GHz) to remove lower harmonic components of the signal. Non linear

effects from the HM can create unwanted signals from combinations of lower harmonics (Mr1 and

M2), Such as My1 + M~2, 2MI, or 2M~2. This can introduce crosstalk effects in the detection

electronics and the use of high pass filters cancels this effect. The signal rej section for the first

harmonic below cutoff for each filter is on the order of 75 dB. For instance, when working in the

frequency range of 23 5 GHz to 344 GHz, we use the 23 5 GHz filter that will attenuate lower

harmonic components of the signal below 23 5 GHz but will pass the desired signal along with

higher harmonic components. These higher harmonic components are of course rej ected by the

narrow band receiver set to detection at the precise beat frequency for the proper harmonic. The

power output in this frequency range is on the order of approximately 1 mW at 300 GHz.

For EPR measurements where we wish to have large powers in short pulses to avoid

sample heating, we remove the multi harmonic multiplier and attach a one stage power amplifier

and fast pin switch. These components operate in a much narrower frequency band (89 GHz -

98 GHz) than the CW components but allow for larger powers (186 mW at 92 GHz) and pulse









microwave signals at incredibly fast time scales (Fast Switching Unit, 4ns from 10% to 90%).

The pin switch has a low insertion loss (~ 1.5 dB) and provides approximately 40 dB of isolation

between the "on" and "off" states. Additionally, we can attach a frequency doubler or tripler to

work at frequencies in the range of 178 GHz -196 GHz or 267 GHz 294 GHz respectively. In

this configuration we also have the option of removing the switch to work in CW mode with

amplified powers (approximately 10 mW at 288 GHz) at the above mentioned frequencies.

We have two commercial superconducting magnet systems available in our lab. The first

is a 17 Tesla, vertical field Oxford Instruments system [53]. The second is a 7 Tesla, horizontal

field, Quantum Design system [54]. These systems allow temperature ranges from 300 K down

to less than 2 K. For a period of time during our research, the 7 Tesla magnet was undergoing

repairs and a replacement 9 Tesla vertical field magnet was loaned to us. Some experiments

done in this magnet will be presented in Ch. 5. Fig. 2-3 shows a schematic of our typical

experimental setup, including a waveguide probe and sample cavity inserted in the bore of a

superconducting magnet immersed in a cryostat.

2.4 Quasi Optical Setup

The use of waveguides begins to become problematic at high frequencies. First and

foremost, the signal losses associated with the finite conductivity of the waveguides are larger

for higher frequencies [7]. This arises from ohmic losses in the conducting waveguides, as the

induced current at the surface of the waveguides increases with increasing frequency.

Additionally, the resonant cavities become completely overmoded and it becomes impossible to

determine the microwave field configuration for a given frequency. An alternative method that

we have employed is to use a quasi optical bridge setup operating in reflection mode. The

constituent components of the setup are all quite low loss and consist of a corrugated reflection

tube (the probe), corrugated horns, wire grid polarizers, and elliptical focusing mirrors. The









corrugated tube and associated homs are optimized for a specific frequency of 250 GHz,

although the horns and tube are broadband, as data has been collected in the frequency range of

180 GHz 350 GHz for these particular components. Expected loss for the probe working at

250 GHz is ~0.01 dB/m [55]. Since the length of the tube is 100.3 cm in accordance with the

sample chamber in our Quantum Design system magnet, it provides a theoretical minimal loss of

approximately 0.02 dB.

The tube, which is based on a corrugated HEn1 waveguide, is made from thin walled (.3

mm) German silver, chosen for its low thermal conductivity and ease of machining. The low

thermal conductivity reduces the heat load on the cryostat and allows the sample to remain at

cryogenic temperatures throughout the experiment. The reflection probe is tapered close to the

field center in order to concentrate the incident power over a 1.7 mm diameter region at the

bottom of the tube. The sample can be attached to a reflecting, copper sample holder at the field

center. Careful alignment of the optical bridge with the probe ensures that most of the signal

reaching the detector comes from the reflecting sample holder at the field center, thus

guaranteeing good coupling to small samples of sub-millimeter dimensions. The top of the tube

has a high density polyethylene (HDPE) window with an antireflection coating, which minimizes

insertion loss. The two, identical, corrugated homs are made from gold plated copper (each

being ~19.7 cm long with a 1.78 cm circular opening). The homs are also tapered at the input

end to match with waveguide couplers that are used for the conversion between Gaussian optics

and the rectangular guided modes that are required for coupling to the source and detector

through external diode attachments (Schottky and Gunn).

The elliptical focusing mirrors are manufactured from aluminum and are machined to a

incredibly high tolerance for optimum performance. The mirrors facilitate the propagation of an









undistorted Gaussian beam. A free space Gaussian beam has a minimum beam waist, wo, that is

greater than half the wavelength of the radiation [56]. A narrower beam waist will constitute a

signal with more intensity reaching the sample, but the divergence of the beam is inversely

proportional to the beam waist. Faster diverging beams necessitate larger surface mirrors in

order to refocus the beam. For distances far from the point of minimum beam width, the beam

width approaches a straight line.



If (Z) = il 1 + (2-4a)




Z= (2-4b)


The angle between the beam propagation axis and the beam radius is given by






The total angular spread of the beam is twice this value. A diagram of a Gaussian beam is shown

in Fig. 2-5. Thus there is a compromise between a narrow beam waist and the practical size of

the mirror surface. For reflecting optics, the mirrors must be machined to tolerances of3l/20 in

order to approach diffraction limited performance [56]. At 300 GHz (32 = 1 mm) this corresponds

to a tolerance of +0.0025 cm. The mirrors have a focal length of 25 cm and all components are

configured such that the Gaussian beam will be focused at the entrance to the probe and detector

horn. Since the reflectivity of aluminum is close to unity for the microwave frequency range of

interest these components introduce minimal loss. We employ two wire grid polarizers to ensure

that the polarization of the signal is correct when returning to the detector. These wire grid

polarizers are necessary in order to propagate the signal through the quasi optical bridge setup.










Fig. 2-6 illustrates the transmission of the signal through the system, and in Fig. 2-7 we show

how the polarization of the signal is changed by the respective components. From Fig. 2-7, it

can be seen that the polarization of the signal reflecting off the first wire grid polarizer must be

changed after interacting with the sample and before entering the detector. If the second wire

grid polarizer was not there, the signal returning to the first wire grid polarizer would be totally

reflected before it enters the detector. The insertion of the second wire grid polarizer changes the

polarization of the signal so that it will pass through the first wire grid polarizer and reach the

detector. All components of the optical setup are mounted securely on a solid stainless steel

bench (87.6 cm x 49.5 cm), held firmly in place with screws and dowel pins, and were

manufactured by Thomas Keating Ltd [57].

Here we discuss the setup in detail, which should be referenced to Fig. 2-6 and Fig. 2-7.

The signal originating from the Gunn diode source is polarized in the manner for a rectangular

waveguide (H field along the longer dimension). The mode of the rectangular waveguide is

launched into a rectangular-to-circular waveguide transition. The transition piece produces the

TEn1 circular waveguide mode, which converts to a HEn1 mode once inside the corrugated part of

the horn. After exiting the horn, the signal becomes a Gaussian profile as it propagates in free

space. First, the signal encounters a 450 wire grid polarizer 12.5 cm from the edge of the horn.

This splits the signal and the reflected beam is polarized at 450 with respect to the horizontal,

while the transmitted signal (900 polarization with respect to the reflected signal) is absorbed by

a pad that is highly absorbent in the microwave region. This helps to minimize the amount of

signal returning to the detector that does not interact with the sample. The reflected signal then

meets the first elliptical focusing mirror, 12.5 cm from the middle of the wire grid polarizer. The

signal has now traveled 25 cm total, and the mirror reflects the diverging beam such that the









minimum beam waist is 25 cm from this point. The next component is a second wire grid

polarizer, with an orientation that is horizontal (450 from the first polarizer). This again changes

the polarization of the signal so that it is now vertically polarized (900 from the horizontal).

A second mirror is located 50 cm from the first mirror (25 cm from the minimum beam

waist). Again the purpose of the mirror is to focus the beam profie so that it will have its

minimum beam waist as it enters the corrugated waveguide probe, 25 cm below the mirror. The

signal then changes to a HEn1 mode as it travels down the tube to the sample and reflects back

up. In principle, the signal retains its polarization as it exits the probe and becomes a Gaussian

profile once again. However, the magnetic response of the sample can alter this polarization

slightly. The beam exiting the probe travels 25 cm back toward the second mirror, where the

diverging profie is refocused. Going from the second mirror back toward the first, it passes

through the second wire grid polarizer. If the polarization of the signal was unchanged by the

sample then it is also unchanged by the polarizer since it is the same as the first time passing

through this component. The beam then encounters the first mirror, where the diverging beam is

focused such that the minimum beam waist will be at the entrance to the detector horn (25 cm

from the first mirror). After being refocused by the first mirror, but before entering the detector,

the signal passes through the first wire grid polarizer, where the polarization is changed for a

final time so that it will be correctly oriented with respect to the detector (450 from vertical).

Finally, the properly focused and polarized signal enters the detector horn and is propagated as a

HEn1 mode and then in the manner for a rectangular waveguide as it enters the Schottky diode

and is mixed down to the proper detection frequency by the MVNA electronics.

Both the tube that inserts into the cryostat and the horns connected to microwave source

and detector diodes are corrugated. In contrast to a smooth walled (no corrugations) circular










waveguide, a corrugated waveguide couples extremely effectively to a free space Gaussian beam

profile. Hence, the main advantage of the corrugated components is the extremely low loss

associated with microwaves at the optimized frequency. First, there is excellent coupling

between the free space Gaussian beam and the HEn1 mode that propagates through the tube and

horns. The efficiency of this coupling can reach 98% with proper matching. Second, the HEn1

mode has little attenuation inside these waveguides since the cylindrical symmetry allows

efficient signal propagation. A non corrugated circular waveguide couples the Gaussian profile

to both the TEn1 and TM11 modes which progress at different phase rates and can result in a

distorted output beam. The reason for the corrugations is to make the wall of the waveguides

reactive which allows the HEn1 mode to propagate with almost no attenuation [56]. An ideal

HEn1 mode has identical E plane and H plane radiation patterns with a main lobe that is

approximately Gaussian, which is the reason for the excellent coupling between this mode and

the free space Gaussian profile. A diagram of the TEn1 and HEn1 modes in a circular, corrugated

waveguide are shown in Fig. 2-8. The loss is proportional to 32 r3, where 32 is the wavelength of

the radiation and r is the radius of the tube. Consequently, the loss decreases for higher

frequencies. The periodicity of the corrugations is chosen to match well with the optimum

frequency of the tube, with a periodicity of about 0.4 mm (31/3) for 250 GHz, and the depth of

each corrugation is 3/4. A larger number of corrugations per wavelength is advantageous for

optimal signal propagation, but becomes difficult to machine. Three corrugations per

wavelength is a good compromise for this frequency region. As for the corrugation depth, it is

reported [58] that the loss for the HEn1 mode is less than the TEol mode loss in a smooth walled

waveguide for corrugation depths in the range of 0.35 31/4 1.75 31/4.










For experiments conducted using this setup, the sample can be placed on a flat copper plate

that seals against the bottom of the corrugated tube such that the sample is enclosed by the small

circular opening at the end of the tube. This maximizes coupling of the microwaves to the

sample. No rotation of the endplate is possible, but the tube is designed for our magnet system

possessing a horizontal Hield. In this orientation, the entire probe (and thus the sample) may be

rotated with respect to the field, allowing one axis of rotation. Similar to the cavity setups, the

temperature is taken by a Cernox thermometer that is secured to the copper plate that holds the

sample.

Another advantage of the quasi optical setup is the possibility to conduct studies that

incorporate microwaves with other devices that are too large to fit inside a cavity. We have

developed a technique that allows devices such as a piezoelectric transducer or Hall

magnetometer to be placed near the end of the corrugated waveguide probe. In such a

configuration, a copper block holds that device, where a sample is placed on the surface. The

sample is then positioned such that it is just below (~ 1 mm) the opening where microwaves exit

the probe in order to maximize coupling of the radiation to the sample. In this manner, we have

conducted two unique experiments.

2.4.1 Piezoelectric Transducer Device

The first experiment involved a technique where a piezoelectric device creates short heat

pulses (pump) to drive the system from equilibrium and the microwaves act as a probe of the

system dynamics as it relaxes back to equilibrium. Hybrid piezoelectric inter-digital transducers

(IDTs) deposited on the 128 YX cut of LiNbO3 Substrates [59] were used in the experiments to

produce the surface acoustic waves. A picture of the device, with a sample placed on the

substrate centered between the IDTs, is shown in Fig. 2-9. We employed a special transducer

design [59] which yields devices capable of generating multiple harmonics with a fundamental










frequency of 1 12 MHz, and up to a maximum frequency of approximately 0.9 GHz. The

reflection coefficient, Sy;, was characterized in the frequency range of 100 500 MHz in order to

determine the optimum frequency to transmit the maximum amount of power to the IDTs. All

the experiments performed were done using the third harmonic at 336 MHz, which was

determined to be the optimum frequency to transmit the maximum power to the IDTs.

The 250 GHz probe was modified in order to incorporate a coax cable [60] that could

transmit the proper power to the IDTs. We used the proper electrical connections (SMA and

coax) for as large a length as possible so that the signal integrity would be maintained and thus,

the maximum amount of power would be transmitted to the IDTs. The improvised wiring

necessary to connect the coax cable to the piezoelectric device only constituted about 2.5% of the

total length the signal had to travel. A hole was machined in the top of the probe head that could

hold a hermetically sealed bulkhead SMA female to female connector. The 1.19 mm diameter

coax cable was run down the length of the tube to a pocket in the copper block holding the

device. A small piece of the outer conductor and dielectric were removed to expose the inner

conductor. From here a small breadboard piece, designed as a transition piece from the coax

cable to the device, was soldered onto the inner and outer conductor of the coax. The final

electrical connection was from the transition piece to the IDTs on the device with two 40 gauge

copper wires. A single crystal sample was placed directly on the IDTs, with a small amount of

commercial silicon vacuum grease used to attach the sample to the IDTs and ensure coupling of

the surface acoustic wave (SAW) to the sample. This piezoelectric device was mounted on a

copper block with a Cernox thermometer fixed approximately 1 cm from the device itself in

order to indirectly monitor the temperature variations of the system. The copper block holding

the device was attached to the end of our 250 GHz corrugated tube. The 1.7 mm opening where









radiation exits the tube was moved as close to the sample as possible (~ .10 mm away) and

centered over the sample to maximize coupling of the microwaves to the crystal. The entire

setup was placed in the Quantum Design superconducting magnet system.

In order to have the IDTs produce SAWs we had to pulse the device at the corresponding

MHz frequency. The device only transmits suitable amounts of power at frequencies that

correspond to multiples of 111 MHz. A Marconi Instruments [61] function generator capable of

frequencies between 80 k
power is needed to produce a SAW that will interact with the crystal. This instrument has two

modes of operation, continuous and pulse. The drawback to the pulse mode is that the output

power is limited to 3 dBm. In contrast, powers of up to 7 dBm are achievable in continuous

mode when working with the amplitude modulation option. This was the mode we used for our

experiments. A diagram of our equipment setup is shown in Fig. 2-10. In order to realize a

pulse with sufficient power we used an Agilent 81104A pulse pattern generator [62] to modulate

the continuous waveform from the Marconi function generator. The 81104A has two

independent channels, which can output separate signals. As an external modulation source,

channel one of the 81 104A supplied a 0.8 V (+0.4 V high, -0.4 V low) bipolar signal to trigger

the Marconi. This would in turn cause the IDTs to create a SAW pulse which would couple to

the sample and push the system out of equilibrium. Pulses from 5 ms to 50 ms were used. In

addition to triggering the SAW pulse, the pulse pattern generator also served another purpose,

which was to trigger the fast data acquisition card (DAQ) simultaneously. Channel two of the

81104A sent a 5 V (+5 V high, O V low) signal to trigger the DAQ card. In this experiment we

were measuring processes with timescales on the order of milliseconds or less. To this end, we

used an Acqiris APS 240 DAQ card [63] to collect data. The Acqiris APS 240 is a fully










programmable, dual channel instrument that is capable of 2 Gs/s real time averaging

performance. The signal that we wished to record originated from the receiver in the MVNA,

which can operate at a frequency of either 9 MHz or 34 MHz. This MHz IF signal was sent to a

Stanford Research Systems (SRS 844) high frequency lock in amplifier [64], which can work at

frequencies up to 200 MHz. The lock in-amplifier converts the MHz signal from the MVNA

into a DC signal that is directed to the data acquisition card. A lock-in amplifier exploits a

technique known as phase sensitive detection. The lock-in amplifier detects the MHz from the

MVNA at the respective frequency, which is the external reference frequency. Additionally, the

lock-in amplifier has an internal reference frequency. The two frequencies are combined and the

resultant signal is the product of the two sine waves. This consists of a component that is the

sum of the two frequencies, and one component that is the difference of the two frequencies.

This output is then passed through an internal low pass filter, which removes the high frequency

component. Under the condition that the two frequencies are equal, the component of the output

signal is the difference of the two frequencies, which is a DC signal. An internal phase locked

loop (PLL) locks the internal reference frequency to the external reference. Time constants for

the lock-in amplifier ranged from 30 Cps to 100 Cps for these specific experiments. After being

processed by the SRS 844, the signal was sent to the APS 240 and the system dynamics were

recorded by changes in the detected microwave radiation.

The simultaneous triggering of the Marconi and APS 240 could be done manually for a

single shot acquisition or repeatedly at some frequency for repeated sequences. For time

resolved measurements, especially when performing repeated sequences where the data are

averaged to a final result, it is critical that all instruments are phase locked to each other. To this

end we used a 10 MHz time-base from the MVNA to connect to an Agilent 33220A function










generator [62], which was in turn connected to the 81140A and SRS 844. It is necessary to

repeat the pulse sequence such that it is commensurate with the IF (receiver) frequency to

achieve a phase lock of the instruments. The MVNA receiver frequency is specified to a

precision of CLHz, and therefore it is necessary to use the function generator to phase lock the

MVNA to the other instruments, which are not capable of such frequency resolution. Since the

APS 240 and Marconi were both linked through the 81140A, this ensured a proper phase lock of

all the instruments to the MVNA.

The APS 240 records data in real time with 8 bit resolution and maximum rate of real time

averaging at 2Gs/s. In averaging mode the newly acquired waveforms are summed in real time

with the corresponding previous waveforms. The DAQ card starts recording data after it is

triggered. The corresponding samples are those with identical delay calculated from the

beginning of the trigger. While the acquisition is running, the previous sum is read from the

averaging memory, added to the incoming sample, and written back to the memory. The

processor is designed to perform on-board averaging at a maximum re-arm rate of 1 MHz. The

clock resynchronization time between successive triggers is ~ 200 ps. With a proper phase lock,

the signal should be the same for every trigger event, and thus the signal gets properly averaged

by the processor. Without a proper phase lock the DAQ card will start recording data at non

synchronous times during the IF signal period and thus the averaged signal would go to zero.

2.4.2 Hall Magnetometer Device

The second experiment investigated the influence of microwave radiation on the

magnetization of a sample, which was recorded with a Hall magnetometer. Like the

piezoelectric device, the Hall magnetometer device was mounted securely in the socket of an

electrical pin connector, which was fastened into a copper block that mounts on the end of our









250 GHz quasi optics probe. A picture of the Hall device, with a sample placed near one of the

Hall crosses, is shown in Fig. 2-11.

A single crystal sample is placed on the surface of the magnetometer such that it sits

slightly off center from the active area (Hall cross) of the device (~50 Cpm x 50 Cpm). This

ensures that the maximum amount of the dipolar field from the sample will pierce the Hall cross

(active area of the device). The 1.7 mm opening where radiation exits the tube was moved as

close to the sample as possible (~ .10 mm away) and centered over the sample to maximize

coupling of the microwaves to the cry stal.

In order to generate a Hall voltage we use a low frequency AC current (~ 500 Hz) on the

order of 1-2 CIA. We use a Stanford SRS 830 lock-in amplifier [64] to both generate the AC

current and detect the Hall voltage at the same measurement frequency. The lock-in technique is

used since it enhances the sensitivity of the detection of small voltage signals (on the order of CLV

or less). This relies on the extremely narrow bandwidth phase sensitive detection mentioned

earlier.

The lock-in amplifier is set to a sinusoidal output voltage of 1 V,, at the desired frequency.

In series with the device is a 500 kR resistor, which helps provide a steady output current of ~ 2

IA. Currents much larger than 1-2 CIA can damage the device, and low frequencies minimize

the capacitive coupling between the leads that can develop at higher frequencies. At low

frequencies a capacitor acts like an open circuit, as no current flows in the dielectric. Driven by

an AC supply a capacitor will only accumulate a limited amount of charge before the potential

difference changes sign and the charge dissipates. The higher the frequency, the less charge will

accumulate and the smaller the opposition to the flow of current, which introduces capacitive









coupling. Twisted wire pairs are used to minimize low frequency inductive coupling. The

expression for the Hall voltage in two dimensions is given by

iB
yH (2-6)


In Eq. 2-6, i is the current, B is the transverse component of the magnetic field, 722D is the

charge carrier density in two dimensions, and e is the charge of the carriers. The basic principle

behind the Hall effect is the Lorentz force on an electron in a magnetic field. When a magnetic

field is applied in a direction perpendicular to a current carrying conductor, the electrons that

constitute the current experience a force due to the magnetic field. This force causes the

electrons to migrate to one side of the conductor, and hence, an electric field and an

accompanying potential difference develop between opposite sides of the conductor.

Consequently, there is also an electric force experienced by the electrons. In equilibrium the

magnetic and electric forces on the electrons will balance. Eq. 2-6 gives the formula for the Hall

voltage in a 2D system such as our device.

2.5 Hall Magnetometer Fabrication

The Hall magnetometers used in our studies were fabricated at the University of Central

Florida in the research lab of Dr. Enrique del Barco using optical lithography and chemical

etching techniques. The device is composed of a Gallium-Arsenide / Gallium-Aluminum-

Arsenide (GaAs/GaA1As) heterostructure, with the active area being a two dimensional electron

gas (2DEG) that resides roughly 100 nm below the surface of the wafer. This material is ideal

for Hall sensors due to the small carrier density, n, which can be manipulated during the growth

process of the heterostructure by changing the number of dopants. Typical values for the carrier









density in the 2DEG is on the order of 5 x 1011 cm-2. Low carrier densities lead to larger Hall

coefficients (proportional to 77- ), and thus larger response voltages from the device.

Additionally, the mobility of the 2DEG is on the order of 6 x 104 Cm2-V- ~-s- at cryogenic

temperatures, which is large compared to most semiconducting materials [65] with the exception

of those containing Indium. Since the GaAs and GaA1As have different band gaps, the carriers

become trapped in an approximately triangular potential barrier at the interface between the

materials. Thus, the conduction band forms a two dimensional sheet at the interface of the GaAs

and the GaA1As [66, 67]. Above the 2DEG sheet is the undoped layer of GaA1As, followed by a

layer of GaA1As that is doped with Silicon. The Si atoms act as electron donors for the 2DEG.

Finally, there is a layer of GaAs above the Si doped GaA1As layer that prevents oxidation of the

Al.

We begin the fabrication process by carefully cutting a square piece (typical size of 8 mm

on a side) from the GaAs/ GaA1As wafer with a diamond scribe. This piece must then be

cleaned in an ultrasound bath. We place it in a plastic beaker filled with acetone and clean for 5

min. This is then repeated for isopropanol and ethanol for 5 min each. Next, we glue the square

wafer to a clean microscope slide with a small drop of Shipley S1813 photoresist [68] for easier

handling during the remainder of the fabrication process. The slide and wafer are baked for 15

minutes at 120o C directly on a hot plate. After cooling, the microscope slide is placed on a

spinner that is designed to coat thin organic films on various substrates and is held securely in

place by a vacuum line underneath it. The entire square wafer is coated with the S1813

photoresist and then spun at 500 rpm for 2 seconds, followed by 5000 rpm for 30 seconds. This

creates an approximately 1.2 Cpm layer of the photoresist on the wafer. The wafer is then baked

again for 2 minutes at 1200 C to harden the photoresist.









The next steps involve the mask that is imprinted with the design for the Hall cross and

ohmic contacts. The layout for our Hall magnetometers consists of three crosses with an active

area of 50 Cpm x 50 Cpm, spaced approximately 1 mm apart. There are eight rectangular areas (~1

mm x 2 mm) for ohmic contact pads. Two pads function as the current (voltage) leads for each

Hall cross, and the other six are pairs for each of the crosses to measure the voltage (current).

This configuration allows not only for measurements of up to three different samples in one

setup, but also for use of any one cross individually should the others become damaged. The

mask pattern is carefully aligned and positioned with respect to the wafer using a microscope and

mask aligner. Then the wafer is exposed to UV radiation (~ 260 Watts) for 9 seconds and then

placed in a solution of CD 26 developer [68] for ~ 45 seconds followed by de-ionized (DI) water.

The UV exposure sets the photoresist with the exception of what was covered by the mask

pattern. The developer then removes the portion that was not exposed, while the DI water stops

the developing of the photoresist. What emerges is the desired Hall cross pattern on the wafer,

and now the wafer is ready to etch. The etching compound is a mixture of DI water, hydrogen

peroxide, and sulfuric acid in a 160:8:1 ratio. We place the wafer in this mixture for 30 seconds

and immediately remove and place in pure DI water to stop the etching process. Now the wafer

has been etched down to the 2DEG region of the structure (~1 10 120 nm), but only in the

regions not covered by the photoresist. We finish the first round of fabrication by removing the

remainder of the photoresist by placing the microscope slide in acetone, isopropanol, and ethanol

for ~ 1 minute each.

The second round of the fabrication involves a double layer photoresist, or undercut,

method. While the single layer process (described in the first round of fabrication) is sufficient

for the final sensor, this procedure allows for a better removal of the metals used for the ohmic









contact pads from the rest of the device after the chemical vapor deposition process. We begin

by placing the slide in the spinner and coating the wafer with LOR-3A photoresist [68]. This is

spun at 500 rpm for 2 seconds followed by 3000 rpm for 30 seconds. We then bake at 1750 C for

5 minutes to harden the photoresist. After cooling we place the slide back in the spinner for the

second layer, which consists of S1813 photoresist, spun at 500 rpm for 2 seconds and 5000 rpm

for 30 seconds. This is baked for 5 minutes at 1200 C. Now the wafer is ready for exposure

again. The same mask and aligner are used to position the Hall cross pattern on the wafer, which

is then exposed to UV radiation for 9 seconds, just like before. The slide is then placed in a

solution of CD 26 developer for ~ 45 seconds followed by DI water. This removes the first layer

of photoresist from the mask pattern area. We next bake the wafer at 1300 C for 5 minutes,

followed by development in the CD 26 solution for ~ 60 seconds and then DI water.

This added step removes the second layer of photoresist, and since the LOR-3A has a higher

dissolution rate in the developer than the S1813, the resulting photoresist profile will have an

undercut.

Now the sample is ready to deposit the metal for the ohmic contacts. This procedure is

carried out inside an ultra high vacuum electron beam evaporation chamber, where we deposit a

5 nm thickness of chromium, followed by a 100 nm thickness of a Au/Ge alloy (88%/12%), and

finally a 50 nm thickness of Au on top. To remove the metal layers residing away from the

ohmic contact pads we immerse the wafer in PG remover [68], which is a solvent stripper

designed for complete removal of resist films on GaAs and many other substrate surfaces.

The process of removing the photoresist can be time consuming, and one should avoid

an ultrasound bath as this can remove the metal from the contact pads. Instead we employ a

careful method of stirring the solution to create friction between it and the photoresist, until all









unwanted material is removed. The wafer is then carefully cleaned in separate baths of acetone,

isopropanol, and ethanol without ultrasound.

The final part of the fabrication process involves annealing the ohmic contacts, so that the

Ga atoms diffuse out of the wafer and Ge atoms diffuse into the conducting 2DEG layer, thus

creating an electrical conducting path. The annealing is done inside a small chamber that has a

continuous flow of helium gas which prevents oxidation of the metals during the annealing

process, and the wafer is placed directly on a highly resistive wire mesh grid. The temperature

of the wafer is monitored by a thermometer that is directly and carefully attached to it. The three

step process consists of heating the wafer to 1 10o C for 60 seconds, followed by 2500 C for 10

seconds, and finally 4100 C for 30 seconds. Once the wafer is cooled, the external wire

connections can be made. For stability and practicality, the entire device was glued onto a 28 pin

connector with an area (15 mm x 15 mm) approximately the size of the wafer using an insulating

commercial epoxy [69] (Stycast 1266). Wire contacts were made between the ohmic contact

pads on the device and the leads on the pin connector with gold wire (0.05 mm diameter) and a

conducting (room temperature resistance of < 4 x 10-4 Ohms) commercial epoxy [70] (EPO-TEK

H20E).

When cooled in zero field, SMMs will have a net magnetic moment of zero. But an

external field will bias the system such that it can develop a significant magnetic moment at low

temperatures. The magnetic moment has an intrinsic dipole field associated with it that

constitutes the magnetic field to induce a Hall voltage. For our experiments conducted with this

device, we are able to position our sample such that its magnetization vector is parallel to the

plane of the sensor and the perpendicular component of the dipole field from the sample pierces

the active area of the device. Additionally, we apply an external magnetic field parallel to the









direction of the sample' s magnetization vector in order to bias the sample. Under ideal

circumstances, the external Hield does not influence the Hall voltage signal and the only

contribution is the perpendicular component of the dipole Hield from the sample. However, in

practice there is always a small background signal coming from the external field due to small

misalignments. Fortunately, this background can easily be subtracted off to isolate the response

from the sample. The background from the external Hield is linear and superimposes itself upon

the true response from the sample. We fit a straight line to the data, and subtract this to get the

sample response. It is not possible to measure absolute values of the magnetization with this

method, but relative values provide sufficient information. By dividing the data set

(normalizing) by the saturated magnetization value (that at + 3 T), we obtain a measure of the

magnetization of the sample relative to the maximum magnetization value. In this way we are

able to measure the magnetization of a sample under the influence of an external magnetic Hield

as well as microwave radiation. Experiments using the techniques outlined in sections 2.4.1 and

2.4.2 are discussed in detail in Ch. 6.

2.6 Summary

In this chapter we discussed the experimental techniques and equipment we use for our

research. First, we briefly discussed EPR in the context of SMM systems. Next, we explained

the two types of cavities we use for a cavity perturbation technique and the instrument that acts

as our microwave source and detector. This source, in conjunction with an external Gunn diode

and complementary components (amplifier, switch, frequency tripler) allow for high power

pulsed microwaves. Additionally, our quasi optical bridge setup provides an incredibly low loss

propagation system that relies on coupling of a free space Gaussian profile to an HEn1 mode in

corrugated horns and a tube. The corrugated sample probe is used to conduct experiments that

integrate other devices into the setup and complement our normal EPR studies by combining










microwaves with either surface acoustic waves or Hall magnetometry to conduct unique

experiments on single crystals. Finally, we outlined the process of fabricating the Hall

magnetometer used in our experiment in section 6.3.









~IC~

X


~;55
O

Q)
3
U '
d)
k
F4


0 1 2 3 4 5 6 7 8 9 10

Magnetic Field (Tesla)


Figure 2-1. Plot of the energy levels in the Ni4 SMM with its easy axis aligned along the
external field. For a fixed frequency of 172 GHz, there is a strong absorption of the
signal each time the resonance condition (energy difference between adjacent levels
matches 172 GHz) is met as the magnetic field is swept.















1;/3 m--4to-3
CS=4

~ rm 0 to I


~~m, -1 to 0o


0 102 11 1






Magnetic Field (Tesla)

Figure 2-2. A normal EPR spectrum from the system in Fig. 2-1. As the magnetic field is
swept, sharp inverted peaks appear in the transmission spectrum. Each peak
corresponds to a transition between spin states within the ground state spin manifold,
and the decrease in transmission signal is due to the absorption of microwave
radiation by the spins within the sample. The peak positions match those shown in
Fig. 2-1. With respect to the larger intensity transitions, we see 2S peaks,
corresponding to 2S transitions between 2S+1 states. The additional, smaller intensity
peaks in the range of 4-7 T, which are attributed to transitions within higher lying
spin multiplets, are also discussed in Ch. 4. The change in the background signal
seen in the 4-7 T range we attribute to broadening of the resonance peaks due to
closely spaced energy levels within multiple higher lying spin multiplets.












il


B1xfor


B
Quantum D~esign


Vertical


Rotating
Cavity


- Solenoid


Figure 2-3. A schematic diagram of our typical experimental setup, including a waveguide
probe and sample cavity inserted in the bore of a superconducting magnet immersed
in a cryostat. Either cavity sits in the center of the magnetic field, which is oriented
either vertically or horizontally, depending upon the system in use. Reused with
permission from Monty Mola, Stephen Hill, Philippe Goy, and Michel Gross, Review
of Scientific Instruments, 71, 186 (2000). Fig. 1, pg. 188. Copyright 2000, American
Institute of Physics. Reused with permission from Susumu Takahashi and Stephen
Hill, Review of Scientific Instruments, 76, 023114 (2005). Fig. 2, pg. 023114-4.
Copyright 2005, American Institute of Physics.









QD System
Two axis rotation

Z aXIS
rotation


Oxford System
One axis rotation


transverse


Figure 2-4. A schematic diagram of the rotational capabilities of each magnet system. A
rotating cavity in the Oxford system allows for one axis rotation, while the same
cavity in the Quantum Design system allows for two axis rotation since the field is
aligned in the xy plane.


B
axial





Figure 2-5. A diagram of a free space Gaussian beam propagating along the : axis. The beam
has a minimum waist, wo, that corresponds to the point of maximum intensity. The
divergence of the beam is inversely proportional to the beam waist, and faster
diverging beams necessitate larger surface mirrors in order to refocus the beam. For
distances far from the point of minimum beam width, the beam width approaches a
straight line. The total angular spread of the beam is 28.














































Figure 2-6. A schematic diagram of our quasi optics equipment. The microwaves are
propagated as a Gaussian profile in free space and as a HEn1 mode in a corrugated
probe. First, the signal encounters a 450 polarizer, which splits the signal and the
reflected beam is polarized at 450 with respect to the horizontal. This meets the first
elliptical focusing mirror, which reflects the diverging beam such that the minimum
beam waist is 25 cm from this point. The 180o wire grid polarizer changes the
polarization of the signal so that it is now vertically polarized. A second mirror is
located 50 cm from the first mirror (25 cm from the minimum beam waist). Again
the purpose of the mirror is to focus the beam profile so that it will have its' minimum
beam waist as it enters the corrugated waveguide probe, 25 cm below the mirror.
Before entering the detector, the signal passes thorough the first wire grid polarizer,
where the polarization is changed for a final time so that it will be correctly oriented
with respect to the detector horn (450 from vertical).










Reflected from
450
wire gIrid po'larite~r (3)


Tran~smnited through
450
wire grid polalrizer (5)


Defe'Cc)tr Horn (6)


Figure 2-7. Illustration of the signal polarization as it changes from interactions with the
respective components of the quasi optical setup. These changes are necessary since
the polarization of the signal reflecting off the first wire grid polarizer must be
changed after interacting with the sample and before entering the detector. If the
second wire grid polarizer was not there, the signal returning to the first wire grid
polarizer would be totally reflected before it enters the detector. The insertion of the
second wire grid polarizer changes the polarization of the signal so that it will pass
through the first wire grid polarizer and reach the detector.


Source Horn (2)


Waveguide (1)

I E I










TIransmlli tted th rouglh
180
wire grid polarizer (4)








t'


1"


JY:


Figure 2-8. A diagram of the TEn1 and HE11 modes in a circular, corrugated waveguide. A non
corrugated circular waveguide couples the Gaussian profile to both the TEn1 and TM11
modes which progress at different phase rates and can result in a distorted output
beam. An ideal HEn1 mode has identical E plane and H plane radiation patterns with
a main lobe that is approximately Gaussian, which is the reason for the excellent
coupling between this mode and the free space Gaussian profile.


HE,, Mode


TEI Mode












































Figure 2-9. The piezoelectric device used in our experiments. A coax cable was run down the
length of the tube to a pocket in the copper block holding the device. A small piece
of the outer conductor and dielectric were removed to expose the inner conductor.
From here a small breadboard piece, designed as a transition piece from the coax
cable to the device, was soldered onto the inner and outer conductor of the coax. The
final electrical connection was from the transition piece to the IDTs on the device
with two 40 gauge copper wires. A single crystal was placed directly on the IDTs,
with a small amount of commercial silicon vacuum grease used to attach the sample
to the IDTs and ensure coupling of the surface acoustic wave (SAW) to the sample.
















SRS 844
External
Input Output Reference


Agilen t 81140A
1-.neanal
Refe2rence Ch. 1 Ch. 2


Marconi

External
M~odulatlio~n Slgnal out


MCVNA

Reference


Agilent 33220A





Reference


APS 240


Clh. 2 -


To IDT's


Figure 2-10. Schematic diagram of the electronic equipment used in our avalanche experiments.
The MHz IF signal was sent to a Stanford Research Systems SRS 844 high frequency
lock in amplifier. After being processed by the SRS 844, the signal was sent to the
APS 240. For optimized performance, we would use the 81104A to trigger the
Marconi and APS 240 simultaneously with a 5 V TTL signal. The trigger would in
turn cause the IDTs to create a SAW pulse which would couple to the sample and
push the system out of equilibrium. The dynamics were measured with the incident
microwave radiation and were recorded with the APS 240. For time resolved
measurements, especially when performing repeated sequences where the data are
averaged to a final result, it is critical that all instruments are phase locked to each
other. To this end we used a 10 MHz time-base from the MVNA to connect to an
Agilent 33220A function generator, which was in turn connected to the 81140A and
SRS 844. Since the APS 240 and Marconi were both linked through the 81140A, this
ensured a proper phase lock of all the instruments to the MVNA.


34 Alliz IF


L


I


-


Tri g er in











































Figure 2-11. The Hall device used for our magnetometry measurement. The sample sits on the
surface of the device and the perpendicular component of the dipole field (piercing
the Hall cross) constitutes the field necessary to induce a Hall voltage.









CHAPTER 3
THEORETICAL BASIS OF THE SPINT HAMILTONIAN

In the previous chapter we have described the experimental techniques and

instrumentation necessary for our HFEPR measurements of SMMs. Now we will focus our

attention on the mathematical basis for modeling our systems of interest and briefly discuss the

construction of a proper mathematical model. We will outline the two most widely used

approaches to describing SMM systems and the approximations and limitations of each

approach.

3.1 Two Versions of the Spin Hamiltonian

In order to properly describe a system of interest, it is necessary to construct the relevant

quantum mechanical Hamiltonian that characterizes the energy spectrum of the system. Once

the Hamiltonian is diagonalized it must produce the proper energy eigenvalues for a given spin

multiple. A simplified approach is used, by using a spin Hamiltonian to model the experimental

data for SMMs. This redefines the orbital momentum contributions needed to characterize the

system, and relies only on spin contributions and symmetry properties of the system. The spin

Hamiltonian considers only the spin angular momentum operators and their interactions with

each other and an external magnetic Hield. The spin orbit interaction is parameterized in

constants of the model, which are essential to the magnetic anisotropy in SMMs.

Any spin Hamiltonian must satisfy certain basic requirements for it to be a valid model.

First, it must properly express the point symmetry of the molecule and the lattice, and must be

invariant under all point symmetry operations [10, 11] and maintain the space group symmetry.

Second, it must be Hermitian since the eigenvalues (energies) of the system are real. Finally, for

cases of practical interest, only even powers of the spin angular momentum operators, starting

with the quadratic term, are included in the zero Hield terms of the Hamiltonian. Thus, the spin









operators are invariant under time reversal. Of course the Zeeman term includes an angular

momentum operator of first order, but this reflects the fact that an external magnetic field breaks

time reversal symmetry [7] since the current density responsible for the external magnetic field

reverses under the time transformation operator.

3.1.1 The Giant Spin Hamiltonian

The giant spin Hamiltonian contains the Stevens operators obtained by using the operator

equivalent method of finite groups [71]. This considers the crystal field potential (from

molecules within a crystal) of appropriate symmetry and expands the operators like spherical

harmonics functions. The spherical harmonic function is expressed in terms of the quantum

mechanical spin angular momentum operator for a given S multiple [11, 12].


VCF= B.Ok~m (S. S (3 -1)
k=0,2,4 m=0


In Eq. 3-1, Gkm is the spin operator of power k and symmetry m (m = 0 is axial, m = 2 is


rhombic, m = 4 is tetragonal, etc.), and B'1 is the coefficient for the respective spin operator.

This is the starting point for the derivation of the spin Hamiltonian commonly used by both

physicists and chemists in the SMM community. As previously stated in Ch. 1, SMMs are

polynuclear complexes consisting of transition metal ions as the magnetic components. For most

transition metal ions the spin orbit coupling interaction is relatively small and quenching of the

orbital contribution to first order in perturbation theory allows one to express the states of the

systems as pure spin multiplets. Thus, we assume that S is a good quantum number. Within a

given S multiple the anisotropy terms lift the degeneracy of the spin proj section states, which are

expressed as m, All interactions that take place in the absence of an external magnetic field

are referred to as zero field splitting. The combined effects of crystal field symmetry and the









mixing of excited levels into the ground state through second order (or higher) perturbation

theory lifts the degeneracy of the spin multiplets and is the source of the zero field splitting [72].

The spin Hamiltonian can be expressed as

H=S*b*9+pB-g-3S (3 -2)

The first term contains all of the zero field interactions within the D tensor. The second term

(the Zeeman interaction) lifts the degeneracy of each Im,) state with an external field, B. The

Lande factor, g, which is a scalar for a free electron, becomes a tensor in the presence of spin-

orbit interactions. In Eq. 3-2 we have neglected the hyperfine interaction term, which can be

excluded in strongly exchange coupled systems like SMMs due to the delocalization of electrons

making this a rather small effect [3].

Eq. 3-2 can be simplified if we assume the principal axes of the D tensor coincide with the

molecular coordinate axes. Then only diagonal terms contribute to the product and

5' i.= o D 2 +D 2' +D 2? (3-3)


Now we introduce a new set of parameters


D = (-D' D + 2D) (3-4)

2""W


E = (D' -D )D (3-5)




So Eq. 3-3 is equivalently expressed as

5' i.9= D~ 2+E(S 2' 2+ K 2 (3-7)









If we express Eq. 3-7 in terms of Eq. 3-1 we get

s f) s = B,:O, + Bf Of + B220~2 (3-8)

by using the explicit form of the Stevens operators [71]. Now we can relate the parameters in

Eq. 3-7 to the new parameters in Eq. 3-8.

D = 3B~ (3-9a)

E 2 (3-9b)

K = -Bi (3-9c)


92 = O (3-9d)


S2- 2 2~ (3-9e)


~2 = OU (3-9f)


Since in Eq. 3-7 KS2 IS a constant that uniformly shifts the energy levels, it can be omitted from

the Hamiltonian because in EPR experiments we measure differences between energy levels.

Finally we arrive at the spin Hamiltonian expressed in its most common form in the SMM

community

fl = D 2 + E( ~S2 2) pUBB J (3-10)

Symmetry plays a critical role in the formulation of the correct Hamiltonian, and

determines which terms may be allowed. For a system with perfectly spherical symmetry, no

zero field anisotropy terms will be allowed since the system is completely isotropic in any

direction. If we take a point at the center of the sphere as the origin of our coordinate system,

then any radial distance from the center, at any angle, is energetically equivalent. Hence, all









anisotropic terms describing the system are zero. For a slightly lower symmetry system, such as

one possessing cubic symmetry, no second order anisotropy terms are allowed. This can be seen

by taking the origin as the center of the cube. All side lengths and angles of a cube are identical

and a rotation of 900 leaves the system unchanged. However, fourth order anisotropy terms are

allowed. Since a rotation of 450 in any one of the three orthogonal planes (xy, x:, yz) will

change the magnitude of the position vector with respect to the center, this permits a fourth order

anisotropy term. In general, angles of z/n will be those that correspond to the periodicity of the

anisotropy term, where n is the order of the term. If we distort the cube slightly such that it

compresses or elongates along one axis, then a second order axial anisotropy term develops.

Now the geometry has changed to a square based rectangle, and along the axis of distortion there

is a two fold symmetry pattern since a rotation of 900 changes the magnitude of the position

vector with respect to the center once again. Finally, we can distort the previous geometry in the

plane perpendicular to the first distortion axis, and now second order transverse anisotropies are

allowed in addition to the fourth order terms.

As discussed above, depending on the symmetry of the system, it may be necessary to


include higher order terms in Eq. 3-10. These terms take the form C EB"'Ok"(S,S.) For
k=4,6,8 #;=0

the sysPtems wei haveP studI;T7ied wenlyr included terms up to thep order of BJ 4(^;(,S+) While


higher order terms may be present, certainly the second and fourth order terms are dominant and

it becomes hard to resolve the effect of these higher order terms. When allowed by symmetry


the transverse anisotropy term (7 (9 ,() is written (S + Other situations can arise where


the E term is forbidden by symmetry and the lowest term allowed is a fourth order term. This is

dependent upon the symmetry of the system in question. The most important requirement for









SMM behavior, however, is a significant D value whose sign is negative. Thus, some kind of

axial elongation of the system symmetry is necessary to have a SMM.

A maj or assumption imbedded in this giant spin model is that the system obeys the strong

exchange limit. In this limit, the isotropic Heisenberg exchange coupling that splits the different

S multiplets is much larger than any anisotropies that split the levels within a given multiple. In

other words, the contributions from the non Heisenberg interactions to the energy spectrum are

much less than those from the Heisenberg exchange interaction. In fact, this model applied to

SMMs assumes that the ground state spin multiple is perfectly isolated from any higher lying

spin multiplets. In this case, the coupling strength between the metal ions in the cluster is

infinite and the magnetic moment vector for the ground state is perfectly rigid. This is

commonly referred to as the giant spin approximation (GSA). The advantage of the GSA is the

relative simplicity. The matrices that constitute the Hamiltonian are formed in a basis where S,

the spin value of the ground state multiple, is a good quantum number. All the eigenstates of the

system are expressed as linear combinations of the basis states (spin proj section states along the

quantization axis) Im, -

Of course in reality the exchange coupling always has some finite value and the energy

levels of the ground state have the possibility to be influenced by higher lying S multiplets. In

situations where the exchange and anisotropy interactions are comparable the GSA starts to

reach its limitations. In Ch. 4 we discuss a system that can be modeled quite well by the GSA,

but it requires higher order terms that are unphysical in the sense that they exceed the highest

order allowed (second) by the single ions constituting the molecule. Therefore, the GSA can

often model a molecule of interest quite well, although the maj or limitation is the fact that it









provides no information as to how the single ions that constitute a molecule influence the

magnetic properties of the cluster as a whole.

3.1.2 Coupled Single lon Hamiltonian

The coupled single ion Hamiltonian includes the exchange interactions between metal ions

and is necessary when considering single ion contributions to the cluster anisotropy. There are

numerous interactions that take place between metal ions within a molecule, such as spin orbit

(both with themselves and between others) and dipolar couplings. Due to the organic ligands

surrounding the magnetic core, dipolar interactions in many SMMs are negligible in comparison

to the isotropic and anisotropic energy scales, and can therefore usually be omitted. However,

exchange interactions within a molecule are dominant since SMMs are polynuclear molecules,

and the nature of the exchange interaction between ions within the molecule cluster is important.

There are a number of different exchange interactions, but the dominant one in SMMs is

superexchange, which is the coupling of localized magnetic moments through diamagnetic

groups in an insulating material [10]. The general form of the superexchange Hamiltonian is

expressed in Eq. 3-11.

H = Hnsoec +Hecag (3-11)

First, we consider the anisotropy contained within a single metal ion due to its own spin

orbit coupling. This is, of course, the same interaction that gives rise to the zero field anisotropy

terms in the GSA Hamiltonian, only now we consider the single ion anisotropies as opposed to

the molecular anisotropies (which are the tensor projections of the single ion anisotropies). The

lowest order terms allowed for a spin i are:


Hi,,,, = dS + e,(S S ) (3-12)









Next, we consider the exchange interactions between two ions in a molecule.

exchange =CCS~ -T -SC (3-13)

In Eq. 3-13, T,, is the general spin-spin interaction tensor. This can be separated into three

different interactions, which we express in Eq. 3-14.

Hadag = Histo +H,,,,en +Han~yaen (3-14)

The first term is the dominant isotropic Heisenberg interaction between spin vectors, which

is given by Eq. 1-1. Here J,, is the magnitude of the exchange interaction (positive for

antiferromagnetic coupling and negative for ferromagnetic coupling) between spin i and spin j.

The final two contributions to the superexchange are the symmetric and antisymmetric

interactions. The symmetric part of the exchange is given by

symte =C S A SI (3-15)


In Eq. 3-15, A, represents the symmetric anisotropy tensor between two respective spins, i and

j. The symmetric exchange is also known as a pseudodipolar effect [73].

The antisymmetric exchange, or Dzyaloshinskii-Moriya (DM) interaction, is expressed as

a vector product between two spins, i and j.


anityll~illeti c~ f E1G ( x S ) (3-16)


In Eq. 3-16, G~ is a constant vector connecting the two spins of interest whose orientation

depends on the symmetry of the molecule. The physical origin of this vector is the same

interactions that give rise to the other anisotropic constants such as D, E, and g. Namely, this is









the mixing of excited states into the ground state due to spin orbit coupling [74, 75]. Moriya [73]

was the first to note that this expression is the antisymmetric part of the most general expression

for a bi-lateral spin-spin interaction.

For the systems discussed in Ch. 4 and Ch. 6, the symmetric and antisymmetric

interactions are negligible with respect to the more dominant interactions and are omitted from

the Hamiltonian. This is true with respect to ions with orbitally nondegenerate ground states

(pure spin multiplets) [76]. However, for the system in Ch. 5, it is clear that the ions have

orbitally degenerate ground states, and therefore will have a contribution to the magnetic

properties from symmetric and antisymmetric exchange interactions.

In the case of the system in Ch. 5, we are dealing with four effective S' = '/ Co+2 ions that

when coupled together produce an energy spectrum showing significant zero field splitting.

Since a system with S' = 1/ has no axial or transverse anisotropy, we turn to the symmetric and

antisymmetric exchange interactions as the dominant source of the observed zero field splitting

anisotropy for this system. Based on the symmetry of the system in question, some components

of the antisymmetric exchange tensor can vanish, which simplifies calculations. A careful

discussion of this interaction will be done in Ch. 5.

Moriya' s contribution was the extension of the Anderson theory [77] of superexchange to

include spin orbit coupling. He calculated that the magnitude of the symmetric (A,) and

antisymmetric (G,) parameters can be related to the isotropic coupling constant (J) by the

following relations [73]


J (3-17a)











~ J (3-17b)


In Eq. 3-17, Ag is the shift in g factor from the free electron value due to the spin orbit

interaction. Moriya argued that since the symmetric part of the exchange is second order in the

spin orbit coupling scheme, it could be neglected. The dominant contribution to the exchange

interaction would then be the antisymmetric part, since it is first order in the spin orbit coupling

scheme. However, this assumes that the g anisotropy of the system is marginal. As we discuss

in section 5.2, we have Co+2 ions with a large amount of g anisotropy. For the z component of

the g tensor, we get a value of 7.8. Inserting this into Eq. 3-17, we see that the symmetric

exchange interaction is ~74% of the antisymmetric exchange interaction, and should not be

neglected.

The coupled single ion approach is favorable to the giant spin approximation since it

provides more information about the exchange interactions between ions within a molecule and

how the single ion anisotropies couple together and proj ect onto the molecular anisotropy.

However, from a practical standpoint, the coupled single ion Hamiltonian is not always readily

solvable. The Hilbert space for a system of interacting ions, and thus the size of the Hamiltonian

matrix, goes like the product of (2S, + 1)" where Nis the number of particles with a given spin,

S,. For systems like Mnl2Ac with twelve ions in mixed valence states, this problem becomes

incredibly complicated due to the size of the Hilbert space (10s x 10 ). This is the limitation of

this model. However, for certain systems, this approach is quite reasonable to use. We present

work in Ch. 4 that models a system consisting of four s = 1 particles using a coupled single ion

Hamiltonian.










3.2 Summary

In this chapter we discussed the theoretical origin of the Hamiltonians used to model SMM

systems. The Hamiltonian corresponding to a giant spin approximation is certainly the most

widely used form and applies to ideal systems where the isotropic exchange interaction is much

larger than the anisotropic interactions. For systems where this assumption starts to break down,

we can employ a coupled single ion Hamiltonian that in spite of being more complicated, can

provide information on the nature of the exchange interactions between metal ions in a molecular

cluster.

In the following chapters we will present work on single molecule magnets where the GSA

is valid and describes the system quite well (Ch. 6), as well as systems where the GSA breaks

down (Ch. 5) or must be modified to account for the observed behavior (Ch. 4).









CHAPTER 4
CHARACTERIZATION OF DISORDER AND EXCHANGE INTTERACTIONS INT
[NI(HMP)(DMB)CL]4

The results presented in this chapter can be found in the articles, Single M~olecule

Magnets: High Field Electron ParamagneticP~~PP~~~PP~~PP Resonance Evaluation of the Single lon Zero-Field

Interaction in a Zn3IINH Complex, E.-C. Yang, C. Kirman, J. Lawrence, L. Zakharov, A.

Rheingold, S. Hill, and D.N. Hendrickson, Inorg. Chem. 44, 3827-3836 (2005), Origin of the

Fast Magnetization Tunneling in the Single M~olecule Magnet[Ni(hmp) (t-BuEtOH)Cl]4, -.

Kirman, J. Lawrence, S. Hill, E.-C. Yang, and D.N. Hendrickson, J. Appl. Phys. 97, 10M501

(2005), Magnetization Tunneling in High Symmetry Single M~olecule Magnets: Limits of the

Giant Spin Approximation, A. Wilson, J. Lawrence, E.-C. Yang, M. Nakano, D. N. Hendrickson,

and S. Hill, Phys. Rev. B 74, 140403 (2006), and Disorder andlntermolecular Interactions in a

FamFFFFFFFF~~~~~~~~~ily of Tetranuclear Ni(II) Complexes Probed by High Frequency Electron Paramagnetic

Resonance, J. Lawrence, E.-C. Yang, R. Edwards, M. Olmstead, C. Ramsey, N. Dalal, P.

Gantzel, S. Hill, and D. N. Hendrickson, Inorganic Chemistry, submitted.

4.1 The Tetranuclear Single Molecule Magnet [Ni(hmp)(dmb)Cl]4

A series of SMM compounds have recently been synthesized, each possessing as its

magnetic core four Ni+2 ions on opposing corners of a slightly distorted cube [25]. Extensive

studies have been conducted to characterize the basic spin Hamiltonian parameters, as well as the

disorder and intermolecular exchange interactions present in each system [25, 78], and the

system that gives the nicest results from an EPR perspective is the compound

[Ni(hmp)(dmb)Cl]4, where dmb is 3, 3-dimethyl-1 -butanol and hmp- is the monoanion of

2-hydroxy-methylpyridine. A diagram of the molecule is shown in Fig. 4-1. The bulky aliphatic

groups that surround the magnetic core help to minimize intermolecular interactions, as will be

discussed.









Each Ni+2 ion has a spin of s = 1 and the four Ni+2 ions couple ferromagnetically [25] to

give a spin ground state multiple of S = 4. The aspects that make this compound quite attractive

are the sharp EPR peaks and high symmetry (S4) Of the molecule. However, in spite of the high

symmetry, this compound displays extremely fast ground state tunneling of the magnetization

[25, 79]. This observation was a maj or motivation for the work done to characterize the single

ion anisotropies [80], the magnitude of the isotropic Heisenberg exchange interaction, and how

relating these two can account for the observed fast quantum tunneling of the magnetization

(QTM) [81].

This complex consists of a diamond-like lattice ofNi4 mOlecules. The ligands associated

with a Ni4 mOlecule are involved in Cl---Cl contacts with four neighboring Ni4 mOlecules. The

shortest Cl---Cl contact distance between neighboring molecules is 6.036 A+, which is the largest

such distance in a family of similar Ni4 COmplexes exhibiting measurable intermolecular

exchange [25, 82]. Since this distance is longer than the 3.6 A+ obtained for the sum of the Van

der Waals radii of two chloride ions, the intermolecular magnetic exchange interactions

propagated by this pathway should be negligible [25]. Although this may be true for elevated

temperatures where these interactions can be unresolved due to thermally averaging, later we

will discuss experimental evidence for intermolecular exchange interactions at low temperatures

(< 6 K) in this system. The ability to resolve these interactions is largely due to the resonance

peaks of high quality in terms of line shape and width. In a system with larger amounts of

disorder, intermolecular exchange will broaden the peaks, but this effect would likely be buried

within the broadening due to the disorder. In this system the amount of disorder is minimal, and

thus the effect of intermolecular exchange on the line widths and peak splitting is able to be

resolved.









Another noticeable feature of this system is the absence of any solvate molecules in the

lattice, which results in a reduced distribution of microenvironments. Hence, interactions

between the magnetic core and lattice solvate molecules are not an issue in this compound. The

lack of lattice solvate molecules makes an ideal candidate to study with HFEPR.

If all molecules posses the same microenvironment they will all undergo transitions at the same

magnetic field, with the absorption peaks having sharp, Lorentzian lineshapes. Conversely, for a

system with large amounts of disorder or strains, there will be multiple microenvironments

associated with the molecules within the crystal. This will cause the EPR peaks to broaden and

deviate from a true Lorentzian shape. Dipolar and hyperfine fields can also contribute to such

effects. Since there is a distribution of microenvironments, different molecules undergo

transitions at slightly different field values which can lead to asymmetric line shapes, increased

peak widths, and peak splitting. Peaks with such characteristics make it difficult to determine

which interactions are dominating the broadening. Both the sizeable Cl---Cl contact distances

helping to minimize intermolecular interactions, and the absence of any solvate molecules in the

lattice, contribute to the EPR spectra exhibiting flat base lines and sharp, narrow peaks. Hence,

we will show that we are able to separate the various contributions (disorder and intermolecular

exchange) to the EPR line shapes.

4.2 HFEPR Measurements of [Ni(hmp)(dmb)Cl]4

Magnetic field sweeps were performed at a number of different temperatures and

frequencies. As explained in section 2.2, when the magnetic field is swept, sharp inverted peaks

appear in the transmission spectrum. Each peak corresponds to a transition between spin states,

and the decrease in cavity transmission signal is due to the absorption of microwave radiation by

the spins within the sample. A normal EPR spectrum from a SMM with its easy axis (c) aligned

along the external field consists of a series of absorption peaks corresponding to transitions









between states within the ground state spin multiple. At sufficient temperatures where all states

within the multiple are populated, one should see 2S peaks, corresponding to 2S transitions

between 2S+1 states.

4.2.1 Characterization of Easy Axis Data

[Ni(hmp)(dmb)Cl]4, CryStallizes in the shape of an approximately square based pyramid.

A single crystal (~1 mm x ~0.8 mm x ~1.1 mm) was mounted on the endplate of a vertical

cavity. We use a minimal amount of silicon vacuum grease to attach the crystal to the copper

endplate. 172.2 GHz HFEPR spectra are displayed in Fig. 4-2 for various temperatures in the

range of 10 K to 59 K. One thing to notice about this figure is that there are two sets of peaks:

the main intensity peaks and the lower intensity peaks. From the temperature dependence, it is

clear that the approximately evenly spaced dominant peaks correspond to transitions within the

S = 4 ground state. Indeed, as labeled in Fig. 4-2, all eight transitions within the S = 4 state are

observed. The other, weaker transitions (marked by vertical dashed lines) are within excited

state multiplets (S < 4) and we discuss these in section 4.4.

One interesting feature of the spectra is the splitting of several of the peaks, particularly

for transitions involving states with larger absolute ms values (at low and high fields). This ms

dependence of the splitting implies at least two distinct Ni4 Species, with slightly different D

values (D strain). Other spectra taken at 30 K and frequencies from 127 GHz to 201 GHz

confirm this conclusion and were used to determine the axial spin Hamiltonian parameters for

each species within the crystal. Fig. 4-3 plots the peak positions in magnetic field for different

frequencies. By considering a giant spin approximation (discussed in Ch.3), the fourth order

spin Hamiltonian is given by Eq. 4-1.

H~ = Dif' + BJ'O + pB f (4-1)










All peak positions in Fig. 4-3 have been fit to Eq. 4-1 assuming the same g, and BqO

values, and two slightly different D values (red and blue lines). The obtained parameters are: gz

= 2.3, D = -0.60 cm- and BY = -0.00012 cm-l for the higher frequency peaks, and D = -0.58

cm-l for the lower frequency peaks. The D values from our HFEPR measurements are quite

close to those obtained from fitting the reduced magnetization data [25], which gives a value of

D = -0.61 cm- At this point we stress that it is not possible to satisfactorily fit the data without

the inclusion of the fourth order term, B From Fig. 4-3 we see that the resonance branches are

not evenly spaced with respect to one another. If they were, then a D term alone would be

sufficient. The energy splitting would go as D(2|ms| -1), where ms is the spin projection state

from which the transition originates. For this system with S = 4, a D term would produce zero

field offsets of 7D, 5D, 3D, and D respectively for the states with ms = 4, ms = 3, ms = 2, and

ms = il. Hence the energy difference between branches goes like 2D. However, the branches

involving transitions between lower lying states (ms = -4 to -3) are spaced further apart with

respect to adj acent branches than those involving transitions between higher lying states (ms = -1

to 0). This kind of nonlinearity is introduced by a negative, axial, fourth order term. This is

illustrated in Fig. 4-4 for an exaggerated value of the fourth order term.

4.2.2 Peak Splittings Arising from Disorder

It is now well documented that disorder associated even with weakly (hydrogen) bonding

solvate molecules can cause significant distributions in the g and D values for SMMs such as Fes

and Mnl2-acetate, leading to so-called g and D strain [83, 84]. Such strains have a pronounced

effect on the line widths and shapes. Fortunately, HFEPR measurements provide the most direct

means for characterizing such distributions, which can ultimately have a profound influence on

the low-temperature quantum dynamics of even the highest symmetry SMMs such as Mnl2Ac.









The fact that the fits in Fig. 4-3 agree so well with the ms dependence of the peak splitting

provides compelling support for the existence of two distinct Ni4 Species. A closer examination

of the 172 GHz EPR spectra (Fig. 4-5a) indicates that the splitting is absent above about 46 K.

In fact, measurements performed at closely spaced temperature intervals reveal that the splitting

appears rather abruptly below a critical temperature of about 46 K, as shown in the inset to

Fig. 4-5a. This suggests a possible structural transition at this temperature, which explains a

lowering of the crystallographic symmetry and, hence, two distinct Ni4 Species. This scenario is

supported by thermodynamic studies and low-temperature X-ray measurements [78] which

suggest that the two fine structure peaks may be explained in terms of a weak static disorder

associated with the dmb ligand which sets in below 46 K.

In order to ascertain the origin of the fine structure splitting observed in the spectra below

46 K, detailed heat capacity measurements in the temperature range from 2-100 K were carried

out. The results are given in Fig. 4-6a, where it can be seen from the plot of heat capacity at

constant pressure versus temperature that there is a peak at 46.6 K which corresponds quite well

to the temperature at which the peaks in the HFEPR spectrum start to split (46 K). Heat capacity

measurements were also performed for a Zn analog (~ 2.3% Ni) in order to determine whether

the phase transition observed at 46.6 K is due to a structural change that causes different micro-

environments, or whether it perhaps arises from a spin related phenomenon such as magnetic

ordering due to intermolecular magnetic exchange interactions. The diamagnetic Zn analog,

which has the same structure, also exhibits a heat capacity peak at a similar temperature (49.6 K,

red data in Fig. 4-6a), suggesting that it cannot be the result of a magnetic phase transition. The

fact that the two structurally analogous complexes have this peak with similar amplitude at about

the same temperature indicates that it is due to a structural phase transition.









In order to further confirm that the 46 K peak splitting observed in the HFEPR spectra

stems from different microenvironments, single-crystal X-ray diffraction data were collected at a

temperature of 12(2) K. The thermal ellipsoid plot comparison of symmetry independent parts

of the molecule at 12 K and 173 K are given in Fig. 4-6b. Interesting thermal ellipsoids in the t-

butyl group from the dmb ligand are evident in both plots. This should be where the order-

disorder activity is taking place. One notable feature is that the thermal ellipsoids have shrunk at

12 K for all the atoms except for the t-butyl group. This can be seen in Fig. 4-6b, where the t-

butyl group is represented by the upper group bonded to the central Ni+2 ion. The lower group

bonded to the central Ni+2 ion is the hmp ligand, whose thermal ellipsoids have also shrunk by a

temperature of 12 K.

These abnormal looking ellipsoids of the t-butyl group are indeed much larger than would

be expected at this temperature. Moreover, it is shown in Fig. 4-7 that a slight clockwise shift of

the structure obtained at 12 K mapped onto the 173 K structure. The pink line in Fig. 4-7 is the

structure at 173 K, while the green dashed line is the structure at 12 K. As can been seen, the

only part of the structure that changes is the part coming from the t-butyl group. This small

change in the structure supports the suggestion that different microenvironments in the dmb

ligand cause the HFEPR peak splitting at low temperature. At high temperatures the motion is

thermally averaged, but below 46 K the motion freezes out and the structure takes on two distinct

orientations. The EPR peaks are then split due to the effects of the disorder.

4.2.3 Peak Splittings Arising from Intermolecular Exchange

Upon cooling the sample, additional broadening and splitting of the EPR spectra are

observed at temperatures below about 6 K, as shown in Fig. 4-5b and Fig. 4-8. At least four

(possibly up to six) ground state fine structure peaks are seen at the lowest temperatures (~1 K)

between ~1.15 T and 1.5 T in Fig. 4-5b. It is quite clear that these fine structure splitting cannot









be attributed to (static) structurally different microenvironments, as we now outline. First of all,

if this was the source of the splitting, then it should be apparent for all transitions, including the

ms = -3 to -2 resonance which is observed down to 2.6 K at around 2.7 T in Fig. 4-5b.

However, there is no broadening and only two fine structure peaks are observed for this

resonance to temperatures well below 6 K, which is where the additional fine structures begin to

emerge in the ground state resonance. Studies to higher frequencies (Fig. 4-8) indicate that the

fine structures in the ground state resonance (ms = -4 to -3) persist to the same field range

(~2.5 T) where the ms = -3 to -2 resonance is seen in Fig. 4-5b. Consequently, one can rule out

field-dependent structural changes. In fact, as seen in Fig. 4-8, the temperature below which the

additional fine structures begin to appear increases with increasing magnetic field/frequency (see

red arrows in Fig. 4-8 as rough guide). For comparison, the dashed line in Fig. 4-8 represents the

energy separation, Ao/kB, between the ms = -4 ground state and the first excited state (ms = +4 for

B < 0.66 T, and ms = -3 for B > 0.66 T). Thus, it appears as though the onset of the additional

fine structures is related to the depopulation of excited states. Indeed, similar evidence for

diverging linewidths has been reported previously for this same temperature regime [83]

(kBT< 0). However, these earlier studies involved SMMs (Mnl2Ac and Fes) with broad EPR

lines compared to the present Ni4 COmplex, making it difficult to clearly resolve additional EPR

fine structures brought on by intermolecular exchange interactions. For this reason, studies of

the present Ni4 COmplex provide an excellent opportunity to better understand the effects of

intermolecular exchange on the EPR spectra of SMMs.

While most aspects of earlier EPR line width studies on Mnl2Ac and Fes have been

understood in terms of competing exchange and dipolar interactions [83, 84], the behavior of the

extra splitting of the ground state resonance (ms = -4 to -3 in the present study) has remained










unexplained for kBT < Ao. We speculate that this is related to the development of short-range

intermolecular magnetic correlations (either ferromagnetic or antiferromagnetic) which are

exchange averaged at higher temperatures. These correlations are due to the off diagonal part of

the intermolecular exchange interaction (J/2[S, S,- + S,-S, ]), which entangles neighboring

molecules at low temperatures [85]. Since this phenomenon involves the interaction between

multiple molecules, of which there are two inequivalent types (the two fine structure peaks

observed from ~6 K to 46 K), one anticipates an increase in the number of fine structure peaks

corresponding to the development of short-range correlations between different combinations of

the two molecular species having slightly different D values. In addition, the fact that there

clearly exist two different molecular species also suggests that there may be differences in the

interaction strengths between various pairs of molecules. Additional evidence for short range

magnetic exchange interactions comes from single-crystal HFEPR studies of a single Ni+2 i08 in

a [Zn3Ni(hmp)4(dmb)4 14] COmplex doped into the crystal of the isostructural diamagnetic

[Zn(hmp)(dmb)Cl]4 COmplex [80]. The doping level (molar ratio of Zn/Ni: 97.7:2.3) was such

that the compound contained small amounts of magnetic molecules which are sufficiently

isolated from one another by the nonmagnetic [Zn(hmp)(dmb)Cl]4 mOlecules. Similar

temperature dependence studies on this compound revealed no extra splitting of the ground state

transition peaks at low temperatures. The same microenvironments due to ligand conformations

are present in this compound as the tetranuclear complex, but no short range intermolecular

exchange is possible. Without this possibility, no extra peak splitting appear. However, in the

tetranuclear complex where short range intermolecular exchange is possible, extra ground state

fine structure peaks are seen to appear at the lowest temperatures.









Without a more detailed understanding of the disorder, and of the nature of the inter-

molecular interactions, it is not possible to give a more precise explanation for the low

temperature spectrum. Nevertheless, the observation of 3D ordering in [Ni(hmp)(dmb)Cl]4 does

signify the relevance of intermolecular interactions, either due to dipolar interactions or weak

superexchange. Regardless of the source of the short range magnetic correlations, their

development will significantly impact the EPR spectrum.

4.3 Physical Origin of the Fast QTM in [Ni(hmp)(dmb)Cl]4

In addition to easy axis measurements, the transverse anisotropy associated with this

complex was also characterized [79]. Here we only quote the main results of this study, where

the obtained fourth-order B,4 term results in a large tunnel splitting (4 x 10-4 Cm-1) within the

ms = f4 ground state. This provides an explanation for the fast magnetization tunneling in this

system andl can be att;~~+ribte to the fct, that the O operator connects the ms = f4 states in


second order of perturbation theory. Consequently, it is exceptionally effective at mixing these

levels, thereby lifting the degeneracy between them. Since a second order transverse anisotropy

is symmetry forbidden, in particular, it is the fourth order transverse anisotropy B,4014 that

connects the ms = f4 states in second order perturbation theory.

The presence of fourth order terms needed to fit both the easy axis and hard plane data to a

giant spin approximation model suggests additional physics that is taking place that can not be

accounted for with this approximation. As we have noted in a previous publication [81], the

fourth order interactions BJ~f and BJ~f are completely unphysical within the context of a rigid

giant spin approximation appropriate for this complex. To understand this, one must recognize

that the molecular cluster anisotropy is ultimately related to the single-ion anisotropies associated

with the individual Ni+2 ions. Since each ion has spin s = 1, their zero field splitting tensors do









not contain terms exceeding second order in the local spin operators. Within a rigid giant spin

(strong exchange) approximation, the molecular zero field splitting tensor corresponds to nothing

more than a proj section of the single-ion zero field splitting tensors onto the S = 4 state, i.e. an

addition of the (3 x 3) single-ion tensors after rotating them into the molecular basis [80].

Consequently, such a procedure results also in a (3 x 3) matrix, and terms of order greater than

two in the molecular spin operators are not generated by this approach. Additionally, the fact

that higher lying spin multiplets become populated at relatively low temperatures (12 K)

indicates that the isotropic exchange interaction and the anisotropic interactions are of

comparable magnitude, which violates the fundamental assumption of the giant spin

approximation (J >> d).

Other experiments [80] on a single Ni+2 ion in a [Zn3Ni(hmp)4(dmb)4 14] COmplex doped

into the crystal of the isostructural diamagnetic [Zn(hmp)(dmb)Cl]4 COmplex, demonstrated that

the single-ion zero field splitting interactions for the single Ni+2 ion give rise to the second order,

axial zero field splitting interaction for the S = 4 ground state of the [Ni(hmp)(dmb)Cl]4 SMM. It

has also been determined in this study that the Ni+2 ion possesses a significant second order

transverse anisotropy (e) term and that the easy axes are tilted 150 away from the

crystallographic c axis (molecular easy axis) for the [Ni(hmp)(dmb)Cl]4 SMM. These factors,

when combined with the four-fold symmetry of the molecule, proj ect fourth order spin

Hamiltonian terms onto the S = 4 ground state.

In order to understand the apparent fourth-order contributions to the EPR spectra, one must

consider the full 81 x 81 Hilbert space associated with the four uncoupled Ni+2 ions, and then

consider also the Heisenberg exchange coupling between them. This procedure is described in

detail in a separate publication [81]. Nevertheless, we briefly summarize the findings of this









analysis here. Solution to the problem of four coupled s = 1 spins results in a spectrum of

(2S + 1)4 = 81 eigenvalues. In the limit in which the exchange coupling constant, J, between the

individual Ni+2 ions exceeds the single-ion second order axial anisotropy parameter (d), the

lowest nine levels are reasonably well isolated from the other seventy two levels, as shown in

Fig. 4-9 for the situation in which the magnetic field is applied parallel to the molecular easy-

axis (actual parameters obtained for this complex [80] were used in this simulation). It is the

magnetic-dipole transitions between these nine low-lying energy levels (red lines in Fig. 4-9) that

dominate the EPR spectrum for the Ni4 COmplex. Furthermore, it is these transitions that can be

well accounted for in terms of the giant spin Hamiltonian with S = 4, albeit that fourth (and

higher) order terms are necessary in order to get the best agreement. Henceforth, we shall refer

to this low-lying cluster of nine levels as the S = 4 ground state.

Roughly 30 cml above the S = 4 ground state in Fig. 4-9 is another reasonably well

isolated cluster of twenty one levels. Significant insight into the nature of the spectrum of this

complex may be obtained by considering the exchange coupling between four isotropic (d = 0)

spin s = 1 entities. Using a single ferromagnetic coupling parameter, J, it is straightforward to

show using a Kambe equivalent operator method [86] that the spectrum consists of a single S = 4

ground state (degeneracy of nine): then, at +8J relative to the ground state there are three S = 3

states (total degeneracy of twenty one); followed by six S = 2 states (degeneracy of 30) at +14J;

followed by six S = 1 states (degeneracy of 18) at +18J; with three S = 0 states (degeneracy of

three) finishing off the spectrum at +20J relative to the ground state. Therefore, it becomes

apparent that the twenty one levels above the S = 4 ground state in Fig. 4-9 correspond to the

three 'effective' S = 3 states found by the Kambe method. The term 'effective' here is meant to

imply that the spin quantum number is approximate, i.e. it is not an exact quantum number. The









zero field splitting associated with these twenty one levels has several origins which are

discussed in detail elsewhere [80, 81]: mainly this is caused by the single-ion zero field splitting

tensor, which contains an axial d parameter and a significant rhombic term e; in addition, the

local magnetic axes associated with the Ni+2 ions are tilted significantly with respect to the

molecular symmetry directions (although they are related by the S4 Symmetry operation). Above

the S = 3 states, it becomes harder to differentiate the various levels. Nevertheless, they may be

thought of as belonging to 'effective' S = 2, 1 and 0 states, though S becomes less and less exact

near the top of the spectrum due to the strong competition between isotropic (J) and anisotropic

(d and e) interactions.

The most important conclusion from the work in Ref. 81 is the fact that the higher-order

contributions to the zero field splitting of the S = 4 ground state arise through S-mixing. In other

words, one may start with a model containing only second order anisotropy terms. However, the

competing isotropic (J) and anisotropic (d and e) interactions mix spin states, generating

corrections to the eigenvalues which show up as fourth (and higher) order terms when one tries

to map the S = 4 ground state onto a giant spin model. The magnitudes of these higher order

corrections depend on the degree of mixing of the S = 4 state with higher lying S < 4 states. In

the extremely strong exchange limit (J >> d), the mixing is essentially absent, and the well

isolated S = 4 multiple maps perfectly onto a giant spin model requiring only a second order D

parameter (E is forbidden in the S4 Symmetry group). It is only when Jis reduced and the

proximity between the S = 4 state and higher lying levels starts to approach the energy scale

associated with the zero field splitting within the S = 4 state that the S-mixing begins to have

measurable consequences in terms of fourth-order terms obtained from a mapping of the S = 4

state onto a giant spin model. Consequently, it is highly desirable to be able to make direct









measurements of the locations of excited states. This provided the motivation to quantify the

isotropic interaction through HFEPR in order to make a direct comparison to the dominant single

ion anisotropic interaction. We achieve this by looking for additional peaks at elevated

temperatures (T~ 8J) that may be associated with transitions within higher lying spin multiplets.

4.4 Measuring the Exchange Interaction with HFEPR

The presence of relatively low lying excited states was first realized by fitting DC

magnetic susceptibility data for this complex [82]. Nevertheless, in order to confirm this and to

gain additional information concerning excited states, we analyzed the higher temperature EPR

spectra from section 4.2 in more detail. Fig. 4-10 shows an expanded view of data obtained at a

frequency of 198 GHz in the range from 4 T to 9 T. In addition to the ground state (S = 4)

transitions, there are other weaker peaks (some marked by vertical dashed lines) which can be

seen to appear in between the stronger peaks as the temperature is increased. Based on our

understanding of the energy spectrum associated with four coupled spin s = 1 particles, it is clear

that the resulting excited S = 3 spin multiplets already become sufficiently affected by S-mixing

for the parameter regime appropriate to this complex, that attempts to assign meaningful labels to

resonance peaks according to a simple S = 3 picture can be problematic. Indeed, it is apparent

that the three effective S = 3 multiplets in Fig. 4-9 are not degenerate. Consequently, the

transitions within the various S = 3 multiplets need not be degenerate either, thus giving rise to

additional fine structures and/or peak broadening. This can be seen in Fig. 2-2, where the change

in the background signal seen in the 4-7 T range we attribute to broadening of the resonance

peaks due to closely spaced energy levels within the multiple higher lying spin multiplets.

Nevertheless, close examination of the 198 GHz data in Fig. 4-10 does reveal six clusters of

peaks, which we tentatively assign to transitions within an S= 3 multiple. These peaks have

been highlighted with vertical dashed lines in Fig. 4-10 and, for convenience, labeled A to F at










the top of the figure. In addition to these peaks, there are other peaks within the data that suggest

transitions within another S = 3 multiple or within higher-lying (S < 3) multiplets. We rule out

that these additional peaks are the result of splitting due to disorder since such splitting

manifest as separations on the order of 0. 1 T, and the peak separations we see here are on the

order of 0.2 T.

The inset to Fig. 4-11 displays the frequency dependence of the positions of the six

resonances labeled A to F in Fig. 4-10. The solid lines are guides to the eye. However, the

slopes of these lines were constrained using the average g-value for the S = 3 multiplets

determined from the Zeeman diagram in Fig. 4-9. It should be noted that the exited S = 3 levels

in Fig. 4-9 exhibit significant non-linearities with respect to the magnetic field. Consequently,

the effective g-values associated with the different transitions between these levels vary

significantly, which may explain why some of the resonances vanish behind the stronger S = 4

peaks at certain frequencies, particularly resonance A, which is not seen at the lowest

frequencies. This can also account for the fact that not all transitions within the additional higher

lying multiple (S = 2) are able to be resolved.

As already mentioned, the zero-field offsets associated with the solid lines in Fig. 4-11

were chosen arbitrarily so as to lie on the data points. However, a comparison between these

offsets and the extrapolated zero field splitting obtained for several of the possible transitions

between S = 3 levels in Fig. 4-9 reveals good agreement. Thus, not only do the calculations

presented in Ref. 81 give excellent agreement in terms of the ground state S = 4 spectrum, but the

agreement appears to extend to the excited states as well, although at this point it is not possible

to make precise comparisons between experiment and theory due to the strong S-mixing among

excited levels.









In order to estimate the location of the S = 3 multiplets relative to the ground state, we used

a simplify ed approach where we consi dered first an i sotropi c Hei senberg coupling, JC s, s, ,


between four spin s = 1 Ni+2 ions. One can easily solve this problem for four spins, assuming a

single ferromagnetic coupling parameter J (< 0), using a Kambe vector coupling scheme [86].

This gives rise to states with total spin, ST, at energies given by [Sr(ST + 1) 8]J. We then added

zero field splitting by hand to the ST = 4 and 3 states using parameters (up to second order)

estimated from our EPR experiments. The values for the ST = 4 state are given in section 4.2 and

elsewhere [78]. For the ST = 3 state, we used values that approximate the zero field splitting

observed in Fig. 4-11, namely g, = 2.30 and D= -0.3 cm l. The corresponding energies of the

various ST states are then given by

E(S,,nz,)= J S, (S, +1)-8 +D~[ ns-S,(S,+1) +g pBna (4-3)

where we set D2 = -0.29 cm- for the Sr = 2 states, and all other Ds~ values equal to zero. We

then used Eq. 4-3 to compute the temperature and field dependence of the populations of each of

the eighty one levels corresponding to all possible ST and no values (taking also into

consideration the degeneracies of the various ST states). This information was then used to

compute the temperature dependence of a given EPR transition observed at a particular magnetic

field strength. In order to evaluate the coupling parameter J, we analyzed the intensity of the

transition labeled B in Fig. 4-11. We used the data obtained at 172 GHz and B = 4.55 T, due to

its superior quality and the fact that this resonance was well isolated from other peaks at this

frequency. We also made the assumption that this transition corresponds to the nas = -2 to -1

transition within the ST = 3 state.









The main panel in Fig. 4-1 1 plots the temperature dependence of the intensity (integrated

area under the resonance) of transition B observed at 172 GHz. The red curve is a simulation of

the data, obtained using only two adjustable parameters: the single coupling parameter J (< 0),

and an arbitrary vertical scale factor. This best simulation gave a value of J= -6.0 cm l. While

one should strictly consider two inequivalent J parameters for a distorted Ni4 cubane complex

having S4 Symmetry, detailed fits to DC magnetization data give good reason to believe that only

a single Jparameter is needed for this complex [25]. Furthermore, such an analysis allows for

the most direct comparison with the work in Ref. 81, where only a single Jparameter was

considered. Indeed, the agreement between the present analysis and previous studies is

excellent: a value of -5.9 cm- was obtained from the rather detailed analysis of the ground state

(S = 4) EPR spectrum in terms of four coupled Ni+2 i0ns, including a rigorous treatment of the

exchange and anisotropic interactions via full matrix diagonalization. We note that the value

obtained from the present analysis involved a number of approximations, so it is likely that there

is considerable uncertainty (maybe +1 cm l) in the obtained value of the coupling parameter J.

Nevertheless, the good agreement with the work in Ref. 81 is not coincidental. We also

comment on the value ofJ obtained from fits to 3&T data. The earliest published values for this

complex considered only the isotropic coupling between Ni+2 ions in the cubane unit, ignoring

the effects of local anisotropic spin-orbit coupling (d and e). Consequently, these values do not

agree so well with the more recent analyses, because the anisotropic terms have an appreciable

influence on the susceptibility to relatively high temperatures. A value ofJ= -7.05 cm- was

more recently obtained from fits to &T data for this complex using precisely the same model

employed in the analysis of the ground state (S = 4) EPR spectrum. Given the level of

approximation we have presently employed, the agreement is excellent.









This characterization used HFEPR to measure the isotropic exchange interaction, which

allows one to precisely determine the locations of higher lying spin multiplets above the ground

state. By analyzing the intensity of transitions within an excited S = 3 multiple we can extract a

value for J, and by collecting data at various frequencies we are able to estimate the zero field

splitting parameters for this multiple. Additionally, we can compare the values of the dominant

isotropic and anisotropic interactions (J/d~- 1.1) to explain the breakdown of the giant spin

model in terms of requiring fourth order terms in the Hamiltonian to fit the data. The relative

magnitudes of these competing interactions cause the S-mixing between the S = 4 state and

higher lying levels and necessitate fourth order (and higher order) terms obtained from a

mapping of the S = 4 state onto a giant spin model.

4.5 Summary

In this chapter we presented HFEPR studies done on a highly symmetric Ni4 SMM. First

we reported on experiments to characterize the spin Hamiltonian parameters for this SMM. Data

reveal vastly sharp, symmetric EPR lines due to the lack of solvate molecules in the crystal

lattice and large intermolecular exchange pathway distances. However, variable frequency,

variable temperature measurements have revealed the presence of two distinct molecular species

within the crystal and we are able to extract the relevant spin Hamiltonian parameters for each

species. Below 46 K the peak splits into two, which we attribute to differences in the molecular

environments arising from different t-butyl group conformations in the dmb ligand. At high

temperatures the motion of these is thermally averaged, but below 46 K the motion freezes out

and the t-butyl group takes on two distinct orientations. These EPR peaks are then split due to

the effects of the disorder.

Additional low temperature data (< 6 K) reveal additional splitting and broadening of

the peaks, which we attribute to short range intermolecular exchange interactions among









neighboring molecules that are averaged out at higher temperatures. It is likely that exchange

interactions provide an additional contribution to the line widths/shapes, i.e. exchange probably

also contributes to the broad lines. However, given the minimal amount of disorder in this

system, we are able to separate the various contributions (disorder and intermolecular exchange)

to the EPR line shapes.

Finally, we are able to measure the magnitude of the isotropic exchange coupling constant,

J, with our HFEPR data. By simulating the intensity of peaks originating from transitions within

a low lying excited state spin multiple to a model that includes both isotropic and anisotropic

interactions we obtain a value of J= -6.0 cm l. This provides insight into the spacing between

the ground state and higher lying spin multiplets, and supports the evidence that the isotropic and

anisotropic parameters (J and d) can cause mixing between states unless J >> d. Such mixing

manifests itself as unphysical higher order terms in the Hamiltonian with the giant spin model.




































Figure 4-1. A molecule of [Ni(hmp)(dmb)Cl]4. Each Ni+2 ion (green sphere) has a spin of S 1
and the four Ni+2 ions couple ferromagnetically to give a spin ground state manifold
of S = 4. The bulky aliphatic groups that surround the magnetic core help to
minimize intermolecular interactions and the absence of H20 solvate molecules in the
lattice results in a reduced distribution of microenvironments. Reprinted from E.- C.
Yang et al., Exchange Bias in Ni4 Single Molecule Magnets, Polyhedron 22, 1727
(2003). Fig. 1, pg. 1728. Copyright 2003, with permission from Elsevier.















~1
O
u~
r;n

E
m
d
k






O


-CI
d)
cn
cH
tcr
O
I

c,
J1



k


0 1 2 -2 to -1 I U. I


Magnetic Field (Tesla)
Figure 4-2. 172.2 GHz HFEPR spectra for temperatures in the range of 10 K to 59 K. The main
intensity peaks correspond to transitions within the ground state manifold, and the
lower intensity peaks (marked by vertical dashed lines) are from transitions within
excited state manifolds. From the temperature dependence, it is clear that the
approximately evenly spaced dominant peaks correspond to transitions within the
S = 4 ground state, as all eight transitions are observed.










200 -r


160 -r



120 i/V / ;1oI
D = -060(5 cm D = 057()
100 -O
B -0.00125) cm gz 230(2

0 24 8 1

Manei Fild(esa
Fiue43 lto h ekpsiin nmgei il o ifeetfeunisfo 2 l
to201 ~ t3 .Tem eedneo h pitn mle tlattodsic
Ni pcewt lgtydifrn aus h ort re pnHmloini
gie yE.41adwsue odtrie h xa pnHmloinprmtr o
eahseiswihntecysa.Rue with060(5 perisio fro.7~m R .Ewrs ora
ofApidPyis 3 80 20) i.2 g 88.Cprgt20,A eia
Intiut of0025)m Physics.2







-D = -0.60 em-1
-D = -0.60 canll BO = -3.34 x 10-4 -1~
300-

250 e'


~3200 1




~50



50

012345678910

Magnetic Field (Tesla)
Figure 4-4. Plot of the influence of the spacing between the resonance branches due to a
negative, axial, fourth order anisotropy term (red lines). The value of this term has
been exaggerated for illustrative purposes. For comparison, the branches are evenly
spaced with only aD term (black lines). The branches involving transitions between
lower lying states (ms = -4 to -3) are spaced further apart with respect to adj acent
branches than those involving transitions between higher lying states (ms = -1 to 0).















O 45.2 4.1









114.2 1. 2.0 1 2. 3C

Manei Fil Tsa
Fiur 4-5.9 Plo of th eksltig safucino eprtr.a A nagd iwo h
pekslitn at 7. eel hti per ut butyblwaciia
temeraur of abu 6K hchsget osil tutrltrniintkn
place Thi cause a oeigo h rsalgahcsm er nhnew
ditntN4Seis ) ddtoa raeigadslttnso h P pcr r





Figur 4 unresotolvted pat h pitigher a tmea fntures.meaur.a.Anelrgdvewo








300
Ni Zn
r 0.09 3.91
L 275

250 -~N'

P~225
T = 46.6 K
200
40 45 50 55
Temperature (kelvin)

12, K, C12 "\ 173 2



i-kMly 01 Nil 01

C5N1 61 je5 l 1
C4~F C5



Figure 4-6. Heat capacity measurements of [Ni(hmp)(dmb)Cl]4. a). Plot of heat capacity at
constant pressure versus temperature [Ni(hmp)(dmb)Cl]4 (blue data) and
[Zn3Ni(hmp)4(dmb)4C 4] (red data). The peak at 46.6 K corresponds rather well to
the temperature at which the peaks in the HFEPR spectrum start to split (46 K). b).
The ORTEP (Oak Ridge Thermal Ellipsoid) plot comparison of symmetry
independent parts of the molecule of [Ni(hmp)(dmb)Cl]4 at 12 K and 173 K. The
thermal ellipsoids shrink at 12 K for all the atoms except for the t-butyl group. These
ellipsoids are indeed much larger than would be expected, and this should be where
the order-disorder activity is taking place.













i
c
i


-i


~"


Figure 4-7. Comparison of the structure of [Ni(hmp)(dmb)Cl]4 at 173 K and 12 K. The pink
line is the structure at 173 K, while the green dashed line is the structure at 12 K. As
can been seen, the only part of the structure that changes is the part coming from the
t-butyl group. The observed small change in the structure supports the claim that
there are different microenvironments, and thus two distinct species of molecule in
the system at low temperatures which cause the HFEPR peak splitting.















* 140 GHz
* 172 GHz
*201 GHz


r****


+ I


* *


* *


~1


Q1






Cd
k
Q1

E
d)


C





* *


I I)

IIIII


II)

Ml+
Ir)
IIIII
II + + +
IIIII


-. *j -


*I ****


0.0


0r


0.5


1.0


1.5


2.0


2.5


Magnetic Field (Tesla)

Figure 4-8. Plot of the temperature dependence of the peak splitting at a given magnetic field
for three frequencies. The temperature below which the additional fine structures
begin to appear increases with increasing magnetic field/frequency (see red arrows as
rough guide). For comparison, the dashed line represents the energy separation,
Ao/kB, between the ms = -4 ground state and the first excited state (ms = 4 for
B < 0.66 T, and ms = 3 for B > 0.66 T). Thus, it appears as though the onset of the
additional fine structures is related to the depopulation of excited states.











S ~ 2, etc..


-S= 4


10

E -0


-d 20
a-30
Crl -40
-50


d -4.72 cml
e t1.19 -1'

J= -5.9 cm'
Tilt = 15o


-60
-70


Magnetic Field (Tesla)

Figure 4-9. Plot of a simulation of four coupled s 1 spins resulting in a spectrum of
(2S + 1)4 81 eigenvalues. For this simulation the magnetic field is applied parallel
to the molecular easy-axis. It is the magnetic-dipole transitions between these nine
low-lying energy levels (red lines) that dominate the EPR spectrum for the Ni4
complex.











I 35 K

30 K

ilV i I i II i i 20 K

I ~16 K


12 K



4 5 6 7 8 9

Magnetic Field (Tesla)
Figure 4-10. Plot of an expanded view of data obtained at a frequency of 198 GHz in the range
from 4 T to 9 T. In addition to the ground state (S = 4) transitions, there are other
weaker peaks (some marked by vertical dashed lines) which can be seen to appear in
between the stronger peaks as the temperature is increased. The cluster of six peaks,
which have been highlighted with vertical dashed lines and labeled A to F at the top
of the figure, we tentatively assign to transitions within the S = 3 states.










\rY J = -o) cm





200

FI 175
JIF1150b m e~

~d 15 125 D
e0 3 4 5 6 7 8 E
Magnetic field (Tesla) F r
10 20 30 40 50 60

Temperature (Kelvin)
Figure 4-11i. Plot of the temperature dependence of the intensity (integrated area under the
resonance) of transition B observed at 172 GHz. The red curve is a best fit to the
data, which gave a value of J= -6 cm l. The inset shows the frequency dependence
of the A F transitions within the S = 3 states for a temperature of 3 5 K.









CHAPTER 5
HFEPR CHARACTERIZATION OF SINGLE CO (II) IONS IN A TETRANUCLEAR
COMPLEX

The results presented in this chapter can be found in the article, High Frequency Electron

Paramagnetic Resonance (HFEPR) Study of a High Spin Co(II) Complex, J. Lawrence, C. C.

Beedle, E.-C. Yang, J. Ma, S. Hill, and D.N. Hendrickson, Polyhedron 26, 2299-2303 (2007).

5.1 Introduction to the [Co(hmp)(dmb)Cl]4 and [Zn3Co(hmp)4(dmb)4 14] COmplexes

The large success of the comprehensive studies on the [Ni(hmp)(dmb)Cl]4 and

[Zn3Ni(hmp)4(dmb)4 14] COmplexes, some of which were presented in Ch. 4, provided

motivation for investigation of two analogous compounds, [Co(hmp)(dmb)Cl]4 and

[Zn3CO(hmp)4(dmb)414] These complexes possess the same structure and molecular

microenvironment as the former, only with Co+2 ions replacing the Ni+2 ions on opposing corners

of the distorted molecular cube. A major difference in these systems is the added complexity of

strong spin orbit coupling inherent in the Co+2 ions that was not part of the Ni+2 ions [9, 87].

This greatly affects the magnetic behavior and necessitates the use of a coupled single ion

Hamiltonian approach as opposed to the usual giant spin Hamiltonian in order to model the

system. The first evidence for SMM behavior in the tetranuclear complex was reported

elsewhere [88, 89]. The out-of-phase ac susceptibility is less than 0.01 cm3 mO1-1 at temperatures

above 3.5 K and substantially increases to 0.16 cm3 mO1-1 as the temperature is decreased to 1.8

K. The increase of the out of phase AC signal suggests that the Co4 mOlecule has an appreciable

energy barrier for magnetization reversal. Additionally, hysteresis loops for temperatures below

about 1.2 K are shown to increase rapidly upon decreasing the temperature. The temperature

dependence and area within the hysteresis loop indicate that this molecule has considerable

negative magnetoanisotropy as expected for a SMM.









5.2 HFEPR Measurements of [Zn3Co(hmp)4(dmb)414]

5.2.1 The [Zn3Co(hmp)4(dmb)4 14] COmplex

The preparation of single crystals of this complex was similar to that described in Ref. 80.

Crystals of [Zn3.98C00.02(hmp)4(dmb)4 14] were prepared and a relatively precise value of the

Zn/Co ratio in the crystals was obtained by Inductively Coupled Plasma-Optical Emission

Spectroscopy (ICPOES) spectra, which gave Zn/Co: 99.46: 0.54. Assuming there is little

difference in the heats of formation for either the Zn4 Of CO4 COmplexes, the Co+2 i0ns should be

randomly distributed in the crystal. On the basis of this assumption, it is a relatively

straightforward exercise to compute the probabilities for the formation of the Zn4, Zn3CO,

Z n2CO2, ZnCo03, and Co04 COmpl exe s, as a functi on x, i n the formula [Z n4xC O4-4x(hmp)4(dmb)4C 4 -

When x = 0.995, it is found that the Zn4 and Zn3CO species make up 98% and 1.97% of the total

population, respectively. Thus, the doped crystal is comprised of some [Zn3CO(hmp)4(dmb)414]

complexes doped randomly into a diamagnetic Zn4 host crystal, with the overwhelming maj ority

of the magnetic spectra coming from the [Zn3CO(hmp)4(dmb)4 14] COmplex. The core of each

molecule is a distorted cube (analogous to the Ni complexes), the c direction being slightly

longer than the equivalent a and b directions. For each molecule there are four possible sites for

the Co+2 ion to reside, related by the S4 Symmetry operators, which are given in matrix form in

section 5.4. This complex crystallizes in the shape of an approximately square based pyramid.

A single crystal (~1.1 mm x ~0.9 mm x ~1 mm) was mounted with the flat base on the endplate

of a rotating cavity. We use a minimal amount of silicon vacuum grease to attach the crystal to

the copper endplate.

5.2.2 Frequency and Temperature Dependent Measurements

At the lowest temperatures (2 K), one can make the assumption that only the ground state

for each Co+2 ion is populated. Furthermore, as we shall see below, the ground state is a well









isolated Kramers' doublet. Thus, each Co+2 ion may be expected to contribute only one peak to

the HFEPR spectrum (containing some Eine structure due to disorder-induced strain). Thus,

assuming the magnetic axes associated with the four possible CO+2 Sites on the Zn3CO molecule

are non-collinear, one should observe four separate HFEPR peaks for arbitrary Hield orientations

(one for each Co+2 Site). For Hield orientations along certain symmetry directions, two or more of

these transitions may become degenerate, thus reducing the number of peaks. Fig. 5-1 displays

frequency-dependent data obtained at 2 K, with the Hield tilted 32o away from the c axis of the

crystal in the (100) plane (see Fig. 5-3). The inset to Fig. 5-1 shows a typical spectrum obtained

at a frequency of 51.8 GHz. As can be seen, absorptions are grouped into three clusters. The

position of each peak is plotted versus frequency in the main part of the Eigure for measurements

performed on higher-order cavity modes. As can be seen, each peak lies on an absolutely

straight line which passes exactly through the origin. Lande g factors have been assigned to each

of the peaks based on the slope of the straight line through each set of data points. The shapes of

the data points (solid and open circles and squares) have been chosen for comparison with Fig. 5-

3 below. We believe the fine structures are caused by disorder induced g strain (as was found

from similar studies of the Ni/Zn analog [80, 90]). Based on the angle-dependent studies

discussed below, we Eind that the g = 4.2 branch in Fig. 5-1 is in fact degenerate, consisting of

two HFEPR transitions. This explains the observation of only three peaks. The relative

intensities of the peaks may be understood in terms of the transition rates [71], which depend on

the respective Lande g factors (smaller g factor, larger transition rate). The fact that each

resonance lies on a perfectly straight line which passes exactly through the origin is a clear

indication that the transitions occur within a well isolated effective spin S' = 1/2 Kramers'

doublet. The anisotropy is contained entirely within the effective Lande g tensor. Further









confirmation comes from the temperature dependence in Fig. 5-2. These data were obtained for

the same frequency and field orientation as the data in Fig. 5-1; one can see that the field

alignment is not precisely the same due to the slight splitting of the degenerate branch.

Nevertheless, all peaks increase in intensity upon lowering the temperature, thus confirming the

assignment of the ground state as a spin S' = 1/2 Kramers' doublet.

5.2.3 Angle Dependent Measurements

By far the most detailed information comes from angle dependent studies. For an arbitrary

rotation plane each spectrum should consist of four peaks. However, for high symmetry rotation

planes there will be degeneracies and some resonance peaks will superimpose upon one another.

The highest symmetry direction corresponds to a magnetic field applied along the c axis of the

crystal, where only one peak should be seen. A magnetic field applied at an arbitrary angle

within the ab plane of the crystal will give rise to two peaks. Any misalignments of the sample

with respect to the applied field will lift this degeneracy and give rise to up to four maj or peaks.

The first experiment (rotation 1) involved rotating the magnetic field in the (110) plane from the

c axis to the ab plane (Fig. 5-3a). The second experiment (rotation 2) involved rotating the

magnetic field within the ab plane (Fig. 5-3b). For rotation 1 we observe three resonance

branches, which collapse into approximately 1 peak for a = Oo (Hield // c axis) and approximately

two peaks for a = 90o (Hield // ab plane, 450 from a and b). Since only three branches are

observed in Fig. 5-3a, we must assume that one of them is degenerate (solid circles). Based on

simple geometrical considerations, we can immediately determine the nature of the anisotropy at

the individual Co+2 Sites. Due to the octahedral coordination, we assume the anisotropy will be

approximately axial (easy-plane or easy-axis [9]). First of all, it is clear from Fig. 5-3a that the

local magnetic axes are tilted with respect to the crystallographic axes (and, therefore, with

respect to each other). Fig. 5-4 shows a diagram of the magnetic core, with the four possible









sites for a single Co+2 ion to reside. The easy axis of each individual ion is tilted at an angle

(found to be ~580) with respect to the c axis of the crystal. Starting with the field along c (Fig. 5-

3a): when one tilts the field away from this symmetry direction, it necessarily tilts away from

three of the local z axes and toward three of the xy planes; conversely, it tilts toward one of the z

axes and away from one of the xy planes. Considering the two-fold degeneracy of the branch

represented by the solid circles, upon tilting the field away from c one sees that three of the

branches move to higher fields (they become harder), whereas only one moves to lower field (it

becomes easier). Therefore, based on the above geometrical consideration, one can conclude

that the planes are hard, while the axes are easy, i.e. the anisotropy is of the easy-axis type.

Upon rotating the field over a 180o interval, one is guaranteed to find all four hard planes. These

directions correspond to the maxima at -900 (two planes), -32o and +32o (the peak at +900 is

equivalent to the one at -900) in Fig. 5-3a, and the maxima at [(n x 90o) + 45o] in Fig. 5-3b,

where n is an integer. Since we observe a degenerate branch in Fig. 5-3a, this implies that

orientation of the field, as it rotates, is the same for both ions contributing to the branch. Thus,

the field rotation plane must be perpendicular to the easy-axis tilt plane for these two ions and,

therefore by symmetry, it must be parallel to the easy axis tilt plane for the other two ions.

Therefore, we conclude that the two minima labeled z correspond precisely to the easy axes.

One can then immediately determine that gz = 7.80 (indicated by blue dashed line). The

minimum at Oo in Fig. 5-3a corresponds to the point of closest approach of the field to the other

two easy axes. From the positions of the maxima and minima, we determine that the easy axes

(hard planes) are tilted about 58o (32o) away from the crystallographic c axis, and the easy axes

are tilted in the (1 10) and (1 1 0) planes. It is notable in Fig. 5-3a that the heights of the maxima

at 90o and 32o are slightly different, corresponding to g values close to 2.00 and 2.20. This









observation implies weak in-plane anisotropy, suggesting a weak orthorhombic distortion at the

individual Co+2 Sites. The lesser of the two g values corresponds to gx ~ 2.00 (the hard axis, red

dashed line) and the intermediate value to g, ~2.20 (the medium axis, green dashed line). From

the hard-plane rotations (Fig. 5-3b), we see that the maxima coincide with g, ~ 2.20 in Fig. 5-3a.

Therefore, we can conclude that the medium axes lie along the intersections of the four hard

planes, within the ab plane of the crystal [along (110) and (1 1 0)]. The hard axes are, therefore,

directed maximally out of the ab plane [with proj sections also along (1 10) and (1 1 0)], tilted 32o

away from c. The minima in Fig. 5-3b correspond to the projection of the easy axis anisotropy

onto the ab plane, i.e. g = gz cos32o = 6.61 (marked by horizontal black dashed line). Finally, the

vertical dashed line in Fig. 5-3a denotes the orientation of the data presented in Fig. 5-1 and Fig.

5-2. As can be seen, the data points are coded similarly to the corresponding 51.8 GHz data in

Fig. 5-1.

As already discussed, the Hamiltonian for a well isolated effective spin S' = 1/2 Kramers'

doublet takes the form


H= pgS' B (5-1)

In Eq. 5-1, J represents an effective Lande g tensor that parameterizes all of the anisotropy

associated with the spin-orbit coupling. The resulting relation between the measurement

frequency, J and the resonance field, Bres, is then

/i = g(B)#nBres (5-2)

Thus, for a fixed frequency measurement, we have the relation in Eq. 5-3



rs -)L~ (5-3)









One can therefore attempt to fit the data in Fig. 5-3 assuming a simple angle dependence of

the form

g(0) = gmin +3g cOS2 ( _8o) (5 -4)

In Eq. 5-4, gmzn is the minimum g value for a particular plane of rotation, 6g is the difference

between the minimum and maximum g values, and 8o is the angle corresponding to gmzn. The

magenta curves in Fig. 5-3 correspond to such fits. As can be seen, the agreement is quite good.

To summarize, the orientations of the magnetic axes, with respect to the crystal lattice,

were determined for single Co+2 ions doped into a nonmagnetic Zn4 COmplex. Frequency and

temperature dependence studies confirm the ground state to be an effective spin S' = 1/2

Kramers' doublet with a highly anisotropic g factor. The anisotropy is found to be of the easy-

axis type, with the single-ion easy axis directions tilted away from the crystallographic c

direction by 58o. The g factor anisotropy (gz = 7.8 and gx ~ 2.0) is close to the maximum

expected for an octahedral Co+2 COmplex [9], suggesting a huge axial zero-field-splitting. The

magnitude of the axial zero-field-splitting parameter of the Kramers doublets was calculated [9]

to be close to 1000 cml in another complex in octahedral coordination geometry for such a large

amount of g factor anisotropy (Ag ~ 6). The information obtained from this study will be used in

an attempt to simulate the data we have obtained from the Co4 System, which we present in

section 5-3.

5.3 HFEPR Measurements of [Co(hmp)(dmb)Cl]4

A crystal of [Co(hmp)(dmb)Cl]4 has the same shape and dimensions as those of

[Zn3.98C00.02(hmp)4(dmb)4 4]]. The setup was placed in the 9 T Quantum Design systems

magnet and oriented such that the c axis of the sample was approximately parallel to the applied

magnetic field. Fixed frequency, fixed temperature magnetic field sweeps were performed for









numerous frequencies and temperatures with the magnetic field oriented parallel to the c axis of

the crystal.

5.3.1 Measurements Along the Crystallographic c Axis

We begin with a discussion of the spectra obtained with the field along the c axis. Fig. 5-

5 plots the temperature dependence of the peaks for two different frequencies. It is clear that this

system is more complicated than its nickel counterpart (discussed in Ch. 4) due to multiple

ground state transitions observed to temperatures as low as 2 K. At a frequency of 288 GHz for

a temperature of 2 K we see a strong peak at 0.75 T and a much weaker one at 1.25 T, indicating

that at least the stronger peak is indeed a ground state transition. However, as the frequency is

increased we start to observe many more transitions even at the lowest temperatures. At 501

GHz and 2 K we see 5-6 peaks with some structure evident in the higher field peaks. This is

radically different than the behavior seen in the Ni4 System, and is contradictory to what one

would expect from a simple SMM. While it is true that disorder in the Ni4 System gave rise to

two distinct species of molecules, this manifested itself in the observation of peak splitting on

the order of 0. 1 T. Since the same ligands are present in the Co4 System, it is unlikely that any

disorder would manifest itself as multiple species with such radically different anisotropy

parameters. Therefore, it is clear that the giant spin model is not physically appropriate to model

this system. As discussed in Ch. 3 the giant spin model, which assumes the quenching of the

orbital contribution to first order in perturbation theory, allows one to express the states of the

systems as pure spin multiplets. Within a given S multiple, the anisotropy terms lift the

degeneracy of the spin proj section states, which are expressed as m, However, this is not the

case for a Co+2 ion. Each ion has an orbitally degenerate ground state, and thus the orbital

angular momentum is not quenched. Hence, the assumption that we can express the eigenstates









of the system as pure spin multiplets is no longer valid. The orbital contribution to the magnetic

behavior is significant and must be taken into account.

This is further illustrated in Fig. 5-6 where we plot the c axis frequency dependence of

the resonance peaks at 2 K. The low temperature data show multiple ground state resonance

branches across a wide frequency range (230 GHz 715 GHz). The solid lines are rough guides

to the eye of the field dependence of the resonance branches. Each one extrapolates to a

significant zero field offset value, indicating a large zero field anisotropy. These zero field

offsets display non-linear behavior that can not be modeled by a simple D term. Such a term

would split the levels in zero field in a linear fashion. The energy splitting would go as D(2|m|-

1), where m is the spin projection state from which the transition originates. As an example, for

a system with S = 2 a D term would produce zero field offsets of 3D and D respectively. Hence

the energy difference between branches goes like 2D. This is clearly not representative of the

data for this system. The colored arrows in the figure represent the frequency difference between

adj acent resonance branches and the corresponding values are displayed. The four lowest

resonance branches are separated by values that vary between ~40-60 GHz, and then there is a

sudden spacing of ~ 230 GHz between the next two resonance branches. Then the next two

branches have a zero field spacing of ~ 80 GHz. Additionally, the four lowest branches exhibit

level repulsion as the magnitude of the magnetic field is increased from zero. This is seen from

the curvature of the lines that are guides to the eye as the field is increased, and is indicative of

transverse anisotropies inducing mixing between states. The individual effective S' = 1/2 ions

should have no inherent zero field anisotropic terms, but the exchange coupling between the four

ions will be shown to contain a term that will mix states in zero field. Additionally, the large

tilting of the magnetic easy axis of each ion means that alignment of the external field with each









ion's easy axis simultaneously is impossible, and thus, an external transverse component is

introduced. Such effects are likely to be a significant source of the state mixing.

5.3.2 Discussion of Spectra Within the ab Plane

We collected angle dependent data by rotating the field within the ab plane of the crystal.

Our 7 T horizontal field magnet offered the perfect geometry for this experiment.

We initially oriented the sample geometry such that the magnetic field would be parallel to

the ab plane of the crystal. After selecting a good frequency, we rotated the magnetic field

within the ab plane of the crystal. Fig. 5-7a plots the 10 K EPR spectra for different angles

(separated by So) at a frequency of 123 GHz. Each data set has been offset for clarity. From the

shifts of the peak positions in field we infer that there is a significant transverse anisotropy in this

system. Fig. 5-7b plots the peak position in field as a function of angle within the ab plane. It is

clear that a four fold modulation of the resonance position is seen as we rotate through 180o,

which reflects the S4 Symmetry of the molecule. The periodicity of the maxima/minima in the

peak position is 900 as shown in Fig. 5-7b. We can also see this from the two red traces in Fig.

5-7a since they are identical, and are separated by 900. Normally this four fold modulation is

parameterized in terms of a B,404 term; in,;, the, gintsinHmlto~nian, and unliken the nicklr


system discussed in Ch. 4, such a term would not be forbidden here since the ground state of a

Co+2 ion is characterized by L = 1, S = 3/2. However, as we mentioned above, using a giant spin

approach to model this system is not sensible.

5.4 Spin Hamiltonian for the Tetranuclear system

Since we can not describe the observed behavior in terms of a giant spin Hamiltonian we

turn to the coupled single ion approach. In section 5.3 we were able to determine that the single

ions have an effective S' = 1/2 ground state. Consequently, the ground state has no zero field









splitting terms (d or e). However, from the studies on the tetranuclear complex (Fig. 5-6) it can

be seen that the ground state of that system clearly has zero field splitting anisotropy. Now we

provide a qualitative explanation as to how the ground state of four coupled ions with no zero

field splitting can proj ect onto the tetranuclear complex to give rise to ground state zero field

splitting. Considering the possible interactions (the usual zero field splitting is a second order

spin orbit effect that is zero for Kramers doublets) we are left with the isotropic exchange,

symmetric exchange, and antisymmetric exchange terms discussed in section 3.1.2. While it is

certain that isotropic exchange is a prevalent effect in this system, it will not produce any zero

field splitting in the coupled system. This interaction only splits energy levels by S, not by m.

From this we will get effective spin multiplets with values S = 2, 1, 0, but no removal of the

degeneracy of the levels within the S = 2 and S = 1 multiplets. Therefore we are left with the

symmetric and antisymmetric exchange interactions as the sources of the zero field splitting in

the ground state of the tetranuclear complex.

Based on the symmetry of the system in question, some components of the antisymmetric

exchange vector can vanish, which simplifies the Hamiltonian, as we now demonstrate. The

symmetry of the molecular cluster will determine the orientation of the antisymmetric exchange

vector, G~ The molecule can be modeled as a distorted cube, with Co+2 ions placed on


opposing corners, and the entire entity possessing S4 Symmetry, as shown in Fig. 5-4. To begin,

we consider two ions joined by a straight line in space. At the midway point between these ions

we can picture a mirror plane oriented perpendicular to the line that intersects at this point. G

will be oriented parallel to this mirror plane, and thus, perpendicular to the line joining the two

respective ion centers. Expressed in component form, the antisymmetric part of the Hamiltonian

for a pair of ions i and j can be written













In order to simplify, we omit the transverse components of the interaction, since Gx GY =o for


tetragonal systems, and both Gx and GY are much less than GZ if the tetragonal symmetry is

slightly distorted [91]. Thus, we use only the axial component of the antisymmetric exchange

interaction, and rewrite Eq. 5-5 as

G' [ SS' S'S' ] (5-6)

This antisymmetric exchange interaction will mix states in zero field.

The other terms contributing will be the isotropic exchange and the Zeeman interaction.

The actual molecular geometry is an elongated cube, so that there are two isotropic exchange

coupling constants. With respect to the Zeeman interaction, for an arbitrary magnetic field

orientation, the field will proj ect differently onto the four ions. In order to properly construct the

Zeeman term for the molecular cluster, we must take into account the tilt angles of the individual

ion axes. Since the system has S4 Symmetry, we can relate each ion to one another with a

combination of the Euler matrix relating the single ion and molecular coordinate systems and the

transformation matrices for S4 Symmetry. The Euler matrix is given by

cos(a> cos(p>cos(7) -sin(a) sin(y) sin(a) cos(p cos(y) +costal sin(y) sin(P cos(y)
R = -cos(a)cos(p)sin(y)- sin(a)cos(y) -sin(a) cos( sin(y) +cos(a)cos(y) sin( sin(y) (5-7)
cos(a> sin(P> sin(a> sin(P cos(p

In Eq. 5-7, a, a, and 7 represent the tilt angles for the individual ions. The convention used here

is that a is a rotation about the z-axis of the initial coordinate system. About the y-axis of this

newly generated coordinate system, a rotation by a is performed, followed by a rotation by 7

about the new z-axis. Additionally, the matrices for S4 Symmetry which relate one ion to another

are given by










I= 0O 1O 0O (5-8a)



14, =-1 0 0 (5-8b)




S24 = 0 -1 0 (5-8c)




34 O -O O (5-8d)


Using these matrices, we can properly proj ect the magnetic Hield onto each individual ion. The
proj section onto the first ion will be R1 = R*I, the second ion R2 =R*S14, the third ion R3 =R*S24,
and the fourth ion R4 =R *S34. Any tilting of the magnetic Hield with respect to the desired
orientation can be accounted for by two additional polar angles, 8 and 9, where the first

represents the angle of the Hield with respect to the z direction and the second represents the
angle of the field in the xy plane. For the proj section of the Hield onto the magnetic center for an


ion, i, we express as M,~~ = R. sin 8 csi y9 .s We write the Einal Hamiltonian for the system as



H =C JS S +S A;l S +G "[S,"S, S,'S,"]+ p,B(M, R-S ) (5-9)









In zero field, the isotropic exchange part of the Hamiltonian splits the energy levels into

different effective S multiplets, with the effective S = 2 multiple being the lowest. Above this

are three effective S= 1 multiplets and finally, two effective S= 0 multiplets. The symmetric

and antisymmetric interactions then split the different levels within a given multiple, analogous

to the usual axial and transverse anisotropy (zero field splitting) parameters. Considering the

data set which was taken with the magnetic field aligned along the c axis of the crystal at a

temperature of 2 K, we assume only the ground state of the system is sufficiently populated. It is

clear from the data in Fig. 5-6 that six resonance branches are observed, which is contradictory to

what one would expect from an isolated effective S = 2 multiple. Normal EPR selection rules

allow transitions between states that differ in m by +1, and under the experimental conditions we

should then observe only one resonance branch! Consequently, we must assume that there is

such strong mixing between states (both between and within a given multiple) that normal EPR

selection rules do not apply. Under such conditions of strong mixing between states S and m are

not good quantum numbers. As mentioned previously, the antisymmetric exchange coupling

will mix states in zero field. Additionally, the large tilting of the magnetic easy axis of each ion

means that alignment of the external field with each ion' s easy axis simultaneously is impossible,

and thus, an external transverse component is introduced. Such effects are likely to be a

significant source of the state mixing.

Attempts to simulate the data for the Co4 System assuming four coupled effective spin V/2

particles have not been successful. The final task of simulating the behavior of the tetranuclear

system is still a work in progress. This has been turned over to Motohiro Nakano a collaborator

in the department of molecular chemistry at Osaka University. His approach is to solve the

problem of four coupled ions each with L = 1, S = 3/2. The difficulty comes from the size of the









Hilbert space, which is [(2L+1)(2S+1)]4 = 20736. He is currently trying to simulate the EPR

spectra using the single ion parameters we have measured. A preliminary simulation is

illustrated in Fig. 5-8, which incorporates our data and assumes two different isotropic exchange

values. The first attempt at simulating the data comes from Fig. 5-8 for a frequency of 501 GHz

at 30 K with the magnetic field aligned along the c axis of the crystal. Our data reveal resonance

peaks at the following approximate field values: 1.6 T, 2.4 T, 3 T, 4 T, 5 T, and 6 T. These

positions are marked by the vertical black lines in Fig. 5-8. Clearly, no one simulation is able to

reproduce all of the peaks. The simulation with the parameters J1 = 0.814, Jz = 2.33 (marked by

a dark green arrow) closely reproduces the peaks at 2.4 T, 3 T, 4 T and 5 T, but not the other

two. On the other hand, the simulation with the parameters J1 = 1.21, Jz = 1.72 (marked by a

blue arrow) begins to reproduce the peaks at 1.6 T and 6 T, but not the other four. Evidently

more work will be necessary before the Co4 System is solved.









100 oc V I vX





C3 CrrAngle = 32o
60-




04
0 1 2 3
MantcFel Tsa
Fiue51 lto ekpsto safnto o rqec.Albace xrplt ozr
filidctn h rudsaei nefetv pnS / rmr'dult
Rerne rm .Lwec e t a. ig Frqec lcrnPrmgei eoac
Std ofaHg pnC(I opePlhdrn2,29 20) i.1 g 31
Coyih 07 ihprmsinfo leir









51.7 GHz
15 K
12 K
10 K


6K


33K
---- 2.5 K


~i`


~1
O
C10
C10

E
cn
~1
ICrS
k
I ,



le~f
O


1.
d)
CI~
CI
O
I
1
~C1
5

k
erS


Figure 5-2. Temperature dependence confirming that these are ground state transitions, since
the intensity of each peak increases as the temperature is lowered. Reprinted from J.
Lawrence et al., High Frequency Electron Paramagnetic Resonance Study of a High
Spin Co(II) Complex, Polyhedron 26, 2299 (2007). Fig. 2, pg. 2301. Copyright
2007, with permission from Elsevier.


Degenerate Angle = 32"
branch


Magnetic Field (Tesla)








-- X


g= 2.03
-- 0 n


1.8
1.6
1.4

1.0
0.8
0.6
0.4
1.8

1.6

1.2
1.0
0.8

0.4


6


Tb
6

F
C


=7.80


-135 -90 -45 0 45 90 135

Angle (degrees)

Figure 5-3. Peak position as a function of angle for two planes of rotation. a). Rotation of the
magnetic field in the (1 10) plane from the c axis to the ab plane. b). Rotation of the
magnetic field within the ab plane. Reprinted from J. Lawrence et al., High
Frequency Electron Paramagnetic Resonance Study of a High Spin Co(II) Complex,
Polyhedron 26, 2299 (2007). Fig. 3, pg. 2301. Copyright 2007, with permission
from Elsevier.


51.6 GHz


=6.60


























Z


\I


Figure 5-4. Diagram of the magnetic core of a CoZn molecule possessing S4 Symmetry, with
the four possible sites for a single Co+2 ion (purple sphere) to reside. The crystal
directions are given by a, b, and c. The easy (z) axis of the individual ions is tilted
with respect to the crystallographic c axis by ~580.












-l1--_7-I. ~~88 G~Iz
12 K
10 K


4K

~2K


F1
O
I;R
V1
rl

UO
F1
k




I)


d)
m
(I 1
Cr
O
I
V1



5

~


1 2 3 4 5 6


-- 501


GIT K
S24 K
- 18 K

10K
7K
2K


1l I I I Isis
1 2 3 4 5 16
Magnetic Field (Tesla)


Figure 5-5. Plot of the temperature dependence of the peaks for two different frequencies. At a
frequency of 288 GHz for a temperature of 2 K we see a single ground state transition
at 0.75 T However, as the frequency is increased we start to observe many more
transitions even at the lowest temperatures. At 501 GHz and 2 K we see 5-6 peaks
with some structure evident in the higher field peaks.







750
675


* r


AE/h = 80 GHz

AE/h = 62 GHz

aE/h = 42 GHz
rallel to c axis


1~600
S525
~5450
~r375
~1300
S225
15 '
F475


0L
-


-


Figure 5-6. Plot of the easy axis frequency dependence of the resonance peaks at 2 K. The
solid lines are rough guides to the eye of the field dependence of the resonance
branches. Each one extrapolates to a significant zero field offset value, indicating a
large axial anisotropy. The zero field offsets display non-linear behavior that can not
be modeled by a simple D term. The colored arrows in the figure represent the
frequency difference between adj acent resonance branches and the corresponding
values are displayed.


B pa~


2K


1 23 45 67 8
Magnetic Field (Tesla)











123.4 GHz 10K Rotation ~vithin ab plane

---- '

--
A
7
C~~--------~y_
I- -(.~~---------
-------; -.------
7CI---i----.-- 700
-
--- -~~

--J1-L__ -
~1 --- --- -- ----- ----
,-~~-------=
V~~--^------ ~_F- -~~-__ -1
_~YT ---.,----- --- --
-I~y. 1600
i. --7,
---
-
1 2 3 4

Magnetic Field ITcsla)
900


d
O
m





k
E-l





U


C1
Q)
CR
4--1
CCI
O
I
CO
C,
F1


q


~c~

Gfl
a)

'L,
C
O

CO
O


ed
a,


11 1


*=

. .

..I

:*


* ",


O 20


Figure 5-7. Data taken with the field aligned within the ab plane of the crystal. A). EPR spectra
for different angles (separated by So) at a frequency of 123 GHz. From the shifts of
the peak positions in field we infer that there is a significant transverse anisotropy in
this system. B). Plot of the peak position in field as a function of angle within the ab
plane. It is clear that a four fold modulation of the resonance position is seen as we
rotate through 180o, which reflects the S4 Symmetry of the molecule. The periodicity
of the maxima/minima in the peak position is 900. We can also see this from the two
red traces in A). since they are identical and are separated by 900.


*==.....I B
-11 ..----...

:**.........*l*::::= -

:**;:** ; I *:!":*
... .- 1.. ......


=.I...,





..,,, .....


40 60 80 100 120 140 160 180

Angle (degrees)











[.(IO.OUEMK 3.636.97
(0.211055 3.28989
-- (0.422111 2.94810

r a --s 0.814070 2.32637
r i (I.02513 1 9980))
(1.20603 1 72155)-
O -v --~~t ~ / v o ->n98 8
~~n a- (IK1.5905 0.R240(79)
-- (2.00000 0.549595)
Cn u,(2.200101 .262947)
(2.401201 4).019162
(2.60302 -0.296736
- ----- -- 12.80402 -0.5691771
O ~(3.00503 -0.838269

-) -~or (.70-1.36165)
(3.GUtlU4 1.61654)
-- (3.80905 -1.86689)
4.01005 2.11270)
(4.21 106 -2.35398)
-4, -- (4Al2 -2.159071)
3 (4.61307 2.82291)

(5.01508 -3.27370)
(5.39698 -3.68514)
-~~~ ~~~~ -- .- .-. 5.59799 -3.89510)
Wm (5.79899 -4.10)053)
(6.00000 4.30143)
(6.2012 -4.50009)J



**-(7.000001 5.23842)

-501L~ilI (lJr2)



S2 8 10
S/ Tesia

Figure 5-8. Plot of the simulation of the Co4 System for a frequency of 501 GHz at 30 K with
the field aligned along the c axis. The resonance peak positions for this data set are
marked by the vertical black lines. The simulation accounts for two isotropic
exchange interactions (J1, Jz). The simulation with the parameters J1 = 0.814, J
2.33 (marked by a dark green arrow) closely reproduces the peaks at 2.4 T, 3 T, 4 T
and 5 T, but not the other two. On the other hand, the simulation with the parameters
J1 = 1.21, Jz = 1.72 (marked by a blue arrow) begins to reproduce the peaks at 1.6 T
and 6 T, but not the other four.









CHAPTER 6
HFEPR STUDIES OF MAGNETIC RELAXATION PROCESSES INT MN12AC

The results presented in this chapter can be found in the articles, Magnetic Quantum

Tunneling in M~nl2 Single-M~olecule Magnets M~easured With High Frequency Electron

Paramagnetic Resonance, J. Lawrence, S.C. Lee, S. Kim, N. Anderson, S. Hill, M. Murugesu,

and G. Christou, AIP Conference Proceedings, 850, 1 133-1 134 (2006), and A Novel Experiment

Combining Surface Acoustic Waves and High Frequency Electron ParamagneticP~~PP~~~PP~~PP Resonance in

Single M~olecule Magnets, F. Macia, J. Lawrence, S. Hill, J. M. Hernandez, J. Tejada, P. V.

Santos, C. Lampropoulos, and G. Christou, Phys. Rev. Lett., submitted.

6.1 Introduction to Mnl2Ac

[Mnl2012(CH3COO)16(H20)4] 2CH3COOH-4H20, hereafter Mnl2Ac, is the most widely

studied SMM. It was first synthesized [92] in 1980 and has a large spin ground state (S =10),

combined with a sizeable D value (D = -0.45 cm- ) that gives a large barrier to magnetization

reversal. The magnetic core of Mnl2Ac consists of four Mn+4 ions (each with S = 3/2) and eight

Mn+3 ions (each with S = 2) that couple antiferromagnetically to give a net spin ground state of S

= 10. This is illustrated in Fig. 1-1. The molecule is modeled as having a spin vector of S = 10

with twenty one possible proj sections (states) between ms=-10 and ms = -10 along a quantization

axis. Since the D value is negative for this (and any) SMM, there is an energy barrier to reversal

of the magnetic moment' s spin proj section. Graphically, this energy barrier can be pictured as a

double potential well, with states possessing opposite spin proj sections residing in opposite wells,

as shown in Fig. 1-2.

Due to the high symmetry of the molecule, transverse anisotropies less than fourth order

should be non existent and the rate of ground state quantum tunneling of the magnetization

(QTM) at low temperatures (kBT<








tunneling splitting (8.2 x 10-16 Cm-1) between the ground states [93]. The tunnel splitting was

calculated assuming that the antiferromagnetic interaction between the Mn+3 and Mn+4 ions are

dominant, and therefore each Mn+3/Mn+4 pair can be treated as a S' = 1/2 dimer. The reduced the

size of the Hamiltonian matrix 104 x 104 allowed a full diagonalziation. While the measured rate

of pure ground state QTM is immeasurably small [25] (< 10- s )~, there are other transverse

anisotropies present that arise from disorder. This is clearly evidenced by observation of

regularly spaced steps in the hysteresis loops. If only fourth order transverse anisotropies

existed, then QTM would only be possible between states that differ in m by multiples of 4. As

discussed in Ch. 1, this is due to the fact that the matrix elements of the states m and m (states

with opposite spin proj sections along the field quantization axis) vanish unless that stated

condition is met. Consequently, this allows one to define a selection rule for tunneling, which

states that tunneling is forbidden unless m + m '= 4n, where n is an integer. However, since

QTM is seen between states that differ in m by any integer, there must be additional terms

present that do not commute with the Sz operator in the Hamiltonian. This exciting discovery

opened the door for an abundance of extensive studies in the years to follow in an attempt to

explain the QTM [40, 94, 95, 96]. Much of the research conducted on this system has been

dedicated to understanding why some of the idealized predictions are not realized by

experimental tests, and what are the sources of these additional transverse anisotropies.

6.1.1 Theory and Effects of Disorder in Mnl2Ac

A theoretical Mnl2Ac molecule has S4 Symmetry. However, disorder in the system lowers

the symmetry of some molecules, which introduces transverse anisotropies [17] that are

forbidden by S4 Symmetry. The disorder manifests itself in the form of a distribution of

molecules with different spin Hamiltonian parameters [97]. In Ch. 4 we presented work that









outlined the influence of disorder on the width and structure of HFEPR resonance peaks. Such

disorder will also have an effect on the QTM in a given SMM system. In section 6.2 we

examine the disorder from a different perspective, primarily how the disorder results in a

distribution of molecules that relax on different time scales and how this manifests itself in

different experiments.

First we discuss the main results [97] from work done to characterize the disorder in this

SMM, since they provided a motivation for the experiments presented in section 6.2 and are

directly applicable to our work. Low temperature x-ray diffraction studies were conducted [98]

on Mnl2Ac and reveal the presence of species that have symmetry lower than the theoretical S4.

A single Mnl2Ac molecule has four acetic acid solvent molecules, each of which is shared by

two neighboring Mnl2Ac molecules. The acetic acid molecules are disordered around two

equally populated positions of the molecule between adj acent magnetic clusters and are involved

in strong hydrogen bonding to the acetate ligands. However, the acetic acid molecule will only

hydrogen bond to the ligand in a specific orientation, leaving the neighboring Mnl2Ac molecule

without such a bond. From this perspective, there are two equivalent ways to place one acetic

acid molecule between two Mnl2Ac molecules. That is, the methyl group (CH3) in the acetic

acid molecule can point toward a Mnl2Ac molecule or away from it [99]. In the latter case, there

is hydrogen bonding between the oxygen atom in the Mnl2Ac molecule and the oxygen atom in

the solvent molecule. Thus, disorder arises from the hydrogen bond interaction with the acetic

acid molecules, reducing the number of hydrogen bonded ligands. In fact, there exist six

different isomers of Mnl2Ac molecules within a crystal, with three different symmetries and five

different D values. The D values are calculated by proj ecting the contributions of the single ion

anisotropies of the Mn+3 ions onto the molecular anisotropy. Fig. 6-1 illustrates the six different










Mnl2 isomers from this model. The black spheres represent the magnetic core of the molecule,

while the cyan arrows represent the acetic acid molecules. The arrow head pointing toward the

molecule represents the hydrogen bond between the oxygen atom in the Mnl2Ac molecule and

the oxygen atom in the solvent molecule. As can be seen, two of the six isomers (D1 and De)

maintain strict S4 Symmetry. However, these constitute only 12.5% (6.25% each) of the total

species within the distribution. For a large number (87.5%) of molecules the S4 Symmetry is

lowered, and an additional second order transverse anisotropy (E term) develops. As can be seen

for the isomers D Ds, the symmetry is lowered, creating a rhombic transverse anisotropy.

Some of those molecules possessing an E term also have their magnetic axes tilted from the

crystallographic axes. Thus, even when the external field is aligned with the crystallographic

easy axis, there will be a misalignment with respect to the easy axes of these molecules. This

introduces an external transverse Hield component. The calculated tilt angles are all less than

0.50. However, later work [100] using HFEPR determined that these tilt angles for molecules in

the distribution can actually be significantly larger (up to 1.70). These transverse Hields provide a

natural explanation for the lack of tunneling selection rules. Without this, tunneling steps would

only be allowed for resonance fields that correspond to states that differ in m by a multiple of

two. And under these conditions, only those molecules with E terms would obey this selection

rule. The other molecules would only tunnel at resonance fields that differ in m by a multiple of

four, due to the S4 Symmetry. However, since tunneling steps are observed at each resonance

field in hysteresis loops of this compound, this work done in Ref. 98 and Ref. 100 y6was a maj or

step forward in understanding the mechanisms that govern the observed QTM in Mnl2Ac, where

QTM in SMMs was first observed [16, 18, 27, 95, 101].









6.1.2 Dipolar and Hyperfine Field Relaxation Mechanisms

Here we qualitatively discuss one of the prominent theories of QTM in SMMs, the results

of which will be used to explain some of the magnetization relaxation observed in our

experiments. The theory of Prokof ev and Stamp [102, 103] (PS) considers a giant spin coupled

to a bath of environmental spins (dipolar and nuclear). The model concentrates on the low

temperature (kBT<< |D|S) relaxation (below 2 K for Mnl2Ac) in the presence of a small external

Hield so that only the two lowest levels (+S) are involved. In this regime is it predicted [104] that

the phonon-mediated spin relaxation (r 1) is proportional to the cube of the energy difference

(bias) between the two lowest energy levels. Ignoring hyperfine and dipolar Hields, this energy

difference will go like 2 g pBS IB and consequently r ~ B I 3 he striking contradiction is that

while this predicts a minimum in the relaxation for B = 0, it is experimentally observed in a

system such as Fes that the relaxation rate is ~ 104 greater for B = 0 than for a non resonant Hield

(B = 0. 1 T). If one now considers the previously ignored hyperfine and dipolar fields we see that

they have little influence on the prediction of a minimum in r for B = 0, as is now outlined.

For resonant tunneling in this low temperature regime, the energy bias must be less than

the tunnel splitting between respective levels. The ground state tunnel splitting (without any

transverse fields) in systems like Mnl2Ac and Fes is on the order of 10-10 cm while the typical

bias caused by dipolar fields is ~ 0.07 .35 cm The bias from dipolar fields then forces the

molecules off resonance and the only way for the molecules to relax is then via spin-phonon

interactions, which have incredibly long relaxation rates for the regime we are considering. Even

if one considers transverse components to the dipolar fields, it would take transverse fields ~100

times larger than the actual hyperfine/dipolar fields to increase the tunnel splitting to the values

needed to explain the maximum in the relaxation rate for B = 0. The key point to these









arguments is that a distribution of static hyperfine and dipolar fields is insufficient to model the

observed relaxation rate behavior. However, a distribution of dynamic hyperfine and dipolar

fields provides a mechanism to account for the observed behavior. Dynamic hyperfine and

dipolar fields in the low field, low temperature regime can vary the bias at each molecular site in

time, which can continually bring more molecules to resonance [105].

In the low temperature limit only the dynamic nuclear fields (i.e., hyperfine and nuclear

dipolar fields) will cause relaxation since dipolar flip-flop processes are frozen out for this

regime, except when molecules tunnel. The nuclear interactions are expected to be important in

Mnl2Ac since all the manganese nuclei are 55Mn with a spin of 5/2. However, although it is

necessary to have rapidly fluctuating hyperfine fields to bring molecules initially to resonance,

the ensuing gradual adjustment of the dipolar fields across the sample, caused by tunneling

relaxation, brings those molecules that have not tunneled further into resonance, and allows

continuous relaxation through this process. The fluctuating nuclear spin field also makes the

tunneling incoherent. On the basis of these assumptions a formula for the magnetization

relaxation as a function of time in this regime was derived, which is shown to follow a tm/

dependence. This model is in principle valid at zero temperature and short times, when the

initial magnetization is close to the saturation magnetization.

It is also shown that for higher temperatures such a tm/ relaxation rate is not expected. This

is due to the influence of dipolar flip-flop processes which are frozen out at low temperatures.

These interactions cause the nuclear spin lattice relaxation time, TI, to become much shorter than

in the low temperature regime, and the magnetization reversal proceeds via higher levels through

a thermally assisted tunneling processes. Coupling to the phonon bath is then crucial, which

alters the arguments in the theory that give rise to a tm/ relaxation rate. When Ti is much less









than the experimental time scale and hyperfine Hields are comparable to or larger than

intermolecular dipolar Hields, exponential relaxation is predicted [106].

While this theory was derived considering only tunneling between the two lowest levels of

a large spin system with a small applied Hield, we will show that the qualitative aspects of

dynamic nuclear and dipolar fields influencing the relaxation can extend to tunneling at non zero

resonance fields as well. Additionally, we will provide data that supports the predictions of a tm/

dependent relaxation rate at in the low temperature regime. However, for the relaxation rate in

the high temperature regime we find that the form deviates from exponential to that of a

stretched exponential. First we outline the results from an experiment [107] on Mnl2Ac

conducted using a magnetometer to measure the magnetization decay as a function of time for a

temperature range of 1.5 K to 3 K.

6.1.3 High and Low Temperature Relaxation Regimes

PS theory predicts a low temperature regime relaxation law for the magnetization that goes

like


2M(t) = 290 [1- (6-1)


This study was done in zero field on a single crystal of Mnl2Ac and showed that in the low

temperature regime (below 2 K) the magnetization relaxation can in fact be well described by a

square-root law such as that in Eq. 6-1. In this regime, the mean relaxation time becomes as

large as 50 years, and it depends weakly on the temperature. However, as the temperature is

increased closer to and above 2 K, the relaxation can only satisfactorily be fit with a stretched

exponential law that goes like


M(t) = Moe'P (6-2)









However, it is found that a is strongly temperature dependent. Below 2.0 K, a is approximately

constant and close to 0.5. It increases linearly with temperature up to 2.4 K to reach a = 1.1, and

then it slightly decreases at higher temperatures (between 2.8 K and 3 K, a = 1). Also, the mean

relaxation time, r, starts to follow the Arrhenius law r = roexp(-E/T) roughly above 2 K, with

[108] the characteristic time, to, having a value of 107 s. Below 1.9 K, with a fixed value B =

0.5, r is weakly dependent on the temperature and tends toward r = 1.5 x 109 S at 1.5 K. This

experiment demonstrates the existence of two relaxation regimes, as predicted by PS theory.

6.2 QTM Studied by HFEPR

Most studies investigating QTM in SMMs have used changes in magnetization to monitor

the tunneling. Such experiments have provided information as to where (in magnetic Hield) the

spins are tunneling and the probability for such tunneling to occur. In contrast, our experiment

provides information on which spins within the distribution of molecules are tunneling and on

what time scales. HFEPR is a technique that can discriminate between different molecules since

each molecule sees the same microwave frequency. However, the shape (width and structure) of

the resonance peak illuminates the different microenvironments within the crystal. Each

molecule with slightly different parameters will transition at slightly different Hields and this is

clearly seen in the character of the observed resonance peak. By selecting how long we allow

the tunneling at a resonance Hield and then measuring those molecules that tunneled with EPR,

we also gain insight into the length of time it takes for different molecules to tunnel while on

resonance.

As stated before, Mnl2Ac is an incredibly well studied system, and thus the spin

Hamiltonian parameters are extremely well characterized [37]. With this information we can

simulate the energy spectrum for the molecule using Eq. 6-1 and make choices on what magnetic

Hield and frequency combinations will be appropriate for a data set. A discussion of the relevant









tunneling processes involved in the experiments in sections 6.2.1 and 6.2.2 is given in section

1.4.2.

6.2.1 Experiment in Stretched Exponential Relaxation Regime

This experiment deals with dynamics on the border of the low temperature (t'/) and high

temperature regimes (stretched exponential) relaxation regimes as demonstrated by the

discussion in section 6.2.3. Mnl2Ac crystallizes in the shape of a long rectangular needle. A

single crystal (~1.5 mm x ~0.3 mm x ~0.3 mm) was mounted with the long, flat edge of the

crystal against the base of a copper plate that mounts on the end of our 250 GHz quasi optics

probe [55]. The crystal is attached to the copper endplate with a minimal amount of silicon

vacuum grease. We cool the sample from room temperature to 2 K in zero Hield and then align

the sample such that its easy axis of magnetization (the c axis) is parallel to the applied magnetic

Hield with angle dependent measurements (not shown).

In order to study QTM using HFEPR, we apply a magnetic Hield of -6 T in order to fully

bias the system such that the spins are in the ground (ms = 10) state. This is illustrated in Fig 6-

2, which plots the energy levels of the system as a function of magnetic field. The blue arrows

represent the biased magnetization of the system for a sufficient (3 T) magnetic field. At each

resonance field (k = integer), spins can reverse their state by tunneling. While applying a

microwave frequency of 286 GHz, we sweep to a magnetic field value of +0.9 T (k = 2), where

levels on either side of the barrier are in resonance, and hold the magnetic field at this value for

different waiting times, which results in tunneling of spins between resonant states. Then we

sweep back to -6 T and observe changes of the EPR intensities.

Fig. 6-3 shows EPR spectra for different waiting times at +0.9 T. Each trace involves

sweeping from +0.9 T to -6 T after a different wait time. The inset to Fig. 6-3 shows the sweep

from -6 T to +0.9 T, which was identical for all data sets. As Fig. 6-3 shows, the area of the









peak occurring on the positive Hield side decreases for longer wait times, while the area of the

peak occurring on the negative Hield side increases accordingly. The peak occurring at positive

Hield involves transitions on the metastable side of the potential energy barrier (ms = 10 to ms=

9). As the Hield is held at +0.9 T for longer times, more spins from the ms = 10 state tunnel

through the anisotropy barrier. After tunneling, they quickly decay back to the ground state,

ms= -10. As the Hield is swept back to -6 T, those spins that tunneled no longer contribute to the

positive field peak, but will instead contribute to the negative Hield peak; i.e. the transition

between the ms = -10 and ms = -9 states.

Fig. 6-4 shows the area of the positive Hield peak as a function of wait time. The peak area

is a measure of the number of spins that have tunneled, which in turn is a measure of the

magnetization relaxation. Thus, we should be able to apply the theory of relaxation to the

change in peak area as a function of time. At the experimental bath temperature, we are above

the regime where a t/2 relaxation should be observed. Long range dipolar Hields will affect the

tunneling [102, 103], i.e. the local magnetic induction will evolve due to the fluctuating dipolar

fields of molecules that have tunneled. Indeed, this curve can not be satisfactorily fit to an

exponential relaxation law, but can be fit to a stretched exponential of the form Moexp[-(t/zRf],

which is in agreement with other work for magnetization relaxation in this temperature regime

[107].

We estimate a mean relaxation time, z, on the order of 500 s. The best matching curve to

the data gives a value of f = 0.70 + 0.05. For the experiment discussed in section 6. 1.3, a was

found to be quite close to 0.5 at T < 2 K and then increase linearly with temperature up to ~ 1.1

for 2.4 K. While our value of f is in quite good agreement with that for the other study, our

value of 7is much lower. One reason for this difference is that our experiment was conducted at









a non zero resonance field, while the other was done in zero field. As has already been

determined (section 1.4.2) the relaxation is much faster between the ground state and higher

lying states than for zero field ground state tunneling (k = 0), and we were waiting at the

resonance field corresponding to k = 2. Another factor is the sample shape, which r has a strong

dependence upon due to differing initial distributions of internal fields [102]. It is reasonable to

assume that the shapes of the samples used in the two experiments were different, and this will

affect the value of the mean relaxation time.

The evolution of the shape and structure of the peaks in Fig. 6-3 reinforces the idea that

there is a distribution of molecules relaxing on different time scales that govern the observed

behavior. Previous studies on this compound have found significant distributions of the D and g

parameters [83, 84]. All of the spins are subjected to a frequency of 286 GHz, but not all spins

see the same microenvironment. Careful inspection of the data reveals that different portions of

the spectrum relax on different time scales. For example, the signal changes most rapidly at

early times on the high-field sides of the peaks. Molecules contributing to these tails of the

spectrum possess larger than average D values. Since these are relaxing quite fast they must

posses an E term, and this would correspond to the D4 and Dg molecules, which have larger than

average D values (-.541 cm-l and -.548 cml respectively) but also have E values (3.27 x 10-3

cm-l and 1.64 x 10-3 Cm-1 TOSpectively). However, it also appears that a significant fraction of the

molecules on the low-field (lower D) sides of the peaks have already tunneled during the first 30

seconds, since one always observes weight in the spectrum at low negative fields. This

observation is consistent with the 25% of the molecules (Dz molecules) that have smaller than

average D values but also have E values.









Once most of the molecules have tunneled, a small fraction of slower relaxing species

remain, giving rise to the much narrower EPR peak at low positive fields. The fact that this

remnant peak is quite narrow indicates that this population is probably rather pure. This finding

agrees with the Cornia model which predicts that the fraction of highest symmetry molecules (D1

and Dsin Fig. 6-1) in a crystal should constitute ~ 6.25% each. These molecules should have the

longest relaxation times, since they maintain S4 Symmetry. Additionally, the Dg molecules have

the largest D value of all the isomers. These molecules must have a narrow distribution of

microenvironments amongst themselves, which manifests as a sharp tail in the emerging

resonance peak. Thus, in addition to measuring the ensemble average of the relaxation, these

EPR measurements demonstrate that one can separately monitor the relaxation from different

parts of the inhomogeneous distribution of molecular environments.

To summarize, at these temperatures, both hyperfine and dipolar fluctuations are fast

enough to maintain the conditions for tunneling. Thus, the observed deviation from exponential

relaxation comes from the time evolution of the local mean field induced by tunneling and the

distribution of molecules with different relaxation rates. Different molecules within the

distribution relax on different time scales, with both the D values and symmetry of the molecules

contributing to the relaxation rates.

6.2.2 Experiment in the t"Z Relaxation Regime

At a later time we decided to re-visit this work in order to repeat the experiment in a lower

temperature regime where we could test the t'/2relaxation law of PS theory. The experiment

discussed in section 6.2.1 was conducted in a system where 2 K was the minimum bath

temperature. The following experiment was conducted in a system where we are able to reach

temperatures as low as 1.4 K. A single crystal was placed with the long, flat edge of the crystal

against the endplate of a rotating copper cavity. The setup was placed in the Oxford systems










magnet and the sample was oriented such that the c axis was parallel to the applied magnetic

Hield. A data acquisition procedure similar to the previous experiment explained in section 6.2.1

was used.

6.2.2a Low Temperature MQT

Fig. 6-5 shows the time evolution of an EPR peak for a frequency of 237.8 GHz and

temperature of 1.4 K after waiting at a Hield of 1.8 T where the levels ms=-10 and ms = -6 were in

resonance. If one compares this to Fig. 6-3, one sees that the same qualitative trends are

observed. For short wait times (< 60 s) a small peak starts to emerge. As we wait longer at the

resonance Hield (< 1200 s) a broad peak develops whose area increases with wait time. And for

the longest wait times (> 1200 s) a second, higher Hield, narrow feature emerges in the spectrum.

One noticeable difference is the ability to fit the peak area to a t'/2relaxation law. The peak area

is proportional to the magnetization of the sample and in this regime the magnetization should

follow a square root relaxation law [102]. Fig. 6-6 plots the peak area as a function of wait time,

and it does indeed fit to a square root relaxation law. At these low temperatures, both the lattice

vibrations and dipolar fluctuations are minimal. As a consequence, the relaxation, which is

induced by nuclear spin fluctuations, takes the square-root form in Eq. 6-1. The square-root

decay arises from the formation of a depletion in the dipolar field distribution at the tunneling

energy due to the spins that have tunneled. The distribution of dipolar fields is close to

equilibrium, but there is a slow shift of its mean value with time [102].

From Fig. 6-6, we extract a mean relaxation time of r=1.5 x 106 S. This value is closer to

the mean relaxation time from the previous study, but is still two orders of magnitude lower even

though the bath temperature in our experiment was lower ( 1.4 K Vs. 1.5 K). Again, we state

that this discrepancy may be explained by the fact that their experiment was conducted in zero

field, while for our experiment we waited at the fourth resonance field (k = 4). Such a field will










bring higher lying states into resonance and any contribution from phonons will excite spins to

higher levels, making it much easier for them to tunnel. Also, we again mention that the sample

shape has a strong effect on the mean relaxation time.

It is known that disorder in this system also plays a role in the observed QTM behavior

and EPR can resolve which molecules within the distribution are tunneling. To this end, we took

the traces from Fig. 6-5 and fit each spectrum to a simulation that combined two different peaks,

each with a Gaussian profie. The main part of Fig. 6-7 shows an example of this fit for a wait

time of 2400 s. As can be seen, the spectrum consists of one broad peak and one narrower peak.

For each trace, we plot the peak position, peak width, and average D value for peak 1 and peak 2.

Fig. 6-8a and Fig. 6-8b plot these quantities as a function of wait time. Since we are observing a

transition taking place in the metastable well, those molecules with larger D values will

transition at higher magnetic Hields, hence the correlation between Fig. 6-8a and the inset to Fig.

6-8a. As the wait time at a resonance field is increased, the average D values of the resulting

EPR peaks become larger. This suggests that those molecules with smaller D values tunnel first,

while those molecules with larger D values take longer before they tunnel. Not only do these

slower relaxing molecules have higher symmetry (no E term) than the faster relaxing molecules,

but they also do not have their magnetic easy axis tilted away from the crystallographic c axis.

With the field aligned parallel to c, these molecules will not have a transverse component to the

external field, while the molecules with tilted axes will have such a term in addition to an E term.

Both factors increase the tunneling rate for the respective molecules. Additionally, as can be

seen from Fig. 6-8b, the width of peak 1 (consisting of molecules with larger D values) decreases

for longer wait times, while the width of peak 2 (consisting of molecules with smaller D values)

increases for longer wait times. This implies that the molecules that have the largest D values









have a rather narrow distribution. This finding is identical to the results of the first study where

we saw evidence for a narrow distribution of slow relaxing molecules for the longest wait times.

One can assume that by the longest wait times (3600 s), most of the molecules will have

tunneled, since ~ 87.5% of the molecules have a rhombic transverse anisotropy (E) term. By

taking the ratio of the area of peak 1 to the total peak area and peak 2 to the total peak area for

the data set at 3600 s, we can reinforce this assumption. Peak 1 represents the higher field (D

value), high symmetry molecules with a narrow distribution, while peak 2 represents the lower

Hield (D value), lower symmetry molecules with a broad distribution. Peak 1 constitutes ~ 11 %

of the total area, while peak 2 constitutes ~ 89 % of the total area. This is in nice agreement with

the Cornia model [97], which estimates that ~ 12.5 % of the molecules maintain S4 Symmetry,

while ~ 87.5 % molecules will have an E term. Even though the % of higher symmetry

molecules is underestimated, these molecules may take a sufficiently longer time before they

ever tunnel, since they are lacking an E term, and the lowest order transverse anisotropy term

allowed would be fourth order. This term will still cause tunneling, but only in higher orders of

perturbation theory.

6.2.2b Magnetic Avalanches

Here we briefly comment on some behavior involving magnetic avalanches observed in

this experiment. A magnetic avalanche is a different kind of relaxation mode than QTM,

exhibited by sufficiently large crystals, where a rapid magnetization reversal takes place that

typically lasts a few ms or less. It was initially studied by Paulsen and Park [109] and attributed

to a thermal runaway or avalanche [110]. In the avalanche, the initial relaxation of the

magnetization toward the direction of the Hield results in the release of heat that further

accelerates the magnetic relaxation. A detailed experiment involving magnetic avalanches will

be discussed in section 6.4.










Fig. 6-9 shows spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to -6 T

from waiting for 600 s at different magnetic (not necessarily resonance) fields. A clear signature

of a magnetic avalanche observed with EPR is the sudden sharp change in absorption of the

microwave radiation. If this occurs in the middle of an absorption peak, the signal will sharply

increase as the spins change state and are no longer absorbing microwaves.

Our data exhibit two regimes, one where no avalanches occur, and the other where

avalanches almost always occur. For wait fields up to 2.45 T, no magnetic avalanches are

observed. But, for all but a few fields from 2.45 T and above, magnetic avalanches are always

observed (fields marked in red in Fig. 6-9). Additionally, avalanches are also always observed

for fields that induce full magnetization biasing (sweeping back and forth from 6 T to -6 T) as is

shown in the inset to Fig. 6-9.

We provide an explanation for the observed behavior by considering which molecules

within the distribution are tunneling. Since we always wait for 600 s at the wait field, only the

faster relaxing molecules will have tunneled during this time period. However, as we increase

the field at which we wait, we decrease the effective barrier for magnetization reversal and

consequently, the slower relaxing molecules have more of an opportunity to tunnel. Eventually,

for large enough wait fields, the effective barrier for magnetization reversal is non existent and

all the molecules tunnel, giving rise to a full biasing of the spin states (full biasing of the

magnetization). Under these conditions, we always observe an avalanche as we sweep back to

the reverse bias field. To summarize, we observe no avalanches for wait fields up to 2.45 T

where only the fastest relaxing species have tunneled, and frequently observe avalanches for wait

fields from 2.45 T to 6 T where the slower relaxing molecules have had a chance to tunnel.









From this perspective, we hypothesize that the slower relaxing molecules are involved in

triggering the avalanches in Mnl2Ac.

6.2.3 Characterization of Minority Species

As a final part of the analysis from this experiment, we are able to characterize the

minority species that constitutes about 5% of the molecules within a crystal of Mnl2Ac, shown in

Fig. 6-10. These exceptionally fast relaxing molecules are attributed to a reorientation of the

Jahn Teller axis of the one or more of the Mn+3 ions, which is oriented ~ 900 from its position in

the maj ority species molecules [1 11]. This will introduce an E term and a resulting tilting of the

molecular easy axis with respect to the crystallographic c axis.

Referring back to the inset to Fig. 6-5, we see that while sweeping from -6 T to our wait

field, we see a small EPR peak at about +1 T. This is an indication that a small population of the

molecules have tunneled already! Multiple frequencies taken at 1.4 K sweeping from 0 T to -6

T allow us to extract the zero field splitting value of the minority species. This is illustrated in

Fig. 6-11. As we repeat these measurements at an elevated temperature (15 K) we see multiple

peaks start to appear as the excited states of the minority species become populated. The peak

spacing allows us to extract aD value. Assuming a ground state spin manifold identical to the

maj ority species (S = 10), we obtain a zero field splitting of 7.23 cm-l and a D value of -0.3 8

cm-l for the minority species. These values are significantly lower than those of the majority

species (zero field splitting of 10.22 cm-l and a D value of -0.45 cm- ), and since the minority

species have significant E terms and tilting of the molecular easy axis due to the reoriented Jahn-

Teller axis, they are exceptionally fast relaxing molecules. Experiments with AC susceptibility

estimate values [1, 112] in the range of 16 cm-l to 24 cm-l for the effective anisotropy barrier

(|D|S2) for these fast relaxing species. From this we can estimate a zero field splitting of 4.6 cml

and D = -.244 cm-l for the 24 cm-l barrier value. Our values are higher than these, since they are









measuring the effective anisotropy barrier (difference between the ground state and highest lying

level where tunneling occurs) and we are dealing with the true anisotropy barrier. More recent

inelastic neutron scattering data [113] have determined values (zero field splitting = 7.23 cml

and D = -.29 cm- ) that are in nice agreement with our work.

6.3 Microwave Induced Tunneling Measured with Hall Magnetometry

Now we turn to an experiment that was conducted in order to explore the influence of

microwaves on the magnetization dynamics of Mnl2Ac. Other work [114, 115, 116] has been

done studying photon assisted tunneling in systems such as Fes and Ni4.

We cool the sample from room temperature to 2 K in zero field and then align the sample

such that its easy axis of magnetization (the c axis) is parallel to the applied magnetic field with

angle dependent measurements (not shown). For all of the following data the bath temperature

was 2 K, although it will be seen that the sample temperature can vary depending upon the

microwave power incident upon the sample.

A typical hysteresis loop is shown in Fig. 1-4. The steps in the magnetization seen at

fields of approximately 0.45 T correspond to the resonance fields where spins are tunneling

through the anisotropy barrier. The flat plateaus in the figure correspond to fields where

tunneling is forbidden, but the sharp steps seen at resonant field values are where the tunneling is

switched on. The data in Fig. 1-4 was taken in the absence of any external radiation. Now we

turn to our studies on how microwave radiation can affect the magnetization dynamics in this

sy stem.

To begin with, we chose a frequency that corresponds to an EPR transition between states

at the same magnetic field where QTM is allowed. For instance, 0.5 T is the first resonance field

and it corresponds to an anti-crossing of the nas = 10 and na, = -9 states. Additionally, for a

frequency of ~ 286 GHz, there will be an EPR transition from the nas = 10 state to the nas = 9 state









at 0.5 T. Consequently, as the field is swept back and forth from the minimum saturation fields

(+3 T) we should observe a change in the magnitude of the step at this resonance field. In Fig. 6-

12 we plot hysteresis loops taken under the influence of a number of different microwave

frequencies. It is clear that for a given frequency, the resonance field step is larger in magnitude

than without any microwaves at all. This is due to a multi step process initiated by the

microwaves. First, EPR transitions are induced between states as the sample absorbs the incident

microwaves. The relaxation of the spin back to the ground state is accompanied by an emission

of phonons. This increases the effective temperature of the lattice, which increases the effective

spin temperature as well, leading to thermal population of higher lying levels. It is easier for

spins to tunnel from higher lying states, so the observed tunneling step is larger since a larger

fraction of the molecules are changing their magnetization state. Additionally, only one step is

affected for each frequency, indicating that it is the EPR transition that is affecting the tunneling.

While the field is swept, the microwaves do not influence the tunneling since they are not being

absorbed by the sample. However, once the proper resonance field is met, the sample absorbs

microwaves during the transition process, which raises the temperature of the lattice.

Another thing to notice about these data sets is the small step at zero field. This is evident

for the sweep without microwaves, and thus, is independent of any external radiation. The

explanation for this step is the fast relaxing minority species of the sample that was discussed in

section 6.2.3. Analogous to the EPR peaks observed on both sides of zero field in the insets to

Fig. 6-9 and Fig. 6-11, the step near zero field in the magnetization indicates that the fast

relaxing species is tunneling at zero field. The height of the step indicates that the minority

species contributes approximately 3 5% of the total sample magnetization, in nice agreement

with other work [112].









All of the data taken in Fig. 6-12 was with continuous wave radiation, which is shown to

have a significant heating effect while on resonance. In an attempt to minimize this, we also

collected data by pulsing the microwaves. We chose to focus on the resonance at ~0.5 T since

this was in a frequency range (286 GHz) where we could work with our amplifier, pin switch,

and tripler components for the Gunn diode. With the amplified power output from the Gunn,

care must be taken not to heat the sample. The copper block holding the device can become

heated due to large microwave powers and this will transfer heat to the sample, which is an

example of non-resonant heating. Additionally, when the sample is on resonance it will absorb

more microwaves for larger powers, and this will lead to heating of the sample. Since we are

pulsing the microwaves we will speak in terms of duty cycle, which is the percentage of time that

the sample is exposed to radiation for a given pulse period. As an example, exposing the sample

to a 100 ns pulse every 1 Cls would correspond to a 10% duty cycle. Fig. 6-13 plots the

percentage of magnetization reversal for the step at 0.5 T under the influence of 286 GHz

microwave radiation for different duty cycles. The inset to Fig. 6-13 shows the data for some of

the duty cycles. As can be seen for the larger duty cycles such as 50% and 20% not only is the

amount of tunneling larger, but the coercive field of the hysteresis loops is smaller. This is an

indication that the sample is being heated while not on resonance since it has been shown that at

elevated temperatures (> 2 K) the coercive Hield (the magnitude of the magnetic field between

the saturated magnetization values) decreases [13] due to thermally assisted tunneling as well as

thermal activation over the anisotropy barrier. The greater magnitude of the step height

(magnetization reversal) has to do with a larger amount of thermally assisted tunneling while on

resonance, coming from the elevated spin and lattice temperature.









As mentioned earlier we wished to eliminate any non resonant heating due to the

microwaves, and to this end we selected the smallest pulse width accessible with our instruments

(12.5 ns) and a small duty cycle (1%). Data confirm that there is an immeasurable effect of non

resonant heating on the sample. To investigate the microwave contribution to the tunneling, we

conducted an experiment similar to those described in section 6.2. We would go to a resonance

field, in this case 0.5 T, and wait for different times (30 s to 3600 s) to see how the magnitude of

the magnetization step changed. We also continuously recorded the decay of the magnetization

signal during the wait time. This was done for the above mentioned duty cycle, as well as

without microwaves. For data taken in the absence of any microwave radiation, the only

contribution to the tunneling can come from the finite lattice temperature. On the other hand,

data taken while exposing the sample to microwaves has contributions to the tunneling from the

finite lattice temperature and the incident microwaves. Since the bath temperature was identical

for both data sets, we can subtract the difference to get the microwave contribution to the

tunneling. For the subsequent discussion, when we mention a "difference", we are specifically

referring to the difference between the data set taken in the absence of any microwave radiation,

and those taken with the 1% duty cycle microwave radiation. Fig. 6-14 plots the difference in

the magnetization reversal for the data sets taken with and without microwaves as a function of

wait time at 0.5 T. The scatter points in Fig. 6-14 plot the difference in percent of magnetization

reversal for different wait times by measuring the step height. Similarly, the curve in Fig. 6-14

plots the difference in percent of magnetization reversal by measuring the decay of the

magnetization for the wait time of 3600 s. The two are qualitatively similar, as would be

expected. The data set taken with microwaves shows larger magnetization reversal for a given

wait time, indicating that the microwaves influence the magnetization dynamics even with such a









small duty cycle. However, it is unlikely that the increased tunneling is solely due to the

microwaves. Such tunneling would take place between the ms = 9 and ms = -8 states, while

tunneling between higher lying states is possible due to increased lattice and spin temperatures

from resonant heating. The latter scenario is much more probable than the former, due to

respective tunnel splitting (Eq. 1-15).

In order to qualitatively describe the effect of microwaves on the tunneling in Fig. 6-14, we

consider the fact that at this bath temperature (2 K), we are in the regime where the relaxation of


the magnetization should begin to follow a stretched exponential decay (M(t) = Moe '). In

this regime the mean relaxation time, r, begins to follow an Arrhenius law (r = roexp(-E/T)) and

hence, it is temperature dependent. Under the influence of the microwaves, the spin and lattice

temperatures will increase, which will lead to a smaller value of r. We can compare the

relaxation for two different values of r, (r; and zz), corresponding to the situation when the

microwaves were on or off, respectively. Since the spin and lattice temperatures are higher for

the case of microwaves on, r; will be smaller than zz. Consequently, the magnetization will

decay quicker for a given time than for the case of the microwaves off (zz). In Fig. 6-14 we are

taking the difference of the magnetization decay between the situation with microwaves on and


off, so the magnetization decay should be ofthe form MV ) M el!~ .' By dgoig

a Taylor expansion of the exponentials and keeping only the lowest order terms, the


magnetization will go like MZ(t) = MotP .rf;R Since the term in brackets is a constant, the


magnetization is proportional to tP, which in this regime is close to 0.7. From this we see that the

influence of the microwaves on the tunneling is consistent with the above mentioned arguments.









We have presented work studying the influence of microwave radiation on the quantum

tunneling in Mnl2Ac. The tunneling at a given resonant field is enhanced when the microwave

frequency corresponds to transitions between states at this resonant field. This resonant

enhancement is due to microwave absorption leading to an increase in the lattice and spin

temperatures, and consequently, thermally assisted tunneling. By reducing the amount of power

incident upon the sample through low duty cycle pulsed radiation, we are able to reduce the

resonant heating and eliminate the non resonant heating of the sample. Comparison of resonant

tunneling for different wait times with and without the presence of external microwaves allows

us to observe the contribution of microwaves to the tunneling. It was shown by comparing the

difference in magnetization reversal with and without the influence of microwave radiation that

the observed behavior can be attributed to different temperature dependent mean relaxation

times. The different temperatures arises from resonant heating of the spin and lattice

temperatures during the microwave absorption process.

6.4 Relaxation in Mnl2Ac Measured with HFEPR

The observed magnetic avalanches from the experiment in section 6.2 were processes

that were randomly occurring and not amenable to manipulation. However, other research has

been conducted studying this phenomenon in Mnl2Ac under quite controlled circumstances. It

was first studied by Paulsen and Park [109] and attributed to a thermal runaway or avalanche

[110]. Recent studies analyzed the stochasticity of the process and the spatial dependence,

dealing with avalanches as a deflagration process [117, 118]. In that set of experiments,

avalanches were triggered in a similar way to how they were triggered in the experiment

discussed in section 6.2.2 (although in our case it was not intentional) by sweeping the magnetic

field back and forth from -6 T to +6T until an avalanche was triggered. Once triggered the

avalanche propagates through the crystal in a similar fashion to a flame propagating through a









flammable substance. Consequently, this phenomenon has been termed magnetic deflagration.

The magnetization reversal occurs inside a narrow interface that propagates through the crystal at

a velocity of a few m/s. The speed of the deflagration is determined by the thermal conductivity

of the crystal, as well as the rate of thermal activation over the anisotropy barrier. From a

thermodynamic perspective, when the spins are biased by a magnetic field such that all are in a

metastable well, this can be considered as a flammable chemical substance. The chemical

energy per molecule that is released during the spin reversal is equivalent to the Zeeman energy,

g~pBB AS, where AS corresponds to the difference between the states of the system that are

parallel and antiparallel to the applied field, B.

Recently, a new technique was developed using surface acoustic waves to ignite the

magnetic deflagration associated with the spin avalanches at a determined value of the applied

magnetic field [119]. In collaboration the group from the University of Barcelona who

pioneered this technique, we were able to combine their controlled avalanche equipment with our

HFEPR setup and perform unique experiments on Mnl2Ac. Magnetic deflagration has been

measured through magnetization with Hall bars, coils, or SQUIDs. Although these methods may

provide spatial resolution, none of them allow analysis of the avalanche in a single energy level.

Our novel technique uses surface acoustic waves (SAWs) from a piezoelectric device to push the

system out of thermal equilibrium and microwaves from our MVNA as a probe to study spin

relaxation dynamics.

Other work has been done to measure relaxation times in SMMs [115, 120, 121, 122] such

as Ni4 and Fes. All of these studies involved measuring magnetization dynamics in the presence

of microwave radiation. In contrast, our technique uses microwave radiation to measure

dynamics in the system after sending it out of equilibrium with a SAW. EPR measurements










probe the spin population differences between energy levels and hence, our data provide energy

resolved information about how spins relax to equilibrium in Mnl2Ac when an avalanche is

triggered with a SAW pulse.

We discuss two experiments where we show how EPR signals can probe the time

evolution of spins as they relax to equilibrium. The first involves measuring the thermal

population difference between single energy levels during the deflagration process. The second

involves measuring the thermal population difference between single energy levels after

application of a short heat pulse to the system. In both cases the system is put into a non

equilibrium state due to coherent acoustic vibrations. These vibrations in turn introduce phonons

into the system, which thermalize with the sample. The electronics and instrumentation for this

experiment is described in detail in Ch. 2.

6.4.1 Triggered Avalanches in Mnl2Ac

We cooled the system in zero field to approximately 2.1 K. The sample was aligned such

that the easy axis of magnetization was approximately parallel (within lo) to the applied

magnetic field. This was deduced from preliminary angle dependent measurements (not shown).

We use a large applied magnetic field (>3 Tesla) to put all of the spins in the ground state (either

ms = +10) of one of the wells. Next we chose which EPR transition to observe and tuned the

frequency to the corresponding value. Once an avalanche was triggered in the system, we would

observe a change in the absorption spectra as the spins move from the ground state in the

metastable well to the ground state in the stable well.

The response of the system is measured by a change in the absorption of the incident

microwaves. Typically, the signal returning to the receiver is a relatively constant value until the

resonance conditions are met, and then a drastic reduction of the signal is seen as the sample









absorbs the microwaves. Once an avalanche was triggered in the system, we would observe a

change in the absorption as the spins move from the ground state in the metastable well to the

ground state in the stable well. This is illustrated in Fig. 6-15. The black arrows represent the

evolution of the spins as they avalanche. The red arrows represent the EPR transition between

states during an avalanche. We can see the avalanche by recording the absorption due to

transitions between levels in either of the energy wells (metastable or stable) and there is a

significant difference between these kinds of transitions. First, it is worth noticing that there is a

significant amount of heat released during an avalanche, and increases in temperatures between 6

and 12 K have been measured [117, 1 19], depending on the value of the applied magnetic field at

which at the avalanches occur. This heating causes the system to be out of equilibrium after all

spins have avalanched, and an elapsed time of hundreds of milliseconds is needed to recover the

initial bath temperature, during which the excited spins relax to equilibrium in the ground state of

the stable well. The main difference, therefore, between the two kinds of transitions is that the

heat dissipation after the appearance of the avalanche hardly effects the spin population within

the metastable well since these transitions take place before all spins have changed to the stable

well. It is true that higher lying states start to become thermally populated, but only before the

spins avalanche over to the stable well. Hence, the lifetimes of these states are mostly decoupled

from the phonons arising from the avalanche propagation. On the other hand, the transitions

within the stable well are greatly influenced by the heat dissipation since both the ground state

and higher lying levels become populated.

For the purposes of discussing the data we shall refer to a reference level of zero

microwave absorption and the initial values of the EPR signal are presented with arbitrary offset

units. The EPR signal that we plot is the transmitted microwaves returning to the detector after









interacting with the sample. When there is no absorption from the sample the signal should be at

a zero baseline (within the oscillations of the noise). On the other hand, the signal will be below

this reference level when the sample is absorbing. Fig. 6-16 shows the time evolution of the

transition between the ms = 10 to 9 (Fig. 6-16a) and ms = 9 to 8 (Fig. 6-16b) states in the

metastable well after igniting an avalanche. The flat dashed line in each graph represents the

level of zero microwave absorption. In (a) the initial conditions were such that all the spins were

in the metastable well and should be absorbing microwaves as they transition to the ms = 9 state.

We can see that this is indeed the case until a few milliseconds after the SAW pulse is applied

when the avalanche appears. Since all of the spins have avalanched to the stable well there is no

more absorption. In (b) we can see that initially, at t = 0, there is no absorption of the signal, but

a sharp absorption peak appears a few milliseconds after the ignition of the avalanche. As the

spins move up from the ground state in the metastable well they spend a brief amount of time in

the ms = 9 state where we see transitions from the ms = 9 to 8 states. This is related to the time

that the spins in the ms = 9 state need to overcome the barrier to get the stable state. Initial and

final EPR absorption values are the same because before the avalanche all spins populated ms

10 level and after the deflagration there are no more spins in the metastable well. As a contrast,

after recording the data from an avalanche we would pulse the IDT in order to heat the sample

with a pure SAW. Neither situation shows any absorption under these circumstances, as would

be expected, since the avalanche has taken place and all the spins are in the stable well while the

frequency and magnetic field are tuned for a transition in the metastable well.

Fig. 6-17a and Fig. 6-17b show the EPR signal of the transition between the ms = -10 to -9

(Fig. 6-17a) and ms = -9 to -8 (Fig. 6-17b) states in the stable well. In Fig. 6-17a we see that

initially there is no absorption of the signal, but once the avalanche occurs there is a strong









absorption. This can be pictured as the reverse of Fig. 6-16a. Since the spins all relax to the

ground state in the stable well we observe continuous absorption after the avalanche as the spins

transition from ms = -10 to -9. Fig. 6-17b shows different behavior. Again, the avalanche

always takes some time on the order of a few ms after the trigger pulse before it is ignited, as can

be seen from the EPR signal and there is no absorption until the avalanche occurs. As the spins

avalanche they pass through the ms = -9 state before relaxing to the ground state. The sharp,

short time side of the absorption is the transition between the ms = -9 to -8 states. However, it is

clear by the behavior of the signal that there is still some absorption for approximately 100 ms

after the avalanche. This can be understood by the fact that the ignition of an avalanche releases

a significant amount of heat into the sample. Heating of the sample during this process causes

the ms = -9 state to be thermally populated and is the reason for the prolonged absorption of the

microwaves. This extra heat takes some time to disperse and re-establish equilibrium in the

system. While this is happening the ms = -9 state will be thermally populated and will continue

to absorb microwaves. Eventually the phonons disperse, the spins only populate the ground state

of the stable well, and the EPR signal returns to its initial value (after ~ 100 ms) as there is no

longer any absorption from the sample and the system comes to thermal equilibrium with the

bath. For comparison, after triggering an avalanche, we would again pulse the piezoelectric

device to produce a SAW to heat the sample. The red traces in the figures show this. It is clear

then that there is a distinct difference in the EPR signal when comparing its evolution after

triggering an avalanche, and after pure heating with a SAW. The slope of the signal after an

avalanche is steeper than that of the signal after pure SAW heating. Additionally, the SAW

directly couples with the sample and the signal begins to change immediately, while the change

of the signal due to an avalanche takes a few ms before the avalanche is ignited.










The phonon bottleneck effect describes the fact that at low temperatures the number of

spins is greater than the number of phonons, due to the lattice temperature [116]. Although the

spins may be able to thermalize with the lattice quite easily, poor coupling between the lattice

and bath causes a prolonged relaxation of the spin temperature. Thus, the spins can not freely

exchange energy with the bath and it takes some additional time to equilibrate the spin and lattice

temperature with the bath temperature.

The data we collected for transitions within the stable well support this assertion. The

spin temperature should follow the lattice temperature quite quickly, but the time for the lattice

to equilibrate with the environment may be rate limited by a bottleneck effect if the coupling

between the lattice and bath is weak. The times for establishing equilibrium in the stable well

agree well with other studies [123] showing times on the order of 100 ms. However, for

transitions within the metastable well, we are able to observe dynamics that are not hampered by

this effect. This is because we take our measurement before the system has fully reversed its

magnetization state, and therefore has not yet released the amount of heat into the system that is

typical of an avalanche. The dynamics associated with population differences in the metastable

well as the avalanche propagates are on the order of 5 ms and we can infer that the time needed

to excite spins to higher levels is on the order of 1 ms. Since we were using a lock in amplifier

with a time constant of 100 Cps, from Fig. 6-14a we can see that if T; were less than 100 Cps, then

we would not observe the resonance peak, which is seen to span a width of ~ 5 ms. Thus, we can

place an upper bound for the spin lattice relaxation time, T;, on the order of 0. 1 1 ms.

6.4.2 Pulsed Heating and Spin Lattice Relaxation

The second kind of experiment we performed involved using short SAW pulses to heat the

lattice, which in turn heats the spins in the ground state of the stable well. This does not trigger

an avalanche but does perturb the system such that higher lying energy states in the stable well










become populated briefly before relaxing back to the ground state. The process of exciting to

higher levels and relaxing back down is probed with our low power microwave radiation. We

would choose a frequency and hold a specific value of magnetic field for the respective transition

that we wished to observe. We then repeat the pulses at a particular frequency and average the

signal in order to improve the signal-to-noise, as discussed in section 2.4.1. Pulses of 1 Cps to 50

ms and a nominal power of 6 dBm were used to induce heating of the sample and this was done

for a number of different bath temperatures.

Fig. 6-18 shows a plot of the EPR signal as a function of time for the ms = -9 to -8

transition. In contrast to the signal after triggering an avalanche, the signal here evolves

smoothly without any sharp changes. As soon as the pulse is applied (t = 0) there is a decrease in

the signal due to the ms = -9 state becoming thermally populated and spins transitioning to the ms

= -8 state. The population difference between these states reaches a maximum after

approximately 10 ms. We can see this by the fact that the microwave absorption is a maximum

at this time, and the amount of absorption is a measure of the spin population difference between

respective levels. Beyond this point the system is no longer heated by the SAW and the phonons

begin to disperse. The time for the excited levels to thermally depopulate is on the order of 100

ms, which is the time for the system to return to equilibrium due to the phonon bottleneck effect.

During this process, the populations of the ms = -9 and ms = -8 states decay and eventually are

zero.

In order to quantify the observed behavior we assume that there are two important

temperatures during the relaxation process: the lattice temperature, TL, and the spin temperature,

Ts with respective relaxation time constantS TL and z,. The time rate of change of the spin and

lattice temperatures is given by [124]









dT
L = W(t)+ P(TL TO) Lr Ts) (6-3a)


dT,
s= a(TL -Ts) (6-3b)


In Eq. 6-3, a and a are constants of heat transport, To is the bath temperature, TL is the lattice

temperature, Ts is the spin temperature, and W(t) is a pulse function that introduces heat into the

system. Ti corresponds to the time related to temperature variations of the lattice when the heat

pulse is switched on and off. rs corresponds to the time the spins need to follow these

temperature variations of the lattice, which is related to the spin lattice relaxation time, Ty. Eq.

6-3 is made linear by neglecting the higher order terms (quadratic, cubic, etc). This assumption

is valid since the differences in the spin and lattice temperatures are small during the period of

the measurement, as the spin temperature can follow the lattice temperature quickly. We observe

lattice decay times (zI) on the order of 100 ms when the heat pulse is switched on and off, and it

is estimated [125] from simulations using Eq. 6-3 that z, is 10-100 times faster than rt. This was

done using Eq. 6-3 to simulate the evolution of the magnetization after a SAW was applied to the

crystal. A reasonable simulation of the data was obtained when the values of ts is 10-100 times

faster than rt. Therefore, from our experiment, we estimate zs is on the order of 0. 1 1 ms. Once

again, in agreement with the data from the avalanche experiments, we can estimate an upper

bound of 0. 1 1 ms for T; in this system. A better agreement with the data may be obtained

from simulations that include higher order terms in Eq. 6-3, especially those including the

difference between the bath and lattice temperatures, which can be significant at longer times.

In this work we have demonstrated a novel technique to monitor spin population dynamics

by combining the techniques of SAWs and HFEPR. We are able to detect how the spins excite

and relax on rather fast times scales for different spin levels. By measuring the lifetimes of states









within both the metastable and stable wells as an avalanche propagates, we can obtain

information about the spin lattice relaxation time in this SMM. Our results indicate an upper

bound of 0. 1 1 ms for T; in a single crystal of Mnl2Ac.

6.5 Summary

In this chapter we discussed a number of unique experiments done on the Mnl2Ac SMM in

order to characterize the quantum tunneling observed in this system. First we reported on work

done using HFEPR to detect quantum tunneling. This technique allows us to observe different

molecules within the distribution tunneling on different time scales. We conducted experiments

in two different temperature regimes in order to make comparisons with the PS theory of

magnetic relaxation in SMMs. Our results show that there is indeed a low temperature regime

where the relaxation goes like t'/ and a higher temperature regime where the relaxation follows a

stretched exponential law, in accordance with predications made by PS theory. In both regimes

we see that the molecules with smaller D values and lower symmetry tunnel much sooner than

those with larger D values and higher symmetry. Additionally, we were able to characterize the

zero field splitting and average D value of the minority species of molecules present in the

crystal. These molecules have lower symmetry, a D value approximately 16% smaller than the

maj ority species of molecules and it is likely that they have a significant tilting of their molecular

easy axis. This is consistent with the rapid zero field tunneling observed. Next, we presented

work studying the influence of microwave radiation on the quantum tunneling. Using both CW

radiation and short pulses of microwaves at low duty cycles we are able to use microwaves to

influence the tunneling. Even without significantly heating the system due to non resonant

effects we see evidence for thermally assisted tunneling to due to increased spin temperatures

from microwave absorption. By comparing the difference in magnetization reversal with and









without the influence of microwave radiation, we attribute the observed behavior to different

temperature dependent mean relaxation times.

Finally, we described a novel experimental technique that was developed in order to

measure the relaxation processes on quite fast times scales during a magnetic avalanche. We are

able to record EPR transitions during the avalanche propagation in both the metastable and stable

wells of the system. The transitions in the metastable well are largely decoupled from the

increased spin and lattice temperatures due to the heat released during the avalanche. We also

are able to heat the system with a short pulse and observe the relaxation back to equilibrium by

recording the magnitude of the EPR signal. These experiments provide information on the

relaxation times associated with individual energy levels and from this we can put an upper

bound of 0. 1 1 ms for the spin lattice relaxation time, T;, in this system.










D,= 535 cm'
EZ = 1.63 x 103 cru
Concentration = 25%






D =-.541 cuf I
E = 3.27 x 10" em'
C~oncentration=- 12.5%


D = -.528 cKI
E = 0

I)


Concentration = 625%





D = -.541 cm"
E5 1.3d x 103 en{'





Concentration = 25% b


D = -.555 enf'
6
E6=0
C'onlcentraltionI


S6.2.5%;


Figure 6-1. Diagram of the six different Mnl2 isomers, each with a different D value and some
with E values. The cyan arrows represent hydrogen-bound acetate ligands.








~ ~\\.,


-5 -4-3 -1 1 2 3 4
MantcFel Tsa
Fiue -. ltofte nry eel nMn2c safucio fmanti ild hebu
arosrpesn h iae antiaino tesse frasficet(3T
mantcfed tec eoanefed( nee) pnscnrvretersaeb
tunneling















~ 2K.







CrS~~~~~~ 18 11 Ii 5~i 00s






-6 -5 4 -3 -2 -1 0 1
MLa~gnetic Field (Tesla)

-3 -2 -1 0 1

Magnetic Field (Tesla)

Figure 6-3. 2 K EPR spectra for different waiting times at 0.9 T. Each trace involves sweeping
from 0.9 T to -6 T after a different waiting time. The inset to Fig. 6-1 shows the
sweep from -6 T to 0.9 T, which was identical for all data sets. The area of the peak
occurring on the positive field side decreases for longer wait times, while the area of
the peak occurring on the negative field side increases accordingly, indicating QTM.
Arrows indicate the direction of increasing wait time. Reused with permission from
J. Lawrence, S.C. Lee, S. Kim, S. Hill, M. Murugesu, and G. Christou, Magnetic
Quantum Tunneling in a Mnl2 Single-Molecule Magnet Measured With High
Frequency Electron Paramagnetic Resonance, AIP Conference Proceedings 850, 1133
(2006). Fig. 1, pg. 1 134. Copyright 2006, American Institute of Physics.








b: 286 GHz
2K

~ 130 .7 5

2=488 s







0 500 1000 1500 2000 2500 3000 3500
Wait Time (s)
Figure 6-4. Plot of the area of the positive field peak as a function of wait time. This curve can
be fit to a stretched exponential of the form Aoexp[(1/z) ]. We estimate an average
relaxation time, r, on the order of 500 s. Reused with permission from J. Lawrence,
Magnetic Quantum Tunneling in a Mnl2 Single-Molecule Magnet Measured With
High Frequency Electron Paramagnetic Resonance, AIP Conference Proceedings 850,
1133 (2006). Fig. 2, pg. 1134. Copyright 2006, American Institute of Physics.













O~6 sl 1500 s
120I s 1800s


600 s' 3600


f; 237.8 GHz i
C 1.4 K ~ l :
C.) wait at 1.8 T
(m 10to 6) -6 -5 -4 -3 -2 -1 0 1 2
(mslq o -) chMagnetic Field (T~sa
-2.8 -2.1 -1.4

Magnetic Field (Tesla)
Figure 6-5. Plot of emerging resonance peak for different wait times at 1.8 T and sweeping
back to -6 T. The inset shows a sweep from -6 T to 1.8 T, which was identical for
all traces. If one compares this to Fig. 6-3, one sees that the same qualitative trends
are observed.








s2 I1 2 37. 8 GH z
~1 1.4 K
56
I, \I I= 1.59 x 106 s







0 500 1000 1500 2000 2500 3000 3500
Wait Time (s)
Figure 6-6. Plot of the peak area in Fig. 6-5 as a function of wait time. The peak area is
proportional to the magnetization of the sample and in this regime the magnetization
is shown to follow a square root relaxation law.



























I I I


V3
~cl
IE~
5
~13
k


O
cn
cn

E


k





U


waiting 2400 s

~ data

peak 1

----peak 2
~ fit to peak 1

+ peak 2


-3.0 -2.5


-2.0


-1.5


Figure 6-7. Plot of the spectrum for a wait time of 2400 s after fitting it to a simulation that
combined two different peaks, each with a Gaussian profile. As can be seen, the
spectrum consists of one broad peak and one narrower peak. Different molecules
within the distribution contribute to the spectrum on different time scales.


Magnetic Field (Tesla)









r~ -2.30
a~~~~ e 2.1501002030040
-0.465~~~ -z1 aiig m
0.470 .

S-0.475 A *3

0 1000 20002M) 3000 4000
L Watm Tim satn Im



~0.154 ea

-0.10 = "
B
0 1000 20001 30001 4000
Waiting Time (s)
Fiur 6-.AeaeDvlean ekwdhv.tme ) lto h veaeDvlefrpa
1 n ek2asafnto o attm f reahtceiFg.65Tein tplste
pea postio asafnto fwi iefrec taei i.65 hs oeue
with~~~~ lagrDvle iltasto thge antcfedhne a tecrlaio

value take logrbfr hytne.b.Po ftepa it safnto fwi




Figre 8largerg D values adecrease fort loge wi times, wh leto the widthg of plea 2o (cosisin


molecules with smaller D values) iunease f ort hl tonger wait ties. wThi imlies tha


the molecules that have the largest D values have a narrow distribution.










237.8 GE;z, 1.4 K, wait 600 s



c1.8 T 2.35 T
.18 T~ 2.40 T
1.9 T .4


-! 1.95 T1 2.50 T
I~2. T, 2.55 T

O Y2.10 T -2.65 T
.1 2.70 T1
~"2.20 T 27
V3~ T to -6 T 2.25 T --2.80 T
o sE II& ',1 2.310T ----- 2.85 T


L0l, -4 -33 -2-- .8


Magnetic Field (Tesla)

Figure 6-9. Plot of spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to -6 T
from waiting for 600 s at different magnetic (not necessarily resonance) fields. A
clear signature of a magnetic avalanche observed with EPR is the sudden sharp
change in absorption of the microwave radiation. Those fields where an avalanche
occurred are marked in red. The inset shows that after full magnetization reversal
(sweeping back and forth from +6T), an avalanche is always observed. We
hypothesize that the slower relaxing molecules (larger D value) are triggering the
avalanches.















orientation








Abnormal Jahan T~eller axis

orientation (~900)

Jahn Teller axis'

pointed towards O 2



Figure 6-10. Diagram of the minority species molecules (bottom) in Mnl2Ac. In comparison to
the maj ority species molecules (top), the minority species has a reorientation (~ 900)
of the Jahn Teller axis of one or more of the Mn+3 ions. This introduces an E term,
which explains the observed fast relaxation. There are two dashed lines because the
molecule has a crystallographic Ct axis disorder. Reprinted from S. M. J. Aubin, Z.
Sun, H. J. Eppley, E. M. Rumberger, I. A. Guzei, K. Folting, P. K. Gantzel, A. L.
Rheingold, G. Christou, D. N. Hendrickson, Single molecule magnets: Jahn-Teller
isomerism and the two magnetization relaxation processes in Mnl2 COmplexes,
Polyhedron 20, 1139-1145 (2001). Fig. 5, pg 1143. Copyright 2001, with permission
from Elsevier.


Normal Jahn Teller axis









300

N 250

200

O 150

~100

F450


data c


.


-g = 2.05


( I ~~Magnetic I-.eld I rteam II
-3 -2 -1

Magnetic Field (Tesla)


Figure 6-11i. Plot of the low temperature frequency dependence of the minority species peak.
Assuming a ground state spin manifold identical to the maj ority species (S = 10), we
obtain a zero field splitting of 7.23 cm-l and a D value of -0.3 8 cm-l for the minority
species.


Frequency Dependenc~e
Minority Species
i------_ 1.4 K


fit to (


- ZFS: 216.5 GHz= 7.23 cm













O
*? 0.6 -1
N 0.4 -1

O)~I 0.2 -no microwaves
tM)3 .C 289.9 GHz 0.45 T
5 276.9 GHz 0.90 T
r" ~1O.Z ~ ----263.3 GHz 1.35 T in
b)k-0.4 '250.9 GHz 1.8 T


I I Il i



3M~agnetic- Fiel d (Tesl~a)

Figure 6-12. Plot of hysteresis loops taken under the influence of a number of different
microwave frequencies. The bath temperature was 2 K for each sweep. It is clear
that for a given frequency, the resonance field step that corresponds is larger in
magnitude than without any microwaves at all. Additionally, only one step is
effected for each frequency, indicating that it is the EPR transition that is effecting the
tunneling. While the field is swept, the microwaves do not influence the tunneling
since they are not being absorbed by the sample. However, once the proper
resonance field is met, the sample rapidly absorbs photons and emits phonons during
the transition process. This emission of phonons heats the sample, which thermally
populates higher lying levels. It is easier for spins to tunnel from higher lying states,
so the observed tunneling step is larger since a larger fraction of the molecules are
changing their magnetization state.










70



5-



60







40



0-


m
S-r
6)
E,
Q)
pi:
d
O





d
t~ZP)



C~,
O


d
O

.r?







o
z


----oCNt
186.1 GHa
~ 50 ~ib
2096
~- 1096
196


-3 -2 -1 0 1
Magnetic Field (Tesla)



DutOy Cycle (%) 5


r III


Figure 6-13. Plots of the percentage of magnetization reversal for the step at 0.5 T under the
influence of 286 GHz microwave radiation for different duty cycles. The inset shows
the data for some of the duty cycles. As can be seen for the larger duty cycles such as
50% and 20% not only is the amount of tunneling larger, but the coercive field of the
hysteresis loops is smaller. This is an indication that the sample is being heated while
not on resonance since it has been shown that at elevated temperatures (> 2 K) the
coercive field decreases due to thermally assisted tunneling as well as thermal
activation over the anisotropy barrier. The greater magnitude of the step height
(magnetization reversal) has to do with a larger microwave absorption on resonance.


286.1 GHz

sweeping through 0.5 T

dB/dt =- 0.6 T/min










2.5-







1.5- r
O



b)* Difference between 12.5ns(1.25us) and
bZ) no uwaves @ 0.5 T
0.5 step hle ght
magnetization decay

O
0 0.0 ,,.....
O 500 1000 1500 20001( 2500 .3000 3500r

Wait Time (s)

Figure 6-14. Plot of the difference in the magnetization reversal for the data sets taken with and
without microwaves as a function of wait time at 0.5 T. The scatter points plot the
difference in percent of magnetization reversal for different wait times by measuring
the step height. Similarly, the curve plots the difference in percent of magnetization
reversal by measuring the decay of the magnetization for the wait time of 3600 s.










Spin evolution during~ avalanche propagation

SSpin evolution during an EPR trnosition


m72=7
m =8

m =9

m= 10 -> 9

m =10


-m=-7
-m -8

-m =-9



m= -10


Figure 6-15. The energy barrier diagram illustrating how spins move during an avalanche. The
black arrows represent the evolution of the spins as they avalanche. The red arrows
represent the EPR transition between states during an avalanche. Spins in the right
well relax to the ground state by emitting phonons, which produces heating of the
system. This heating is fed back into the avalanche as it is absorbed by the spins in
the left well.


M~etastable well


Stable well











O I biP 237.315 GJHz
0).581 T"
I ~t = 1.0 m~S Avalanche

r 1JB m~ = 9 to 8

d0 10 20 30 40 50




F1 270.0 GHz
b]D t =200 ms
v 1.1 T1



S0 40 80 120
Time (ms)

Figure 6-16. Plots of transitions within the metastable well during an avalanche. a). All the
spins begin in the metastable well and should be absorbing microwaves as they
transition to the nas= 9 state. We can see that this is indeed the case until a few
milliseconds after the SAW pulse is applied when avalanche appears. Since all of the
spins have avalanched to the stable well there is no more absorption. b). At t = 0,
there is no absorption of the signal, but a sharp absorption peak appears a few
milliseconds after the ignition of the avalanche, which is the transition from the nas
9 to 8 states. Initial and final EPR absorption values are the same. As a contrast,
after recording the data from an avalanche we would pulse the IDT in order to heat
the sample with a pure SAW. Neither situation shows any absorption under these
circumstances, as would be expected.






















320.4 GJHz
m = -10 to -9 0.54 T
S -- Avalanche
SAW heating


A t tlo m
i IP I I I I


I



..0


;I


0


------- -- -- --P. I~
L
'' '-~ V1


SAvalanche
---- SAW hleatin >


40 80


120


J


40

Time (ms)


80


120


Figure 6-17. Plots of transitions within the stable well during an avalanche. a). Initially there is
no absorption of the signal, but once the avalanche occurs there is a strong absorption.
Since the spins all relax to the ground state in the stable well we continuous
absorption after the avalanche as the spins transition from nas= -10 to -9. b). As the
spins avalanche they pass through the nas= -9 state before relaxing to the ground
state. The sharp, short time side of the absorption is the transition between the nas
-9 to -8 states. However, it is clear by the behavior of the signal that there is still
some absorption for approximately 100 ms after the avalanche. This can be
understood by the fact that the ignition of an avalanche releases a significant amount
of heat into the sample, which causes the nas= -9 state to be thermally populated and
this extra heat takes some time to disperse.










0n 0.00


-0.05 -


a,-0.10 -


-0.15 --
9K
S-0.20 6 K-


i -0.25--

0 10 20 30 40 50

t (m s)
Figure 6-18. Plot of the EPR signal as a function of time for the ms = -9 to -8 transition for a
frequency of 269 GHz and a heat pulse of 5 ms. As soon as the pulse is applied (t =
0) there is a decrease in the signal due to the ms = -9 becoming thermally populated
and spins transitioning to the ms = -8 state. The population difference between these
states reaches a maximum after approximately 10 ms. Beyond this point the system
is no longer heated by the SAW and the phonons begin to disperse. The time for the
excited levels to thermally depopulate is on the order of a few hundred ms, which is
the time for the system to return to equilibrium. The dashed curve is a simulation
using Eq. 6-3.









CHAPTER 7
SUMMARY

This chapter gives a summary of the work presented in the previous chapters of this

dissertation. This Ph.D. dissertation is focused on low temperature, high Hield, high frequency

magnetic resonance spectroscopic studies of various SMMs.

Chapter 1 is an introduction to the class of compounds known as single molecule magnets.

We explained the sources and importance of anisotropy to the magnetic behavior. We also

described a unique feature of SMMs: the observed quantum tunneling in hysteresis loops of a

classical object. Quantum tunneling is only allowed when sources of transverse anisotropy

and/or transverse Hields are present. Two tunneling regimes were outlined and we stressed that

the main goal of studying these materials is to gain a deeper understanding of the magnetic

quantum tunneling behavior.

Chapter 2 presents the experimental techniques and equipment that we use in our

research. First, we briefly discussed EPR in the context of SMM systems. Next, we explained

the two types of cavities we use for a cavity perturbation technique and our main instrument that

acts dual microwave source and network analyzer. This source, in conjunction with an external

Gunn diode and complimentary components (amplifier, switch, frequency tripler) allow for high

power pulsed microwaves. Additionally, our quasi optical bridge setup provides a low loss

propagation system that relies on coupling of a free space Gaussian profie to an HEn1 mode in

corrugated horns and a tube. The corrugated sample probe is used to conduct experiments that

integrate other devices into the setup and compliment our normal EPR studies by combining

microwaves with either surface acoustic waves or Hall magnetometry to conduct unique

experiments on single crystals. Finally, we outlined the process of fabricating the Hall

magnetometer used in our experiment in section 6.3.










Chapter 3 gives the basic theoretical foundations for the origin of the spin Hamiltonian that

is used to model SMM systems. The Hamiltonian corresponding to a giant spin approximation is

certainly the most widely used form and applies to ideal systems where the isotropic exchange

interaction is much larger than the anisotropic interactions. For systems where this assumption

starts to break down, we can employ a coupled single ion Hamiltonian that in spite of being more

complicated of a model, can provide information on the nature of the exchange interactions

between metal ions in a molecular cluster.

Chapter 4 outlines HFEPR studies done on a tetranuclear Nickel SMM,

[Ni(hmp)(dmb)Cl]4. First we reported on experiments to characterize the spin Hamiltonian

parameters for this SMM. Data reveal sharp, symmetric EPR lines due to the lack of solvate

molecules in the crystal lattice and large intermolecular exchange pathway distances. However,

variable frequency, variable temperature measurements have revealed the presence of two

distinct molecular species within the crystal and we are able to extract the relevant spin

Hamiltonian parameters for each species. Below 46 K the peak splits into two, which we

attribute to differences in the molecular environments arising from different t-butyl group

conformations in the dmb ligand. At high temperatures the motion of these is thermally

averaged, but below 46 K the motion freezes out and the t-butyl group takes on two distinct

orientations. These EPR peaks are then split due to the effects of the disorder. Additional low

temperature data (< 6 K) reveal additional splitting and broadening of the peaks, which we

attribute to short range intermolecular exchange interactions among neighboring molecules that

are averaged out at higher temperatures. It is likely that exchange interactions provide an

additional contribution to the line widths/shapes, i.e. exchange probably also contributes to the

broad lines. However, given the minimal amount of disorder in this system, we are able to









separate the various contributions (disorder and intermolecular exchange) to the EPR line shapes.

Finally, we are able to measure the magnitude of the isotropic exchange coupling constant, J,

with our HFEPR data. By simulating the intensity of peaks originating from transitions within a

low lying excited state spin multiple to a model that includes both isotropic and anisotropic

interactions we obtain a value of J= -6.0 cm- This provides insight into the spacing between

the ground state and higher lying spin multiplets, and supports the evidence that the isotropic and

anisotropic parameters (J and d) can cause mixing between states unless J >> d. Such mixing

manifests itself as unphysical higher order terms in the Hamiltonian with the giant spin model.

Chapter 5 deals with experiments done on two Cobalt systems ([Co(hmp)(dmb)Cl]4 and

[Zn3CO(hmp)4(dmb)4 14]) that are similar to the Nickel systems discussed in chapter 4. The

CoZn study provides crucial information as to the orientation and magnitudes of the anisotropy

parameters in the tetranuclear system. We then present HFEPR data taken on the tetranuclear

system, which exhibits complicated spectra that can not be modeled by a giant spin Hamiltonian.

We mention the possible contributions of the symmetric and antisymmetric exchange

interactions to the observed zero field splitting anisotropy and conclude by remarking that this is

still a work in progress by a collaborator.

Chapter 6 presents numerous studies done on a manganese based SMM,

[Mnl2012(CH3COOH)16(H20)4] 2CH3COOH-4H20, or just Mnl2Ac. First, we showed research

that we did using HFEPR to measure QTM. This allowed us to investigate how different

molecules within a crystal relax on different time scales. These studies were done in two

temperature regimes that exhibit different relaxation laws. We also characterized the ZFS and D

value of the minority species. Next, we talked about a study that combined Hall magnetometry

with HFEPR to measure the effects of microwaves on the magnetization dynamics. The










tunneling at a given resonant field is enhanced when the microwave frequency corresponds to

transitions between states at this resonant field. This resonant enhancement is due to microwave

absorption leading to an increase in the lattice and spin temperatures, and consequently,

thermally assisted tunneling. By reducing the amount of power incident upon the sample

through low duty cycle pulsed radiation, we are able to reduce the resonant heating and eliminate

the non resonant heating of the sample. Comparison of resonant tunneling for different wait

times with and without the presence of external microwaves allows us to observe the

contribution of microwaves to the tunneling. Finally, we discussed a unique study combining

SAWs and HFEPR in order to measure fast relaxation dynamics after the system was pushed out

of equilibrium. In addition to being a novel technique, we are able to get time resolved

information about the how spins relax in a single energy level in this system. From this research

we are able to estimate an upper bound of 0. 1-1 ms for the spin lattice relaxation time, Ty.









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BIOGRAPHICAL SKETCH

Jon Lawrence was born in Kalamazoo, Michigan. After graduating high school in June of

1997 he enrolled at Grand Valley State University in Allendale, Michigan where he received his

bachelors of physics degree in May of 2002. For his senior proj ect he was involved in the

synthesis of LaCaMnO and the characterization of its colossal magnetoresistive properties.

In August of 2002 he started graduate school at the University of Florida in Gainesville,

Florida. In May of 2003 he joined the research group of Dr. Stephen Hill. For his career as a

graduate student he learned an enormous amount about being an experimental physicist and

worked on many different proj ects, including the magnetic resonance spectroscopy studies

discussed in this dissertation, as well as the design and construction of a low temperature probe

and cryostat. After 4.5 years of hard work, he completed all of the proj ects necessary for his

Ph.D. in experimental physics. He received his degree in December of 2007.

After graduation he began the next stage of his life, using all of the vast experience and

knowledge obtained from his physics degree. He promises to only use these powers for good.





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1 COMPREHENSIVE HIGH FREQUENCY ELE CTRON PARAMAGNE TIC RESONANCE STUDIES OF SINGLE MOLECULE MAGNETS By JONATHAN D. LAWRENCE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Jonathan D. Lawrence

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3 To my family

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4 ACKNOWLEDGMENTS This thesis would not have been possible wi thout the help and guida nce of a number of people. First I need to express my thanks to my advisor, Dr. Stephen Hill. For the past 4.5 years Steve has provided a world class environment for cutting edge research in physics. He has always been a strong mentor, guiding me toward ch allenging but rewarding pr ojects. I have also had the privilege to travel to numerous conferen ces to present my work and interact with my peers in the same field of res earch. Steve has financially suppor ted all of these endeavors, and for that, I am quite grateful. I am truly i ndebted to Steve for teaching me how to be a professional scientist, and gi ving me the opportunity to lear n what being an independent researcher is all about. I would also like to thank my other committee members, Dr George Christou, Dr. Mark Meisel, Dr. Yasumasa Takano, and Dr. Selman Hershfield, for overseeing the completion of my research work. I wish to express my deepest gratitude to ever yone in the machine shop here in the physics department: Marc Link, Bill Malphurs, Ed Storc h, Skip Frommeyer, Mike Herlevich, and John Van Leer. Every experiment that I conducted us ed equipment built or modified by the machine shop. They always provided helpful design sugg estions and world class craftsmanship on every project. Without the help of these kind, talent ed gentlemen, none of the research presented in this dissertation would have been possible. I am grateful to the technical staff in the electronics shop (Larry Phelps, Pete Axson, and Rob Hamersma) for helpful discussions, advice, a nd assistance with regards to any electronic equipment problems or design issues. They we re always willing to help, and contributed significantly to a number of my projects.

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5 I want to thank the cryogenic engineer s, Greg Labbe and John Graham, who are responsible for the recovery and re-liquification of helium, as we ll as all the aspe cts of cryogenic equipment implemented in our lab. Throughout my research there were many technical challenges dealing with low temperature equipm ent. They were vastly knowledgeable and always willing to help with any matter. Th eir generous assistance is much appreciated. I want to thank Dr. Enrique del Barco and Dr Chris Ramsey for their generous help in fabrication of the Hall magnetometers used for ce rtain experiments. They allowed the use of their facilities and were patient enough to go th rough the fabrication proc ess twice so that my experiment could be successful. Fo r this, I am deeply indebted. For a great collaborative effort with a unique experiment, I want to thank Ferran Maci from the Tejada group at the University of Barcel ona. This work was some of the best I have done as a graduate student, and he was a large part of the success. I wish to thank Christos Lampropoulos fr om the Christou group in the chemistry department here at the University of Florida. He supplied the Mn12Ac samples used in numerous experiments. I wish to thank En-Che Yang and Chris Beedle of the Hendrickson group in the chemistry department at the University of Cali fornia at San Diego. They supplied Ni4, NiZn, Co4, and CoZn samples used in many studies. I want to recognize the UF Alumni Associa tion for providing me with a $500 scholarship in order to travel to Denver, CO for the A PS 2007 meeting where I was able to network with scientists working in similar areas of research and present some of my work to a diverse audience.

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6 Finally, I want to thank all of the wonderful friends I have made during my time here. The few hours each week not spent doing physics were made all the more enjoyable by you.

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7 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION TO SINGLE MOLECULE MAGNETS.................................................14 1.1 Basic Properties of Single Molecule Magnets.............................................................14 1.2 The Magnetic Anisotropy Barrier in Single Molecule Magnets..................................15 1.3 The Role of Magnetic Anisotropy...............................................................................17 1.4 Magnetic Hysteresis and Magnetic Quantum Tunneling.............................................19 1.4.1 Zero Field Tunneling.........................................................................................20 1.4.2 Magnetic Field Induced Tunneling....................................................................21 1.5 Applications.................................................................................................................28 1.6 Summary......................................................................................................................29 2 EXPERIMENTAL INSTRUMENTATION AND TECHNIQUES.......................................36 2.1 Measurement Techniques............................................................................................36 2.2 Electron Paramagnetic Resonance in the Context of Single Molecule Magnets.........36 2.3 Cavity Perturbation......................................................................................................37 2.3.1 Technical Challenges.........................................................................................37 2.3.2 Equipment..........................................................................................................38 2.4 Quasi Optical Setup.....................................................................................................45 2.4.1 Piezoelectric Transducer Device........................................................................51 2.4.2 Hall Magnetometer Device................................................................................55 2.5 Hall Magnetometer Fabrication...................................................................................57 2.6 Summary......................................................................................................................62 3 THEORETICAL BASIS OF THE SPIN HAMILTONIAN..................................................75 3.1 Two Versions of the Spin Hamiltonian.......................................................................75 3.1.1 The Giant Spin Hamiltonian..............................................................................76 3.1.2 Coupled Single Ion Hamiltonian.......................................................................81 3.2 Summary......................................................................................................................85 4 CHARACTERIZATION OF DISORDER AND EXCHANGE INTERACTIONS IN [Ni(hmp)(dmb)Cl]4.................................................................................................................86 4.1 The Tetranuclear Single Molecule Magnet [Ni(hmp)(dmb)Cl]4.................................86 4.2 HFEPR Measurements of [Ni(hmp)(dmb)Cl]4............................................................88

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8 4.2.1 Characterization of Easy Axis Data...................................................................89 4.2.2 Peak Splittings Arising from Disorder...............................................................90 4.2.3 Peak Splittings Arising from Intermolecular Exchange....................................92 4.3 Physical Origin of the Fast QTM in [Ni(hmp)(dmb)Cl]4............................................95 4.4 Measuring the Exchange Interaction with HFEPR......................................................99 4.5 Summary....................................................................................................................103 5 HFEPR CHARACTERIZATION OF SING LE Co (II) IONS IN A TETRANUCLEAR COMPLEX........................................................................................................................ ...116 5.1 Introduction to the [Co(hmp)(dmb)Cl]4 and [Zn3Co(hmp)4(dmb)4Cl4] Complexes..116 5.2 HFEPR Measurements of [Zn3Co(hmp)4(dmb)4Cl4].................................................117 5.2.1 The [Zn3Co(hmp)4(dmb)4Cl4] Complex..........................................................117 5.2.2 Frequency and Temperature Dependent Measurements..................................117 5.2.3 Angle Dependent Measurements.....................................................................119 5.3 HFEPR Measurements of [Co(hmp)(dmb)Cl]4.........................................................122 5.3.1 Measurements Along the Crystallographic c Axis..........................................123 5.3.2 Discussion of Spectra Within the ab Plane......................................................125 5.4 Spin Hamiltonian for the Tetranuclear system..........................................................125 6 HFEPR STUDIES OF MAGNETIC RELAXATION PROCESSES IN Mn12Ac...............139 6.1 Introduction to Mn12Ac..............................................................................................139 6.1.1 Theory and Effects of Disorder in Mn12Ac......................................................140 6.1.2 Dipolar and Hyperfine Fiel d Relaxation Mechanisms.....................................143 6.1.3 High and Low Temperature Relaxation Regimes............................................145 6.2 QTM Studied by HFEPR...........................................................................................146 6.2.1 Experiment in Stretched E xponential Relaxation Regime...............................147 6.2.2 Experiment in the t1/2 Relaxation Regime........................................................150 6.2.2a Low Temperature MQT...........................................................................151 6.2.2b Magnetic Avalanches...............................................................................153 6.2.3 Characterization of Minority Species..............................................................155 6.3 Microwave Induced Tunneling Meas ured with Hall Magnetometry.........................156 6.4 Relaxation in Mn12Ac Measured with HFEPR..........................................................161 6.4.1 Triggered Avalanches in Mn12Ac....................................................................163 6.4.2 Pulsed Heating and Spin Lattice Relaxation....................................................167 6.5 Summary....................................................................................................................170 7 SUMMARY...................................................................................................................... ....190 REFERENCES..................................................................................................................... .......194 BIOGRAPHICAL SKETCH.......................................................................................................202

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9 LIST OF FIGURES Figure page 1-1 Molecule of Mn12Ac. ............................................................................................................31 1-2 Energy barrier for a molecule of Mn12Ac in zero field. .......................................................32 1-3 Energy barrier for a molecule of Mn12Ac with an external magnetic field applied parallel to the easy axis. .................................................................................................... 33 1-4 Low temperature hysteresis loop for a single crystal of Mn12Ac with an external magnetic field applied para llel to the easy axis. ...............................................................34 1-5 Two energy levels in a system of Mn12Ac as they pass through the first non zero resonance field................................................................................................................ ...35 2-1 Energy levels in the Ni4 SMM with its easy axis aligned al ong the external field. .........64 2-2 Normal EPR spectrum for the Ni4 SMM. .............................................................................65 2-3 Typical experimental setup. ............................................................................................. .....66 2-4 Rotational capabilities of each magnet system. ....................................................................67 2-5 Free space Gaussian beam. ............................................................................................... ....68 2-6 Quasi optics equipment. ................................................................................................. ......69 2-7 Signal polarization as it changes from interactions with the respective components of the quasi optical setup. ...................................................................................................... 70 2-8 TE11 and HE11 modes in a circular, corrugated waveguide. .................................................71 2-9 Piezoelectric device us ed in our experiments. ......................................................................72 2-10 Electronic equipment used in our avalanche experiments.....................................................73 2-11 Hall device used for our magnetometry measurement. ........................................................14 4-1 Molecule of [Ni(hmp)(dmb)Cl]4. .......................................................................................105 4-2 172.2 GHz HFEPR spectra of [Ni(hmp)(dmb)Cl]4.for different temperatures. .................106 4-3 Peak positions of [Ni(hmp)(dmb)Cl]4.in magnetic field for different frequencies. ...........107 4-4 Influence of the spacing between the re sonance branches due to a negative, axial, fourth order anisotropy term. ..........................................................................................108

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10 4-5 Peak splittings at 172.2 GHz in [Ni(hmp)(dmb)Cl]4. .........................................................109 4-6 Heat capacity measur ements of [Ni(hmp)(dmb)Cl]4. .........................................................110 4-7 Comparison of the st ructure of [Ni(hmp)(dmb)Cl]4 at 173 K and 12 K. ............................111 4-8 Temperature dependence of the peak splittings in Ni(hmp)(dmb)Cl]4.at a given magnetic field for three frequencies................................................................................112 4-9 Simulation of four coupled s = 1 spins. ..............................................................................113 4-10 Expanded view of data obtained at a fre quency of 198 GHz in the range from 4 T to 9 T. .......................................................................................................................... ........114 4-11 Temperature dependence of the intens ity of transition B observed at 172 GHz. ...............115 5-1 Peak position of [Zn0.995Co0.005(hmp)(dmb)Cl]4.as a function of frequency. .....................131 5-2 Temperature dependence of [Zn0.995Co0.005(hmp)(dmb)Cl]4. .............................................132 5-3 Peak position as a function of angle in [Zn0.995Co0.005(hmp)(dmb)Cl]4for two planes of rotation. ..................................................................................................................... ......133 5-4 Magnetic core of a CoZn molecule. ...................................................................................134 5-5 Temperature dependence of the peaks in [Co(hmp)(dmb)Cl]4 for two different frequencies. .................................................................................................................. ...135 5-6 Easy axis frequency dependence of the resonance peaks in [Co(hmp)(dmb)Cl]4 at 2 K. ..136 5-7 Data taken from [Co(hmp)(dmb)Cl]4 with the field aligned within the ab plane of the crystal. ...................................................................................................................... .......137 5-8 Simulation of the Co4 system for a frequency of 501 GH z at 30 K with the field aligned along the c axis. ...............................................................................................................138 6-1 Six different Mn12 isomers. ................................................................................................172 6-2 Energy levels in Mn12Ac as a function of magnetic field. ..................................................173 6-3 2 K EPR spectra for different waiting times at 0.9 T. ........................................................174 6-4 Area of the positive field peak as a function of wait time. .................................................175 6-5 Emerging resonance peak for different wait times at 1.8 T and sweeping back to 6 T. ..176 6-6 Peak area in Fig. 6-5 as a function of wait time. ................................................................177

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11 6-7 Spectrum for a wait time of 2400 s after fi tting it to a simulation that combined two different Gaussian peaks. ................................................................................................178 6-8 Average D value and peak width vs. time. .........................................................................179 6-9 Spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to 6 T from waiting for 600 s at differe nt magnetic fields. ................................................................180 6-10 Minority species molecules in Mn12Ac. .............................................................................181 6-11 Low temperature frequency dependen ce of the minority species peak. .............................182 6-12 Hysteresis loops taken under the influe nce of a number of different microwave frequencies. .................................................................................................................. ...183 6-13 Percentage of magnetizati on reversal for the step at 0.5 T under the influence of 286 GHz microwave radiation for different duty cycles. ......................................................184 6-14 Difference in the magnetization reversal for the data sets taken with and without microwaves as a function of wait time at 0.5 T. .............................................................185 6-15 Energy barrier diagram illustrating how spins move during an avalanche. .......................186 6-16 Transitions within the metastable well during an avalanche. .............................................187 6-17 Transitions within the stab le well during an avalanche. .....................................................188 6-18 EPR signal as a function of time for the ms = 9 to 8 transition. .....................................189

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12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPREHENSIVE HIGH FREQUENCY ELE CTRON PARAMAGNE TIC RESONANCE STUDIES OF SINGLE MOLECULE MAGNETS By Jonathan D. Lawrence December 2007 Chair: Stephen O. Hill Major: Physics This dissertation presents research on a number of single molecule magnet (SMM) compounds conducted using high frequency, low te mperature magnetic resonance spectroscopy of single crystals. By developing a new techniqu e that incorporated other devices such as a piezoelectric transducer or Hall magnetometer w ith our high frequency microwaves, we were able to collect unique measurements on SMMs. This class of materials, which possess a negative, axial anisotropy barri er, exhibit unique magnetic prope rties such as quantum tunneling of a large magnetic moment vector. There are a number of spin Hamiltonians used to model these systems, the most common one being the giant spin approximation. Work done on two nickel systems with identical symmetry and microenvironments indicates that this model can contain terms that lack any physical significance. In this case, one must tu rn to a coupled single ion approach to model the system. This provides information on the nature of the exchange interactions between the constituent ions of the molecule. Additional st udies on two similar cobalt systems show that, for these compounds, one must use a coupled single i on approach since the assumptions of the giant spin model are no longer valid.

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13 Finally, we conducted a collection of studies on the most famous SMM, Mn12Ac. Three different techniques were used to study magnetiz ation dynamics in this system: stand-alone HFEPR in two different magnetization rela xation regimes, HFEPR combined with magnetometry, and HFEPR combined with surface ac oustic waves. All of this research gives insight into the relaxation mechanisms in Mn12Ac.

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14 CHAPTER 1 INTRODUCTION TO SINGLE MOLECULE MAGNETS 1.1 Basic Properties of Si ngle Molecule Magnets Progressive research over the last decade in the inorganic chemistry community has allowed the synthesis of an ex citing class of materials with unique magnetic properties [1]. These new materials were given the name “si ngle molecule magnets” (SMMs) for reasons that will soon be clear. All SMMs have transition metal ions such as Fe, Mn, Ni and Co as the source of their magnetic properties. The magnetic core of each complex comprises multiple ions with unpaired electrons, which are strongly coupled to each other through intramolecular exchange interactions (isotropic being the mo st dominant). This isotropic Heisenberg exchange interaction is expressed as ˆˆ ˆisotropicijij iji H JSS (1-1) In Eq. 1-1, Jij is the magnitude of the isotropic Heisenberg interaction (positive for antiferromagnetic coupling and negative fo r ferromagnetic coupling) between spin i and spin j S represents the spin operator for an individual ion within the molecule. This coupling leads to a large magnetic moment for each molecule, and se parates the energy spectrum into respective spin multiplets. For traditional SMM systems the isotropic exchange interaction is sufficient to isolate the ground state spin multiplet from higher lying multiplets. The ground state multiplet is then modeled as a state with a perfectly rigid ma gnetic moment vector. Intramolecular exchange is the dominant interaction between ions within a molecule, and the bulky organic ligands that surround the magnetic core serve to isolate each molecule from surrounding neighbors. Thus, intermolecular exchange interactions are rather weak and there is no long range ordering. Each molecule can be considered as an independent magnetic nanocluster possessing a large magnetic

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15 moment. Additionally, to a first approximation, a ll molecules within a cr ystalline sample are identical. Since each molecule behaves like an isolated magnetic moment, the term single molecule magnet was appropriately coined for these compounds [2]. Fig. 1-1 illustrates a molecule of the most famous SMM, [Mn12O12(CH3COO)16(H2O)4] 2CH3COOH4H2O, hereafter Mn12Ac. Eight Mn+3 ions and four Mn+4 ions are antiferromagneti cally coupled through the oxygen atoms, giving rise to a large S = 10 ground state for the molecule. The ligands that surround the magnetic core minimize the interactions th at each molecule experiences from its neighbors by increasing the distance between effec tive dipole centers (~ 14 for this particular compound [3]). We present stud ies done on this SMM in Ch. 6. 1.2 The Magnetic Anisotropy Barri er in Single Molecule Magnets The essential feature of all SMM systems is their signifi cant negative axial anisotropy that creates a barrier to reve rsal of the magnetization vector From a quantum mechanical perspective, the Hamiltonian in its s implest form can be expressed [1] as 2ˆ ˆzDS (1-2) In Eq. 1-2, D represents the dominant, ax ial anisotropy of the molecule which must be negative for any SMM. ˆzS is the spin operator for the magnetic moment of the molecule. For simplicity we will consider each state to have no orbital angular momentum contribution and can therefore be expressed as a pure spin multiplet. The wa vefunction for each energy state can be expressed as the spin projection, s m. The number of spin projections (e nergy states) for any molecule is given by 2S + 1, where S is the spin ground state of the mo lecule. In the absence of any transverse anisotropies or transv erse fields, energy states with equal but opposite spin projections are degenerate in zero magnetic field. From the Hamiltonian in Eq. 1-2 we obtain the energy eigenvalues for each eigenstate:

PAGE 16

16 2() s s E mDm (1-3) where ms is the spin projection along the z axis. The ground state for the system is the largest magnitude of the spin projection ( ms = S ), since D is negative. The height of the energy barrier in Fig. 1-2, and given by Eq. 1-4, is governed by both the magnitude of D and the magnitude of the spin ground state multiplet, S 2 E DS (1-4) This can be pictured as an inverse parabola poten tial, as shown in Fig. 1-2, which represents the energy barrier for a S = 10 system like Mn12Ac or the Fe8 SMM [1]. This dominant axial anisotropy allows one to define a quantiz ation axis, traditionally labeled as the z axis, for the energy levels of the system, which are quanti zed. The energy barrier makes the magnetic moment bi-stable due to the fact that energe tically it prefers to poi nt in one direction ( +z ) or a direction that is antiparallel to the first (– z ). The above mentioned source of anisotropy occurs in the absence of an external magnetic field. Any phenomenon such as this is known as zer o field splitting, referri ng to the fact that the energy degeneracy is lifted in the absence of any external field [4]. However, anisotropies can manifest themselves in the presence of an exte rnal field as well, in terms of the Zeeman interaction. In an external magnetic field, th e Zeeman energy of an electron now depends on the orientation of the field with respect to its spin projection along the fiel d. The energy of the Zeeman interaction is related to the strength of the external magnetic field by a proportionality factor [5], g Eq. 1-5a gives the general expression for the Zeeman energy ˆBzeeman E BgS (1-5a) z Bszeeman E gBm (1-5b)

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17 where B is the Bohr magneton, B is the external field strength, g is the Land tensor and S is the spin operator for the molecule. For a free electron g is a scalar (2 ), but in a molecule g becomes a tensor due to spin orbit induced an isotropies. In equati on Eq. 1-5b we give the expression for the case of the magnetic field aligne d with the spin vector of the molecule, where gz is the component of the Land tensor along the exte rnal field direction and ms is the projection of the spin vector along the ex ternal field direction. The zero field mechanism mentioned above contributes to anisotropic electron distributions among electroni c orbitals [6]. This manifests itsel f in the lifting of degeneracies of energy levels through energy splittings. Measur ing the energy splittings with high frequency electron paramagnetic resonance (HFEPR) spectro scopy is a way to prob e the anisotropies present in the material of interest. 1.3 The Role of Magnetic Anisotropy Anisotropy plays a fundamental role in the ma gnetic properties of all SMM systems. Since they comprise multiple transition metal ions, SMMs constitute an exchange coupled system. The anisotropy can come from many so urces, including spin-spin dipol ar and exchange coupling of electrons, hyperfine interactions of electrons with the nuclei of the constituent atoms of the molecule, and most notably spin orbit coupling of the electrons. All of th ese sources can lead to anisotropic electron distributions on the molecule. This coupling is the dominant mechanism responsible for the essential f eature of SMMs, which is their negative axial anisotropy that creates a barrier to magnetization reversal. Spin orbit coupling results from the interaction of one electron’s orbital an gular momentum with its own spin a ngular momentum [7]. The orbital momentum of an electron creates a magnetic fi eld, which will couple to the spin magnetic moment. This type of coupling is also present between the spin of the electron and the magnetic field of a proton, which, from the electron’s perspective, constitutes an orbital momentum. In

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18 this scenario, the individual momenta are not separately conserved. The sum of the two momenta, however, is conserved. One source of anisotropy is the Jahn-Teller elongations of electronic orbitals. Such a distortion of molecular orbitals of the Mn+3 ions is the significant contribution to the axial anisotropy in the Mn12Ac SMM system. The Jahn-Teller theorem states that “for a non-linear molecule in an electronically degenerate state, a distortion must occur to lower the symmetry, to remove the degeneracy, and lower the energy [8].” An atom in free space with a symmetric distribution of electronic orbitals will have no anisotropy and, thus, the energy levels of a certain orbital will be degenerate. But atoms in a molecule will have their molecular orbitals distorted. This distortion can be a geomet ric compression or elongation and it causes degeneracies to be lifted. In the case of a compression, the sign of the axial anisotropy parameter, D, is positive and conversely, D is negative in the case of an elongation [9]. If the distortion occurs along one axis only, then only axial anisotropies will develop. If the distortion occurs along multiple axes, then both axial and transverse anisotropies will develop. Spin-spin dipolar coupling between electrons arises from the interaction of one electron’s spin in another electron’s dipolar field. Simila r interactions between electrons and nuclei can give rise to hyperfine couplings as well. The nuclei in most materials have magnetic moments which couple to the orbital and spin angular mome ntum of the electrons. However, in exchange coupled systems, like SMMs, the delocalization of electrons throughout the molecule usually makes this a weak effect [3]. Although, we will show in Ch. 6 that the interactions between a molecule and the fields from nuclear moments as well as other molecules play an important role in the tunneling process.

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19 Exchange interactions are classified by many different types, such as direct exchange, indirect exchange, superexcha nge, itinerant exchange, and doubl e exchange [10]. The common feature for all of these is a weak bond between magnetic moment centers within the molecule [11, 12] which can be spread over distances on the order of a few These exchange interactions can manifest as bot h isotropic and anisotropic quanti ties. For the systems discussed in this dissertation the dominant type of exch ange is superexchange, which is the coupling of localized magnetic moments in insulating mate rials through diamagnetic groups. In many SMM systems the isotropic Heisenberg interaction, gi ven by Eq. 1-1, is of a much larger magnitude than the anisotropic interactions or any anisot ropic or antisymmetric exchange. Under these circumstances dipolar interactions, anisotropic exchange, and antisymmetric exchange are usually ignored. However, in Ch. 5 we w ill discuss a system where both anisotropic and antisymmetric exchange are considered. 1.4 Magnetic Hysteresis and M agnetic Quantum Tunneling In the absence of any transverse anisotropies or transverse fields, energy states with equal but opposite spin projections are degenerate in zer o magnetic field. Application of an external magnetic field shifts the energy levels of the poten tial well with respect to one another. At low temperatures ( kBT DS2) where only the ground state of a mo lecule is significantly populated, one can observe magnetic hysteresis. Many studies have been done reporting magnetic hysteresis loops in various systems [13, 14]. In most systems, magnetic hysteresis occurs due to the formation of domain boundaries within the la ttice. The exchange interaction between spins is shorter ranged than the dipolar interactions, and the magnitude of the dipolar interactions can become significant in bulk systems where large numb ers of spins are involved [15]. The dipolar energy can be reduced by dividing the system into uniformly magnetized domains, with each domain having a different directi on. This costs energy in terms of the exchange interaction due

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20 to the fact that the individual spins within a dom ain will have their energy increased by spins in neighboring domains with different orientations. However, this energy cost is minimal since only those spins at the domain walls have this in crease in energy and hence this effect is short ranged. Conversely, the dipolar interaction is longer, and cons equently the dipolar energy of every spin in the system is lowered. Thus it is energetically favorable for the system to adopt this configuration. However, in SMM systems it is the anisotropy ba rrier of each individual molecule that is responsible for the observed hysteresis, in cont rast to the collective effect of the dipolar interaction in domain wall formation. At lo w temperatures, the anisotropy barrier makes it energetically favorable for all the spins to populat e the ground state of the system. The barrier prevents spins from directly reversing their spin state projection from parallel to antiparallel with respect to the quantization axis. Perhaps most interesting is the appearance of steps in the hysteresis loops, which is an indication of quantum tunneling of the magnetic moment through the anisotropy barrier [16]. After biasing the system with a magnetic field such that all the spins are in one of the wells, it is possible for the spins to tunnel through the barrier at resonanc e fields. By measuring the magnetization of a single crystal, it is possible to observe drastic changes in the magnetization as the external field is swept from a large biasi ng field through resonance fields. At resonance fields the magnetization is seen to have step s where it moves toward the opposite saturation value. This indicates that spins are reversi ng their projection state by tunneling through the energy barrier at thes e resonance fields. 1.4.1 Zero Field Tunneling For magnetic quantum tunneling to occur, there need to exist terms that break the axial symmetry and mix the energy states on opposite sides of the energy barrier, such as transverse

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21 anisotropies and transverse magnetic fields, bot h internal and external [17]. Transverse anisotropies arise due to symmetries of the molecu les, which is discussed in detail in Ch. 3. Internal transverse fields aris e due to dipolar interactions between neighboring molecules and fields due to nuclear magnetic moments. External transverse fields arise due to misalignment of the sample’s easy axis with respect to the magnetic field. All of these can be expressed in the Hamiltonian as terms that do not commute with Sz, and these off diagonal terms cause mixing between states. As an example, a second order transverse anisotropy can be written as22ˆˆ () x y E SS Any term such as this can cause the wa vefunctions for the stat es on each side of the energy barrier to become mixed and extend to the opposite side of the energy barrier. Without a transverse anisotropy, states w ith equal but opposite spin projections are completely degenerate, but with such a term there is an energy difference between the new states, which are symmetric and antisymmetric comb inations of the unmixed states. This energy difference is known as the tunnel splitting ( ), and since the wavefunc tions become mixed into linear superposition states there is a probability for the projection of the spin vector to be measured in either state. 1.4.2 Magnetic Field Induced Tunneling An external magnetic field will bias the energy levels with respect to each other, and at certain magnetic field values, energy states w ith different and opposite spin projections can become degenerate due to the Zeeman interactio n. These are known as resonance fields and as shown in Fig. 1-3, these are the non zero fields where magnetic quantum tunneling occurs [18]. With the magnetic field applied parallel to the easy magnetization axis of the molecule ( B parallel to z ) and considering only the do minant second order term, D, and the Zeeman term, the spin Hamiltonian will be

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22 2ˆ ˆszB zHDS g BS (1-6) The energies given by Eq. 1-6 are 2 s Bs zEDmgBm (1-7) We can solve for the values of resonance fields with the expression for a resonance field where quantum tunneling of the magnetiz ation (QTM) occurs given by res B zkD B g (1-8) In Eq. 1-8, k = m + m’ and can take on integer values, while m and m’ represent the states with opposite spin projections along the field quantiz ation axis, with the lo west two states being m = S ( k = 0 resonance). At resonance fields, whic h occur in integer steps of approximately B z D g states with opposite spin projections become mixed by terms in the spin Hamiltonian that do not commute with the ˆzSoperator. From Eq. 1-8 we can calculate the values of the longitudinal field where tunneling can occur for a given system. For Mn12Ac, resonance fields appear approximately every 0.45 T. In this re spect, the magnetic field can be used to switch tunneling “on” and “off”. When the field does not correspond to a resonance value, tunneling of the magnetic moment is forbidden. However, when the field matches a resonance value, tunneling is allowed. Fig. 1-4 shows a typical hysteresis loop for the Mn12Ac SMM. The flat plateaus in the figure correspond to fields where no tunneling is a llowed, but the sharp steps seen at resonance fields are where the tunneling is switched on. The steps correspond to spins changing their magnetization state by tunneling through the energy barrier, which changes the value of the sample magnetization being measured Thus the steps are the relaxation of the magnetization of the spins toward the opposite saturated magnetization value.

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23 For simplicity we will consider only the previously mentioned second order transverse anisotropy. By rewriting this in term s of the raising and lowering operators ( 22 22ˆˆ ˆˆ 2xySS SS ), we see that this term will mix states that differ in m by two, and hence for an integer spin system this will cause mixing of the pure spin multiplet states, s m. However, there is a symmetry imposed by this term such that tunneling is possi ble only between certain st ates. To illustrate this we start with the general e xpression for the eigenvectors [1]. '(')'mmmm (1-9) In Eq. 1-9, ( m’ ) is the wavefunction for a given state m’ and the sum is carried out over all possible m’ The overlap, or amount of mixing, of two states is quantified by the matrix elements between the respective states. 22ˆˆ ˆ '()' 2TE mHmmSSm (1-10) Eq. 1-10 is zero unless the values of the states m and m’ differ by a multiple of two. The eigenvectors of each state can take on one of two forms [1] 0(2)2S pmSpSp (1-11a) or 1 0(21)21S pmSpSp (1-11b) where p is an integer. Eq. 1-11a applies to even integer states and Eq 1-11b applies to odd integer states. Since the states m and m’ are mixed, the wavefuncti on describing each state will have a component of both m and m’ From Eq. 1-11a it follows that m = S 2 p and m’ = S 2 p’ or from Eq. 1-11b it follows that m = S 2 p 1 and m’ = S 2 p’ 1. Now we introduce a

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24 longitudinal external field and formulate the selection rule fo r tunneling between states at resonance fields. By taking the difference between m and m’ we get a value of 2( p p’) Since p is an integer we write the selection rule as m+m’ = 2 n Therefore, tunneling can only occur between states that satisfy this condition and this occurs at nth order in perturbation theory. For example, for a system with a given S tunneling can occur between states ms = S and ms = ( S +2) in first order, but tunneling between states ms = S and ms = S occurs in Sth order. Consequently, tunneling between the lowest states is less probab le than tunneling between higher energy states. The above mentioned selection ru le was obtained for a second order transverse anisotropy. Similar selection rules are applicab le for higher order transv erse anisotropies as well. Another quite common one is the fourth order term 4 44 4ˆˆ () 2 B SS The selection rule for this symmetry would be m+m’ = 4 n, otherwise no tunneling is allowed. At resonance fields, the non-commuting term s create an energy difference between the symmetric and antisymmetric combinations of th e spin projection states (tunnel splitting). As the magnetic field is swept through a resona nce value there is a probability for a spin to change its projection state, which is given by th e Landau-Zener formula fo r tunneling [19]. Fig. 1-5 illustrates the energy levels of Mn12Ac close to the first ( k = 1) resonance field. The effect of transverse anisotropies manifests itself as the re pulsion of energy levels (re d lines in Fig. 1-5) close to the resonance field. This is known as an “anti-crossing” since without any transverse anisotropies the energy levels cr oss (grey lines in Fig. 1-5) at the resonance field. While on resonance the spins have a non zero tunneling frequency that is given by Eq. 1-12 [1]. ˆ 'TmHm (1-12)

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25 Thus, the tunneling frequency depends on the matr ix elements between the respective tunneling states. The relation between the tunnel splitting and t unneling frequency is given by '2mm mm T (1-13) Hence, small tunnel splittings generate low t unneling frequencies and minimal tunneling. In the absence of any decoherence effects the spin would oscillate between states m and m’ at the tunneling frequency given by Eq. 1-13, with a probabi lity to find the spin in either state. In Fig. 1-3, we illustrate a situation for the k = 2 resonance field, wh ere the spins could tunnel back and forth through the anisotropy barrier. Ho wever, finite lifetimes of excited state energy levels arise from the possibility of phonon emissi on, and the oscillations between two states ( m and m’ where one or both are excited states) in re sonance are damped since the spin can relax from the excited state to the ground state by em itting phonons after tunneling. Consequently the spins that tunnel through the energy barrier quic kly decay back to the ground state and do not tunnel back again to the other side. The solution to Eq. 1-12, for the case of a second order transverse anisotropy ( E ), as given by Eq. 1-10 is presented in Eq. 1-14. 222k s k Tg E D (1-14a) 2222 (2)!(2)! 2[(/21)!]k SkD SkS g k sk (1-14b) In Eq. 1-14, k must be an even integer, which is imposed by the second order transverse anisotropy term. It is easily se en that for smaller values of k (tunneling between the lowest lying levels), the tunneling frequency is small. However, as k increases, the tunneling frequency can quickly increase by many orders of magnitude. Fr om a physical standpoint, this demonstrates

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26 that while tunneling between lowe r lying energy states is weak or even negligible, tunneling between higher lying levels can be quite significant. For the studies on the incohere nt tunneling processes presen ted in section 6.2 we can separate the tunneling into two regimes. As we sweep the magnetic field back and forth from 3 T, we pass through resonance fi elds where there is a probability for QTM. Once the system is fully biased and we sweep the field back thr ough zero toward the revers e saturation field, the spins have a chance to tunnel as we pass through each resonance field. The amount of spins that tunnel increases as the field increas es due to two effects. First, the tunnel splitting increases as the difference between m and m’ decreases and k increases, as can be seen Eq. 1-14. Second, the effective energy barrier is lowere d as the field increases and it is more probable for spins to tunnel through the lowered effectiv e barrier due to the finite latt ice temperature. Consequently, spins can be excited to higher states, which in creases the amount of tunneling as we sweep through a resonance field since t unneling between higher lying states is more probable (as shown in Eq. 1-14). Eventually as the biasing field becomes large enough ( gzBBms > | D | S2) the energy barrier becomes non existent and al l spins have reversed their proj ection state. This corresponds to a field of about 5 T for Mn12Ac. On the other hand, for much of the data colle cted, we would wait at a resonance field for a fixed amount of time as opposed to sweeping th rough the resonance at a given rate. In this case, there exists a tunneling probability per uni t time for each spin. Approximate formulas for the tunneling probability per unit ti me from the ground state into an excited state have been derived [20, 21, 22] and take into account the lifetime of each state (ground and excited) calculated without tunneling and th e tunneling frequency calculated fo r an isolated spin. These are directly applicable to our experiments, since they relate to conditions where two states ( m and

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27 m’ ) are in resonance. We find it necessary to only mention the qualitati ve aspect, which shows that the total amount of spins that tunnel will in crease for longer wait times. Of course, within a distribution, different mo lecules will have differe nt probabilities and cons equently will tunnel on different time scales. This can be seen through Eq. 1-14, since there ar e distributions of the D and E values among different molecules. The fact that we can observe which molecules are tunneling on certain time scales is the main point of the HFEP R studies done to monitor the QTM presented in Ch. 6. Interesting phenomena such as quantum phase interference of spin tunneling trajectories have been observed in single molecule magnets [23, 24]. This manifests itself in oscillations of the value of the tunnel sp litting at a longitudinal resonance fiel d while the value of the transverse field is varied. While a signi ficant amount of research has be en done to characterize this phenomenon, in this dissertation we will focus on quantum tunneling in the absence of an externally applied transverse fiel d. For our interests, there are two tunneling regimes that can be considered: Thermally assisted regime. In this regime spins can populate excited energy levels above the ground state. The probability for spin t unneling increases for resonant levels higher up the barrier. Thus, more tunne ling takes place between higher ly ing states in resonance. This can take place with or without an extern al magnetic field. An application of an external magnetic field shifts the zero field ener gy levels with respect to each other. At certain values of the field (g iven by Eq. 1-8) energy levels with opposite spin projections along the quantization axis re siding on opposite sides of the energy well become nearly degenerate. At these resonance fields, the spins can tunnel through the energy barrier and relax back to the ground state through phonon emi ssion. This process is illustrated in Fig. 1-3. Any spins on the left side of the barri er (ground state or exci ted states) that tunnel through the barrier to an excited state (right side of the barrier) relax to the other ground state by the process of phonon emission. Pure quantum tunneling regime. In the abse nce of an external magnetic field and at extremely low temperatures ( kBT | D | S2) spins will only populate the ground state energy level. For an integer spin system the tunnel splitting between the symmetric and antisymmetric linear combina tions of the two ground state wa vefunctions allows for spins to tunnel through the energy barrier and reverse their spin projection. No phonon

PAGE 28

28 absorption or emission is involved in this process. The rate of this process depends on the system, and can vary over many orde rs of magnitude [25]. For Mn12Ac, the rate is immeasurably small (< 10 8 s 1), while for the Ni4 system discussed in Ch. 4, the rate is quite fast (2 10 1 s 1). For all of the QTM presented in this dissertation the observed tunneling is a resonant quantum tunneling process in the ther mally assisted regime. In addition to tunneling, spins can also cha nge their energy state by thermal activation over the anisotropy barrier. At elevated temperatures where the thermal energy is comparable to the anisotropy barrier ( kBT | D | S2) the spins can relax by essentia lly going over the barrier through a thermal activation process. U nder these conditions the reversal of the magnetization can be an ongoing process as spins move back and forth over the top of the barrier, which is pure thermal relaxation. The activation energy is expressed as | D | S2 by considering a relaxation time that follows an Arrhenius law [1]. 1.5 Applications SMMs offer opportunities for exciting research in both the physical chemistry and physics communities. From a physics perspective, they allow investigation of the quantum mechanical properties of individual nanopart icles and how these properties are influenced by the surrounding environment. Additionally, they are unique in that they lie on the border of classical and quantum mechanical physics [26]. Even though th e magnetic moment of each molecule exhibits quantum mechanical behavior by tunneling through a magnetic an isotropy barrier, the magnetic moment is larger than a nor mal quantum mechanical system [16, 27]. Quantum phase interference effects have also been predicted and reported in certai n systems [24, 28, 29]. In the sense that they are macroscopic systems exhib iting quantum mechanical behavior, SMMs lie in both the classical and quantum regimes.

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29 The fact that each molecule is magnetically bi -stable due to the negative axial anisotropy barrier has created excitement for using these sy stems in possible technol ogical applications. Each molecule could potentially act as an individu al computing cluster, with the entangled states of the molecule acting as a qubit. In this wa y they could be used as a quantum computing device, with the speed for a single computation on the order of 10–10 s in a system such as Mn12Ac [30, 31]. Another promising application w ould be as a magnetic memory storage device [32]. The isolated magnetic moment of each mol ecule would represent one bit of information, and an incredibly high density (30 109 molecules / cm2 [30 terabits]) of da ta storage would be possible. Regardless of whether or not the realiz ation of these applications is ever achieved, SMMs are interesting to study from a purely scientific aspect. 1.6 Summary In this chapter we gave an introduction to the SMM systems which are the focus of the research presented in this disse rtation. We explained the sour ces and importance of anisotropy to the magnetic behavior and also described a unique feature of SMMs: quantum tunneling of the large magnetic moment vector of a single molecule The main goal of studying these materials is to gain a deeper understanding of the magnetic be havior. There are various parameters that control this behavior and in this dissertation we will present experiments and analysis done to determine such parameters and use this to explain the observed phenomena. In Ch. 4 we outline studies done on two nickel based compounds in order to determin e the disorder and isotropic exchange terms in the system that is a SMM. It is shown that the di sorder and intermolecular exchange interactions in this system have a dramatic effect on the low temperature magnetic spectrum. In Ch. 5 we outline studies done on two cobalt based compounds in order to determine the anisotropy and exchange terms in the system that is reported to be a SMM [33]. It is shown that anisotropic exchange is a ma jor component of the low temperature magnetic

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30 properties. Finally, in Ch. 6 we outline studies done on Mn12Ac that combine high frequency electron paramagnetic resonance (HFEPR), Ha ll magnetometry, and surface acoustic waves (SAWs) in order to characterize the effects of disorder on the QTM in addition to the relaxation processes in this SMM.

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31 Figure 1-1. An illustration of a molecule of Mn12Ac. Eight Mn+3 ions and four Mn+4 ions are antiferromagnetically coupled through the oxygen atoms, giving rise to a large S = 10 ground state for the molecule. The cent ral blue circles represent the giant magnetization vector pointing out of the page, along the S4 symmetry axis of the molecule.

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32 Figure 1-2. The energy barrier for a molecule of Mn12Ac in zero field. The axial anisotropy forces the magnetic moment to point either parallel (“up”) or an itparallel (“down”) to the quantization (easy) axis. The energy ba rrier to magnetization reversal is given by | D | S2.

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33 Figure 1-3. The energy barrier for a molecule of Mn12Ac with an external magnetic field applied parallel to the quantization (easy) axis. At values of the resonance fields the magnetic moment (blue arrows) can change its projection state by tunneling through the energy barrier. As an example we show a spin tunneling from the ms = –10 to ms = 8 state. The spin then relaxes back to the ground state ( ms = 10) by emitting phonons.

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34 Figure 1-4. A hysteresis loop for a single crystal of Mn12Ac at a temperature of 2 Kelvin and with an external magnetic field applied para llel to the quantization (easy) axis. The flat plateaus in the figure co rrespond to fields where tunne ling is switched off, but the sharp steps seen at field values that correspond to multiples of approximately 0.45 Tesla are resonance fields wher e the tunneling is switched on.

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35 Figure 1-5. A diagram of two energy levels in a system of Mn12Ac as they pass through the first non zero resonance field. The dashed lines show how the energy levels would behave in the absence of any tunnel splitting term. The red lines show how the energy levels are repelled due to the tunnel spli tting term. A spin in the state ms = 10 state has a probability, dependent upon to tunnel to the ms = –9 state as the field is swept through the resonance field.

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36 CHAPTER 2 EXPERIMENTAL INSTRUMENTATION AND TECHNIQUES 2.1 Measurement Techniques There exist many different expe rimental techniques used to study SMMs. Some of these include x-ray crystallography to measure the molecular struct ure [34, 35], susceptibility measurements for basic magnetic characterization [17], neutron scattering to measure exchange couplings and spin multiplet excitations [36, 37, 38, 39], high frequency electron paramagnetic resonance (HFEPR) to measure spin multiplet en ergy spectra [40, 41, 42], and Hall or SQUID magnetometry to measure magnetic hysteresis and observe quantum t unneling steps in the magnetization [43, 44]. All of th e studies presented in this di ssertation were done using HFEPR as the main experimental technique. Some wo rk, presented in Ch. 6, combined HFEPR with surface acoustic waves or magnetometry to study the influence of microwaves on non equilibrium processes in single crystals of Mn12Ac. 2.2 Electron Paramagnetic Resonance in the Context of Single Molecule Magnets In Ch. 3 where we formulate the spin Hamiltonian for SMM systems, the Zeeman interaction is expressed as ˆBzeemangBS (2-1) For a magnetic field given by 0ˆ BBz the energy of this interaction is 0BzeemanzsgEBm (2-2) where ms is the projection of the elec tron’s magnetic moment along the z axis. In order to induce a transition between states one must s upply an energy of 0. Therefore, the resonance condition is such that this energy must match the en ergy difference between respective states, or equivalently, 0 = E The energy comes in the form of microwave radiation at a

PAGE 37

37 frequency 0. This is the basic principal behind which all of our EPR measurements lie. As an example of this, in Fig. 2-1, we plot the energy levels in the Ni4 SMM (discussed in Ch. 4) with its easy axis aligned along the ex ternal field. For a fixed freque ncy, there is a strong absorption of the signal each time the resonance condition (energy difference between adjacent levels matches the microwave radiation) is met as th e magnetic field is swept. The black arrows represent the magnetic fields where absorption of the microwaves causes transitions between states. In a typical experiment, fixed temperature, fixed frequency magne tic field sweeps are performed at a number of different temperatures and freque ncies. As the magnetic field is swept, sharp inverted peaks appear in the spectrum when the resonance condition is met, which corresponds to the transmission of the microwave signal through the experimental setup. Each peak corresponds to a transition between spin projection states, and the decrease in transmission signal is due to the absorption of microwave radiation by the spin s within the sample. A normal EPR spectrum from a SMM with its easy axis aligned along the external field consists of a series of absorption peaks corresponding to transitions between states within the ground state spin multiplet. At sufficient temperatures where al l states within the multiplet are populated, one should see 2 S peaks, corresponding to 2 S transitions between 2 S +1 states. This is illustrated in Fig. 2-2 for the same system in Fig. 2-1. The larger intensity peaks correspond to transitions within the ground state. The a dditional, smaller intensity peak s, which are attributed to transitions within a higher lying spin mu ltiplet, are also discussed in Ch. 4. 2.3 Cavity Perturbation 2.3.1 Technical Challenges The basis for any EPR measurement is to monitor the response of the sample to the application of external microwav e radiation and a magnetic field. This presents many obstacles

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38 to overcome in terms of experimental technique s and instrumentation. First, there are space restrictions in terms of positioning the sample in the center of the magnetic field. High field, high homogeneity magnets possess rather narrow bores which leave little room for a sample stage. Even more problematic is the propaga tion of microwaves thr ough a probe and ensuring proper coupling of the microwav es to the sample. Unwanted standing waves arise due to multiple reflections from impedance mismatch ed components [45]. Another obstacle is attenuation of the signal due to the finite condu ctivity of the waveguide s, especially at high frequencies and/or where the prop agation is along considerable distances. The need for high frequencies stems from the large zero field splitting energies (discussed in section 1.2) present in SMMs. These materials would be EPR silent with typical X band (~ 9 GHz) spectroscopy. Detecting a signal from a tiny si ngle crystal, where th e typical sample dimension used for our experiments is on the order of 1 mm, is not trivial and requires extremely precise methods. 2.3.2 Equipment An ideal method suited for fixed frequency magnetic resonance sp ectroscopy is known as the cavity perturbati on technique [46]. This technique re lies on the resonant modes of a high quality factor cavity to ensure good coupling of th e radiation to the sample and compensate for the small filling factor of the sample. The draw back to this method is that one is limited to working at frequencies that correspond to the m odes of the cavity. We use cylindrical copper cavities that have a fundamental mode, TE011, at close to 50 GHz, but can operate at frequencies up to 400+ GHz. We have two types of cylindri cal cavities: one is a st andard vertical cavity where the z axis of the cylinder is oriented vertically, and the other is a rotating cavity that allows rotation of the sample with respect to the external magnetic field, where the z axis of the cylinder is oriented horizontally [47, 48].

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39 All experiments were done using a single crys tal of the respective compound. If the mass of the crystal is measured, and the molar ratio of each constituent compound is known, then the number of spins participating in a given transition can be estimated from the area of the respective resonance peak. For th e purposes of our experiments, this was not a crucial number, and therefore the mass of a given crystal was never measured. Additionally, the data was recorded as a relative voltage signal, making an absolute measurement of the number of spins contributing to a resonance peak impractical. Co nsequently, the transmission signal is quoted in terms of arbitrary (arb.) units. We now describe the orientation of a sample in the cavities in the interest of subsequent chapters when we discuss experiments using thes e cavities. For experiments conducted using the standard vertical cavity, the sample was placed on a circular endplate of a copper cavity at a position halfway between the center and the edge of the endplate. For th e fundamental mode of the cavity, the magnetic component of the microw aves has a maximum at this position. Thus, we maximize coupling of the microwaves to the sa mple. This endplate then fastens onto the end of the cylindrical cavity. This does not allow fo r sample rotation within the cavity itself, but is convenient when the magnetic axes of the sample are known from x ray crystallography measurements and can be deduced by merely look ing at the crystal. The sample can then be placed in the proper orientation before insertion into the cryostat. For experiments conducted using the rotating cavity, the sample was mounted in a similar manner, except the endplate mounts in a horizontal fashion. The initial samp le orientation depends upon the nature of the experiment, but when used in our magnet system possessing a horizontal fi eld, the rotating cavity allows for two axis rotation, and thus any sample alignment. The cavities and the rotational capabilities of each magnet system are illustrated in Fig. 2-3 and Fig. 2-4. A detailed discussion

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40 of the rotating cavity is given else where [47]. The capability of th e rotating cavity is critical for studies of anisotropic systems such as SMMs and must be used for systems where the orientations of the magnetic axes with respec t to the crystal are not known beforehand. The sample temperature is always determined from a Cernox thermometer that is securely fastened to the outside of the respective copper cavity. Th e waveguide probe used for measurements with a cavity is also described in detail elsewhere [45]. Now we discuss a simple model of the resona nt cavity where the samp le is placed. An ideal resonator is made from a continuous, perf ectly conducting material and is filled with a lossless dielectric. In this case the cavity modes are perfectly disc rete, have the shape of a delta function peak, and can be calculated consider ing the correct boundary conditions for the appropriate geometry. Our cavitie s are cylindrical resonators, w ith the resonant modes of an ideal resonator being given by [49] 221 ()()mn mnpx p rd (2-3) In Eq. 2-3, is the magnetic permeability of the cavity dielectric, is the electric permittivity of the cavity dielectric, m n and p are integers, r is the radius of the cavity, d is the height of the cavity, and xmn is the nth root of the first derivative of the Bessel function. Of course our cavities are not ideal resonators since they are made of a finite conducting material (copper), and filled with a finite loss dielectric (helium thermal ex change gas and the sample ). Additionally, we must allow the microwaves from the waveguide s to couple to the cavity which is done through two small coupling holes with a diameter of 0.97 mm ( /6 for a frequency of approximately 50 GHz). The dimensions (diameter and thickness) of the apertures are wh at determine the amount of coupling to the cavity. There is an important trade off between: (i) strong coupling (large

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41 apertures), which ensures good power throughput from the source to the detector and, hence, a large dynamic range, and (ii) w eak coupling (small apertures), which limits radiation losses from the cavity, resulting in higher cavity Q values and increased sensitivit y, at the expense of some dynamic range. It is found empirically that the optimum coupling apertures should be small for our setup, hence the choice of a diameter of /6. Additionally, it is necessary for the coupling plate to be sufficiently thin (~ /20), since the signal is obviously attenuated as it passes through the apertures, which are way below cutoff. The critical instrument for all of our EPR m easurements is a Millimeter Vector Network Analyzer (MVNA), which acts as both a microwave source and detector [50]. The instrument contains two YIG (Yttriu m Iron Garnet) oscillators, which ar e continuously tunable in the range of 8 to 18.5 GHz. These sources have an intrin sic stability that drifts by only a few MHz per hour when working around 100 GHz. Although the cm sources are phase locked to each other, their absolute frequencies must also be stab ilized, which can be achie ved through phase locking the MVNA sources to an external frequency counter with an internal quartz reference. The first oscillator produces a signal F1 and is locked to an external frequency counter to stabilize the frequency with a feedback loop. This become s useful for improving frequency resolution and stability when measuring high Q factors (> 104) of modes of our resonance cavities. To this end, a Phase Matrix 575 source locking frequency coun ter [51] is used, which provides both the stability and precision necessary for our measurements. The second oscillator produces a signal F2 – Fbeat and is phase locked to the first oscillator, and consequently the frequency counter as well. The beat frequency is derived from an internal reference oscillator operating at 50 MHz and is programmed by the software controlling the MVNA hardware. The MVNA receiver can oper ate at one of two exceptionally precise

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42 frequencies (either 9.01048828125 MHz or 34. 01048828125 MHz), which corresponds to the harmonic multiplication of the frequency difference between F1 and F2 ( N [ F1 – F2]). The signal from the first oscillat or is sent to the harmonic generator (HG) along low loss, flexible coax cable. The signal from the second oscillator is sent to the harmonic mixer (HM), also along low loss, flexible coax cable. These cables introduce a combined insertion loss of 4-5 dB [52]. The two frequencies are beat together, and the amplitude and phase of this signal are what is processed by the MVNA r eceiver. The beat frequency, Fbeat, is given by N F1 – N’ F2, where N and N’ are the harmonic numbers that multiply the respective fundamental frequencies. These values will be 1 for frequencies up to 18.5 GHz; for frequencies above 18.5 GHz we employ external Schottky diodes to work on harm onics of the fundamental frequencies. The Schottky diodes are passive, non-linear devi ces that can generate harmonics from N = 3 (V band) up to N = 15+ (D band). We use pairs of Schott ky diodes that will produce the same harmonic ( N = N’ ) for the HG and HM. In this respect, Fbeat = N ( F1 – F2). Additionally the phase difference, beat, between the two frequencies is given by N 1 – N’ 2. With the above mentioned configuration ( N = N’ ), the two phases are locked to each other ( 1 = 2), which cancels any phase noise associated with the beat signal and allows for a incredibly low noise level. However, any phase diffe rence between the two frequencies is upheld in the beat signal. Thus, we can measure both the amplitude and ph ase of the beat frequency which constitutes a vector measurement. The choice of the beat fre quency (receiver frequency) is determined by the software, and is made depending on the harmonic number. For harmonics up to three, the 9.01048828125 MHz receiver is chosen, and for ha rmonics greater than this the 34.01048828125 MHz receiver is chosen. For si gnal detection, these frequencie s are down converted by beating intermediate frequencies with the receiv er frequency. If using the 34.01048828125 MHz

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43 receiver, the first stage consists of beating this signal with a 25 MHz signal from the internal 50 MHz oscillator. This signal (9.01048828125 MHz) is then beat w ith a 9 MHz signal from the 50 MHz oscillator which leaves a 10.488 kHz signal. The same procedure applies when using the 9.01048828125 MHz receiver, except th e first step (beating with 25 MHz) is obviously not needed. Since all signals used in the down conversion proces s are correlated to the same reference oscillator, the phase in formation is maintained during th e process. Finally, the 10.488 kHz signal is sent to a lock in amplifier where it is converted to a DC signal which maintains the amplitude and phase information. As the harmonic number increases, the power output from the HG decreases. For harmonics greater than nine (or frequencies gr eater than 160 GHz) we use a Gunn diode as an external microwave source that works in conjunction with the MVNA and Schottky diodes that can produce higher frequencies with slightly in creased power outputs. The non-linear effects from the HG and HM create a comb of frequencies ( NF2), which mix with the frequencies from the Gunn source, creating an inte rmediate frequency such that FIF = M FGunn –N F2, where M and N are integers. The frequency of the Gunn source is locked to the first YIG oscillator (and by default the second, as well), with an offset equal to the 50 MHz reference oscillator, such that FGunn = kF1 – 50 MHz, where k is the harmonic number of the first oscillator. The relation between the harmonics of the YIG osci llators and Gunn source is given by N = kM. The intermediate frequency can now be expressed as FIF = N(F1 – F2) – M (50 MHz). As mentioned previously, the MVNA receiver operates at one of two precise frequencies, FR. Therefore, in order to properly tune the hardware, the software programs the MVNA electronics such that difference frequency between the YIG oscillators is F1 – F2 = FR / N + 50 MHz / k. Since the frequency of the Gunn source is phase locked to the internal oscillat ors of the MVNA, we

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44 achieve the same low noise level as when we ar e solely using the MVNA oscillators. We have two Gunn diodes that cover funda mental frequencies from 69 GHz to 102.5 GHz with power outputs of ~40 mW at 92 GHz. The Gunn diodes have two attachment options depending on the type of measurement desired. If we are doing a continuous wave (CW) EPR experiment, we use a multi harmonic multiplier to achieve frequencies up to 500+ GHz. The multi harmonic multiplier has four mechanical tuning knobs that allow precise optimization of the signal for a given frequency. Two of the knobs are for optim ization of the incoming power from the Gunn at the frequency FGunn. The other two knobs are for optim ization of the output power at a frequency M FGunn. Various filters are used at differe nt frequencies (138 GHz, 235 GHz, 345 GHz, 460 GHz, 560 GHz) to remove lower harmoni c components of the signal. Non linear effects from the HM can creat e unwanted signals from combin ations of lower harmonics ( M1 and M2), such as M1 + M2, 2 M1, or 2 M2. This can introduce crosstalk effects in the detection electronics and the use of high pass filters cancels this effect. The signal rejection for the first harmonic below cutoff for each filter is on the orde r of 75 dB. For instance, when working in the frequency range of 235 GHz to 344 GHz, we use the 235 GHz filter that will attenuate lower harmonic components of the signal below 235 GHz but will pass the desi red signal along with higher harmonic components. These higher harmonic components are of course rejected by the narrow band receiver set to detec tion at the precise beat frequenc y for the proper harmonic. The power output in this fr equency range is on the order of approximately 1 mW at 300 GHz. For EPR measurements where we wish to ha ve large powers in short pulses to avoid sample heating, we remove the multi harmonic mu ltiplier and attach a one stage power amplifier and fast pin switch. These components operate in a much narrower frequency band (89 GHz – 98 GHz) than the CW components but allow for larger powers (186 mW at 92 GHz) and pulse

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45 microwave signals at incredibly fast time scales (Fast Switching Unit, 4ns from 10% to 90%). The pin switch has a low insertion loss (~ 1.5 dB ) and provides approximate ly 40 dB of isolation between the “on” and “off” states. Additionally, we can attach a frequency doubler or tripler to work at frequencies in the range of 178 GHz –19 6 GHz or 267 GHz – 294 GHz respectively. In this configuration we also have the option of removing the switch to work in CW mode with amplified powers (approximately 10 mW at 288 GHz) at the above mentioned frequencies. We have two commercial superconducting magnet systems available in our lab. The first is a 17 Tesla, vertical field Oxford Instruments sy stem [53]. The second is a 7 Tesla, horizontal field, Quantum Design system [54]. These sy stems allow temperature ranges from 300 K down to less than 2 K. For a period of time during our research, the 7 Tesla magnet was undergoing repairs and a replacement 9 Tesla vertical fiel d magnet was loaned to us. Some experiments done in this magnet will be presented in Ch. 5. Fig. 2-3 shows a schematic of our typical experimental setup, including a waveguide probe and sample cavity inserted in the bore of a superconducting magnet immersed in a cryostat. 2.4 Quasi Optical Setup The use of waveguides begins to become probl ematic at high frequencies. First and foremost, the signal losses associ ated with the finite conductivity of the waveguides are larger for higher frequencies [7]. This arises from ohmic losses in the conducting waveguides, as the induced current at the surface of the waveguides increases wi th increasing frequency. Additionally, the resonant cavities become comp letely overmoded and it becomes impossible to determine the microwave field conf iguration for a given frequency. An alternative method that we have employed is to use a quasi optical bridge setup opera ting in reflection mode. The constituent components of the se tup are all quite low loss and c onsist of a corrugated reflection tube (the probe), corrugated horns wire grid polarizers, and e lliptical focusing mirrors. The

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46 corrugated tube and associated horns are opt imized for a specific frequency of 250 GHz, although the horns and tube are broa dband, as data has been collect ed in the frequency range of 180 GHz – 350 GHz for these particular components. Expected loss for the probe working at 250 GHz is ~0.01 dB/m [55]. Sin ce the length of the tube is 1 00.3 cm in accordance with the sample chamber in our Quantum Design system magnet, it provide s a theoretical minimal loss of approximately 0.02 dB. The tube, which is based on a corrugated HE11 waveguide, is made fr om thin walled (.3 mm) German silver, chosen for its low therma l conductivity and ease of machining. The low thermal conductivity reduces the heat load on the cryostat and allows the sample to remain at cryogenic temperatures throughout th e experiment. The reflection probe is tapered close to the field center in order to concentrate the incident power over a 1.7 mm diameter region at the bottom of the tube. The sample can be attached to a reflecting, copper sa mple holder at the field center. Careful alignment of th e optical bridge with the probe ensures that most of the signal reaching the detector comes from the reflecti ng sample holder at the field center, thus guaranteeing good coupling to small samples of submillimeter dimensions. The top of the tube has a high density polyethylene ( HDPE) window with an antireflect ion coating, which minimizes insertion loss. The two, identical, corrugated horns are made from gold plated copper (each being ~19.7 cm long with a 1.78 cm circular opening). The horns are also tapered at the input end to match with waveguide couplers that are used for the conversion between Gaussian optics and the rectangular guided mode s that are required for coupling to the source and detector through external diode attach ments (Schottky and Gunn). The elliptical focusing mirrors are manufactur ed from aluminum and are machined to a incredibly high tolerance for optimum performance. The mirrors facilitate the propagation of an

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47 undistorted Gaussian beam. A free space Gaus sian beam has a minimum beam waist, w0, that is greater than half the wavelength of the radiation [56]. A narrower beam waist will constitute a signal with more intensity reaching the sample, but the divergence of the beam is inversely proportional to the beam waist. Faster diverg ing beams necessitate la rger surface mirrors in order to refocus the beam. For distances far fr om the point of minimum beam width, the beam width approaches a straight line. 2 0 0()1 z wzw z (2-4a) 2 0 0w z (2-4b) The angle between the beam propagation ax is and the beam radius is given by 0w (2-5) The total angular spread of the b eam is twice this value. A diag ram of a Gaussian beam is shown in Fig. 2-5. Thus there is a compromise betw een a narrow beam waist and the practical size of the mirror surface. For reflecting optics, the mi rrors must be machined to tolerances of /20 in order to approach diffraction limited performance [56]. At 300 GHz ( = 1 mm) this corresponds to a tolerance of 0.0025 cm. The mirrors have a focal length of 25 cm and all components are configured such that the Gaussian beam will be focused at the entrance to the probe and detector horn. Since the reflectivity of aluminum is clos e to unity for the microwave frequency range of interest these components introduce minimal loss. We employ two wire grid polarizers to ensure that the polarization of the signal is correct when returning to the detector. These wire grid polarizers are necessary in order to propagate the signal through the quasi optical bridge setup.

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48 Fig. 2-6 illustrates the transmission of the signal through the system, and in Fig. 2-7 we show how the polarization of the signa l is changed by the respective components. From Fig. 2-7, it can be seen that the polarization of the signal reflecting off the firs t wire grid polarizer must be changed after interacting with the sample and befo re entering the detector. If the second wire grid polarizer was not there, the signal returning to the first wire grid polarizer would be totally reflected before it enters the detector. The inse rtion of the second wire gr id polarizer changes the polarization of the signal so that it will pass through the first wire grid polarizer and reach the detector. All components of the optical setup are mounted securely on a solid stainless steel bench (87.6 cm x 49.5 cm), held firmly in place with screws and dowel pins, and were manufactured by Thomas Keating Ltd [57]. Here we discuss the setup in detail, which shoul d be referenced to Fig. 2-6 and Fig. 2-7. The signal originating from the Gunn diode source is polarized in the manner for a rectangular waveguide ( H field along the longer dime nsion). The mode of the rectangular waveguide is launched into a rectangular-to-circular waveguide transition. The transition piece produces the TE11 circular waveguide mode which converts to a HE11 mode once inside th e corrugated part of the horn. After exiting the horn, the signal becomes a Gaussian pr ofile as it propagates in free space. First, the signal encounter s a 45 wire grid polar izer 12.5 cm from the edge of the horn. This splits the signal and the reflected beam is polarized at 45 with re spect to the horizontal, while the transmitted signal (90 polarization with respect to the reflected signal) is absorbed by a pad that is highly absorbent in the microwave region. This helps to minimize the amount of signal returning to the detector that does not inte ract with the sample. The reflected signal then meets the first elliptical focusing mirror, 12.5 cm fr om the middle of the wire grid polarizer. The signal has now traveled 25 cm total, and the mirr or reflects the diverging beam such that the

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49 minimum beam waist is 25 cm from this point. The next component is a second wire grid polarizer, with an orientation that is horizontal (45 from the firs t polarizer). This again changes the polarization of the signal so that it is now ve rtically polarized (90 from the horizontal). A second mirror is located 50 cm from the fi rst mirror (25 cm from the minimum beam waist). Again the purpose of the mirror is to fo cus the beam profile so that it will have its minimum beam waist as it enters the corrugated waveguide probe, 25 cm below the mirror. The signal then changes to a HE11 mode as it travels down the tube to the sample and reflects back up. In principle, the signal retains its polariz ation as it exits the probe and becomes a Gaussian profile once again. However, the magnetic respon se of the sample can alter this polarization slightly. The beam exiting the probe travels 25 cm back toward the second mirror, where the diverging profile is refocused. Going from th e second mirror back towa rd the first, it passes through the second wire grid pol arizer. If the polarization of the signal was unchanged by the sample then it is also unchanged by the polarizer since it is the same as the first time passing through this component. The beam then encounters the first mirror, where the diverging beam is focused such that the minimum beam waist will be at the entrance to the detector horn (25 cm from the first mirror). After being refocused by th e first mirror, but before entering the detector, the signal passes th rough the first wire grid polarizer, wh ere the polarization is changed for a final time so that it will be correctly oriented with respect to the detector (45 from vertical). Finally, the properly focused and pol arized signal enters the detect or horn and is propagated as a HE11 mode and then in the manner for a rectangular waveguide as it enters the Schottky diode and is mixed down to the proper detection frequency by the MVNA electronics. Both the tube that inserts into the cryostat and the horns connected to microwave source and detector diodes are corrugated. In contra st to a smooth walled ( no corrugations) circular

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50 waveguide, a corrugated waveguide couples extremel y effectively to a free space Gaussian beam profile. Hence, the main advantage of the co rrugated components is the extremely low loss associated with microwaves at the optimized fr equency. First, there is excellent coupling between the free space Gaussian beam and the HE11 mode that propagates through the tube and horns. The efficiency of this coupling can reach 98% with pr oper matching. Second, the HE11 mode has little attenuation insi de these waveguides since the cylindrical symmetry allows efficient signal propagation. A non corrugated circular waveguide couples the Gaussian profile to both the TE11 and TM11 modes which progress at different phase rates and can result in a distorted output beam. The reas on for the corrugations is to ma ke the wall of the waveguides reactive which allows the HE11 mode to propagate with almost no attenuation [56]. An ideal HE11 mode has identical E plane and H plane radiation patterns w ith a main lobe that is approximately Gaussian, which is the reason for the excellent coupling between this mode and the free space Gaussian profile. A diagram of the TE11 and HE11 modes in a circular, corrugated waveguide are shown in Fig. 2-8. The loss is proportional to 2/r3, where is the wavelength of the radiation and r is the radius of the tube. Cons equently, the loss de creases for higher frequencies. The periodicity of the corrugati ons is chosen to match well with the optimum frequency of the tube, with a periodicity of about 0.4 mm ( /3) for 250 GHz, and the depth of each corrugation is /4. A larger number of corrugations per wavelength is advantageous for optimal signal propagation, but becomes diff icult to machine. Three corrugations per wavelength is a good compromise for this frequency region. As for the corrugation depth, it is reported [58] that the loss for the HE11 mode is less than the TE01 mode loss in a smooth walled waveguide for corrugation depths in the range of 0.35 /4 – 1.75 /4.

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51 For experiments conducted using this setup, th e sample can be placed on a flat copper plate that seals against the bottom of th e corrugated tube such that the sample is enclosed by the small circular opening at the end of the tube. This maximizes coupling of the microwaves to the sample. No rotation of the endplate is possible, but the tube is designed for our magnet system possessing a horizontal field. In this orientation, the entire pr obe (and thus the sample) may be rotated with respect to the field, allowing one axis of rotation. Similar to the cavity setups, the temperature is taken by a Cernox th ermometer that is secured to the copper plate that holds the sample. Another advantage of the quasi optical setup is the possibility to conduct studies that incorporate microwaves with othe r devices that are too large to fit inside a cavity. We have developed a technique that allo ws devices such as a piezoel ectric transducer or Hall magnetometer to be placed near the end of the corrugated waveguide probe. In such a configuration, a copper block holds that device, where a sample is placed on the surface. The sample is then positioned such that it is just below (~ 1 mm) the openi ng where microwaves exit the probe in order to maximize coupl ing of the radiation to the samp le. In this manner, we have conducted two unique experiments. 2.4.1 Piezoelectric Transducer Device The first experiment involved a technique wh ere a piezoelectric devi ce creates short heat pulses (pump) to drive the system from equilibrium and the microwaves act as a probe of the system dynamics as it relaxes back to equilibrium. Hybrid piezoelectric inter-digital transducers (IDTs) deposited on the 128 YX cut of LiNbO3 substrates [59] were used in the experiments to produce the surface acoustic waves. A picture of the device, with a sample placed on the substrate centered between the IDTs, is shown in Fig. 2-9. We employed a special transducer design [59] which yields devices capable of ge nerating multiple harmonics with a fundamental

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52 frequency of 112 MHz, and up to a maximum frequency of approximately 0.9 GHz. The reflection coefficient, S11, was characterized in the frequenc y range of 100 – 500 MHz in order to determine the optimum frequency to transmit the maximum amount of power to the IDTs. All the experiments performed were done using the third harmonic at 336 MHz, which was determined to be the optimum frequency to transmit the maximum power to the IDTs. The 250 GHz probe was modified in order to incorporate a coax cable [60] that could transmit the proper power to the IDTs. We us ed the proper electrical connections (SMA and coax) for as large a length as possible so that the signal integrity would be maintained and thus, the maximum amount of power would be tran smitted to the IDTs. The improvised wiring necessary to connect the coax cable to the piezoelectric device onl y constituted about 2.5% of the total length the signal had to travel. A hole was m achined in the top of the probe head that could hold a hermetically sealed bulkhead SMA female to female connector. The 1.19 mm diameter coax cable was run down the leng th of the tube to a pocket in the copper block holding the device. A small piece of the outer conductor and dielectric were removed to expose the inner conductor. From here a small breadboard piece, designed as a transition piece from the coax cable to the device, was soldered onto the inne r and outer conductor of the coax. The final electrical connection was from the transition piece to the IDTs on the device with two 40 gauge copper wires. A single crystal sa mple was placed directly on th e IDTs, with a small amount of commercial silicon vacuum grease used to attach the sample to the IDTs and ensure coupling of the surface acoustic wave (SAW) to the sample. This piezoelectric device was mounted on a copper block with a Cernox thermometer fixed a pproximately 1 cm from the device itself in order to indirectly monitor the temperature variations of the system. The copper block holding the device was attached to the end of our 250 GHz corrugated tube. Th e 1.7 mm opening where

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53 radiation exits the tube was moved as close to the sample as possible (~ .10 mm away) and centered over the sample to maximize coupling of the microwaves to the crystal. The entire setup was placed in the Quantum De sign superconducting magnet system. In order to have the IDTs produce SAWs we had to pulse the devi ce at the corresponding MHz frequency. The device only transmits su itable amounts of power at frequencies that correspond to multiples of 111 MHz. A Marconi Instruments [61] function generator capable of frequencies between 80 kHz and 520 MHz was used for this purpose. A minimum amount of power is needed to produce a SAW that will intera ct with the crystal. This instrument has two modes of operation, continuous and pulse. The drawback to the pulse m ode is that the output power is limited to 3 dBm. In contrast, powers of up to 7 dBm are achievable in continuous mode when working with the amplitude modulation option. This was the mode we used for our experiments. A diagram of our equipment setup is shown in Fig. 2-10. In order to realize a pulse with sufficient power we used an Agilent 81104A pulse pattern gene rator [62] to modulate the continuous waveform from the Marconi function generator. The 81104A has two independent channels, which can output separate signals. As an exte rnal modulation source, channel one of the 81104A supplied a 0.8 V (+0.4 V high, –0.4 V low) bipol ar signal to trigger the Marconi. This would in turn cause the IDTs to create a SAW pulse which would couple to the sample and push the system out of equilibrium. Pulses from 5 ms to 50 ms were used. In addition to triggering the SAW pul se, the pulse pattern generator also served another purpose, which was to trigger the fast data acquisition card (DAQ) simultaneously. Channel two of the 81104A sent a 5 V (+5 V high, 0 V low) signal to trigger the DAQ card. In this experiment we were measuring processes with timescales on the or der of milliseconds or less. To this end, we used an Acqiris APS 240 DAQ card [63] to co llect data. The Acqiris APS 240 is a fully

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54 programmable, dual channel instrument that is capable of 2 Gs/s real time averaging performance. The signal that we wished to record originated from the receiver in the MVNA, which can operate at a frequency of either 9 MHz or 34 MHz. This MHz IF signal was sent to a Stanford Research Systems (SRS 844) high frequency lock in amp lifier [64], which can work at frequencies up to 200 MHz. The lock in–amp lifier converts the MHz signal from the MVNA into a DC signal that is directed to the data acquisition card. A lock–in amplifier exploits a technique known as phase sensitiv e detection. The lock-in amplifier detects the MHz from the MVNA at the respective frequency, which is the ex ternal reference frequency. Additionally, the lock-in amplifier has an internal reference freque ncy. The two frequencies are combined and the resultant signal is the product of the two sine waves. This consis ts of a component that is the sum of the two frequencies, and one component th at is the difference of the two frequencies. This output is then passed thr ough an internal low pass filter, which removes the high frequency component. Under the condition th at the two frequencies are equa l, the component of the output signal is the difference of the two frequencies, which is a DC signal. An internal phase locked loop (PLL) locks the internal refe rence frequency to the external reference. Time constants for the lock–in amplifier ranged from 30 s to 100 s for these specific experiments. After being processed by the SRS 844, the signal was sent to the APS 240 and the system dynamics were recorded by changes in the dete cted microwave radiation. The simultaneous triggering of the Marconi and APS 240 could be done manually for a single shot acquisition or repeat edly at some frequency for repeated sequences. For time resolved measurements, especially when perf orming repeated sequences where the data are averaged to a final result it is critical that all instruments are phase locked to each other. To this end we used a 10 MHz time-base from the MV NA to connect to an Agilent 33220A function

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55 generator [62], which was in tu rn connected to the 81140A and SRS 844. It is necessary to repeat the pulse sequence such that it is co mmensurate with the IF (receiver) frequency to achieve a phase lock of the instruments. The MVNA receiver frequency is specified to a precision of Hz, and therefore it is necessary to use the functi on generator to phase lock the MVNA to the other instruments, which are not capa ble of such frequency resolution. Since the APS 240 and Marconi were both lin ked through the 81140A, this ensu red a proper phase lock of all the instruments to the MVNA. The APS 240 records data in r eal time with 8 bit resolution an d maximum rate of real time averaging at 2Gs/s. In averaging mode the ne wly acquired waveforms ar e summed in real time with the corresponding previous waveforms. Th e DAQ card starts recording data after it is triggered. The corresponding samples are those with identical delay calculated from the beginning of the trigger. While the acquisition is running, the pr evious sum is read from the averaging memory, added to the incoming sample, and written back to the memory. The processor is designed to perform on-board averaging at a maximum re-arm rate of 1 MHz. The clock resynchronization time between successive tr iggers is ~ 200 ps. With a proper phase lock, the signal should be the same for every trigger event, and thus th e signal gets properly averaged by the processor. Without a proper phase lock the DAQ card will start recording data at non synchronous times during the IF signal period an d thus the averaged signal would go to zero. 2.4.2 Hall Magnetometer Device The second experiment investigated the influence of microwave radiation on the magnetization of a sample, which was recorded with a Hall magnetometer. Like the piezoelectric device, the Hall magnetometer device was mounted securely in the socket of an electrical pin connector, which was fastened into a copper bloc k that mounts on the end of our

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56 250 GHz quasi optics probe. A picture of the Hall device, with a sample placed near one of the Hall crosses, is shown in Fig. 2-11. A single crystal sample is placed on the surface of the magnetomet er such that it sits slightly off center from the active area (Hall cross) of the device (~50 m 50 m). This ensures that the maximum amount of the dipolar field from the sample will pierce the Hall cross (active area of the device). The 1.7 mm opening where radiation exits the tube was moved as close to the sample as possible (~ .10 mm away) and centered over the sample to maximize coupling of the microwaves to the crystal. In order to generate a Hall voltage we use a low frequency AC current (~ 500 Hz) on the order of 1–2 A. We use a Stanford SRS 830 lock–in am plifier [64] to both generate the AC current and detect the Hall voltage at the same m easurement frequency. Th e lock–in technique is used since it enhances the sensi tivity of the detection of small vo ltage signals (on the order of V or less). This relies on the extremely narro w bandwidth phase sensit ive detection mentioned earlier. The lock–in amplifier is set to a sinusoidal output voltage of 1 Vpp at the desired frequency. In series with the device is a 500 k resistor, which helps provide a steady output current of ~ 2 A. Currents much larger than 1–2 A can damage the device, and low frequencies minimize the capacitive coupling between th e leads that can develop at higher frequencies. At low frequencies a capacitor acts like an open circuit, as no current flows in th e dielectric. Driven by an AC supply a capacitor will only accumulate a li mited amount of charge before the potential difference changes sign and the char ge dissipates. The higher the fr equency, the less charge will accumulate and the smaller the opposition to the flow of current, which introduces capacitive

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57 coupling. Twisted wire pairs are used to mi nimize low frequency induc tive coupling. The expression for the Hall voltage in two dimensions is given by 2 H DiB V ne (2-6) In Eq. 2-6, i is the current, Bis the transverse component of the magnetic field, n2D is the charge carrier density in two dimensions, and e is the charge of the carri ers. The basic principle behind the Hall effect is the Lorentz force on an electron in a magnetic field. When a magnetic field is applied in a direction perpendicular to a current carrying conducto r, the electrons that constitute the current experience a force due to the magnetic field. This force causes the electrons to migrate to one si de of the conductor, and hence, an electric field and an accompanying potential differe nce develop between opposite sides of the conductor. Consequently, there is also an electric force experienced by the electrons. In equilibrium the magnetic and electric forces on the electrons will ba lance. Eq. 2-6 gives the formula for the Hall voltage in a 2D system such as our device. 2.5 Hall Magnetometer Fabrication The Hall magnetometers used in our studies were fabricated at the University of Central Florida in the research lab of Dr. Enrique del Barco using op tical lithography and chemical etching techniques. The device is composed of a Gallium-Arsenide / Gallium-AluminumArsenide (GaAs/GaAlAs) heterostructure, with the active area being a two dimensional electron gas (2DEG) that resides roughly 100 nm below the su rface of the wafer. This material is ideal for Hall sensors due to the small carrier density, n which can be manipulated during the growth process of the heterostru cture by changing the number of dopants Typical values for the carrier

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58 density in the 2DEG is on the order of 5 x 1011 cm-2. Low carrier densities lead to larger Hall coefficients (proportional to n-1), and thus larger respons e voltages from the device. Additionally, the mobility of the 2DEG is on the order of 6 104 cm2V-1s-1 at cryogenic temperatures, which is large compared to most semiconducting materials [65] with the exception of those containing Indium. Since the GaAs a nd GaAlAs have different band gaps, the carriers become trapped in an approximately triangular potential barrier at th e interface between the materials. Thus, the conduction band forms a tw o dimensional sheet at th e interface of the GaAs and the GaAlAs [66, 67]. Above the 2DEG sheet is the undoped layer of GaAlAs, followed by a layer of GaAlAs that is doped with Silicon. Th e Si atoms act as electron donors for the 2DEG. Finally, there is a layer of GaAs above the Si dop ed GaAlAs layer that prevents oxidation of the Al. We begin the fabrication process by carefully cutting a square pi ece (typical size of 8 mm on a side) from the GaAs/ GaAlAs wafer with a diamond scribe. This piece must then be cleaned in an ultrasound bath. We place it in a plastic beaker f illed with acetone and clean for 5 min. This is then repeated for isopropanol and ethanol for 5 min each. Next, we glue the square wafer to a clean microscope slide with a sma ll drop of Shipley S1813 phot oresist [68] for easier handling during the remainder of the fabrication process. The slide and wafer are baked for 15 minutes at 120 C directly on a hot plate. Afte r cooling, the microscope slide is placed on a spinner that is designed to coat thin organic films on various subs trates and is held securely in place by a vacuum line underneath it. The entire square wafer is coated with the S1813 photoresist and then spun at 500 rpm for 2 seconds, followed by 5000 rpm for 30 seconds. This creates an approximately 1.2 m layer of the photoresist on the wafer. The wafer is then baked again for 2 minutes at 120 C to harden the photoresist.

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59 The next steps involve the mask that is im printed with the design for the Hall cross and ohmic contacts. The layout for our Hall magnetometers consists of three crosses with an active area of 50 m 50 m, spaced approximately 1 mm ap art. There are eight rectangular areas (~1 mm 2 mm) for ohmic contact pa ds. Two pads function as the current (voltage) leads for each Hall cross, and the other six are pairs for each of the crosses to measure the voltage (current). This configuration allows not only for measurem ents of up to three different samples in one setup, but also for use of any one cross individually should the others become damaged. The mask pattern is carefully aligned and positioned with respect to the wafer using a microscope and mask aligner. Then the wafer is exposed to UV radiation (~ 260 Watts) for 9 seconds and then placed in a solution of CD 26 developer [68] for ~ 45 seconds followed by de-ionized (DI) water. The UV exposure sets the photor esist with the exception of wh at was covered by the mask pattern. The developer then removes the portion that was not exposed, while the DI water stops the developing of the photoresist. What emerges is the desired Hall cros s pattern on the wafer, and now the wafer is ready to etch. The etch ing compound is a mixture of DI water, hydrogen peroxide, and sulfuric acid in a 160:8:1 ratio. We place the wafer in this mixture for 30 seconds and immediately remove and place in pure DI wate r to stop the etching process. Now the wafer has been etched down to the 2DEG region of the structure (~110 – 120 nm), but only in the regions not covered by the photores ist. We finish the first ro und of fabrication by removing the remainder of the photoresist by placing the micr oscope slide in acetone, isopropanol, and ethanol for ~ 1 minute each. The second round of the fabrication involve s a double layer photoresist, or undercut, method. While the single layer pr ocess (described in th e first round of fabrication) is sufficient for the final sensor, this proce dure allows for a better removal of the metals used for the ohmic

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60 contact pads from the rest of the device after the chemical vapor deposition process. We begin by placing the slide in the spinner and coating the wafer with LOR-3A photoresist [68]. This is spun at 500 rpm for 2 seconds followed by 3000 rpm fo r 30 seconds. We then bake at 175 C for 5 minutes to harden the photoresis t. After cooling we place the slide back in the spinner for the second layer, which consists of S1813 photoresis t, spun at 500 rpm for 2 seconds and 5000 rpm for 30 seconds. This is baked for 5 minutes at 120 C. Now the wafer is ready for exposure again. The same mask and aligner are used to position the Hall cross pattern on the wafer, which is then exposed to UV radiation for 9 seconds, just like before. The slide is then placed in a solution of CD 26 developer for ~ 45 seconds followe d by DI water. This removes the first layer of photoresist from the mask pattern area. We next bake the wafer at 130 C for 5 minutes, followed by development in the CD 26 solution for ~ 60 seconds and then DI water. This added step removes the second layer of photoresist, and since th e LOR-3A has a higher dissolution rate in the developer than the S1813, the resulting phot oresist profile will have an undercut. Now the sample is ready to deposit the metal for the ohmic contacts. This procedure is carried out inside an ultra high vacuum electr on beam evaporation chamber, where we deposit a 5 nm thickness of chromium, followed by a 100 nm thickness of a Au/Ge alloy (88%/12%), and finally a 50 nm thickness of Au on top. To re move the metal layers residing away from the ohmic contact pads we immerse the wafer in PG remover [68], which is a solvent stripper designed for complete removal of resist film s on GaAs and many other substrate surfaces. The process of removing the photoresist can be time consuming, and one should avoid an ultrasound bath as this can remove the meta l from the contact pads. Instead we employ a careful method of stirring the solution to create friction between it and the photoresist, until all

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61 unwanted material is removed. The wafer is then carefully cleaned in se parate baths of acetone, isopropanol, and ethanol without ultrasound. The final part of the fabrication process invol ves annealing the ohmic contacts, so that the Ga atoms diffuse out of the wa fer and Ge atoms diffuse into the conducting 2DEG layer, thus creating an electrical conducting path. The annea ling is done inside a small chamber that has a continuous flow of helium gas which prevents oxidation of the metals during the annealing process, and the wafer is placed directly on a high ly resistive wire mesh grid. The temperature of the wafer is monitored by a thermometer that is directly and carefully att ached to it. The three step process consists of heating the wafer to 110 C for 60 seconds, followed by 250 C for 10 seconds, and finally 410 C for 30 seconds. On ce the wafer is cooled, the external wire connections can be made. For stability and practi cality, the entire device was glued onto a 28 pin connector with an area (15 mm 15 mm) approximately the size of the wafer using an insulating commercial epoxy [69] (Stycast 1266). Wire c ontacts were made between the ohmic contact pads on the device and the leads on the pin connector with gold wire (0.05 mm diameter) and a conducting (room temperature resistance of < 4 10-4 Ohms) commercial epoxy [70] (EPO-TEK H20E). When cooled in zero field, SMMs will have a net magnetic moment of zero. But an external field will bias the system such that it can develop a significant magnetic moment at low temperatures. The magnetic moment has an in trinsic dipole field a ssociated with it that constitutes the magnetic field to induce a Hall voltage. For our experiments conducted with this device, we are able to position our sample such that its magnetization vector is parallel to the plane of the sensor and the perpendicular compon ent of the dipole field from the sample pierces the active area of the device. A dditionally, we apply an external magnetic field parallel to the

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62 direction of the sample’s magne tization vector in order to bi as the sample. Under ideal circumstances, the external field does not in fluence the Hall voltage signal and the only contribution is the perpendicular component of the dipole field fr om the sample. However, in practice there is always a small background signal coming from the external field due to small misalignments. Fortunately, this background can easily be subtract ed off to isolate the response from the sample. The background from the external field is linear and superimposes itself upon the true response from the sample. We fit a straight line to the data, and subtract this to get the sample response. It is not possible to measure absolute values of the magnetization with this method, but relative values pr ovide sufficient information. By dividing the data set (normalizing) by the saturated magnetization value (that at 3 T), we obt ain a measure of the magnetization of the sample relative to the maximum magnetization value. In this way we are able to measure the magnetization of a sample unde r the influence of an external magnetic field as well as microwave radiation. Experiments us ing the techniques outlined in sections 2.4.1 and 2.4.2 are discussed in detail in Ch. 6. 2.6 Summary In this chapter we discussed the experiment al techniques and equipment we use for our research. First, we briefly discussed EPR in th e context of SMM systems. Next, we explained the two types of cavities we use for a cavity perturbation technique and the instrument that acts as our microwave source and detector. This sour ce, in conjunction with an external Gunn diode and complementary components (amplifier, swit ch, frequency tripler) allow for high power pulsed microwaves. Additionally, our quasi optical bridge setup provides an incredibly low loss propagation system that relies on coupling of a free space Gaussian profile to an HE11 mode in corrugated horns and a tube. The corrugated samp le probe is used to conduct experiments that integrate other devices into the setup and co mplement our normal EPR studies by combining

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63 microwaves with either surface acoustic wa ves or Hall magnetometry to conduct unique experiments on single crystals. Finally, we outlined the process of fabricating the Hall magnetometer used in our experiment in section 6.3.

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64 Figure 2-1. Plot of the energy levels in the Ni4 SMM with its easy ax is aligned along the external field. For a fixed frequency of 172 GHz, there is a strong absorption of the signal each time the resonance condition (e nergy difference between adjacent levels matches 172 GHz) is met as the magnetic field is swept.

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65 Figure 2-2. A normal EPR spectrum from the sy stem in Fig. 2-1. As the magnetic field is swept, sharp inverted peaks appear in the transmission spectrum. Each peak corresponds to a transi tion between spin states within the ground state spin manifold, and the decrease in transmission signal is due to the absorption of microwave radiation by the spins within the sample. The peak positions match those shown in Fig. 2-1. With respect to the larger intensity transitions, we see 2 S peaks, corresponding to 2 S transitions between 2 S +1 states. The additional, smaller intensity peaks in the range of 4-7 T, which are attr ibuted to transitions within higher lying spin multiplets, are also discussed in Ch. 4. The change in the background signal seen in the 4-7 T range we attribute to broadening of the resonance peaks due to closely spaced energy levels within multiple higher lying spin multiplets.

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66 Figure 2-3. A schematic diagram of our t ypical experimental set up, including a waveguide probe and sample cavity inserted in th e bore of a superconducting magnet immersed in a cryostat. Either cavity sits in the center of the magnetic field, which is oriented either vertically or horizontally, depending upon the system in use. Reused with permission from Monty Mola, Stephen Hill, Philippe Goy, and Michel Gross, Review of Scientific Instruments, 71, 186 (2000). Fi g. 1, pg. 188. Copyright 2000, American Institute of Physics. Reus ed with permission from Su sumu Takahashi and Stephen Hill, Review of Scientific Instruments, 76, 023114 (2005). Fig. 2, pg. 023114-4. Copyright 2005, American Institute of Physics.

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67 Figure 2-4. A schematic diagram of the ro tational capabilities of each magnet system. A rotating cavity in the Oxford system allows for one axis rotation, while the same cavity in the Quantum Design system allows for two axis rotation since the field is aligned in the xy plane.

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68 Figure 2-5. A diagram of a free sp ace Gaussian beam propagating along the z axis. The beam has a minimum waist, w0, that corresponds to the point of maximum intensity. The divergence of the beam is inversely proportional to the beam waist, and faster diverging beams necessitate larger surface mi rrors in order to re focus the beam. For distances far from the point of minimum be am width, the beam width approaches a straight line. The total angular spread of the beam is 2

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69 GUNNH M 4 5 D E G R E EP O L A R I Z E R180 DEGREE POLARIZER MIRROR MIRROR PROBE Figure 2-6. A schematic diagram of our quasi optics equipment. The microwaves are propagated as a Gaussian profile in free space and as a HE11 mode in a corrugated probe. First, the signal en counters a 45 polarizer, which splits the signal and the reflected beam is polarized at 45 with resp ect to the horizontal. This meets the first elliptical focusing mirror, which reflects the diverging beam such that the minimum beam waist is 25 cm from this point. The 180 wire grid polarizer changes the polarization of the signal so that it is now vertically pol arized. A second mirror is located 50 cm from the first mirror (25 cm from the minimum beam waist). Again the purpose of the mirror is to focus the beam profile so that it will have its’ minimum beam waist as it enters th e corrugated waveguide probe, 25 cm below the mirror. Before entering the detector, the signal pa sses thorough the first wi re grid polarizer, where the polarization is changed for a final time so that it will be correctly oriented with respect to the detector horn (45 from vertical).

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70 Figure 2-7. Illustration of the signal polarization as it changes from interactions with the respective components of the quasi optical se tup. These changes are necessary since the polarization of the signal reflecting off the first wire grid polarizer must be changed after interacting with the sample and before entering the detector. If the second wire grid polarizer was not there, the sign al returning to th e first wire grid polarizer would be totally reflected before it enters the detector. The insertion of the second wire grid polarizer changes the polariz ation of the signal so that it will pass through the first wire grid polari zer and reach the detector.

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71 Figure 2-8. A diagram of the TE11 and HE11 modes in a circular, corrugated waveguide. A non corrugated circular waveguide couples the Gaussian profile to both the TE11 and TM11 modes which progress at different phase ra tes and can result in a distorted output beam. An ideal HE11 mode has identical E plane and H plane radiation patterns with a main lobe that is approximately Gaussi an, which is the reason for the excellent coupling between this mode and th e free space Gaussian profile.

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72 Figure 2-9. The piezoelectric device used in our experiments. A coax cable was run down the length of the tube to a pocket in the c opper block holding the de vice. A small piece of the outer conductor and di electric were removed to expose the inner conductor. From here a small breadboard piece, designed as a transition piece from the coax cable to the device, was soldered onto the inner and outer conductor of the coax. The final electrical connection was from the transition piece to the IDTs on the device with two 40 gauge copper wires. A single crystal was placed directly on the IDTs, with a small amount of commercial silicon vacuum grease used to attach the sample to the IDTs and ensure c oupling of the surface acoustic wave (SAW) to the sample. To inner conductor of coax cable Conducting stripe LiNbO3 substrate Sample IDT To outer conductor of coax cable Conducting stripe

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73 Figure 2-10. Schematic diagram of the electroni c equipment used in our avalanche experiments. The MHz IF signal was sent to a Stanford Research Systems SRS 844 high frequency lock in amplifier. After being processed by the SRS 844, the signal was sent to the APS 240. For optimized performance, we would use the 81104A to trigger the Marconi and APS 240 simultaneously with a 5 V TTL signal. The trigger would in turn cause the IDTs to cr eate a SAW pulse which would couple to the sample and push the system out of equilibrium. The dyna mics were measured with the incident microwave radiation and were recorded with the APS 240. For time resolved measurements, especially when performing repeated sequences where the data are averaged to a final result, it is critical that a ll instruments are phase locked to each other. To this end we used a 10 MHz time-base from the MVNA to connect to an Agilent 33220A function generator, which wa s in turn connected to the 81140A and SRS 844. Since the APS 240 and Marconi we re both linked through the 81140A, this ensured a proper phase lock of a ll the instruments to the MVNA.

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74 Figure 2-11. The Hall device used for our magnet ometry measurement. The sample sits on the surface of the device and the perpendicular component of the dipole field (piercing the Hall cross) constitutes the field necessary to induce a Hall voltage. Voltage Lead (V ) Current Lead ( I ) Voltage Lead (V+) Sample HallCross Current Lead ( I+)

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75 CHAPTER 3 THEORETICAL BASIS OF THE SPIN HAMILTONIAN In the previous chapter we have desc ribed the experiment al techniques and instrumentation necessary for our HFEPR meas urements of SMMs. Now we will focus our attention on the mathematical basis for modeling our systems of interest and briefly discuss the construction of a proper mathematical model. We will outline the two most widely used approaches to describing SMM systems and the approximations and limitations of each approach. 3.1 Two Versions of the Spin Hamiltonian In order to properly describe a system of interest, it is necessary to construct the relevant quantum mechanical Hamiltonian that character izes the energy spectrum of the system. Once the Hamiltonian is diagonalized it must produce the proper energy eigenvalues for a given spin multiplet. A simplified approach is used, by using a spin Hamiltonian to model the experimental data for SMMs. This redefines the orbital mome ntum contributions needed to characterize the system, and relies only on spin contributions a nd symmetry properties of the system. The spin Hamiltonian considers only the spin angular momentum operators and their inte ractions with each other and an external magnetic field. Th e spin orbit interaction is parameterized in constants of the model, which are essent ial to the magnetic anisotropy in SMMs. Any spin Hamiltonian must sati sfy certain basic requi rements for it to be a valid model. First, it must properly express th e point symmetry of the molecule and the lattice, and must be invariant under all point symmetry operations [ 10, 11] and maintain the space group symmetry. Second, it must be Hermitian since the eigenvalues (energie s) of the system ar e real. Finally, for cases of practical interest, onl y even powers of the spin angu lar momentum operators, starting with the quadratic term, are included in the zero field terms of the Hamiltonian. Thus, the spin

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76 operators are invariant under time reversal. Of course the Zeeman term includes an angular momentum operator of first order, but this reflects the fact that an external magnetic field breaks time reversal symmetry [7] since the current dens ity responsible for the external magnetic field reverses under the time tr ansformation operator. 3.1.1 The Giant Spin Hamiltonian The giant spin Hamiltonian contains the St evens operators obtained by using the operator equivalent method of finite gr oups [71]. This considers the crystal field potential (from molecules within a crystal) of appropriate sy mmetry and expands the ope rators like spherical harmonics functions. The spherical harmonic fu nction is expressed in terms of the quantum mechanical spin angular mo mentum operator for a given S multiplet [11, 12]. 0,2,4...0ˆˆˆ ˆ (,)k mm CFkkz kmVBOSS (3-1) In Eq. 3-1, ˆm kO is the spin operator of power k and symmetry m ( m = 0 is axial, m = 2 is rhombic, m = 4 is tetragonal, etc.), and m kB is the coefficient for the respective spin operator. This is the starting point for the derivation of the spin Ham iltonian commonly used by both physicists and chemists in the SMM community. As previously stated in Ch. 1, SMMs are polynuclear complexes consisting of transition metal ions as the magnetic components. For most transition metal ions the spin orbit coupling intera ction is relatively small and quenching of the orbital contribution to first order in perturbation theory allows one to express the states of the systems as pure spin multiplets. Thus, we assume that S is a good quantum number. Within a given S multiplet the anisotropy terms lift the degener acy of the spin projection states, which are expressed as s m. All interactions that ta ke place in the absence of an external magnetic field are referred to as zero field splittings. The co mbined effects of crystal field symmetry and the

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77 mixing of excited levels into the ground stat e through second order (o r higher) perturbation theory lifts the degeneracy of the spin multiplets and is the source of the zero field splitting [72]. The spin Hamiltonian can be expressed as ˆBSDSBgS (3-2) The first term contains all of the zero field interactions within the D tensor. The second term (the Zeeman interaction) lif ts the degeneracy of each s m state with an external field, B The Land factor, g which is a scalar for a free electron, b ecomes a tensor in the presence of spinorbit interactions. In Eq. 3-2 we have neglected the hyperf ine interaction term, which can be excluded in strongly exchange c oupled systems like SMMs due to the delocalization of electrons making this a rather small effect [3]. Eq. 3-2 can be simplified if we assume the principal axes of the D tensor coincide with the molecular coordinate axes. Then only diagonal terms contribute to the product and '2'2'2ˆˆˆxxxyyyzzzSDSDSDSDS (3-3) Now we introduce a new set of parameters '''1 (2) 2xxyyzzDDDD (3-4) ''1 () 2xxyyEDD (3-5) '''1 () 3xxyyzz K DDD (3-6) So Eq. 3-3 is equivalently expressed as 2222ˆˆˆˆ ()zxySDSDSESSKS (3-7)

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78 If we express Eq. 3-7 in terms of Eq. 3-1 we get 000022 002222ˆˆˆSDSBOBOBO (3-8) by using the explicit form of the Stevens operators [71]. Now we can relate the parameters in Eq. 3-7 to the new parameters in Eq. 3-8. 0 23DB (3-9a) 2 2 E B (3-9b) 0 0 K B (3-9c) 20 2ˆˆzSO (3-9d) 222 2ˆˆˆxySSO (3-9e) 20 0ˆˆ SO (3-9f) Since in Eq. 3-7 2ˆ K Sis a constant that uniformly shifts the energy levels, it can be omitted from the Hamiltonian because in EPR experiments we measure differences between energy levels. Finally we arrive at the spin Hamiltonian e xpressed in its most common form in the SMM community 222ˆˆˆ ˆ ()zxyBDSESSBgS (3-10) Symmetry plays a critical role in the fo rmulation of the correct Hamiltonian, and determines which terms may be allowed. For a system with perfectly spherical symmetry, no zero field anisotropy terms will be allowed since the system is completely isotropic in any direction. If we take a point at the center of the sphere as the origin of our coordinate system, then any radial distance from the center, at any angle, is energetically equivalent. Hence, all

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79 anisotropic terms describing the system are zero. For a slightly lower symmetry system, such as one possessing cubic symmetry, no second order anis otropy terms are allowed. This can be seen by taking the origin as the center of the cube. All side lengths and angles of a cube are identical and a rotation of 90 leaves the system unchanged. However, fourth order anisotropy terms are allowed. Since a rotation of 45 in any one of the three orthogonal planes ( xy xz yz ) will change the magnitude of the positio n vector with respect to the center, this permits a fourth order anisotropy term. In general, angles of / n will be those that correspond to the periodicity of the anisotropy term, where n is the order of the term. If we di stort the cube slightly such that it compresses or elongates along one ax is, then a second order axial anisotropy term develops. Now the geometry has changed to a square based rectangle, and along the ax is of distortion there is a two fold symmetry pattern since a rotati on of 90 changes the magnitude of the position vector with respect to the center once again. Finally, we can dist ort the previous geometry in the plane perpendicular to th e first distortion axis, and now second order transverse anisotropies are allowed in addition to the fourth order terms. As discussed above, depending on the symmetry of the system, it may be necessary to include higher order terms in Eq. 3-10. These terms take the form 4,6,8...0ˆˆˆ (,)k mm kkz km B OSS For the systems we have studied, we only include terms up to the order of 44 44ˆˆˆ (,)z B OSS. While higher order terms may be presen t, certainly the second and four th order terms are dominant and it becomes hard to resolve the effect of these higher order terms. When allowed by symmetry the transverse anisotropy term 4 4ˆˆˆ (,)zOSS is written 44ˆˆ () 2 SS. Other situations can arise where the E term is forbidden by symmetry and the lowest term allowed is a fourth order term. This is dependent upon the symmetry of the system in question. The most important requirement for

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80 SMM behavior, however is a significant D value whose sign is negati ve. Thus, some kind of axial elongation of the system symmetry is necessary to have a SMM. A major assumption imbedded in this giant spin mode l is that the system obeys the strong exchange limit. In this limit, the isotropic Heise nberg exchange coupling th at splits the different S multiplets is much larger than a ny anisotropies that split the leve ls within a given multiplet. In other words, the contributions from the non Heise nberg interactions to the energy spectrum are much less than those from the Heisenberg exchange interaction. In fact, this model applied to SMMs assumes that the ground state spin multiple t is perfectly isolated from any higher lying spin multiplets. In this case, the coupling strength between the metal ions in the cluster is infinite and the magnetic moment vector for th e ground state is perfectly rigid. This is commonly referred to as the giant spin approximation (G SA). The advantage of the GSA is the relative simplicity. The matrices that constitute the Hamiltonian are formed in a basis where S the spin value of the ground state multiplet, is a good quantum number. A ll the eigenstates of the system are expressed as linear combinations of the basis states (spin pr ojection states along the quantization axis) s m. Of course in reality the exchange coupli ng always has some finite value and the energy levels of the ground state have the possibi lity to be influenced by higher lying S multiplets. In situations where the exchange and anisotropy in teractions are comparable the GSA starts to reach its limitations. In Ch. 4 we discuss a system that can be modeled quite well by the GSA, but it requires higher order terms that are unphysica l in the sense that they exceed the highest order allowed (second) by the single ions cons tituting the molecule. Therefore, the GSA can often model a molecule of interest quite well, although the major limitatio n is the fact that it

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81 provides no information as to how the single ions that constitute a molecule influence the magnetic properties of the cluster as a whole. 3.1.2 Coupled Single Ion Hamiltonian The coupled single ion Hamiltoni an includes the exchange interactions between metal ions and is necessary when consideri ng single ion contributi ons to the cluster anisotropy. There are numerous interactions that take place between me tal ions within a molecule, such as spin orbit (both with themselves and betw een others) and dipolar couplings Due to the organic ligands surrounding the magnetic core, dipolar interactions in many SMMs are negligible in comparison to the isotropic and anisotropic energy scales, an d can therefore usually be omitted. However, exchange interactions within a molecule are dominant since SMMs are polynuclear molecules, and the nature of the exchange interaction between ions within the molecule cluster is important. There are a number of different exchange in teractions, but the dominant one in SMMs is superexchange, which is the coupling of lo calized magnetic moment s through diamagnetic groups in an insulating material [10]. The gene ral form of the superexchange Hamiltonian is expressed in Eq. 3-11. ˆˆˆanisotropic exchangeHHH (3-11) First, we consider the anisot ropy contained within a single me tal ion due to its own spin orbit coupling. This is, of cour se, the same interaction that give s rise to the zero field anisotropy terms in the GSA Hamiltonian, only now we consider the single ion anisot ropies as opposed to the molecular anisotropies (which are the tensor projections of the single ion anisotropies). The lowest order terms allowed for a spin i are: 222ˆˆˆ ˆ ()iiianisotropicizixy i H dSeSS (3-12)

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82 Next, we consider the exchange interactions between two ions in a molecule. ˆˆˆiijj ijiexchangeSTSH (3-13) In Eq. 3-13, Tij is the general spin-spin interaction tens or. This can be separated into three different interactions, which we express in Eq. 3-14. ˆˆˆˆisotropicsymmetricantisymmetric exchangeHHHH (3-14) The first term is the dominant isotropic Heisen berg interaction between spin vectors, which is given by Eq. 1-1. Here Jij is the magnitude of the exch ange interaction (positive for antiferromagnetic coupling and negative fo r ferromagnetic coupling) between spin i and spin j The final two contributions to the superexc hange are the symmetric and antisymmetric interactions. The symmetric part of the exchange is given by ˆˆ ˆij iji s ymmetricijA H SS (3-15) In Eq. 3-15, ij A represents the symmetric anisotropy tensor between two respective spins, i and j The symmetric exchange is also know n as a pseudodipolar effect [73]. The antisymmetric exchange, or Dzyaloshinskii –Moriya (DM) interacti on, is expressed as a vector product between two spins, i and j ()ˆˆ ˆijianitsymmetricijij H GSS (3-16) In Eq. 3-16,ijG is a constant vector c onnecting the two spins of interest whose orientation depends on the symmetry of the molecule. The physical origin of this vector is the same interactions that give rise to the other anisotropic constants such as D E and g Namely, this is

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83 the mixing of excited states into the ground state due to spin orbit coupling [74, 75]. Moriya [73] was the first to note that this expression is the antisymmetric part of th e most general expression for a bi-lateral spin-spin interaction. For the systems discussed in Ch. 4 and Ch. 6, the symmetric and antisymmetric interactions are negligible with respect to the more dominant interactions and are omitted from the Hamiltonian. This is true with respect to ions with orbitally nonde generate ground states (pure spin multiplets) [76]. However, for the syst em in Ch. 5, it is clear that the ions have orbitally degenerate ground states, and theref ore will have a contribution to the magnetic properties from symmetric and antisymmetric exchange interactions. In the case of the system in Ch. 5, we are dealing with four effective S' = Co+2 ions that when coupled together produce an energy spectr um showing significant ze ro field splitting. Since a system with S' = has no axial or transverse anisotropy, we turn to the symmetric and antisymmetric exchange interact ions as the dominant source of the observed zero field splitting anisotropy for this system. Based on the symmetry of the system in question, some components of the antisymmetric exchange tensor can vanish, which simplifies calculations. A careful discussion of this interaction will be done in Ch. 5. Moriya’s contribution was th e extension of the Anderson theo ry [77] of superexchange to include spin orbit coupling. He calculate d that the magnitude of the symmetric ( Aij) and antisymmetric ( Gij) parameters can be related to the isotropic coupling constant ( J ) by the following relations [73] ijg GJ g (3-17a)

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84 2ijg A J g (3-17b) In Eq. 3-17, g is the shift in g factor from the free electron value due to the spin orbit interaction. Moriya argued that since the symmetric part of the exchange is second order in the spin orbit coupling scheme, it could be neglecte d. The dominant contri bution to the exchange interaction would then be the antisymmetric part, since it is first order in the spin orbit coupling scheme. However, this assumes that the g anisotropy of the system is marginal. As we discuss in section 5.2, we have Co+2 ions with a large amount of g anisotropy. For the z component of the g tensor, we get a value of 7.8. Inserting th is into Eq. 3-17, we see that the symmetric exchange interaction is ~74% of the antisymme tric exchange interac tion, and should not be neglected. The coupled single ion approach is favorable to the giant spin approximation since it provides more information about the exchange inte ractions between ions within a molecule and how the single ion anisotropies couple together and project onto the molecular anisotropy. However, from a practical standpoint, the couple d single ion Hamiltonian is not always readily solvable. The Hilbert space for a system of inter acting ions, and thus the size of the Hamiltonian matrix, goes like the product of (2 Si + 1)N where N is the number of partic les with a given spin, Si. For systems like Mn12Ac with twelve ions in mixed va lence states, this problem becomes incredibly complicated due to th e size of the Hilbert space (108 108). This is the limitation of this model. However, for certain systems, this approach is quite reasonable to use. We present work in Ch. 4 that models a system consisting of four s = 1 particles using a coupled single ion Hamiltonian.

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85 3.2 Summary In this chapter we discussed the theoretical origin of the Hamiltonians used to model SMM systems. The Hamiltonian corresponding to a gi ant spin approximation is certainly the most widely used form and applies to ideal systems wh ere the isotropic exchange interaction is much larger than the anisotropic interactions. For sy stems where this assumpti on starts to break down, we can employ a coupled single i on Hamiltonian that in spite of being more complicated, can provide information on the nature of the exchange interactions between metal ions in a molecular cluster. In the following chapters we will present work on single molecule magnets where the GSA is valid and describes the system quite well (C h. 6), as well as systems where the GSA breaks down (Ch. 5) or must be modified to account for the observed behavior (Ch. 4).

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86 CHAPTER 4 CHARACTERIZATION OF DISORDER AND EXCHANGE INTERACTIONS IN [NI(HMP)(DMB)CL]4 The results presented in this chap ter can be found in the articles, Single Molecule Magnets: High Field Electron Paramagnetic Resonan ce Evaluation of the Si ngle Ion Zero-Field Interaction in a Zn3 IINiII Complex E.-C. Yang, C. Kirman, J. Lawrence, L. Zakharov, A. Rheingold, S. Hill, and D.N. Hendrickson, Inorg. Chem. 44, 3827-3836 (2005), Origin of the Fast Magnetization Tunneling in the Single Molecule M agnet[Ni(hmp)(t-BuEtOH)Cl]4, C. Kirman, J. Lawrence, S. Hill, E.-C. Yang, and D.N. Hendrickson, J. Appl. Phys. 97, 10M501 (2005), Magnetization Tunneling in Hi gh Symmetry Single Molecule Magnets: Limits of the Giant Spin Approximation A. Wilson, J. Lawrence, E.-C. Ya ng, M. Nakano, D. N. Hendrickson, and S. Hill, Phys. Rev. B 74, 140403 (2006), and Disorder and Intermolecular Interactions in a Family of Tetranuclear Ni(II) Complexes Probed by High Fr equency Electron Paramagnetic Resonance J. Lawrence, E.-C. Yang, R. Edwards, M. Olmstead, C. Ramsey, N. Dalal, P. Gantzel, S. Hill, and D. N. Hendrickson, Inorganic Chemistry, submitted. 4.1 The Tetranuclear Single Molecule Magnet [Ni(hmp)(dmb)Cl]4 A series of SMM compounds have recently been synthesized, each possessing as its magnetic core four Ni+2 ions on opposing corners of a sligh tly distorted cube [25]. Extensive studies have been conducted to characterize the ba sic spin Hamiltonian parameters, as well as the disorder and intermolecular exch ange interactions present in each system [25, 78], and the system that gives the nicest results from an EPR perspective is the compound [Ni(hmp)(dmb)Cl]4, where dmb is 3, 3-dimethyl-1-butanol and hmpis the monoanion of 2-hydroxy-methylpyridine. A diagram of the molecule is shown in Fig. 4-1. The bulky aliphatic groups that surround the magnetic core help to mi nimize intermolecular interactions, as will be discussed.

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87 Each Ni+2 ion has a spin of s = 1 and the four Ni+2 ions couple ferromagnetically [25] to give a spin ground state multiplet of S = 4. The aspects that make this compound quite attractive are the sharp EPR peaks and high symmetry ( S4) of the molecule. However, in spite of the high symmetry, this compound displa ys extremely fast ground state tunneling of the magnetization [25, 79]. This observation was a major motiva tion for the work done to characterize the single ion anisotropies [80], the magnitude of the isot ropic Heisenberg exchange interaction, and how relating these two can account for the observed fast quant um tunneling of the magnetization (QTM) [81]. This complex consists of a diamond-like lattice of Ni4 molecules. The ligands associated with a Ni4 molecule are involved in ClCl contacts w ith four neighboring Ni4 molecules. The shortest ClCl contact distance between neighbori ng molecules is 6.036 which is the largest such distance in a family of similar Ni4 complexes exhibiting measurable intermolecular exchange [25, 82]. Since this distance is longer than the 3.6 obtained for the sum of the Van der Waals radii of two chloride ions, the intermolecular magnetic exchange interactions propagated by this pathway should be negligible [25]. Although th is may be true for elevated temperatures where these interactions can be unr esolved due to thermally averaging, later we will discuss experimental evidence for intermolecu lar exchange interactions at low temperatures (< 6 K) in this system. The ab ility to resolve these interactions is largely due to the resonance peaks of high quality in terms of line shape and width. In a system with larger amounts of disorder, intermolecular exchange will broaden the peaks, but this effect would likely be buried within the broadening due to the di sorder. In this system the amount of disorder is minimal, and thus the effect of intermolecular exchange on the line widths and peak splittings is able to be resolved.

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88 Another noticeable feature of this system is the absence of any solvate molecules in the lattice, which results in a reduced distributi on of microenvironments. Hence, interactions between the magnetic core and latt ice solvate molecules are not an issue in this compound. The lack of lattice solvate molecules makes an ideal candidate to study with HFEPR. If all molecules posses the same microenvironment they will all undergo transitions at the same magnetic field, with the absorpti on peaks having sharp, Lorentzian lineshapes. Conversely, for a system with large amounts of disorder or stra ins, there will be multiple microenvironments associated with the molecules within the crystal. This will cause the EPR peaks to broaden and deviate from a true Lorentzian shape. Dipolar and hyperfine fiel ds can also contribute to such effects. Since there is a distribution of microenvironmen ts, different molecules undergo transitions at slightly different field values wh ich can lead to asymmetric line shapes, increased peak widths, and peak splittings. Peaks with su ch characteristics make it difficult to determine which interactions are dominating the broadening. Both the sizeable ClCl contact distances helping to minimize intermolecular interactions, a nd the absence of any solvate molecules in the lattice, contribute to the EPR spect ra exhibiting flat base lines a nd sharp, narrow peaks. Hence, we will show that we are able to separate the various contributions (disorder and intermolecular exchange) to the EPR line shapes. 4.2 HFEPR Measurements of [Ni(hmp)(dmb)Cl]4 Magnetic field sweeps were performed at a number of different temperatures and frequencies. As explained in se ction 2.2, when the magnetic field is swept, sharp inverted peaks appear in the transmission spectru m. Each peak corresponds to a transition between spin states, and the decrease in cavity transmission signal is due to the absorption of microwave radiation by the spins within the sample. A normal EPR spectrum from a SMM with its easy axis ( c ) aligned along the external field consists of a series of absorption p eaks corresponding to transitions

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89 between states within the ground st ate spin multiplet. At sufficien t temperatures where all states within the multiplet are populated, one should see 2 S peaks, corresponding to 2 S transitions between 2 S +1 states. 4.2.1 Characterization of Easy Axis Data [Ni(hmp)(dmb)Cl]4, crystallizes in the shape of an a pproximately square based pyramid. A single crystal (~1 mm ~0.8 mm ~1.1 mm) was mounted on the endplate of a vertical cavity. We use a minimal amount of silicon vacu um grease to attach th e crystal to the copper endplate. 172.2 GHz HFEPR spectra are displayed in Fig. 4-2 for various temperatures in the range of 10 K to 59 K. One thing to notice about this figure is th at there are two sets of peaks: the main intensity peaks and the lower intensity peaks. From the temperature dependence, it is clear that the approximately evenly spaced domin ant peaks correspond to transitions within the S = 4 ground state. Indeed, as labeled in Fig. 4-2, all eight transitions within the S = 4 state are observed. The other, weaker transitions (marke d by vertical dashed line s) are within excited state multiplets ( S < 4) and we discuss these in section 4.4. One interesting feature of the spectra is the splitting of several of the peaks, particularly for transitions involving stat es with larger absolute ms values (at low and high fields). This ms dependence of the splitting implies at least two distinct Ni4 species, with slightly different D values ( D strain). Other spectra taken at 30 K and frequencies from 127 GHz to 201 GHz confirm this conclusion and were used to determine the axial spin Hamiltonian parameters for each species within the crystal. Fig. 4-3 plots the peak positions in magnetic field for different frequencies. By considering a giant spin appr oximation (discussed in Ch.3), the fourth order spin Hamiltonian is given by Eq. 4-1. 200 44ˆˆˆ ˆzBHDSBOBgS (4-1)

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90 All peak positions in Fig. 4-3 have be en fit to Eq. 4-1 assuming the same g and 0 4B values, and two s lightly different D values (red and blue lines). The obtained parameters are: gz = 2.3, D = 0.60 cm-1, and 0 4B= 0.00012 cm-1 for the higher frequency peaks, and D = 0.58 cm-1 for the lower frequency peaks. The D values from our HFEPR measurements are quite close to those obtained from fitting the reduced magnetization data [25], which gives a value of D = 0.61 cm-1. At this point we stress th at it is not possible to satisfactorily fit the data without the inclusion of the fourth order term, 0 4B. From Fig. 4-3 we see that the resonance branches are not evenly spaced with respect to on e another. If they were, then a D term alone would be sufficient. The energy splittings would go as D (2| ms| –1), where ms is the spin projection state from which the transition originates. For this system with S = 4, a D term would produce zero field offsets of 7 D 5 D 3 D and D respectively for the states with ms = 4, ms = 3, ms = 2, and ms = 1. Hence the energy difference between branches goes like 2 D However, the branches involving transitions between lower lying states ( ms = –4 to –3) are spaced further apart with respect to adjacent branches than those involvi ng transitions between higher lying states ( ms = –1 to 0). This kind of nonlinearity is introduced by a negative, axial, fourth order term. This is illustrated in Fig. 4-4 for an exaggerated value of the fourth order term. 4.2.2 Peak Splittings Arising from Disorder It is now well documented that disorder asso ciated even with weak ly (hydrogen) bonding solvate molecules can cause signi ficant distributions in the g and D values for SMMs such as Fe8 and Mn12-acetate, leading to so-called g and D strain [83, 84]. Such strains have a pronounced effect on the line widths and shapes. Fortunately HFEPR measurements pr ovide the most direct means for characterizing such distributions, which can ultimately have a profound influence on the low-temperature quantum dynamics of even the highest symmetry SMMs such as Mn12Ac.

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91 The fact that the fits in Fi g. 4-3 agree so well with the ms dependence of the peak splitting provides compelling support for the existence of two distinct Ni4 species. A closer examination of the 172 GHz EPR spectra (Fig. 45a) indicates that the splitting is absent above about 46 K. In fact, measurements performed at closely spaced temperature intervals reveal that the splitting appears rather abruptly below a critical temperature of about 46 K, as shown in the inset to Fig. 4-5a. This suggests a possible structural tr ansition at this temperature, which explains a lowering of the crystallographic symme try and, hence, two distinct Ni4 species. This scenario is supported by thermodynamic studies and low-te mperature X-ray measurements [78] which suggest that the two fine structure peaks may be explained in terms of a weak static disorder associated with the dmb ligand which sets in below 46 K. In order to ascertain the origin of the fine st ructure splitting observed in the spectra below 46 K, detailed heat capacity measurements in the temperature range fr om 2-100 K were carried out. The results are given in Fig. 4-6a, where it can be seen from the pl ot of heat capacity at constant pressure versus temper ature that there is a peak at 46.6 K which corresponds quite well to the temperature at which the peaks in the HF EPR spectrum start to split (46 K). Heat capacity measurements were also performed for a Zn anal og (~ 2.3% Ni) in orde r to determine whether the phase transition observed at 46. 6 K is due to a structural cha nge that causes different microenvironments, or whether it perhaps arises fr om a spin related phenomenon such as magnetic ordering due to intermolecular magnetic exchange interactions. The diamagnetic Zn analog, which has the same structure, also exhibits a heat capacity peak at a similar temperature (49.6 K, red data in Fig. 4-6a), suggesting that it cannot be the result of a magnetic phase transition. The fact that the two structur ally analogous complexes have this peak with similar amplitude at about the same temperature indicates that it is due to a structural phase transition.

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92 In order to further confirm that the 46 K peak splitting observed in the HFEPR spectra stems from different microenvironm ents, single-crystal X-ray diffraction data were collected at a temperature of 12(2) K. The th ermal ellipsoid plot comparison of symmetry independent parts of the molecule at 12 K and 173 K are given in Fig. 4-6b. Interesting thermal ellipsoids in the tbutyl group from the dmb ligand ar e evident in both plots. Th is should be where the orderdisorder activity is taking place. One notable f eature is that the thermal ellipsoids have shrunk at 12 K for all the atoms except for the t-butyl group. This can be seen in Fig. 4-6b, where the tbutyl group is represented by the upp er group bonded to the central Ni+2 ion. The lower group bonded to the central Ni+2 ion is the hmp ligand, whose ther mal ellipsoids have also shrunk by a temperature of 12 K. These abnormal looking ellipsoids of the t-butyl group are indeed much larger than would be expected at this temperature. Moreover, it is shown in Fig. 4-7 that a slight clockwise shift of the structure obtaine d at 12 K mapped onto the 173 K structure. The pink line in Fig. 4-7 is the structure at 173 K, while the green dashed line is the structure at 12 K. As can been seen, the only part of the structure that changes is the part coming from the t-but yl group. This small change in the structure supports the suggestion that different microenvironments in the dmb ligand cause the HFEPR peak splittings at low te mperature. At high temperatures the motion is thermally averaged, but below 46 K the motion freeze s out and the structure takes on two distinct orientations. The EPR peaks are then split due to the effects of the disorder. 4.2.3 Peak Splittings Arising fr om Intermolecular Exchange Upon cooling the sample, additional broadeni ng and splittings of the EPR spectra are observed at temperatures below a bout 6 K, as shown in Fig. 4-5b and Fig. 4-8. At least four (possibly up to six) ground state fine structure peaks are seen at the lowest temperatures (~1 K) between ~1.15 T and 1.5 T in Fig. 4-5b. It is quit e clear that these fine structure splittings cannot

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93 be attributed to (static) structurally different mi croenvironments, as we now outline. First of all, if this was the source of the splitting, then it should be apparent for al l transitions, including the ms = 3 to 2 resonance which is observed down to 2.6 K at around 2.7 T in Fig. 4-5b. However, there is no broadeni ng and only two fine structure peaks are observed for this resonance to temperatures well below 6 K, which is where the additional fine structures begin to emerge in the ground state resonanc e. Studies to higher frequencie s (Fig. 4-8) indi cate that the fine structures in the ground state resonance ( ms = 4 to 3) persist to the same field range (~2.5 T) where the ms = 3 to 2 resonance is seen in Fig. 4-5b. Consequently, one can rule out field-dependent structural changes. In fact, as seen in Fig. 4-8, the temperature below which the additional fine structures begin to appear increases with increa sing magnetic field/frequency (see red arrows in Fig. 4-8 as rough guide). For compar ison, the dashed line in Fig. 4-8 represents the energy separation, 0/ kB, between the ms = 4 ground state and the first excited state ( ms = +4 for B < 0.66 T, and ms = 3 for B > 0.66 T). Thus, it appears as though the onset of the additional fine structures is related to the depopulation of excited states Indeed, similar evidence for diverging linewidths has been reported previously for this same temperature regime [83] ( kBT < 0). However, these earlier studies involved SMMs (Mn12Ac and Fe8) with broad EPR lines compared to the present Ni4 complex, making it difficult to clearly resolve additional EPR fine structures brought on by intermolecular excha nge interactions. For th is reason, studies of the present Ni4 complex provide an excellent opportuni ty to better understa nd the effects of intermolecular exchange on the EPR spectra of SMMs. While most aspects of earlie r EPR line width studies on Mn12Ac and Fe8 have been understood in terms of competing exchange and dipol ar interactions [83, 84], the behavior of the extra splittings of the ground state resonance ( ms = 4 to 3 in the present study) has remained

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94 unexplained for kBT < 0. We speculate that this is relate d to the development of short-range intermolecular magnetic correlations (either ferromagnetic or antiferromagnetic) which are exchange averaged at higher temper atures. These correlations are due to the off diagonal part of the intermolecular exchange interaction ( J/2 [ Si +Sj + Si -Sj +]), which entangles neighboring molecules at low temperatures [85]. Since th is phenomenon involves the interaction between multiple molecules, of which there are two ine quivalent types (the two fine structure peaks observed from ~6 K to 46 K), one anticipates an increase in the number of fine structure peaks corresponding to the development of short-range correlations betw een different combinations of the two molecular species having slightly different D values. In addition, the fact that there clearly exist two different molecu lar species also suggests that th ere may be differences in the interaction strengths between various pairs of molecules. Additional evidence for short range magnetic exchange interactions comes from single-crystal HFEPR studies of a single Ni+2 ion in a [Zn3Ni(hmp)4(dmb)4Cl4] complex doped into the crystal of the isostructural diamagnetic [Zn(hmp)(dmb)Cl]4 complex [80]. The doping level (molar ratio of Zn/Ni: 97.7:2.3) was such that the compound contained small amounts of magnetic molecules which are sufficiently isolated from one another by the nonmagnetic [Zn(hmp)(dmb)Cl]4 molecules. Similar temperature dependence studies on this compound revealed no extr a splittings of the ground state transition peaks at low temperat ures. The same microenvironments due to ligand conformations are present in this compound as the tetranucle ar complex, but no shor t range intermolecular exchange is possible. Without th is possibility, no extra peak split tings appear. However, in the tetranuclear complex where short range intermol ecular exchange is possible, extra ground state fine structure peaks are seen to app ear at the lowest temperatures.

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95 Without a more detailed understanding of the disorder, and of the nature of the intermolecular interactions, it is not possible to give a more precise e xplanation for the low temperature spectrum. Nevertheless, the obs ervation of 3D orderi ng in [Ni(hmp)(dmb)Cl]4 does signify the relevance of intermolecular interactions either due to dipolar interactions or weak superexchange. Regardless of the source of the short range magne tic correlations, their development will significantly impact the EPR spectrum. 4.3 Physical Origin of the Fa st QTM in [Ni(hmp)(dmb)Cl]4 In addition to easy axis measurements, the tr ansverse anisotropy associated with this complex was also characterized [79]. Here we only quote the main results of this study, where the obtained fourth-order 4 4 B term results in a larg e tunnel splitting (4 10-4 cm-1) within the ms = 4 ground state. This provides an explanation for the fast magnetization tunneling in this system and can be attributed to the fact that the 4 4ˆ O operator connects the ms = 4 states in second order of perturbation theo ry. Consequently, it is excepti onally effective at mixing these levels, thereby lifting th e degeneracy between them. Since a second order transverse anisotropy is symmetry forbidden, in particular, it is the fourth order transverse anisotropy 44 44ˆ B O that connects the ms = 4 states in second or der perturbation theory. The presence of fourth order terms needed to f it both the easy axis and hard plane data to a giant spin approximation model suggests additional physics that is taking place that can not be accounted for with this approximation. As we have noted in a previ ous publication [81], the fourth order interactions 44 44ˆ B O and 00 44ˆ B O are completely unphysical wi thin the context of a rigid giant spin approximation appropriate for this co mplex. To understand this, one must recognize that the molecular cl uster anisotropy is ultimate ly related to the single-ion anisotropies associated with the individual Ni+2 ions. Since each ion has spin s = 1, their zero field splitting tensors do

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96 not contain terms exceeding second order in the local spin operators. Within a rigid giant spin (strong exchange) approximation, the molecular zero field splitting tensor corresponds to nothing more than a projection of the singleion zero field splitti ng tensors onto the S = 4 state, i.e. an addition of the (3 3) single-ion tensors after rotating them into the molecular basis [80]. Consequently, such a procedure resu lts also in a (3 3) matrix, and terms of order greater than two in the molecular spin operato rs are not generated by this a pproach. Additi onally, the fact that higher lying spin multiplets become populat ed at relatively low temperatures (12 K) indicates that the isotropic ex change interaction and the anis otropic interactions are of comparable magnitude, which violates the fundamental assumption of the giant spin approximation ( J >> d ). Other experiments [80] on a single Ni+2 ion in a [Zn3Ni(hmp)4(dmb)4Cl4] complex doped into the crystal of the isostructural diamagnetic [Zn(hmp)(dmb)Cl]4 complex, demonstrated that the single-ion zero field splitti ng interactions for the single Ni+2 ion give rise to the second order, axial zero field splitting interaction for the S = 4 ground state of the [Ni(hmp)(dmb)Cl]4 SMM. It has also been determined in this study that the Ni+2 ion possesses a signif icant second order transverse anisotropy ( e ) term and that the easy axes are tilted 15away from the crystallographic c axis (molecular easy axis) for the [Ni(hmp)(dmb)Cl]4 SMM. These factors, when combined with the four-fold symmetry of the molecule, projec t fourth order spin Hamiltonian terms onto the S = 4 ground state. In order to understand the appa rent fourth-order contributions to the EPR spectra, one must consider the full 81 81 H ilbert space associated w ith the four uncoupled Ni+2 ions, and then consider also the Heisenberg exchange coupling between them. This pro cedure is described in detail in a separate publication [81]. Neverthe less, we briefly summari ze the findings of this

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97 analysis here. Solution to the problem of four coupled s = 1 spins results in a spectrum of (2 S + 1)4 = 81 eigenvalues. In the limit in which the exchange coupling constant, J between the individual Ni+2 ions exceeds the single-ion second order axial anisotropy parameter ( d ), the lowest nine levels are reasonabl y well isolated from the other se venty two levels, as shown in Fig. 4-9 for the situation in which the magnetic field is applied parallel to the molecular easyaxis (actual parameters obtained fo r this complex [80] were used in this simulation). It is the magnetic-dipole transitions between th ese nine low-lying energy levels (red lines in Fig. 4-9) that dominate the EPR spectrum for the Ni4 complex. Furthermore, it is these transitions that can be well accounted for in terms of the giant spin Hamiltonian with S = 4, albeit that fourth (and higher) order terms are necessary in order to get the best agreemen t. Henceforth, we shall refer to this low-lying cluster of nine levels as the S = 4 ground state. Roughly 30 cm-1 above the S = 4 ground state in Fig. 49 is another reasonably well isolated cluster of twenty one le vels. Significant insight into the nature of the spectrum of this complex may be obtained by considering the ex change coupling between four isotropic ( d = 0) spin s = 1 entities. Using a single ferromagnetic coupling parameter, J it is straightforward to show using a Kambe equivalent operator method [86] that the spectrum consists of a single S = 4 ground state (degeneracy of nine): then, at +8 J relative to the ground state there are three S = 3 states (total degeneracy of twenty one); followed by six S = 2 states (degeneracy of 30) at +14 J ; followed by six S = 1 states (degeneracy of 18) at +18 J ; with three S = 0 states (degeneracy of three) finishing off the spectrum at +20 J relative to the ground state. Therefore, it becomes apparent that the twenty one levels above the S = 4 ground state in Fig. 4-9 correspond to the three ‘effective’ S = 3 states found by the Kambe method. Th e term ‘effective’ here is meant to imply that the spin quantum number is approximate i.e. it is not an exac t quantum number. The

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98 zero field splitting associated w ith these twenty one levels has several origins which are discussed in detail elsewhere [ 80, 81]: mainly this is caused by the single-ion zero field splitting tensor, which contains an axial d parameter and a significant rhombic term e ; in addition, the local magnetic axes associated with the Ni+2 ions are tilted significantly with respect to the molecular symmetry directions (although they are related by the S4 symmetry operation). Above the S = 3 states, it becomes harder to differentiate the various leve ls. Nevertheless, they may be thought of as belonging to ‘effective’ S = 2, 1 and 0 states, though S becomes less and less exact near the top of the spectrum due to th e strong competition between isotropic ( J ) and anisotropic ( d and e ) interactions. The most important conclusion from the work in Ref. 81 is the fact that the higher-order contributions to the zero field splitting of the S = 4 ground state arise through S -mixing. In other words, one may start with a model containing only second order anisotropy terms. However, the competing isotropic ( J ) and anisotropic ( d and e ) interactions mix sp in states, generating corrections to the eigenvalues which show up as fourth (and higher) order terms when one tries to map the S = 4 ground state onto a giant spin model. The magnitudes of these higher order corrections depend on the degree of mixing of the S = 4 state with higher lying S < 4 states. In the extremely strong exchange limit ( J >> d ), the mixing is essentia lly absent, and the well isolated S = 4 multiplet maps perfectly onto a giant spin model requiring only a second order D parameter ( E is forbidden in the S4 symmetry group). It is only when J is reduced and the proximity between the S = 4 state and higher lying levels st arts to approach the energy scale associated with the zero field splitting within the S = 4 state that the S -mixing begins to have measurable consequences in terms of fourth-order terms obtained from a mapping of the S = 4 state onto a giant spin model. Consequently, it is highly desirabl e to be able to make direct

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99 measurements of the locations of excited states. This provided the motivation to quantify the isotropic interaction th rough HFEPR in order to make a dir ect comparison to the dominant single ion anisotropic interaction. We achieve th is by looking for additional peaks at elevated temperatures ( T ~ 8 J ) that may be associated with transitions within higher lying spin multiplets. 4.4 Measuring the Exchange Interaction with HFEPR The presence of relatively low lying excite d states was first realized by fitting DC magnetic susceptibility data for this complex [82]. Nevertheless, in order to confirm this and to gain additional information concerning excited st ates, we analyzed the higher temperature EPR spectra from section 4.2 in more detail. Fig. 4-10 shows an expanded view of data obtained at a frequency of 198 GHz in the range from 4 T to 9 T. In addition to the ground state ( S = 4) transitions, there are other weaker peaks (some marked by vertical dashed lines) which can be seen to appear in between the stronger peaks as the temperatur e is increased. Based on our understanding of the energy spectrum a ssociated with four coupled spin s = 1 particles, it is clear that the resulting excited S = 3 spin multiplets already become sufficiently affected by S -mixing for the parameter regime appropriate to this complex, that attempts to assign meaningful labels to resonance peaks according to a simple S = 3 picture can be problematic Indeed, it is apparent that the three effective S = 3 multiplets in Fig. 4-9 are not degenerate. Consequently, the transitions within the various S = 3 multiplets need not be degenerate either, thus giving rise to additional fine structures and/or pe ak broadening. This can be seen in Fig. 2-2, where the change in the background signal seen in the 4-7 T range we attribute to broadening of the resonance peaks due to closely spaced energy levels with in the multiple higher lying spin multiplets. Nevertheless, close examination of the 198 GHz data in Fig. 410 does reveal six clusters of peaks, which we tentatively as sign to transitions within an S = 3 multiplet. These peaks have been highlighted with vertical dashed lines in Fig. 4-10 and, for convenience, labeled A to F at

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100 the top of the figure. In addition to these peaks, there are other peaks within the data that suggest transitions within another S = 3 multiplet or within higher-lying ( S < 3) multiplets. We rule out that these additional peaks are the result of spl ittings due to disorder since such splittings manifest as separations on the or der of 0.1 T, and the peak sepa rations we see here are on the order of 0.2 T. The inset to Fig. 4-11 displays the freque ncy dependence of the positions of the six resonances labeled A to F in Fig. 4-10. The solid lines are guides to the eye. However, the slopes of these lines were c onstrained using the average g -value for the S = 3 multiplets determined from the Zeeman diagram in Fig. 4-9. It should be noted that the exited S = 3 levels in Fig. 4-9 exhibit significant non -linearities with respect to the magnetic field. Consequently, the effective g -values associated with the different transitions between these levels vary significantly, which may explain why some of the resonances vanish behind the stronger S = 4 peaks at certain frequencie s, particularly resonance A which is not seen at the lowest frequencies. This can also account for the fact th at not all transitions within the additional higher lying multiplet ( S = 2) are able to be resolved. As already mentioned, the zero-field offsets a ssociated with the solid lines in Fig. 4-11 were chosen arbitrarily so as to lie on the da ta points. However, a comparison between these offsets and the extrapolated zero field splitting obtained for several of the possible transitions between S = 3 levels in Fig. 4-9 reveals good agreem ent. Thus, not only do the calculations presented in Ref. 81 give excellent agreement in terms of the ground state S = 4 spectrum, but the agreement appears to extend to the excited states as well, although at this point it is not possible to make precise comparisons between e xperiment and theory due to the strong S -mixing among excited levels.

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101 In order to estimate the location of the S = 3 multiplets relative to the ground state, we used a simplified approach where we considered first an isotropic Heisenberg coupling, ij ijJss between four spin s = 1 Ni+2 ions. One can easily solve this problem for four spins, assuming a single ferromagnetic coupling parameter J (< 0), using a Kambe vect or coupling scheme [86]. This gives rise to states with total spin, ST, at energies given by [ST(ST + 1) 8]J. We then added zero field splitting by hand to the ST = 4 and 3 states using para meters (up to second order) estimated from our EPR experiments. The values for the ST = 4 state are given in section 4.2 and elsewhere [78]. For the ST = 3 state, we used values that approximate the zero field splitting observed in Fig. 4-11, namely gz = 2.30 and D = 0.3 cm 1. The corresponding energies of the various ST states are then given by 2 1 3,181TTsTTSsTTzBsESmJSSDmSSgBm (4-3) where we set D2 = 0.29 cm 1 for the ST = 2 states, and all other DST values equal to zero. We then used Eq. 4-3 to compute the temperature and field dependence of the populations of each of the eighty one levels corresponding to all possible ST and ms values (taking also into consideration the degene racies of the various ST states). This information was then used to compute the temperature dependence of a given EPR transition observed at a particular magnetic field strength. In order to evaluate the coupling parameter J we analyzed the intensity of the transition labeled B in Fig. 4-11. We used the data obtained at 172 GHz and B = 4.55 T, due to its superior quality and the fact that this reso nance was well isolated fr om other peaks at this frequency. We also made the assumption that this transition corresponds to the ms = 2 to 1 transition within the ST = 3 state.

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102 The main panel in Fig. 4-11 plots the temperature dependence of the intensity (integrated area under the resonance) of transition B observed at 172 GHz. The red curve is a simulation of the data, obtained using only two adjustable parameters: the single coupling parameter J (< 0), and an arbitrary vertical scale factor. This best simulation gave a value of J = 6.0 cm 1. While one should strictly consider two inequivalent J parameters for a distorted Ni4 cubane complex having S4 symmetry, detailed fits to DC magnetization data give good reason to believe that only a single J parameter is needed for this complex [25]. Furthermore, such an analysis allows for the most direct comparison with the work in Ref. 81, where only a single J parameter was considered. Indeed, the agreem ent between the present analysis and previous studies is excellent: a value of 5.9 cm 1 was obtained from the rather deta iled analysis of the ground state ( S = 4) EPR spectrum in terms of four coupled Ni+2 ions, including a rigorous treatment of the exchange and anisotropic interactions via full ma trix diagonalization. We note that the value obtained from the present analysis involved a numb er of approximations, so it is likely that there is considerable uncertainty (maybe 1 cm 1) in the obtained value of the coupling parameter J Nevertheless, the good agreement with the work in Ref. 81 is not coincidental. We also comment on the value of J obtained from fits to MT data. The earliest published values for this complex considered only the isotropic coupling between Ni+2 ions in the cubane unit, ignoring the effects of local anisotropic spin-orbit coupling ( d and e ). Consequently, these values do not agree so well with the more recent analyses, b ecause the anisotropic terms have an appreciable influence on the susceptibility to relatively high temperatures. A value of J = 7.05 cm 1 was more recently obtained from fits to MT data for this complex using precisely the same model employed in the analysis of the ground state ( S = 4) EPR spectrum. Given the level of approximation we have presently employed, the agreement is excellent.

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103 This characterization used HFEPR to measure the isotropic exchange interaction, which allows one to precisely determine the locations of higher lying spin multiplets above the ground state. By analyzing the intensity of transitions within an excited S = 3 multiplet we can extract a value for J and by collecting data at va rious frequencies we are able to estimate the zero field splitting parameters for this multiplet. Additionally, we can compare the values of the dominant isotropic and anisotropic interactions ( J / d ~ 1.1) to explain the breakdown of the giant spin model in terms of requiring fourth order terms in the Hamiltonian to fit the data. The relative magnitudes of these competing interactions cause the S -mixing between the S = 4 state and higher lying levels and necessitate fourth or der (and higher order) terms obtained from a mapping of the S = 4 state onto a giant spin model. 4.5 Summary In this chapter we presented HFEPR studies done on a highly symmetric Ni4 SMM. First we reported on experiments to characterize the spin Hamiltonian parameters for this SMM. Data reveal vastly sharp, symmetric EPR lines due to the lack of solvate molecules in the crystal lattice and large intermolecular exchange pathwa y distances. However, variable frequency, variable temperature measurements have revealed the presence of two distinct molecular species within the crystal and we are able to extract th e relevant spin Hamiltonian parameters for each species. Below 46 K the peak splits into two, wh ich we attribute to differences in the molecular environments arising from different t-butyl grou p conformations in the dmb ligand. At high temperatures the motion of these is thermally averaged, but below 46 K the motion freezes out and the t-butyl group takes on two distinct orient ations. These EPR peaks are then split due to the effects of the disorder. Additional low temperature data (< 6 K) reveal additional splittings and broadening of the peaks, which we attribute to short rang e intermolecular excha nge interactions among

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104 neighboring molecules that are averaged out at higher temperatures. It is likely that exchange interactions provide an additional contribution to the line widths/shapes, i.e. exchange probably also contributes to the broad lines. However, given the minimal amount of disorder in this system, we are able to separate the various co ntributions (disorder and intermolecular exchange) to the EPR line shapes. Finally, we are able to measure the magnitude of the isotropic excha nge coupling constant, J with our HFEPR data. By simulating the intensity of peaks originating from transitions within a low lying excited state spin multiplet to a model that includes both isotropic and anisotropic interactions we obtain a value of J = 6.0 cm 1. This provides insight into the spacing between the ground state and higher lying spin multiplets, and supports the evidence that the isotropic and anisotropic parameters ( J and d ) can cause mixing between states unless J >> d Such mixing manifests itself as unphysical high er order terms in the Hamiltonian with the giant spin model.

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105 Figure 4-1. A molecu le of [Ni(hmp)(dmb)Cl]4. Each Ni+2 ion (green sphere) has a spin of S = 1 and the four Ni+2 ions couple ferromagnetically to give a spin ground state manifold of S = 4. The bulky aliphatic groups that surround the magnetic core help to minimize intermolecular interact ions and the absence of H2O solvate molecules in the lattice results in a reduced distribution of microenvironments. Reprinted from E.C. Yang et al ., Exchange Bias in Ni4 Single Molecule Magnets, Polyhedron 22, 1727 (2003). Fig. 1, pg. 1728. Copyright 2003, with permission from Elsevier.

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106 Figure 4-2. 172.2 GHz HFEPR spectra for temper atures in the range of 10 K to 59 K. The main intensity peaks correspond to transitions within the ground state manifold, and the lower intensity peaks (marked by vertical dashed lines) are from transitions within excited state manifolds. From the temper ature dependence, it is clear that the approximately evenly spaced dominant peak s correspond to transitions within the S = 4 ground state, as all eight transitions are observed.

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107 Figure 4-3. Plot of the peak positions in magnetic field for different frequencies from 127 GHz to 201 GHz at 30 K. The ms dependence of the splitting implies at least two distinct Ni4 species, with slightly different D values. The fourth order spin Hamiltonian is given by Eq. 4-1 and was used to determine the axial spin Hamiltonian parameters for each species within the crystal. Reused with permission from R. S. Edwards, Journal of Applied Physics, 93, 7807 (2003). Fig. 2, pg. 7808. Copyright 2003, American Institute of Physics.

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108 Figure 4-4. Plot of the influence of the sp acing between the resonance branches due to a negative, axial, fourth order anisotropy term (red lines). The value of this term has been exaggerated for illustrative purposes. For comparison, the branches are evenly spaced with only a D term (black lines). The branches involving transitions between lower lying states ( ms = –4 to –3) are spaced further ap art with respect to adjacent branches than those involving transi tions between higher lying states ( ms = –1 to 0).

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109 Figure 4-5. Plot of the peak spl ittings as a function of temperature. a). An enlarged view of the peak splitting at 172.2 GHz reveals that it appears quite abruptly below a critical temperature of about 46 K, which suggest s a possible structural transition taking place. This causes a lowering of the crys tallographic symmetry and, hence, two distinct Ni4 species. b). Additional broadening and splittings of the EPR spectra are observed at temperatures below about 6 K. We speculate that this behavior is related to the development of short-range inter-m olecular magnetic correlations which are unresolved at higher temperatures.

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110 Figure 4-6. Heat capacity meas urements of [Ni(hmp)(dmb)Cl]4. a). Plot of heat capacity at constant pressure versus temperature [Ni(hmp)(dmb)Cl]4 (blue data) and [Zn3Ni(hmp)4(dmb)4Cl4] (red data). The peak at 46.6 K corresponds rather well to the temperature at which the peaks in the HFEPR spectrum start to split (46 K). b). The ORTEP (Oak Ridge Thermal Ellipsoid) plot comparison of symmetry independent parts of the molecule of [Ni(hmp)(dmb)Cl]4 at 12 K and 173 K. The thermal ellipsoids shrink at 12 K for all the atoms except for the t-butyl group. These ellipsoids are indeed much larger than wo uld be expected, and this should be where the order-disorder activity is taking place.

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111 Figure 4-7. Comparison of the structure of [Ni(hmp)(dmb)Cl]4 at 173 K and 12 K. The pink line is the structure at 173 K, while the green dashed line is the structure at 12 K. As can been seen, the only part of the structure that changes is the part coming from the t-butyl group. The observed small change in the structure supports the claim that there are different microenvironments, and th us two distinct species of molecule in the system at low temperatures whic h cause the HFEPR peak splittings.

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112 Figure 4-8. Plot of the temper ature dependence of the peak split tings at a given magnetic field for three frequencies. The temperature below which the additional fine structures begin to appear increases with increasing magnetic field/frequency (see red arrows as rough guide). For comparison, the dashed line represents the energy separation, 0/ kB, between the ms = –4 ground state and the first excited state ( ms = 4 for B < 0.66 T, and ms = 3 for B > 0.66 T). Thus, it appears as though the onset of the additional fine structures is related to the depopulation of excited states.

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113 Figure 4-9. Plot of a simulation of four coupled s = 1 spins resulting in a spectrum of (2 S + 1)4 = 81 eigenvalues. For this simulation th e magnetic field is applied parallel to the molecular easy-axis. It is the ma gnetic-dipole transitions between these nine low-lying energy levels (red lines) th at dominate the EPR spectrum for the Ni4 complex.

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114 Figure 4-10. Plot of an expanded view of data obtained at a frequency of 198 GHz in the range from 4 T to 9 T. In addition to the ground state ( S = 4) transitions, there are other weaker peaks (some marked by vertical dashed lines) which can be seen to appear in between the stronger peaks as the temperature is increased. The cl uster of six peaks, which have been highlighted with vertical dashed lines and labeled A to F at the top of the figure, we tentatively assign to transitions within the S = 3 states.

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115 Figure 4-11. Plot of the temperature dependen ce of the intensity (integrated area under the resonance) of transition B observed at 172 GHz. The red curve is a best fit to the data, which gave a value of J = 6 cm-1. The inset shows the frequency dependence of the A F transitions within the S = 3 states for a temperature of 35 K.

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116 CHAPTER 5 HFEPR CHARACTERIZATION OF SINGLE CO (II) IONS IN A TETRANUCLEAR COMPLEX The results presented in this chapter can be found in the article, High Frequency Electron Paramagnetic Resonance (HFEPR) Study of a High Spin Co(II) Complex, J. Lawrence, C. C. Beedle, E.-C. Yang, J. Ma, S. Hill, and D.N. Hendrickson, Polyhedron 26, 2299-2303 (2007). 5.1 Introduction to the [Co(hmp)(dmb)Cl]4 and [Zn3Co(hmp)4(dmb)4Cl4] Complexes The large success of the comprehensiv e studies on the [Ni(hmp)(dmb)Cl]4 and [Zn3Ni(hmp)4(dmb)4Cl4] complexes, some of which were presented in Ch. 4, provided motivation for investigation of two analogous compounds, [Co(hmp)(dmb)Cl]4 and [Zn3Co(hmp)4(dmb)4Cl4]. These complexes possess the same structure and molecular microenvironment as the former, only with Co+2 ions replacing the Ni+2 ions on opposing corners of the distorted molecular cube. A major difference in these systems is the added complexity of strong spin orbit coupling inherent in the Co+2 ions that was not part of the Ni+2 ions [9, 87]. This greatly affects the magnetic behavior and necessitates the use of a coupled single ion Hamiltonian approach as opposed to the usual giant spin Hamiltonian in order to model the system. The first evidence for SMM behavior in the tetranuclear complex was reported elsewhere [88, 89]. The out-of-phase ac susceptibility is less than 0.01 cm3 mol–1 at temperatures above 3.5 K and substantially increases to 0.16 cm3 mol–1 as the temperature is decreased to 1.8 K. The increase of the out of pha se AC signal suggests that the Co4 molecule has an appreciable energy barrier for magnetization re versal. Additionally, hysteresi s loops for temperatures below about 1.2 K are shown to increase rapidly upon decreasing the temperatur e. The temperature dependence and area within the hysteresis loop in dicate that this molecule has considerable negative magnetoanisotropy as expected for a SMM.

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117 5.2 HFEPR Measurements of [Zn3Co(hmp)4(dmb)4Cl4] 5.2.1 The [Zn3Co(hmp)4(dmb)4Cl4] Complex The preparation of single crystals of this complex was similar to that described in Ref. 80. Crystals of [Zn3.98Co0.02(hmp)4(dmb)4Cl4] were prepared and a relatively precise value of the Zn/Co ratio in the crystals was obtained by Inductively Coupled Plasma-Optical Emission Spectroscopy (ICPOES) spectra, which gave Zn/Co: 99.46: 0.54. Assuming there is little difference in the heats of formation for either the Zn4 or Co4 complexes, the Co+2 ions should be randomly distributed in the crystal. On the basis of this assumpti on, it is a relatively straightforward exercise to compute the probabilities for the formation of the Zn4, Zn3Co, Zn2Co2 ZnCo3, and Co4 complexes, as a function x in the formula [Zn4xCo4-4x(hmp)4(dmb)4Cl4]. When x = 0.995, it is found that the Zn4 and Zn3Co species make up 98% and 1.97% of the total population, respectively. Thus, the dope d crystal is comprised of some [Zn3Co(hmp)4(dmb)4Cl4] complexes doped randomly into a diamagnetic Zn4 host crystal, with th e overwhelming majority of the magnetic spectra coming from the [Zn3Co(hmp)4(dmb)4Cl4] complex. The core of each molecule is a distorted cube (analogous to the Ni complexes), the c direction being slightly longer than the equivalent a and b directions. For each molecule there are four possible sites for the Co+2 ion to reside, related by the S4 symmetry operators, which are given in matrix form in section 5.4. This complex crystallizes in the shape of an approximately square based pyramid. A single crystal (~1.1 mm ~0.9 mm ~1 mm) was mounted with the flat base on the endplate of a rotating cavity. We use a minimal amount of silicon vacuum grease to attach the crystal to the copper endplate. 5.2.2 Frequency and Temperature Dependent Measurements At the lowest temperatures (2 K), one can ma ke the assumption that only the ground state for each Co+2 ion is populated. Furthermore, as we shall see below, the ground state is a well

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118 isolated Kramers’ doublet. Thus, each Co+2 ion may be expected to contribute only one peak to the HFEPR spectrum (containing some fine struct ure due to disorder-ind uced strain). Thus, assuming the magnetic axes associated with the four possible Co+2 sites on the Zn3Co molecule are non-collinear, one should observe four separate HFEPR peaks for arbitrary field orientations (one for each Co+2 site). For field orientations along cer tain symmetry directions, two or more of these transitions may become degenerate, thus reduc ing the number of peaks. Fig. 5-1 displays frequency-dependent data obtained at 2 K, with the field tilted 32 away from the c axis of the crystal in the (100) plane (see Fig. 5-3). The in set to Fig. 5-1 shows a typical spectrum obtained at a frequency of 51.8 GHz. As can be seen, absorptions are grouped into three clusters. The position of each peak is plotted versus frequency in the main part of the figure for measurements performed on higher-order cavity modes. As can be seen, each peak lies on an absolutely straight line which passes exact ly through the origin. Land g factors have been assigned to each of the peaks based on the slope of the straight line through each set of data points. The shapes of the data points (solid and open circles and squares) have been chosen for comparison with Fig. 53 below. We believe the fine struct ures are caused by disorder induced g strain (as was found from similar studies of the Ni/Zn analog [80, 90]). Based on the angle-dependent studies discussed below, we find that the g = 4.2 branch in Fig. 5-1 is in fact degenerate, consisting of two HFEPR transitions. This explains the observation of only three peaks. The relative intensities of the peaks may be understood in te rms of the transition rate s [71], which depend on the respective Land g factors (smaller g factor, larger transition ra te). The fact that each resonance lies on a perfectly straight line whic h passes exactly through the origin is a clear indication that the transitions occur w ithin a well isolated effective spin S' = 1/2 Kramers’ doublet. The anisotropy is contained entirely within the effective Land g tensor. Further

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119 confirmation comes from the temperature dependence in Fig. 5-2. These data were obtained for the same frequency and field orie ntation as the data in Fig. 51; one can see that the field alignment is not precisely the same due to the slight splitting of th e degenerate branch. Nevertheless, all peaks increase in intensity upo n lowering the temperature, thus confirming the assignment of the ground state as a spin S' = 1/2 Kramers’ doublet. 5.2.3 Angle Dependent Measurements By far the most detailed information comes from angle dependent studies. For an arbitrary rotation plane each spectrum should consist of four peaks. However, for high symmetry rotation planes there will be degeneracies and some re sonance peaks will superimpose upon one another. The highest symmetry direction corresponds to a magnetic field applied along the c axis of the crystal, where only one peak should be seen. A magnetic field applied at an arbitrary angle within the ab plane of the crystal will give rise to tw o peaks. Any misalignments of the sample with respect to the applied field will lift this degene racy and give rise to up to four major peaks. The first experiment (rotation 1) involved rotating the magnetic field in the (110) plane from the c axis to the ab plane (Fig. 5-3a). The second experime nt (rotation 2) involved rotating the magnetic field within the ab plane (Fig. 5-3b). For rotati on 1 we observe three resonance branches, which collapse into approximately 1 peak for = 0 (field // c axis) and approximately two peaks for = 90 (field // ab plane, 45 from a and b ). Since only three branches are observed in Fig. 5-3a, we must assume that one of them is degenerate (solid circles). Based on simple geometrical considerations, we can immediat ely determine the nature of the anisotropy at the individual Co+2 sites. Due to the octahedral coordination, we assume the anisotropy will be approximately axial (easy-plane or easy-axis [9]). First of all, it is clear from Fig. 5-3a that the local magnetic axes are tilted with respect to the crystallogra phic axes (and, therefore, with respect to each other). Fig. 5-4 shows a diagra m of the magnetic core, with the four possible

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120 sites for a single Co+2 ion to reside. The easy axis of each individual ion is tilted at an angle (found to be ~58) with respect to the c axis of the crystal. Starting with the field along c (Fig. 53a): when one tilts the field away from this sy mmetry direction, it necessarily tilts away from three of the local z axes and toward three of the xy planes; conversely, it tilts toward one of the z axes and away from one of the xy planes. Considering the two-fo ld degeneracy of the branch represented by the solid circles, upon tilting the field away from c one sees that three of the branches move to higher fields (they become harder), whereas on ly one moves to lower field (it becomes easier). Therefore, based on the abov e geometrical considerat ion, one can conclude that the planes are hard, while th e axes are easy, i.e. the anisot ropy is of the easy-axis type. Upon rotating the field over a 180 interval, one is guaranteed to find all four hard planes. These directions correspond to the maxima at 90 (two planes), 32 and +32 (the peak at +90 is equivalent to the one at 90) in Fig. 5-3a, and the maxima at [( n 90) + 45] in Fig. 5-3b, where n is an integer. Since we observe a degenera te branch in Fig. 5-3a, this implies that orientation of the field, as it rotates, is the same for both ions contributing to the branch. Thus, the field rotation plane must be perpendicular to the easy-axis tilt plane for these two ions and, therefore by symmetry, it must be parallel to th e easy axis tilt plane fo r the other two ions. Therefore, we conclude th at the two minima labeled z correspond precisely to the easy axes. One can then immediately determine that gz = 7.80 (indicated by blue dashed line). The minimum at 0 in Fig. 5-3a corresponds to the po int of closest approach of the field to the other two easy axes. From the positions of the maxima and minima, we determine that the easy axes (hard planes) are tilted about 58 (3 2) away from the crystallographic c axis, and the easy axes are tilted in the (110) and (1 10) planes. It is notable in Fig. 5-3a that the heights of the maxima at 90 and 32 are slightly different, corresponding to g values close to 2.00 and 2.20. This

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121 observation implies weak in-plane anisotropy, s uggesting a weak orthorho mbic distortion at the individual Co+2 sites. The lesser of the two g values corresponds to gx ~ 2.00 (the hard axis, red dashed line) and the intermediate value to gy ~2.20 (the medium axis, green dashed line). From the hard-plane rotations (Fig. 5-3b), we see that the maxima coincide with gy ~ 2.20 in Fig. 5-3a. Therefore, we can conclude that the medium axes lie along the intersecti ons of the four hard planes, within the ab plane of the crystal [along (110) and (1 10)]. The hard axes are, therefore, directed maximally out of the ab plane [with projections also along (110) and (1 10)], tilted 32 away from c The minima in Fig. 5-3b correspond to the projection of the easy axis anisotropy onto the ab plane, i.e. g = gz cos32 = 6.61 (marked by horizontal black dashed line). Finally, the vertical dashed line in Fig. 5-3a denotes the orientation of the data presented in Fig. 5-1 and Fig. 5-2. As can be seen, the data points are code d similarly to the corres ponding 51.8 GHz data in Fig. 5-1. As already discussed, the Hamiltonian for a well isolated effective spin S' = 1/2 Kramers’ doublet takes the form ˆBSgB (5-1) In Eq. 5-1,grepresents an effective Land g tensor that parameterizes all of the anisotropy associated with the spin-orbit coupling. Th e resulting relation between the measurement frequency, f, and the resonance field, Bres, is then ()Bres f gB (5-2) Thus, for a fixed frequency measurement, we have the relation in Eq. 5-3 ()res B f B g (5-3)

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122 One can therefore attempt to f it the data in Fig. 5-3 assumi ng a simple angle dependence of the form 2 min()cos()oggg (5-4) In Eq. 5-4, gmin is the minimum g value for a particular plane of rotation, g is the difference between the minimum and maximum g values, and o is the angle corresponding to gmin. The magenta curves in Fig. 5-3 correspond to such fits. As can be seen, the agreement is quite good. To summarize, the orientations of the magnetic axes, with respect to the crystal lattice, were determined for single Co+2 ions doped into a nonmagnetic Zn4 complex. Frequency and temperature dependence studies confirm the ground state to be an effective spin S' = 1/2 Kramers’ doublet with a highly anisotropic g factor. The anisotropy is found to be of the easyaxis type, with the single-ion easy axis direc tions tilted away from the crystallographic c direction by 58. The g factor anisotropy (gz = 7.8 and gx ~ 2.0) is close to the maximum expected for an octahedral Co+2 complex [9], suggesting a huge axial zero-field-splitting. The magnitude of the axial zero-field-splitting parameter of the Kramers doublets was calculated [9] to be close to 1000 cm-1 in another complex in octahedral co ordination geometry for such a large amount of g factor anisotropy ( g ~ 6). The information obtained from this study will be used in an attempt to simulate the data we have obtained from the Co4 system, which we present in section 5-3. 5.3 HFEPR Measurements of [Co(hmp)(dmb)Cl]4 A crystal of [Co(hmp)(dmb)Cl]4 has the same shape and dimensions as those of [Zn3.98Co0.02(hmp)4(dmb)4Cl4]. The setup was placed in the 9 T Quantum Design systems magnet and oriented such that the c axis of the sample was approx imately parallel to the applied magnetic field. Fixed frequency, fixed temperat ure magnetic field sweeps were performed for

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123 numerous frequencies and temperatures with the magnetic field oriented parallel to the c axis of the crystal. 5.3.1 Measurements Along the Crystallographic c Axis We begin with a discussion of the spectra obtained with the field along the c axis. Fig. 55 plots the temperature dependence of the peaks for tw o different frequencies. It is clear that this system is more complicated than its nickel c ounterpart (discussed in Ch. 4) due to multiple ground state transitions observed to temperatures as low as 2 K. At a frequency of 288 GHz for a temperature of 2 K we see a strong peak at 0.75 T and a much weaker one at 1.25 T, indicating that at least the stronger peak is indeed a ground stat e transition. However, as the frequency is increased we start to observe many more transi tions even at the lowest temperatures. At 501 GHz and 2 K we see 5-6 peaks with some structure evident in the higher field peaks. This is radically different than the behavior seen in the Ni4 system, and is contradictory to what one would expect from a simple SMM. While it is true that disorder in the Ni4 system gave rise to two distinct species of molecules, this manifest ed itself in the observation of peak splittings on the order of 0.1 T. Since the same ligands are present in the Co4 system, it is unlikely that any disorder would manifest itself as multiple spec ies with such radically different anisotropy parameters. Therefore, it is clear that the gian t spin model is not physically appropriate to model this system. As discussed in Ch. 3 the giant spin model, which assumes the quenching of the orbital contribution to first order in perturbation theory, allows one to express the states of the systems as pure spin multiplets. Within a given S multiplet, the anisotropy terms lift the degeneracy of the spin projecti on states, which are expressed as s m. However, this is not the case for a Co+2 ion. Each ion has an orbitally degene rate ground state, an d thus the orbital angular momentum is not quenched. Hence, the assumption that we can express the eigenstates

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124 of the system as pure spin multiplets is no longer valid. The orbital contribution to the magnetic behavior is significant and mu st be taken into account. This is further illustrated in Fig. 5-6 where we plot the c axis frequency dependence of the resonance peaks at 2 K. The low temperat ure data show multiple ground state resonance branches across a wide frequency range (230 GHz – 715 GHz). The solid lines are rough guides to the eye of the field dependence of the reso nance branches. Each one extrapolates to a significant zero field offset value, indicating a large zero field anisotropy. These zero field offsets display non-linear behavior that can not be modeled by a simple D term. Such a term would split the levels in zero fi eld in a linear fashion. The energy splittings would go as D(2|m|– 1), where m is the spin projection state from which the transition originates. As an example, for a system with S = 2 a D term would produce zero field offsets of 3D and D respectively. Hence the energy difference between branches goes like 2D. This is clearly not representative of the data for this system. The colored arrows in th e figure represent the frequency difference between adjacent resonance branches and the correspondi ng values are displayed. The four lowest resonance branches are separated by values that vary between ~4 0-60 GHz, and then there is a sudden spacing of ~ 230 GHz between the next two resonance branches. Then the next two branches have a zero field spacing of ~ 80 GHz. Additionally, the four lowest branches exhibit level repulsion as the magnitude of the magnetic field is increased from zero. This is seen from the curvature of the lines that ar e guides to the eye as the field is increased, and is indicative of transverse anisotropies in ducing mixing between states. The individual effective S' = 1/2 ions should have no inherent zero fiel d anisotropic terms, but the exch ange coupling between the four ions will be shown to contain a te rm that will mix states in zero field. Additionally, the large tilting of the magnetic easy axis of each ion means that alignment of the ex ternal field with each

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125 ion’s easy axis simultaneously is impossible, an d thus, an external transverse component is introduced. Such effects are likely to be a significant source of the state mixing. 5.3.2 Discussion of Spectra Within the ab Plane We collected angle dependent data by rotating the field within the ab plane of the crystal. Our 7 T horizontal field magnet offered the perfect geometry for this experiment. We initially oriented the sample geometry such that the magnetic field would be parallel to the ab plane of the crystal. After selecting a go od frequency, we rotated the magnetic field within the ab plane of the crystal. Fig. 5-7a plot s the 10 K EPR spectra for different angles (separated by 5) at a frequency of 123 GHz. Each data set has been offset for clarity. From the shifts of the peak positions in fi eld we infer that there is a signif icant transverse anisotropy in this system. Fig. 5-7b plots the peak position in field as a function of angle within the ab plane. It is clear that a four fold modulation of the resonanc e position is seen as we rotate through 180, which reflects the S4 symmetry of the molecule. The periodicity of the maxima/minima in the peak position is 90 as shown in Fig. 5-7b. We ca n also see this from the two red traces in Fig. 5-7a since they are identical, and are separated by 90. Normally this four fold modulation is parameterized in terms of a 44 44ˆ B Oterm in the giant spin Hamiltonian, and unlike the nickel system discussed in Ch. 4, such a term would not be forbidden here since the ground state of a Co+2 ion is characterized by L = 1, S = 3/2. However, as we men tioned above, using a giant spin approach to model this system is not sensible. 5.4 Spin Hamiltonian for the Tetranuclear system Since we can not describe the observed behavi or in terms of a giant spin Hamiltonian we turn to the coupled single ion approach. In sectio n 5.3 we were able to determine that the single ions have an effective S' = 1/2 ground state. Consequently the ground state has no zero field

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126 splitting terms (d or e). However, from the studies on the tetranuclear complex (Fig. 5-6) it can be seen that the ground state of that system cl early has zero field split ting anisotropy. Now we provide a qualitative explanation as to how the ground state of four coupled ions with no zero field splitting can project onto th e tetranuclear complex to give rise to ground state zero field splitting. Considering the possible interactions (the usual zero field splitting is a second order spin orbit effect that is zero for Kramers doubl ets) we are left with the isotropic exchange, symmetric exchange, and antisymmetric exchange te rms discussed in section 3.1.2. While it is certain that isotropic exchange is a prevalent e ffect in this system, it will not produce any zero field splittings in the coupled system. Th is interaction only splits energy levels by S, not by m. From this we will get effective spin multiplets with values S = 2, 1, 0, but no removal of the degeneracy of the levels within the S = 2 and S = 1 multiplets. Therefore we are left with the symmetric and antisymmetric exchange interactions as the sources of th e zero field splitting in the ground state of the tetranuclear complex. Based on the symmetry of the system in que stion, some components of the antisymmetric exchange vector can vanish, which simplifies the Hamiltonian, as we now demonstrate. The symmetry of the molecular cluster will determine the orientation of the antisymmetric exchange vector, ijG The molecule can be modeled as a distorted cube, with Co+2 ions placed on opposing corners, and the en tire entity possessing S4 symmetry, as shown in Fig. 5-4. To begin, we consider two ions joined by a straight line in space. At the midway point between these ions we can picture a mirror plane oriented perpendicula r to the line that intersects at this point. ijG will be oriented parallel to this mirror plane, and thus, perpendicular to the line joining the two respective ion centers. Expressed in component form, the antisymmetric part of the Hamiltonian for a pair of ions i and j can be written

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127 [][][]xyz ijijxijijyijijzGSSGSSGSS (5-5) In order to simplify, we omit the transverse components of the interaction, since x ijG=y ijG=0 for tetragonal systems, and bothx ijG and y ijG are much less than z ijG if the tetragonal symmetry is slightly distorted [91]. Thus, we use only th e axial component of the antisymmetric exchange interaction, and rewrite Eq. 5-5 as []zxyyx ijijijGSSSS (5-6) This antisymmetric exchange interaction will mix states in zero field. The other terms contributing will be the isotr opic exchange and the Zeeman interaction. The actual molecular geometry is an elongated c ube, so that there are two isotropic exchange coupling constants. With respect to the Zeeman interaction, for an arbitrary magnetic field orientation, the field will project differently onto the four ions In order to properly construct the Zeeman term for the molecular cluster, we must take into account th e tilt angles of the individual ion axes. Since the system has S4 symmetry, we can relate each ion to one another with a combination of the Euler matrix relating the single ion and molecular coordinate systems and the transformation matrices for S4 symmetry. The Euler matrix is given by cos()cos()cos()sin()sin()sin()cos()cos()cos()sin()sin()cos() cos()cos()sin()sin()cos()sin()cos()sin()cos()cos()sin()sin() cos()sin()sin()sin()cos() R (5-7) In Eq. 5-7, and represent the tilt angles for the indivi dual ions. The convention used here is that is a rotation about the z-axis of the initial coordinate system. About the y-axis of this newly generated coordinate system, a rotation by is performed, followed by a rotation by about the new z-axis. Additionally, the matrices for S4 symmetry which relate one ion to another are given by

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128 100 010 001 I (5-8a) 14010 100 001 S (5-8b) 24100 010 001 S (5-8c) 34010 100 001 S (5-8d) Using these matrices, we can properly project the magnetic field onto each individual ion. The projection onto the first ion will be R1 = R*I, the second ion R2 =R*S14, the third ion R3 =R*S24, and the fourth ion R4 =R*S34. Any tilting of the magnetic fiel d with respect to the desired orientation can be accounted for by two additional polar angles, and where the first represents the angle of the field with respect to the z direction and the second represents the angle of the field in the xy plane. For the projection of the field onto the magnetic center for an ion, i, we express as sincos *sinsin siiMR co We write the final Hamiltonian for the system as ˆˆˆ ˆ []()ˆˆzxyyx ijijijijijijBii ijij H JSSAGSSSSBMgSSS (5-9)

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129 In zero field, the isotropic exchange part of the Hamiltonian splits the energy levels into different effective S multiplets, with the effective S = 2 multiplet being the lowest. Above this are three effective S = 1 multiplets and finally, two effective S = 0 multiplets. The symmetric and antisymmetric interactions th en split the different levels w ithin a given multiplet, analogous to the usual axial and transverse anisotropy (zer o field splitting) parameters. Considering the data set which was taken with the magnetic field aligned along the c axis of the crystal at a temperature of 2 K, we assume only the ground stat e of the system is suffici ently populated. It is clear from the data in Fig. 5-6 that six resonanc e branches are observed, wh ich is contradictory to what one would expect from an isolated effective S = 2 multiplet. Normal EPR selection rules allow transitions between states that differ in m by 1, and under the experimental conditions we should then observe only one resonance branch! Consequently, we must assume that there is such strong mixing between states (both between and within a given multiplet) that normal EPR selection rules do not apply. Under such conditions of strong mixing between states S and m are not good quantum numbers. As mentioned pr eviously, the antisymmetric exchange coupling will mix states in zero field. Additionally, th e large tilting of the magnetic easy axis of each ion means that alignment of the external field with ea ch ion’s easy axis simultaneously is impossible, and thus, an external transverse component is introduced. Such effects are likely to be a significant source of the state mixing. Attempts to simulate the data for the Co4 system assuming four coupled effective spin particles have not been successful. The final task of simulating the behavior of the tetranuclear system is still a work in progress. This has b een turned over to Motohi ro Nakano a collaborator in the department of molecular chemistry at Osaka University. His approach is to solve the problem of four coupled ions each with L = 1, S = 3/2. The difficulty comes from the size of the

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130 Hilbert space, which is [(2L+1)(2S+1)]4 = 20736. He is currently trying to simulate the EPR spectra using the single ion parameters we have measured. A preliminary simulation is illustrated in Fig. 5-8, which incorporates our data and assumes two different isotropic exchange values. The first attempt at simulating the data comes from Fig. 5-8 fo r a frequency of 501 GHz at 30 K with the magnetic field aligned along the c axis of the crystal. Our data reveal resonance peaks at the following approximate field values: 1. 6 T, 2.4 T, 3 T, 4 T, 5 T, and 6 T. These positions are marked by the vertical black lines in Fig. 5-8. Clearly, no one simulation is able to reproduce all of the peaks. The simulation with the parameters J1 = 0.814, J2 = 2.33 (marked by a dark green arrow) closely reproduces the peaks at 2.4 T, 3 T, 4 T and 5 T, but not the other two. On the other hand, the si mulation with the parameters J1 = 1.21, J2 = 1.72 (marked by a blue arrow) begins to reproduce the peaks at 1.6 T and 6 T, but not the other four. Evidently more work will be necessary before the Co4 system is solved.

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131 Figure 5-1. Plot of peak position as a function of frequency. All branches extrapolate to zero field, indicating the ground state is an effective spin S’ = 1/2 Kramers’ doublet. Reprinted from J. Lawrence et al., High Frequency Electron Paramagnetic Resonance Study of a High Spin Co(II) Complex, Polyhedron 26, 2299 (2007). Fig. 1, pg. 2301. Copyright 2007, with permission from Elsevier.

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132 Figure 5-2. Temperature dependence confirming that these are ground st ate transitions, since the intensity of each peak increases as the te mperature is lowered. Reprinted from J. Lawrence et al., High Frequency Electron Parama gnetic Resonance Study of a High Spin Co(II) Complex, Polyhedron 26, 2299 (2007). Fig. 2, pg. 2301. Copyright 2007, with permission from Elsevier.

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133 Figure 5-3. Peak position as a function of angl e for two planes of rotation. a). Rotation of the magnetic field in the (110) plane from the c axis to the ab plane. b). Rotation of the magnetic field within the ab plane. Reprinted from J. Lawrence et al., High Frequency Electron Paramagnetic Resonance Study of a High Spin Co(II) Complex, Polyhedron 26, 2299 (2007). Fig. 3, pg. 2301. Copyright 2007, with permission from Elsevier.

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134 Figure 5-4. Diagram of the magnetic core of a CoZn molecule possessing S4 symmetry, with the four possible sites for a single Co+2 ion (purple sphere) to reside. The crystal directions are given by a, b, and c. The easy (z) axis of the individual ions is tilted with respect to the crystallographic c axis by ~58.

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135 Figure 5-5. Plot of the temper ature dependence of the peaks for two different frequencies. At a frequency of 288 GHz for a temperature of 2 K we see a single ground state transition at 0.75 T However, as the frequency is increased we start to observe many more transitions even at the lowest temperatures. At 501 GHz and 2 K we see 5-6 peaks with some structure evident in the higher field peaks.

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136 Figure 5-6. Plot of the easy axis frequency dependence of the resonance peaks at 2 K. The solid lines are rough guides to the eye of the field dependence of the resonance branches. Each one extrapolates to a sign ificant zero field offset value, indicating a large axial anisotropy. The zero field offset s display non-linear behavior that can not be modeled by a simple D term. The colored arrows in the figure represent the frequency difference between adjacent resonance branches and the corresponding values are displayed.

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137 Figure 5-7. Data taken with the field aligned within the ab plane of the crystal. A). EPR spectra for different angles (separated by 5) at a frequency of 123 GHz. From the shifts of the peak positions in field we infer that th ere is a significant tran sverse anisotropy in this system. B). Plot of the peak position in field as a function of angle within the ab plane. It is clear that a four fold modulat ion of the resonance position is seen as we rotate through 180, which reflects the S4 symmetry of the molecule. The periodicity of the maxima/minima in the peak position is 90. We can also see this from the two red traces in A). since they are identical and are separated by 90.

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138 Figure 5-8. Plot of the simulation of the Co4 system for a frequency of 501 GHz at 30 K with the field aligned along the c axis. The resonance peak positions for this data set are marked by the vertical black lines. The simulation accounts for two isotropic exchange interactions (J1, J2). The simulation with the parameters J1 = 0.814, J2 = 2.33 (marked by a dark green arrow) closely reproduces the peaks at 2.4 T, 3 T, 4 T and 5 T, but not the other two. On the ot her hand, the simulation with the parameters J1 = 1.21, J2 = 1.72 (marked by a blue arrow) begins to reproduce the peaks at 1.6 T and 6 T, but not the other four.

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139 CHAPTER 6 HFEPR STUDIES OF MAGNETIC RELAXATION PROCESSES IN MN12AC The results presented in this chap ter can be found in the articles, Magnetic Quantum Tunneling in Mn12 Single-Molecule Magnets Measured With High Frequency Electron Paramagnetic Resonance, J. Lawrence, S.C. Lee, S. Kim, N. Anderson, S. Hill, M. Murugesu, and G. Christou, AIP Conference Proceedings, 850, 1133-1134 (2006), and A Novel Experiment Combining Surface Acoustic Waves and High Fr equency Electron Paramagnetic Resonance in Single Molecule Magnets, F. Maci, J. Lawrence, S. Hill, J. M. Hernandez, J. Tejada, P. V. Santos, C. Lampropoulos, and G. Christou, Phys. Rev. Lett., submitted. 6.1 Introduction to Mn12Ac [Mn12O12(CH3COO)16(H2O)4] 2CH3COOH4H2O, hereafter Mn12Ac, is the most widely studied SMM. It was first synthesized [92] in 1980 and has a large spin ground state (S =10), combined with a sizeable D value (D = 0.45 cm-1) that gives a large barrier to magnetization reversal. The magnetic core of Mn12Ac consists of four Mn+4 ions (each with S = 3/2) and eight Mn+3 ions (each with S = 2) that couple antiferromagnetically to give a net spin ground state of S = 10. This is illustrated in Fig. 1-1. The mo lecule is modeled as having a spin vector of S = 10 with twenty one possible pr ojections (states) between ms =10 and ms = 10 along a quantization axis. Since the D value is negative for this (and any) S MM, there is an energy barrier to reversal of the magnetic moment’s spin projection. Grap hically, this energy barrier can be pictured as a double potential well, with states possessing oppos ite spin projections re siding in opposite wells, as shown in Fig. 1-2. Due to the high symmetry of the molecule, transverse anisotropies less than fourth order should be non existent and the rate of grou nd state quantum tunneling of the magnetization (QTM) at low temperatures (kBT DS2) should be non existent. This stems from the small

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140 tunneling splitting (8.2 10 16 cm) between the ground states [ 93]. The tunnel splitting was calculated assuming that the antiferr omagnetic interaction between the Mn+3 and Mn+4 ions are dominant, and therefore each Mn+3/Mn+4 pair can be treated as a S' = 1/2 dimer. The reduced the size of the Hamiltonian matrix 104 104 allowed a full diagonalziation. While the measured rate of pure ground state QTM is immeasurably small [25] (< 10 8 s 1), there are other transverse anisotropies present that arise from disorder. This is clearly evidenced by observation of regularly spaced steps in the hysteresis loops. If only fourth order transverse anisotropies existed, then QTM would only be possi ble between states that differ in m by multiples of 4. As discussed in Ch. 1, this is due to the fact that the matrix elements of the states m and m’ (states with opposite spin projections along the field quantization axis) vanish unless that stated condition is met. Consequently, this allows on e to define a selection rule for tunneling, which states that tunneling is forbidden unless m + m’ = 4n, where n is an integer. However, since QTM is seen between states that differ in m by any integer, there must be additional terms present that do not commute with the ˆzSoperator in the Hamiltonian. This exciting discovery opened the door for an abundance of extensive stud ies in the years to follow in an attempt to explain the QTM [40, 94, 95, 96] Much of the research conducted on this system has been dedicated to understanding w hy some of the idealized predictions are not realized by experimental tests, and what are the sources of these additional transverse anisotropies. 6.1.1 Theory and Effects of Disorder in Mn12Ac A theoretical Mn12Ac molecule has S4 symmetry. However, disorder in the system lowers the symmetry of some molecules, which introd uces transverse anisotropies [17] that are forbidden by S4 symmetry. The disorder manifests itse lf in the form of a distribution of molecules with different spin Hamiltonian parameters [97]. In Ch. 4 we presented work that

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141 outlined the influence of disorder on the width and structure of HFEPR resonance peaks. Such disorder will also have an e ffect on the QTM in a given SMM system. In section 6.2 we examine the disorder from a different perspect ive, primarily how the disorder results in a distribution of molecules that relax on different time scales and how this manifests itself in different experiments. First we discuss the main results [97] from work done to characterize the disorder in this SMM, since they provided a motivation for the experiments presented in section 6.2 and are directly applicable to our work. Low temperat ure x-ray diffraction studies were conducted [98] on Mn12Ac and reveal the presence of species that have symmetry lower than the theoretical S4. A single Mn12Ac molecule has four acetic acid solvent molecules, each of which is shared by two neighboring Mn12Ac molecules. The acetic acid mo lecules are disordered around two equally populated positions of the molecule betw een adjacent magnetic clusters and are involved in strong hydrogen bonding to the acetate ligands However, the acetic acid molecule will only hydrogen bond to the ligand in a specific orientation, leaving the neighboring Mn12Ac molecule without such a bond. From this perspective, th ere are two equivalent wa ys to place one acetic acid molecule between two Mn12Ac molecules. That is, the methyl group (CH3) in the acetic acid molecule can point toward a Mn12Ac molecule or away from it [99]. In the latter case, there is hydrogen bonding between the oxygen atom in the Mn12Ac molecule and the oxygen atom in the solvent molecule. Thus, disorder arises fr om the hydrogen bond interaction with the acetic acid molecules, reducing the number of hydrogen bonded ligands. In fact, there exist six different isomers of Mn12Ac molecules within a crystal, with three different symmetries and five different D values. The D values are calculated by projecting the contributions of the single ion anisotropies of the Mn+3 ions onto the molecular anisotropy. Fig. 6-1 illustrate s the six different

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142 Mn12 isomers from this model. The black spheres represent the magnetic core of the molecule, while the cyan arrows represent the acetic acid molecules. The arrow head pointing toward the molecule represents the hydrogen bond between the oxygen atom in the Mn12Ac molecule and the oxygen atom in the solvent molecule. As can be seen, two of the six isomers (D1 and D6) maintain strict S4 symmetry. However, these constitute only 12.5% (6.25% each) of the total species within the distribution. For a large number (87.5%) of molecules the S4 symmetry is lowered, and an additional second order transverse anisotropy (E term) develops. As can be seen for the isomers D2 – D5, the symmetry is lowered, creating a rhombic transverse anisotropy. Some of those molecules possessing an E term also have their magnetic axes tilted from the crystallographic axes. Thus, even when the extern al field is aligned with the crystallographic easy axis, there will be a misalignment with respect to the easy axes of these molecules. This introduces an external transverse field componen t. The calculated tilt an gles are all less than 0.5. However, later work [100] using HFEPR determined that these tilt angles for molecules in the distribution can actually be significantly larger (up to 1.7). These transverse fields provide a natural explanation for the lack of tunneling selection rules. Without this, tunneling steps would only be allowed for resonance fields that correspond to states that differ in m by a multiple of two. And under these conditions, only those molecules with E terms would obey this selection rule. The other molecules would only tunn el at resonance fields that differ in m by a multiple of four, due to the S4 symmetry. However, since tunneling steps are observed at each resonance field in hysteresis loops of this compound, this work done in Ref. 98 and Ref. 100 y6was a major step forward in understanding the mechan isms that govern the observed QTM in Mn12Ac, where QTM in SMMs was first observed [16, 18, 27, 95, 101].

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143 6.1.2 Dipolar and Hyperfine Field Relaxation Mechanisms Here we qualitatively discuss one of the prominent theories of QTM in SMMs, the results of which will be used to explain some of the magnetization relaxation observed in our experiments. The theory of Prokof’ev and Stamp [102, 103] (PS) considers a giant spin coupled to a bath of environmental spins (dipolar a nd nuclear). The model concentrates on the low temperature (kBT |D|S2) relaxation (below 2 K for Mn12Ac) in the presence of a small external field so that only the two lowest levels (S) are involved. In this regime is it predicted [104] that the phonon-mediated spin relaxation ( –1) is proportional to the cube of the energy difference (bias) between the two lowest energy levels. Ig noring hyperfine and dipolar fields, this energy difference will go like 2BgSBand consequently –1 ~ 3 B The striking contradiction is that while this predicts a minimum in the relaxation for B = 0, it is experimentally observed in a system such as Fe8 that the relaxation rate is ~ 104 greater for B = 0 than for a non resonant field (B = 0.1 T). If one now considers the previously ignored hyperfine and dipol ar fields we see that they have little influence on the prediction of a minimum in – 1 for B = 0, as is now outlined. For resonant tunneling in this low temperatur e regime, the energy bias must be less than the tunnel splitting between resp ective levels. The ground state tunnel splitting (without any transverse fields) in systems like Mn12Ac and Fe8 is on the order of 10– 10 cm– 1, while the typical bias caused by dipolar fields is ~ 0.07 .35 cm– 1. The bias from dipolar fields then forces the molecules off resonance and the only way for th e molecules to relax is then via spin-phonon interactions, which have incredibly long relaxation rates for the regime we are considering. Even if one considers transverse components to the dipol ar fields, it would take transverse fields ~100 times larger than the actual hype rfine/dipolar fields to increase the tunnel splitting to the values needed to explain the maximum in the relaxation rate for B = 0. The key point to these

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144 arguments is that a distribution of static hyperfine and dipolar fields is insufficient to model the observed relaxation rate behavior. However, a distribution of dynamic hyperfine and dipolar fields provides a mechanism to account for th e observed behavior. Dynamic hyperfine and dipolar fields in the low fi eld, low temperature regime can vary the bias at each molecular site in time, which can continually bring more molecules to resonance [105]. In the low temperature limit only the dynamic nuclear fi elds (i.e., hyperfine and nuclear dipolar fields) will cause relaxation since dipola r flip-flop processes are frozen out for this regime, except when molecules tunnel. The nuclear interactions are expected to be important in Mn12Ac since all the manganese nuclei are 55Mn with a spin of 5/2. However, although it is necessary to have rapidly fluctuating hyperfine fiel ds to bring molecules initially to resonance, the ensuing gradual adjustment of the dipolar fields across the sample, caused by tunneling relaxation, brings those molecules that have no t tunneled further into resonance, and allows continuous relaxation through this process. The fluctuating nuclear spin field also makes the tunneling incoherent. On the basis of thes e assumptions a formula for the magnetization relaxation as a function of time in this regime was derived, which is shown to follow a t1/2 dependence. This model is in principle valid at zero temperature an d short times, when the initial magnetization is close to the saturation magnetization. It is also shown that for higher temperatures such a t1/2 relaxation rate is not expected. This is due to the influence of dipolar flip-flop pro cesses which are frozen out at low temperatures. These interactions cause the nucl ear spin lattice relaxation time, T1, to become much shorter than in the low temperature regime, and the magnetizat ion reversal proceeds via higher levels through a thermally assisted tunneling processes. Coupli ng to the phonon bath is then crucial, which alters the arguments in the th eory that give rise to a t1/2 relaxation rate. When T1 is much less

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145 than the experimental time scale and hyperfine fields are compar able to or larger than intermolecular dipolar fields, exponen tial relaxation is predicted [106]. While this theory was derived considering only tunneling between the two lowest levels of a large spin system with a small applied field, we will show that the qualitative aspects of dynamic nuclear and dipolar fields influencing the relaxation can extend to tunneling at non zero resonance fields as well. Additionally, we will provide data that supports the predictions of a t1/2 dependent relaxation rate at in the low temperat ure regime. However, for the relaxation rate in the high temperature regime we find that the form deviates from exponential to that of a stretched exponential. First we outline th e results from an experiment [107] on Mn12Ac conducted using a magnetometer to measure the magnetization decay as a function of time for a temperature range of 1.5 K to 3 K. 6.1.3 High and Low Temperature Relaxation Regimes PS theory predicts a low temperature regime re laxation law for the magnetization that goes like 0()[1] t MtM (6-1) This study was done in zero fi eld on a single crystal of Mn12Ac and showed that in the low temperature regime (below 2 K) the magnetizati on relaxation can in fact be well described by a square-root law such as that in Eq. 6-1. In this regime, the mean re laxation time becomes as large as 50 years, and it depends weakly on the temperature. However, as the temperature is increased closer to and above 2 K, the relaxation can only satisfactorily be fit with a stretched exponential law that goes like () 0()tMtMe (6-2)

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146 However, it is found that is strongly temperature dependent. Below 2.0 K, is approximately constant and close to 0.5. It increases lin early with temperature up to 2.4 K to reach = 1.1, and then it slightly decreases at higher temperatures (between 2.8 K and 3 K, = 1). Also, the mean relaxation time, starts to follow the Arrhenius law = 0exp(–E/T) roughly above 2 K, with [108] the characteristic time, 0, having a value of 10-7 s. Below 1.9 K, with a fixed value = 0.5, is weakly dependent on the temperature and tends toward = 1.5 x 109 s at 1.5 K. This experiment demonstrates the existence of two relaxation regimes, as predicted by PS theory. 6.2 QTM Studied by HFEPR Most studies investigating QTM in SMMs have used changes in magnetization to monitor the tunneling. Such experiments have provided information as to where (in magnetic field) the spins are tunneling and the probability for such tu nneling to occur. In contrast, our experiment provides information on which spins within th e distribution of molecules are tunneling and on what time scales. HFEPR is a technique that can discriminate between different molecules since each molecule sees the same microwave frequency. However, the shape (width and structure) of the resonance peak illuminates the different microenvironments within the crystal. Each molecule with slightly different parameters will tr ansition at slightly different fields and this is clearly seen in the char acter of the observed resonance peak. By selecting how long we allow the tunneling at a resonance fiel d and then measuring those mole cules that tunneled with EPR, we also gain insight into the length of time it takes for differe nt molecules to tunnel while on resonance. As stated before, Mn12Ac is an incredibly well studied system, and thus the spin Hamiltonian parameters are extremely well characterized [37]. With this information we can simulate the energy spectrum for the molecule using Eq. 6-1 and make choices on what magnetic field and frequency combinations will be appropriat e for a data set. A discussion of the relevant

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147 tunneling processes involved in the experiments in sections 6.2.1 and 6.2.2 is given in section 1.4.2. 6.2.1 Experiment in Stretched Exponential Relaxation Regime This experiment deals with dynamics on the border of the low temperature (t1/2) and high temperature regimes (stretched exponential) relaxation regimes as demonstrated by the discussion in section 6.2.3. Mn12Ac crystallizes in the shape of a long rectangular needle. A single crystal (~1.5 mm ~0.3 mm ~0.3 mm) was mounted with the long, flat edge of the crystal against the base of a copper plate that mounts on the end of our 250 GHz quasi optics probe [55]. The crystal is attached to the co pper endplate with a minimal amount of silicon vacuum grease. We cool the sample from room temperature to 2 K in zero field and then align the sample such that its easy axis of magnetization (the c axis) is parallel to the applied magnetic field with angle dependent measurements (not shown). In order to study QTM using HFEPR, we apply a magnetic field of 6 T in order to fully bias the system such that the spins are in the ground (ms = 10) state. This is illustrated in Fig 62, which plots the energy levels of the system as a function of magnetic field. The blue arrows represent the biased magnetization of the system for a sufficient ( 3 T) magnetic field. At each resonance field (k = integer), spins can reverse their state by tunneling. While applying a microwave frequency of 286 GHz, we sweep to a magnetic field value of +0.9 T (k = 2), where levels on either side of the barrier are in resona nce, and hold the magnetic field at this value for different waiting times, which results in tunneling of spins between resonant states. Then we sweep back to 6 T and observe changes of the EPR intensities. Fig. 6-3 shows EPR spectra for different waiting times at +0.9 T. Each trace involves sweeping from +0.9 T to 6 T after a different wait time. The inset to Fig. 6-3 shows the sweep from 6 T to +0.9 T, which was identical for all data sets. As Fig. 6-3 shows, the area of the

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148 peak occurring on the positive field side decreas es for longer wait times, while the area of the peak occurring on the negative field side increa ses accordingly. The peak occurring at positive field involves transitions on the metastable side of the potential energy barrier (ms = 10 to ms = 9). As the field is held at +0.9 T for longer times, more spins from the ms = 10 state tunnel through the anisotropy barrier. After tunneling, they quickly decay back to the ground state, ms= 10. As the field is swept back to 6 T, those spins that tunnel ed no longer contribute to the positive field peak, but will instead contribute to the negative field peak; i.e. the transition between the ms = 10 and ms = 9 states. Fig. 6-4 shows the area of the positive field pe ak as a function of wait time. The peak area is a measure of the number of spins that have tunneled, which in turn is a measure of the magnetization relaxation. Thus, we should be able to apply the theory of relaxation to the change in peak area as a function of time. At the experimental bath temperature, we are above the regime where a t1/2 relaxation should be observed. Long range dipolar fields will affect the tunneling [102, 103], i.e. the local magnetic indu ction will evolve due to the fluctuating dipolar fields of molecules that have tunneled. Indeed this curve can not be satisfactorily fit to an exponential relaxation law, but can be fit to a stretched exponen tial of the form Moexp[ (t/)], which is in agreement with other work for magn etization relaxation in this temperature regime [107]. We estimate a mean relaxation time, on the order of 500 s. The best matching curve to the data gives a value of = 0.70 0.05. For the experime nt discussed in section 6.1.3, was found to be quite close to 0.5 at T < 2 K and then increase linearly with temperature up to ~ 1.1 for 2.4 K. While our value of is in quite good agreement with that for the other study, our value of is much lower. One reason for this diffe rence is that our expe riment was conducted at

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149 a non zero resonance field, while the other was done in zero field. As has already been determined (section 1.4.2) the relaxation is mu ch faster between the ground state and higher lying states than for zero fi eld ground state tunneling (k = 0), and we were waiting at the resonance field corresponding to k = 2. Another factor is the sample shape, which has a strong dependence upon due to differing initi al distributions of internal fiel ds [102]. It is reasonable to assume that the shapes of the samples used in the two experiments were different, and this will affect the value of the mean relaxation time. The evolution of the shape and structure of th e peaks in Fig. 6-3 reinforces the idea that there is a distribution of molecules relaxing on different time scales that govern the observed behavior. Previous studies on this compound have found significant distributions of the D and g parameters [83, 84]. All of the spins are subj ected to a frequency of 286 GHz, but not all spins see the same microenvironment. Careful inspection of the data reveals th at different portions of the spectrum relax on different time scales. Fo r example, the signal changes most rapidly at early times on the high-field sides of the peaks. Molecules contributing to these tails of the spectrum possess larger than average D values. Since these are relaxing quite fast they must posses an E term, and this would correspond to the D4 and D5 molecules, which have larger than average D values (–.541 cm-1 and –.548 cm-1 respectively) but also have E values (3.27 10-3 cm-1 and 1.64 10-3 cm-1 respectively). However, it also appears that a significant fraction of the molecules on the low-field (lower D) sides of the peaks have alrea dy tunneled during the first 30 seconds, since one always observes weight in the spectrum at low negative fields. This observation is consistent with the 25% of the molecules (D2 molecules) that have smaller than average D values but also have E values.

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150 Once most of the molecules have tunneled, a small fraction of sl ower relaxing species remain, giving rise to the much narrower EPR peak at low positive fields. The fact that this remnant peak is quite narrow indicates that this population is probably rath er pure. This finding agrees with the Cornia model which predicts th at the fraction of highe st symmetry molecules (D1 and D6 in Fig. 6-1) in a crystal should constitute ~ 6.25% each. These molecules should have the longest relaxation times, since they maintain S4 symmetry. Additionally, the D6 molecules have the largest D value of all the isomers. These molecu les must have a narrow distribution of microenvironments amongst themselves, which manifests as a sharp tail in the emerging resonance peak. Thus, in additi on to measuring the ensemble average of the relaxation, these EPR measurements demonstrate that one can sepa rately monitor the relaxation from different parts of the inhomogeneous distributi on of molecular environments. To summarize, at these temperatures, both hy perfine and dipolar fluctuations are fast enough to maintain the conditions for tunneling. Thus, the observed deviation from exponential relaxation comes from the time evolution of the local mean field induced by tunneling and the distribution of molecules with different relaxation rates. Different molecules within the distribution relax on different time scales, with both the D values and symmetry of the molecules contributing to the relaxation rates. 6.2.2 Experiment in the t1/2 Relaxation Regime At a later time we decided to re-visit this work in order to repeat the experiment in a lower temperature regime where we could test the t1/2 relaxation law of PS theory. The experiment discussed in section 6.2.1 was conducted in a system where 2 K was the minimum bath temperature. The following experiment was con ducted in a system where we are able to reach temperatures as low as 1.4 K. A single crystal was placed with the long, flat edge of the crystal against the endplate of a rotating copper cavity. The setup was placed in the Oxford systems

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151 magnet and the sample was oriented such that the c axis was parallel to the applied magnetic field. A data acquisition procedure similar to th e previous experiment explained in section 6.2.1 was used. 6.2.2a Low Temperature MQT Fig. 6-5 shows the time evolution of an EPR peak for a frequency of 237.8 GHz and temperature of 1.4 K after waiting at a field of 1.8 T where the levels ms =10 and ms = 6 were in resonance. If one compares this to Fig. 6-3, one sees that the same qualitative trends are observed. For short wait times (< 60 s) a small peak starts to emerge. As we wait longer at the resonance field (< 1200 s) a broad peak develops whose area increases with wait time. And for the longest wait times (> 1200 s) a second, higher field, narrow feature emerges in the spectrum. One noticeable difference is the abil ity to fit the peak area to a t1/2 relaxation law. The peak area is proportional to the magnetization of the samp le and in this regime the magnetization should follow a square root relaxation law [102]. Fig. 66 plots the peak area as a function of wait time, and it does indeed fit to a square root relaxation law. At these low temperatures, both the lattice vibrations and dipolar fluctuations are minimal. As a consequence, the relaxation, which is induced by nuclear spin fluctuat ions, takes the square-root form in Eq. 6-1. The square-root decay arises from the formation of a depletion in the dipolar field distribution at the tunneling energy due to the spins that have tunneled. Th e distribution of dipola r fields is close to equilibrium, but there is a slow shift of its mean value with time [102]. From Fig. 6-6, we extract a mean relaxation time of =1.5 x 106 s. This value is closer to the mean relaxation time from the previous study, but is still two orders of magnitude lower even though the bath temperature in our experiment wa s lower ( 1.4 K Vs. 1.5 K). Again, we state that this discrepancy may be e xplained by the fact that their e xperiment was conducted in zero field, while for our experiment we wa ited at the fourth resonance field (k = 4). Such a field will

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152 bring higher lying states into resonance and an y contribution from phonons will excite spins to higher levels, making it much easier for them to tu nnel. Also, we again mention that the sample shape has a strong effect on the mean relaxation time. It is known that disorder in this system al so plays a role in the observed QTM behavior and EPR can resolve which molecules within the distri bution are tunneling. To this end, we took the traces from Fig. 6-5 and fit each spectrum to a simulation that combined two different peaks, each with a Gaussian profile. The main part of Fig. 6-7 shows an example of this fit for a wait time of 2400 s. As can be seen, the spectrum c onsists of one broad peak and one narrower peak. For each trace, we plot the peak position, peak width, and average D value for peak 1 and peak 2. Fig. 6-8a and Fig. 6-8b plot th ese quantities as a function of wait time. Since we are observing a transition taking place in the metastable well, those molecules with larger D values will transition at higher magnetic fields, hence the co rrelation between Fig. 6-8a and the inset to Fig. 6-8a. As the wait time at a resonance field is increased, the average D values of the resulting EPR peaks become larger. This suggests that those molecules with smaller D values tunnel first, while those molecules with larger D values take longer before th ey tunnel. Not only do these slower relaxing molecules have higher symmetry (no E term) than the faster relaxing molecules, but they also do not have their magnetic easy axis tilted away from the crystallographic c axis. With the field aligned parallel to c, these molecules will not have a transverse component to the external field, while the molecules with tilted axes will have such a term in addition to an E term. Both factors increase the tunneling rate for th e respective molecules. Additionally, as can be seen from Fig. 6-8b, the width of peak 1 (consisting of molecules with larger D values) decreases for longer wait times, while the width of peak 2 (consisting of molecules with smaller D values) increases for longer wait times. This implie s that the molecules that have the largest D values

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153 have a rather narrow distribution. This finding is identical to the results of the first study where we saw evidence for a narrow distribution of slow relaxing molecules for the longest wait times. One can assume that by the l ongest wait times (3600 s), most of the molecules will have tunneled, since ~ 87.5% of the molecules have a rhombic transverse anisotropy (E) term. By taking the ratio of the area of peak 1 to the tota l peak area and peak 2 to the total peak area for the data set at 3600 s, we can reinforce this a ssumption. Peak 1 represents the higher field (D value), high symmetry molecules with a narrow di stribution, while peak 2 represents the lower field (D value), lower symmetry molecules with a broa d distribution. Peak 1 constitutes ~ 11 % of the total area, while peak 2 constitutes ~ 89 % of the total area. This is in nice agreement with the Cornia model [97], which estimates th at ~ 12.5 % of the molecules maintain S4 symmetry, while ~ 87.5 % molecules will have an E term. Even though the % of higher symmetry molecules is underestimated, these molecules may take a sufficiently longer time before they ever tunnel, since they are lacking an E term, and the lowest order transverse anisotropy term allowed would be fourth order. This term will still cause tunneling, but only in higher orders of perturbation theory. 6.2.2b Magnetic Avalanches Here we briefly comment on some behavior involving magne tic avalanches observed in this experiment. A magnetic avalanche is a different kind of relaxa tion mode than QTM, exhibited by sufficiently large crystals, where a rapid magnetization reversal takes place that typically lasts a few ms or less. It was initially studied by Paulsen and Park [109] and attributed to a thermal runaway or avalanche [110]. In the avalanche, the initial relaxation of the magnetization toward the direction of the field results in the release of heat that further accelerates the magnetic relaxation. A detailed experiment involving magnetic avalanches will be discussed in section 6.4.

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154 Fig. 6-9 shows spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to 6 T from waiting for 600 s at different magnetic (not necessarily resonance) fields. A clear signature of a magnetic avalanche observed with EPR is the sudden sharp change in absorption of the microwave radiation. If this oc curs in the middle of an absorp tion peak, the signal will sharply increase as the spins change state and are no longer absorbing microwaves. Our data exhibit two regimes, one where no avalanches occur, and the other where avalanches almost always occur. For wait fi elds up to 2.45 T, no magnetic avalanches are observed. But, for all but a few fields from 2. 45 T and above, magnetic avalanches are always observed (fields marked in red in Fig. 6-9). Additionally, avalanches are also always observed for fields that induce full magnetization bi asing (sweeping back and forth from 6 T to 6 T) as is shown in the inset to Fig. 6-9. We provide an explanation for the observed behavior by considering which molecules within the distribution are tunnel ing. Since we always wait for 600 s at the wait field, only the faster relaxing molecules will have tunneled during this time period. However, as we increase the field at which we wait, we decrease the effective barrier for ma gnetization reversal and consequently, the slower relaxing molecules have more of an opportunity to tunnel. Eventually, for large enough wait fields, the effective barrie r for magnetization reversal is non existent and all the molecules tunnel, giving rise to a full bi asing of the spin states (full biasing of the magnetization). Under these conditions, we always observe an avalanche as we sweep back to the reverse bias field. To summarize, we observe no avalanches for wait fields up to 2.45 T where only the fastest relaxing species have tunn eled, and frequently observe avalanches for wait fields from 2.45 T to 6 T where the slower re laxing molecules have had a chance to tunnel.

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155 From this perspective, we hypothesize that the slower relaxing molecules are involved in triggering the avalanches in Mn12Ac. 6.2.3 Characterization of Minority Species As a final part of the analysis from this experiment, we are able to characterize the minority species that constitutes about 5% of the molecules within a crystal of Mn12Ac, shown in Fig. 6-10. These exceptionally fast relaxing mo lecules are attributed to a reorientation of the Jahn Teller axis of the one or more of the Mn+3 ions, which is oriented ~ 90 from its position in the majority species molecules [111]. This will introduce an E term and a resulting tilting of the molecular easy axis with respect to the crystallographic c axis. Referring back to the inset to Fig. 6-5, we see that while sweeping from 6 T to our wait field, we see a small EPR peak at about 1 T. This is an indication that a small population of the molecules have tunneled already! Multiple frequencies taken at 1.4 K sweeping from 0 T to 6 T allow us to extract the zero field splitting value of the minority species. This is illustrated in Fig. 6-11. As we repeat these measurements at an elevated temperature (15 K) we see multiple peaks start to appear as the excited states of the minority species become populated. The peak spacing allows us to extract a D value. Assuming a ground state spin manifold identical to the majority species (S = 10), we obtain a zero field splitting of 7.23 cm-1 and a D value of 0.38 cm-1 for the minority species. These values are significantly lower than those of the majority species (zero field splitting of 10.22 cm-1 and a D value of 0.45 cm-1), and since the minority species have significant E terms and tilting of the molecular easy axis due to the reoriented JahnTeller axis, they are exceptionally fast relaxing molecules. Experiments with AC susceptibility estimate values [1, 112] in the range of 16 cm-1 to 24 cm-1 for the effective anisotropy barrier (|D|S2) for these fast relaxing species. From this we can estimate a zero field splitting of 4.6 cm-1 and D = .244 cm-1 for the 24 cm-1 barrier value. Our values are hi gher than these, since they are

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156 measuring the effective anisotropy barrier (difference between th e ground state and highest lying level where tunneling occurs) and we are dealing with the true an isotropy barrier. More recent inelastic neutron scattering data [113] have de termined values (zero field splitting = 7.23 cm-1 and D = .29 cm-1) that are in nice agreement with our work. 6.3 Microwave Induced Tunneling Meas ured with Hall Magnetometry Now we turn to an experiment that was condu cted in order to expl ore the influence of microwaves on the magnetization dynamics of Mn12Ac. Other work [114, 115, 116] has been done studying photon assisted tu nneling in systems such as Fe8 and Ni4. We cool the sample from room temperature to 2 K in zero field and then align the sample such that its easy axis of magnetization (the c axis) is parallel to the applied magnetic field with angle dependent measurements (not shown). For all of the following data the bath temperature was 2 K, although it will be seen that the sa mple temperature can vary depending upon the microwave power incident upon the sample. A typical hysteresis loop is shown in Fig. 1-4. The steps in the magnetization seen at fields of approximately 0.45 T correspond to th e resonance fields where spins are tunneling through the anisotropy barrier. The flat plat eaus in the figure corre spond to fields where tunneling is forbidden, but the sh arp steps seen at resonant field values are where the tunneling is switched on. The data in Fig. 1-4 was taken in the absence of any external radiation. Now we turn to our studies on how micr owave radiation can affect the magnetization dynamics in this system. To begin with, we chose a frequency that corre sponds to an EPR transition between states at the same magnetic field where QTM is allowed. For instance, 0.5 T is the first resonance field and it corresponds to an anti-crossing of the ms = 10 and ms = 9 states. Additionally, for a frequency of ~ 286 GHz, there will be an EPR transition from the ms = 10 state to the ms = 9 state

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157 at 0.5 T. Consequently, as the field is swept back and forth from the minimum saturation fields (3 T) we should observe a change in the magnitude of the step at this resonance field. In Fig. 612 we plot hysteresis loops taken under the in fluence of a number of different microwave frequencies. It is clear that fo r a given frequency, the resonance fi eld step is larger in magnitude than without any microwaves at all. This is due to a multi step process initiated by the microwaves. First, EPR transitions are induced between states as the sample absorbs the incident microwaves. The relaxation of the spin back to the ground state is accompanied by an emission of phonons. This increases the effective temperat ure of the lattice, whic h increases the effective spin temperature as well, leading to thermal popul ation of higher lying leve ls. It is easier for spins to tunnel from higher lyi ng states, so the observed tunnelin g step is larger since a larger fraction of the molecules are changing their magne tization state. Additionally, only one step is affected for each frequency, indicating that it is th e EPR transition that is affecting the tunneling. While the field is swept, the microwaves do not influence the tunneling si nce they are not being absorbed by the sample. However, once the proper resonance field is met, the sample absorbs microwaves during the transition process, whic h raises the temperature of the lattice. Another thing to notice about these data sets is the small step at zero field. This is evident for the sweep without microwaves, and thus, is independent of any external radiation. The explanation for this step is the fast relaxing mino rity species of the sample that was discussed in section 6.2.3. Analogous to the EPR peaks obser ved on both sides of zero field in the insets to Fig. 6-9 and Fig. 6-11, the step near zero fiel d in the magnetization indicates that the fast relaxing species is tunneling at zero field. The height of the step indicates that the minority species contributes approximately 3 5% of th e total sample magnetization, in nice agreement with other work [112].

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158 All of the data taken in Fig. 6-12 was with continuous wave radiation, which is shown to have a significant heating effect while on resonan ce. In an attempt to minimize this, we also collected data by pulsing the microwaves. We chose to focus on the resonance at ~0.5 T since this was in a frequency range ( 286 GHz) where we could work w ith our amplifier, pin switch, and tripler components for the Gunn diode. With the amplified power output from the Gunn, care must be taken not to heat the sample. The copper block holding the device can become heated due to large microwave powers and this will transfer heat to the sample, which is an example of non-resonant heating. Additionally, when the sample is on resonance it will absorb more microwaves for larger powers, and this will lead to heating of the sample. Since we are pulsing the microwaves we will speak in terms of du ty cycle, which is the percentage of time that the sample is exposed to radiation for a given pulse period. As an example, exposing the sample to a 100 ns pulse every 1 s would correspond to a 10% duty cycle. Fig. 6-13 plots the percentage of magnetization reversal for the step at 0.5 T under the influence of 286 GHz microwave radiation for different duty cycles. Th e inset to Fig. 6-13 shows the data for some of the duty cycles. As can be seen for the larger duty cycles such as 50% and 20% not only is the amount of tunneling larger, but the coercive field of the hysteresis l oops is smaller. This is an indication that the sample is being heated while not on resonance since it has been shown that at elevated temperatures (> 2 K) the coercive field (the magnitude of the magnetic field between the saturated magnetization values ) decreases [13] due to thermally assisted tunneling as well as thermal activation over the anisotropy barrier. The greater magnitude of the step height (magnetization reversal) has to do with a larger amount of therma lly assisted tunneling while on resonance, coming from the elevated spin and lattice temperature.

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159 As mentioned earlier we wished to elimin ate any non resonant heating due to the microwaves, and to this end we selected the smal lest pulse width accessible with our instruments (12.5 ns) and a small duty cycle (1%). Data confir m that there is an immeasurable effect of non resonant heating on the sample. To investigate the microwave contribution to the tunneling, we conducted an experiment similar to those describe d in section 6.2. We would go to a resonance field, in this case 0.5 T, and wait for different times (30 s to 3600 s) to see how the magnitude of the magnetization step changed. We also contin uously recorded the decay of the magnetization signal during the wait time. This was done for the above mentioned duty cycle, as well as without microwaves. For data taken in the absence of any microwave radiation, the only contribution to the tunneling can come from the fi nite lattice temperature. On the other hand, data taken while exposing the sample to microwav es has contributions to the tunneling from the finite lattice temperature and the incident microw aves. Since the bath temperature was identical for both data sets, we can subtract the diffe rence to get the microwave contribution to the tunneling. For the subsequent discussion, when we mention a “difference”, we are specifically referring to the difference between the data set ta ken in the absence of any microwave radiation, and those taken with the 1% duty cycle microwave radiation. Fig. 6-14 plots the difference in the magnetization reversal for the data sets take n with and without microwaves as a function of wait time at 0.5 T. The scatter points in Fig. 614 plot the difference in percent of magnetization reversal for different wait times by measuring the st ep height. Similarly, the curve in Fig. 6-14 plots the difference in percent of magnetiza tion reversal by measuring the decay of the magnetization for the wait time of 3600 s. Th e two are qualitatively similar, as would be expected. The data set taken with microwaves shows larger magnetizati on reversal for a given wait time, indicating that the microwaves influen ce the magnetization dynamics even with such a

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160 small duty cycle. However, it is unlikely th at the increased tunneling is solely due to the microwaves. Such tunneling would take place between the ms = 9 and ms = –8 states, while tunneling between higher lying states is possible due to increased lattice and spin temperatures from resonant heating. The latter scenario is much more probable than the former, due to respective tunnel splittings (Eq. 1-15). In order to qualitatively describe the effect of microwaves on the tunneling in Fig. 6-14, we consider the fact that at this bath temperature (2 K), we are in the regime where the relaxation of the magnetization should begin to fo llow a stretched exponential decay ( () 0()tMtMe ). In this regime the mean relaxation time, begins to follow an Arrhenius law ( = 0exp(–E/T)) and hence, it is temperature dependent. Under the influence of the microwav es, the spin and lattice temperatures will increase, which will lead to a smaller value of We can compare the relaxation for two different values of (1 and 2), corresponding to the situation when the microwaves were on or off, respectively. Sin ce the spin and lattice temperatures are higher for the case of microwaves on, 1 will be smaller than 2. Consequently, the magnetization will decay quicker for a given time than for the case of the microwaves off (2). In Fig. 6-14 we are taking the difference of the magnetization decay between the situation w ith microwaves on and off, so the magnetization decay should be of the form 12()() 0()ttMtMee By doing a Taylor expansion of the exponentials an d keeping only the lowest order terms, the magnetization will go like 12 0 12() MtMt Since the term in brackets is a constant, the magnetization is proportional to t, which in this regime is close to 0.7. From this we see that the influence of the microwaves on the tunneling is consistent with the above mentioned arguments.

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161 We have presented work stud ying the influence of microw ave radiation on the quantum tunneling in Mn12Ac. The tunneling at a given resonant field is enhanced when the microwave frequency corresponds to transiti ons between states at this res onant field. This resonant enhancement is due to microwave absorption lead ing to an increase in the lattice and spin temperatures, and consequently, thermally assisted tunneling. By reducing the amount of power incident upon the sample through low duty cycle pulsed radiation, we are able to reduce the resonant heating and eliminate the non resonant heating of the sample. Comparison of resonant tunneling for different wait times with and withou t the presence of extern al microwaves allows us to observe the contribution of microwaves to the tunneling. It was shown by comparing the difference in magnetization reversal with and wi thout the influence of microwave radiation that the observed behavior can be attributed to different temperature dependent mean relaxation times. The different temperatures arises from resonant heating of the spin and lattice temperatures during the microwave absorption process. 6.4 Relaxation in Mn12Ac Measured with HFEPR The observed magnetic avalanches from the experiment in section 6.2 were processes that were randomly occurring and not amenable to manipulation. Howeve r, other research has been conducted studying this phenomenon in Mn12Ac under quite controlled circumstances. It was first studied by Paulsen and Park [109] and attributed to a thermal runaway or avalanche [110]. Recent studies analyzed the stochasticity of the process and the spatial dependence, dealing with avalanches as a deflagration proce ss [117, 118]. In that set of experiments, avalanches were triggered in a similar way to how they were triggered in the experiment discussed in section 6.2.2 (although in our case it was not intentional) by sweeping the magnetic field back and forth from 6 T to +6T until an avalanche was triggered. Once triggered the avalanche propagates through the crystal in a similar fashion to a flame propagating through a

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162 flammable substance. Consequently, this phenom enon has been termed magnetic deflagration. The magnetization reversal occurs inside a narro w interface that propagates through the crystal at a velocity of a few m/s. The speed of the de flagration is determined by the thermal conductivity of the crystal, as well as the rate of thermal activation over the anisotropy barrier. From a thermodynamic perspective, when the spins are biased by a magnetic field such that all are in a metastable well, this can be considered as a flammable chemical substance. The chemical energy per molecule that is releas ed during the spin reversal is equivalent to the Zeeman energy, gzBBzS, where S corresponds to the difference between the states of the system that are parallel and antiparallel to the applied field, Bz. Recently, a new technique was developed us ing surface acoustic waves to ignite the magnetic deflagration associated with the spin av alanches at a determined value of the applied magnetic field [119]. In collaboration the group from the University of Barcelona who pioneered this technique, we were able to comb ine their controlled avalanche equipment with our HFEPR setup and perform unique experiments on Mn12Ac. Magnetic deflagration has been measured through magnetization with Hall bars coils, or SQUIDs. Although these methods may provide spatial resolution, none of them allow analysis of the avalanche in a single energy level. Our novel technique uses surface acoustic waves (SAWs) from a piezoelectric device to push the system out of thermal equilibri um and microwaves from our M VNA as a probe to study spin relaxation dynamics. Other work has been done to measure relaxati on times in SMMs [115, 120, 121, 122] such as Ni4 and Fe8. All of these studies involved measuring magnetization dynamics in the presence of microwave radiation. In contrast, our t echnique uses microwave radiation to measure dynamics in the system after sending it out of equilibrium with a SAW. EPR measurements

PAGE 163

163 probe the spin population differences between ener gy levels and hence, our data provide energy resolved information about how spins relax to equilibrium in Mn12Ac when an avalanche is triggered with a SAW pulse. We discuss two experiments where we sh ow how EPR signals can probe the time evolution of spins as they relax to equilib rium. The first involves measuring the thermal population difference between single energy levels during the defl agration process. The second involves measuring the thermal population diffe rence between single energy levels after application of a short heat pulse to the system. In both cases the system is put into a non equilibrium state due to coherent acoustic vibrations. These vibr ations in turn introduce phonons into the system, which thermalize with the sample The electronics and instrumentation for this experiment is described in detail in Ch. 2. 6.4.1 Triggered Avalanches in Mn12Ac We cooled the system in zero field to approximately 2.1 K. The sample was aligned such that the easy axis of magnetization was appr oximately parallel (within 1) to the applied magnetic field. This was deduced from prelimin ary angle dependent measurements (not shown). We use a large applied magnetic field (>3 Tesla) to put all of the spins in the ground state (either ms = 10) of one of the wells. Next we chose which EPR transition to observe and tuned the frequency to the corresponding value. Once an avalanche was triggered in the system, we would observe a change in the abso rption spectra as the spins m ove from the ground state in the metastable well to the ground st ate in the stable well. The response of the system is measured by a change in the absorption of the incident microwaves. Typically, the signal returning to the receiver is a relatively constant value until the resonance conditions are met, and then a drastic reduction of the signal is seen as the sample

PAGE 164

164 absorbs the microwaves. Once an avalanche was triggered in the system, we would observe a change in the absorption as the spins move from the ground state in the metastable well to the ground state in the stable well. This is illustrate d in Fig. 6-15. The black arrows represent the evolution of the spins as they avalanche. Th e red arrows represent the EPR transition between states during an avalanche. We can see the avalanche by recording the absorption due to transitions between levels in either of the ener gy wells (metastable or stable) and there is a significant difference between these kinds of transitio ns. First, it is worth noticing that there is a significant amount of heat released during an aval anche, and increases in temperatures between 6 and 12 K have been measured [117, 119], depending on the value of the applied magnetic field at which at the avalanches occur. This heating cause s the system to be out of equilibrium after all spins have avalanched, and an elapsed time of hu ndreds of milliseconds is needed to recover the initial bath temperature, during which the excited sp ins relax to equilibrium in the ground state of the stable well. The main difference, therefore, between the two kinds of transitions is that the heat dissipation after the appear ance of the avalanche hardly effects the spin population within the metastable well since these transitions take pl ace before all spins have changed to the stable well. It is true that higher lying states start to become ther mally populated, but only before the spins avalanche over to the stable well. Hence, th e lifetimes of these states are mostly decoupled from the phonons arising from the avalanche propagation. On the other hand, the transitions within the stable well are greatly influenced by the heat dissipation since both the ground state and higher lying levels become populated. For the purposes of discussing the data we shall refer to a reference level of zero microwave absorption and the initial values of th e EPR signal are presented with arbitrary offset units. The EPR signal that we plot is the transm itted microwaves returning to the detector after

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165 interacting with the sample. When there is no absorption from the sample the signal should be at a zero baseline (within the oscillations of the noise) On the other hand, the signal will be below this reference level when the sample is absorb ing. Fig. 6-16 shows th e time evolution of the transition between the ms = 10 to 9 (Fig. 6-16a) and ms = 9 to 8 (Fig. 6-16b) states in the metastable well after igniting an avalanche. Th e flat dashed line in each graph represents the level of zero microwave absorption In (a) the initial conditions we re such that all the spins were in the metastable well and should be absorb ing microwaves as they transition to the ms = 9 state. We can see that this is indeed the case until a few milliseconds after the SAW pulse is applied when the avalanche appears. Since all of the spins have avalanched to the stable well there is no more absorption. In (b) we can see that initially, at t = 0, there is no absorp tion of the signal, but a sharp absorption peak appears a few milliseconds after the ignition of the avalanche. As the spins move up from the ground state in the metast able well they spend a brief amount of time in the ms = 9 state where we see transitions from the ms = 9 to 8 states. This is related to the time that the spins in the ms = 9 state need to overcome the barrier to get the stable state. Initial and final EPR absorption values are the same becau se before the avalanche all spins populated ms = 10 level and after the deflagration there are no more spins in the metastable well. As a contrast, after recording the data from an avalanche we w ould pulse the IDT in order to heat the sample with a pure SAW. Neither si tuation shows any absorption under these circumstances, as would be expected, since the avalanche has taken place and all the spins ar e in the stable well while the frequency and magnetic field are tuned fo r a transition in the metastable well. Fig. 6-17a and Fig. 6-17b show the EP R signal of the transition between the ms = 10 to 9 (Fig. 6-17a) and ms = 9 to 8 (Fig. 6-17b) states in the stable well. In Fig. 6-17a we see that initially there is no absorption of the signal, but once the avalanche occurs there is a strong

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166 absorption. This can be pictured as the reverse of Fig. 6-16a. Since the spins all relax to the ground state in the stable well we observe contin uous absorption after the avalanche as the spins transition from ms = 10 to 9. Fig. 6-17b shows different behavior. Again, the avalanche always takes some time on the orde r of a few ms after the trigger pulse before it is ignited, as can be seen from the EPR signal and there is no abso rption until the avalanche occurs. As the spins avalanche they pass through the ms = 9 state before relaxing to th e ground state. The sharp, short time side of the absorption is the transition between the ms = 9 to 8 states. However, it is clear by the behavior of the signal that there is still some absorption for approximately 100 ms after the avalanche. This can be understood by th e fact that the ignition of an avalanche releases a significant amount of heat into the sample. H eating of the sample during this process causes the ms = 9 state to be thermally populated and is th e reason for the prolonged absorption of the microwaves. This extra heat takes some time to disperse and re-establish equilibrium in the system. While this is happening the ms = 9 state will be thermally populated and will continue to absorb microwaves. Eventually the phonons disperse, the spins only populate the ground state of the stable well, and the EPR signal returns to its initial value (after ~ 100 ms) as there is no longer any absorption from the sample and the system comes to thermal equilibrium with the bath. For comparison, after triggering an av alanche, we would again pulse the piezoelectric device to produce a SAW to heat the sample. The re d traces in the figures show this. It is clear then that there is a distinct difference in th e EPR signal when comparing its evolution after triggering an avalanche, and after pure heating with a SAW. The slope of the signal after an avalanche is steeper than that of the signal af ter pure SAW heating. Additionally, the SAW directly couples with the sample and the signal begins to change immediately, while the change of the signal due to an avalanche takes a fe w ms before the avalanche is ignited.

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167 The phonon bottleneck effect describes the f act that at low temperatures the number of spins is greater than the number of phonons, due to the lattice temperature [116]. Although the spins may be able to thermalize with the latti ce quite easily, poor coupling between the lattice and bath causes a prolonged relaxation of the spin temperature. Thus, the spins can not freely exchange energy with the bath and it takes some ad ditional time to equilibr ate the spin and lattice temperature with the bath temperature. The data we collected for tran sitions within the stable well support this assertion. The spin temperature should follow the lattice temper ature quite quickly, but the time for the lattice to equilibrate with the environment may be ra te limited by a bottleneck effect if the coupling between the lattice and bath is weak. The times for establishing equilibrium in the stable well agree well with other studies [123] showing times on the order of 100 ms. However, for transitions within the metastable well, we are able to observe dynamics that are not hampered by this effect. This is because we take our measurement before the system has fully reversed its magnetization state, and therefore has not yet releas ed the amount of heat in to the system that is typical of an avalanche. The dynamics associated with population differences in the metastable well as the avalanche propagates are on the order of 5 ms and we can infer that the time needed to excite spins to higher levels is on the order of 1 ms. Since we were using a lock in amplifier with a time constant of 100 s, from Fig. 6-14a we can see that if T1 were less than 100 s, then we would not observe the resonance peak, which is seen to span a width of ~ 5 ms. Thus, we can place an upper bound for the spin lattice relaxation time, T1, on the order of 0.1 1 ms. 6.4.2 Pulsed Heating and Spin Lattice Relaxation The second kind of experiment we performed involved using short SAW pulses to heat the lattice, which in turn heats the spins in the ground state of the stable well. This does not trigger an avalanche but does perturb the system such th at higher lying energy states in the stable well

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168 become populated briefly before relaxing back to the ground state. The process of exciting to higher levels and relaxing back down is probed with our low power microwave radiation. We would choose a frequency and hold a specific valu e of magnetic field for the respective transition that we wished to observe. We then repeat th e pulses at a particular frequency and average the signal in order to improve the signal-to-noise, as di scussed in section 2.4.1. Pulses of 1 s to 50 ms and a nominal power of 6 dBm were used to induce heating of the sa mple and this was done for a number of different bath temperatures. Fig. 6-18 shows a plot of the EPR signal as a function of time for the ms = 9 to 8 transition. In contrast to the signal after tr iggering an avalanche, the signal here evolves smoothly without any sharp changes. As soon as the pulse is applied (t = 0) there is a decrease in the signal due to the ms = 9 state becoming thermally populated and spins transitioning to the ms = 8 state. The population difference between these states reaches a maximum after approximately 10 ms. We can see this by the fa ct that the microwave absorption is a maximum at this time, and the amount of absorption is a measure of the spin population difference between respective levels. Beyond this point the system is no longer heated by the SAW and the phonons begin to disperse. The time for the excited leve ls to thermally depopulate is on the order of 100 ms, which is the time for the system to return to equilibrium due to the phonon bottleneck effect. During this process, the populations of the ms = 9 and ms = 8 states decay and eventually are zero. In order to quantify the observed behavior we assume that there are two important temperatures during the relaxation process: the lattice temperature, TL, and the spin temperature, Ts with respective relaxation time constants L and s. The time rate of change of the spin and lattice temperatures is given by [124]

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169 0()()()L L LsdT WtTTTT dt (6-3a) ()s L sdT TT dt (6-3b) In Eq. 6-3, and are constants of heat transport, T0 is the bath temperature, TL is the lattice temperature, TS is the spin temperature, and W(t) is a pulse function that introduces heat into the system. l corresponds to the time related to temperat ure variations of the lattice when the heat pulse is switched on and off. s corresponds to the time the spins need to follow these temperature variations of the lattice, which is related to the spin lattice relaxation time, T1. Eq. 6-3 is made linear by neglecting the higher order terms (quadratic, cubic, etc). This assumption is valid since the differences in the spin and la ttice temperatures are small during the period of the measurement, as the spin temperature can foll ow the lattice temperatur e quickly. We observe lattice decay times (l) on the order of 100 ms when the heat pulse is switched on and off, and it is estimated [125] from simulations using Eq. 6-3 that s is 10-100 times faster than l. This was done using Eq. 6-3 to simulate th e evolution of the magnetization after a SAW was applied to the crystal. A reasonable simulation of the data was obtained when the values of s is 10-100 times faster than l. Therefore, from our experiment, we estimate s is on the order of 0.1 1 ms. Once again, in agreement with the data from the avalanche experiments, we can estimate an upper bound of 0.1 1 ms for T1 in this system. A better agreement with the data may be obtained from simulations that include higher order terms in Eq. 6-3, especially those including the difference between the bath and lattice temperatur es, which can be significant at longer times. In this work we have demonstrated a novel technique to monitor sp in population dynamics by combining the techniques of SAWs and HFEPR. We are able to detect how the spins excite and relax on rather fast times scales for different spin levels. By measuring the lifetimes of states

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170 within both the metastable and stable wells as an avalanche propagates, we can obtain information about the spin latti ce relaxation time in this SMM. Our results indicate an upper bound of 0.1 1 ms for T1 in a single crystal of Mn12Ac. 6.5 Summary In this chapter we discussed a numb er of unique experiments done on the Mn12Ac SMM in order to characterize the quantum tunneling observed in this system. First we reported on work done using HFEPR to detect quantum tunneling. This technique allows us to observe different molecules within the distribution tunneling on di fferent time scales. We conducted experiments in two different temperature regimes in order to make comparisons with the PS theory of magnetic relaxation in SMMs. Our results show th at there is indeed a low temperature regime where the relaxation goes like t1/2 and a higher temperature regime where the relaxation follows a stretched exponential law, in accordance with predications made by PS theory. In both regimes we see that the molecules with smaller D values and lower symmetry tunnel much sooner than those with larger D values and higher symmetry. Additionally, we were able to characterize the zero field splitting and average D value of the minority species of molecules present in the crystal. These molecules have lower symmetry, a D value approximately 16% smaller than the majority species of molecules and it is likely that they have a significant tilting of their molecular easy axis. This is consistent with the rapid ze ro field tunneling observed. Next, we presented work studying the influence of microwave radi ation on the quantum tunneling. Using both CW radiation and short pulses of microwaves at low duty cycles we are able to use microwaves to influence the tunneling. Even without signifi cantly heating the system due to non resonant effects we see evidence for thermally assisted tu nneling to due to increased spin temperatures from microwave absorption. By comparing the difference in magnetizati on reversal with and

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171 without the influence of microwave radiation, we attribute the observed behavior to different temperature dependent mean relaxation times. Finally, we described a novel experimental technique that was developed in order to measure the relaxation processes on quite fast times scales during a magnetic avalanche. We are able to record EPR transitions during the avalanch e propagation in both the metastable and stable wells of the system. The transitions in the me tastable well are largely decoupled from the increased spin and lattice temperatures due to th e heat released during the avalanche. We also are able to heat the system with a short pulse and observe the relaxation back to equilibrium by recording the magnitude of the EPR signal. These experiments provide information on the relaxation times associated with individual energy levels and from this we can put an upper bound of 0.1 1 ms for the spin lattice relaxation time, T1, in this system.

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172 Figure 6-1. Diagram of the six different Mn12 isomers, each with a different D value and some with E values. The cyan arrows repr esent hydrogen-bound acetate ligands.

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173 Figure 6-2. Plot of the energy levels in Mn12Ac as a function of magnetic field. The blue arrows represent the biased magnetization of the system for a sufficient (3 T) magnetic field. At each resonance field (k = integer), spins can reverse their state by tunneling.

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174 Figure 6-3. 2 K EPR spectra for different waiting times at 0.9 T. Each trace involves sweeping from 0.9 T to 6 T after a different waiting time. The inset to Fig. 6-1 shows the sweep from 6 T to 0.9 T, which was identical for all data sets. The area of the peak occurring on the positive fiel d side decreases for longer wa it times, while the area of the peak occurring on the negative field side increases accordingly, indicating QTM. Arrows indicate the direction of increasing wait time. Reused with permission from J. Lawrence, S.C. Lee, S. Kim, S. Hill, M. Murugesu, and G. Christou, Magnetic Quantum Tunneling in a Mn12 Single-Molecule Magnet Measured With High Frequency Electron Paramagnetic Resonance, AIP Conference Proceedings 850, 1133 (2006). Fig. 1, pg. 1134. Copyright 2006, American Inst itute of Physics.

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175 Figure 6-4. Plot of the area of the positive fi eld peak as a function of wait time. This curve can be fit to a stretched exponential of the form Aoexp[(1/ )]. We estimate an average relaxation time, on the order of 500 s. Reused with permission from J. Lawrence, Magnetic Quantum Tunneling in a Mn12 Single-Molecule Magnet Measured With High Frequency Electron Paramagnetic Re sonance, AIP Conference Proceedings 850, 1133 (2006). Fig. 2, pg. 1134. Copyright 2006, American Institute of Physics.

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176 Figure 6-5. Plot of emerging resonance p eak for different wait times at 1.8 T and sweeping back to 6 T. The inset shows a sweep from 6 T to 1.8 T, which was identical for all traces. If one compares this to Fig. 6-3, one sees that the same qualitative trends are observed.

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177 Figure 6-6. Plot of the peak area in Fig. 6-5 as a function of wait time. The peak area is proportional to the magnetization of the sample and in this regime the magnetization is shown to follow a square root relaxation law.

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178 Figure 6-7. Plot of the sp ectrum for a wait time of 2400 s after fitting it to a simulation that combined two different peaks, each with a Gaussian profile. As can be seen, the spectrum consists of one broad peak and one narrower peak. Different molecules within the distribution contribute to the spectrum on different time scales.

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179 Figure 6-8. Average D value and peak width vs. time. a). Plot of the average D value for peak 1 and peak 2 as a function of wait time for each trace in Fig. 6-5. The inset plots the peak position as a function of wait time fo r each trace in Fig. 6-5. Those molecules with larger D values will transition at higher magnetic fields, hence the correlation between Fig. 6-8a and the inset to Fig. 68a. Fig. 6-8a also suggests that those molecules with smaller D values tunnel first, while those molecules with larger D values take longer before they tunnel. b). Plot of the peak width as a function of wait time for each trace in Fig. 6-5. The width of peak 1 (consisting of molecules with larger D values) decreases for longer wait times, while the width of peak 2 (consisting of molecules with smaller D values) increases for longer wait times. This implies that the molecules that have the largest D values have a narrow distribution.

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180 Figure 6-9. Plot of spectra taken at 237.8 GHz and 1.4 K after sweeping the field back to 6 T from waiting for 600 s at different magnetic (not necessarily resonance) fields. A clear signature of a magne tic avalanche observed with EPR is the sudden sharp change in absorption of the microwave radiation. Those fields where an avalanche occurred are marked in red. The inset shows that after full magnetization reversal (sweeping back and forth from 6T), an avalanche is always observed. We hypothesize that the slower relaxing molecules (larger D value) are triggering the avalanches.

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181 Figure 6-10. Diagram of the minority species molecules (bottom) in Mn12Ac. In comparison to the majority species molecules (top), the mi nority species has a reorientation (~ 90) of the Jahn Teller axis of one or more of the Mn+3 ions. This introduces an E term, which explains the observed fast relaxation. There are two dashed lines because the molecule has a crystallographic C2 axis disorder. Reprinted from S. M. J. Aubin, Z. Sun, H. J. Eppley, E. M. Rumberger, I. A. Guzei, K. Folting, P. K. Gantzel, A. L. Rheingold, G. Christou, D. N. Hendrickson, Single molecule magnets: Jahn-Teller isomerism and the two magnetization relaxation processes in Mn12 complexes, Polyhedron 20, 1139-1145 (2001). Fig. 5, pg 1143. Copyright 2001, with permission from Elsevier.

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182 Figure 6-11. Plot of the low temperature freq uency dependence of the minority species peak. Assuming a ground state spin manifold identical to the majority species (S = 10), we obtain a zero field splitting of 7.23 cm-1 and a D value of 0.38 cm-1 for the minority species.

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183 Figure 6-12. Plot of hysteres is loops taken under the influe nce of a number of different microwave frequencies. The bath temperat ure was 2 K for each sweep. It is clear that for a given frequency, the resonance field step that corresponds is larger in magnitude than without any microwaves at all. Additionally, only one step is effected for each frequency, indicating that it is the EPR transition that is effecting the tunneling. While the field is swept, the microwaves do not influence the tunneling since they are not being absorbed by th e sample. However, once the proper resonance field is met, the sample rapi dly absorbs photons and emits phonons during the transition process. This emission of phonons heats the sample, which thermally populates higher lying levels. It is easier for spins to tunnel from higher lying states, so the observed tunneling step is larger since a larger fraction of the molecules are changing their magnetization state.

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184 Figure 6-13. Plots of the per centage of magnetization reversal for the step at 0.5 T under the influence of 286 GHz microwave radiation for different duty cycles. The inset shows the data for some of the duty cycles. As can be seen for the larger duty cycles such as 50% and 20% not only is the amount of tunnelin g larger, but the coer cive field of the hysteresis loops is smaller. This is an in dication that the sample is being heated while not on resonance since it has been shown that at elevated temperatures (> 2 K) the coercive field decreases due to thermally assisted tunneling as well as thermal activation over the anisotropy barrier. The greater magnitude of the step height (magnetization reversal) has to do with a larger microwave absorption on resonance.

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185 Figure 6-14. Plot of the difference in the magnetization reversal for the data sets taken with and without microwaves as a function of wait time at 0.5 T. The scatter points plot the difference in percent of magnetization reversal for different wait times by measuring the step height. Similarly, the curve plot s the difference in percent of magnetization reversal by measuring the decay of the magnetization for the wait time of 3600 s.

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186 Figure 6-15. The energy barrier diagram illustra ting how spins move during an avalanche. The black arrows represent the evolution of the spins as they avalanche. The red arrows represent the EPR transition be tween states during an avalanche. Spins in the right well relax to the ground state by emitting phonons, which produces heating of the system. This heating is fed back into the avalanche as it is absorbed by the spins in the left well.

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187 Figure 6-16. Plots of transitions within the me tastable well during an avalanche. a). All the spins begin in the metastable well and sh ould be absorbing microwaves as they transition to the ms = 9 state. We can see that this is indeed the case until a few milliseconds after the SAW pulse is applied when avalanche appears. Since all of the spins have avalanched to the stable well there is no more absorption. b). At t = 0, there is no absorption of the signal, bu t a sharp absorption peak appears a few milliseconds after the ignition of the avalan che, which is the transition from the ms = 9 to 8 states. Initial and final EPR absorp tion values are the same. As a contrast, after recording the data from an avalanche we would pulse the IDT in order to heat the sample with a pure SAW. Neither si tuation shows any absorption under these circumstances, as would be expected.

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188 Figure 6-17. Plots of transitions within the stable well during an avalanche. a). Initially there is no absorption of the signal, but once the aval anche occurs there is a strong absorption. Since the spins all relax to the ground st ate in the stable well we continuous absorption after the avalanche as the spins transition from ms = 10 to 9. b). As the spins avalanche they pass through the ms = 9 state before relaxing to the ground state. The sharp, short time side of the absorption is the transition between the ms = 9 to 8 states. However, it is clear by the beha vior of the signal that there is still some absorption for approximately 100 ms after the avalanche. This can be understood by the fact that the ignition of an avalanche releases a significant amount of heat into the sample, which causes the ms = 9 state to be thermally populated and this extra heat takes some time to disperse.

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189 Figure 6-18. Plot of the EPR si gnal as a function of time for the ms = 9 to 8 transition for a frequency of 269 GHz and a heat pulse of 5 ms. As soon as the pulse is applied (t = 0) there is a decrease in the signal due to the ms = 9 becoming thermally populated and spins transitioning to the ms = 8 state. The population difference between these states reaches a maximum after approximately 10 ms. Beyond this point the system is no longer heated by the SAW and the phon ons begin to disperse. The time for the excited levels to thermally depopulate is on the order of a few hundred ms, which is the time for the system to return to equ ilibrium. The dashed curve is a simulation using Eq. 6-3.

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190 CHAPTER 7 SUMMARY This chapter gives a summary of the work pr esented in the previous chapters of this dissertation. This Ph.D. dissertation is focuse d on low temperature, hi gh field, high frequency magnetic resonance spectroscopic studies of various SMMs. Chapter 1 is an introduction to the class of compounds known as single molecule magnets. We explained the sources and importance of an isotropy to the magnetic behavior. We also described a unique feature of SMMs: the observ ed quantum tunneling in hysteresis loops of a classical object. Quantum tunneling is only a llowed when sources of transverse anisotropy and/or transverse fields are present. Two tunneling regimes were outlined and we stressed that the main goal of studying these materials is to gain a deeper understanding of the magnetic quantum tunneling behavior. Chapter 2 presents the experimental techni ques and equipment that we use in our research. First, we briefly discussed EPR in th e context of SMM systems. Next, we explained the two types of cavities we use for a cavity perturbation technique and our main instrument that acts dual microwave source and network analyzer. This source, in conjunction with an external Gunn diode and complimentary components (amplifie r, switch, frequency tripler) allow for high power pulsed microwaves. Additionally, our qu asi optical bridge setup provides a low loss propagation system that relies on coupling of a free space Gaussian profile to an HE11 mode in corrugated horns and a tube. The corrugated sample probe is used to conduct experiments that integrate other devices into the setup and co mpliment our normal EPR studies by combining microwaves with either surface acoustic wa ves or Hall magnetometry to conduct unique experiments on single crystals. Finally, we outlined the process of fabricating the Hall magnetometer used in our experiment in section 6.3.

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191 Chapter 3 gives the basic theoretical foundations for the origin of the spin Hamiltonian that is used to model SMM systems. The Hamiltonian corresponding to a giant spin approximation is certainly the most widely used form and applies to ideal system s where the isotropic exchange interaction is much larger than the anisotropic interactions. For systems where this assumption starts to break down, we can employ a coupled single ion Hamiltonian that in spite of being more complicated of a model, can provide information on the nature of the exchange interactions between metal ions in a molecular cluster. Chapter 4 outlines HFEPR studies done on a tetranuclear Nickel SMM, [Ni(hmp)(dmb)Cl]4. First we reported on experiments to characterize the spin Hamiltonian parameters for this SMM. Data reveal sharp, symmetric EPR lines due to the lack of solvate molecules in the crystal lattice and large interm olecular exchange pathway distances. However, variable frequency, variable temperature meas urements have revealed the presence of two distinct molecular species within the crystal an d we are able to extract the relevant spin Hamiltonian parameters for each species. Below 46 K the peak splits into two, which we attribute to differences in the molecular envi ronments arising from different t-butyl group conformations in the dmb ligand. At high temperatures the motion of these is thermally averaged, but below 46 K the motion freezes out and the t-butyl group takes on two distinct orientations. These EPR peaks are then split due to the effects of the disorder. Additional low temperature data (< 6 K) reveal additional splittings and broa dening of the peaks, which we attribute to short range intermolecular exchange interactions among neighboring molecules that are averaged out at higher temperatures. It is likely that exchange interactions provide an additional contribution to the line widths/shapes, i.e. exchange probably also contributes to the broad lines. However, given the minimal amount of disorder in this system, we are able to

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192 separate the various contributions (disorder and intermolecular exchange) to the EPR line shapes. Finally, we are able to measure the magnitude of the isotropic excha nge coupling constant, J, with our HFEPR data. By simulating the intensity of peaks originating from transitions within a low lying excited state spin multiplet to a mode l that includes both isotropic and anisotropic interactions we obtain a value of J = 6.0 cm1. This provides insight into the spacing between the ground state and higher lying spin multiplets, and supports the evidence that the isotropic and anisotropic parameters (J and d) can cause mixing between states unless J >> d. Such mixing manifests itself as unphysical higher order terms in the Hamiltonian with the giant spin model. Chapter 5 deals with experiments done on two Cobalt systems ([Co(hmp)(dmb)Cl]4 and [Zn3Co(hmp)4(dmb)4Cl4]) that are similar to the Nickel sy stems discussed in chapter 4. The CoZn study provides crucial information as to the orientation and magnitudes of the anisotropy parameters in the tetranuclear system. We then present HFEPR data taken on the tetranuclear system, which exhibits complicated spectra that can not be modeled by a giant spin Hamiltonian. We mention the possible contributions of the symmetric and antisymmetric exchange interactions to the observed zero field splitting anisotropy and concl ude by remarking that this is still a work in progress by a collaborator. Chapter 6 presents numerous stud ies done on a manganese based SMM, [Mn12O12(CH3COOH)16(H2O)4] 2CH3COOH4H2O, or just Mn12Ac. First, we showed research that we did using HFEPR to measure QTM. Th is allowed us to investigate how different molecules within a crystal relax on different time scales. These studies were done in two temperature regimes that exhibit different relaxa tion laws. We also characterized the ZFS and D value of the minority species. Next, we talk ed about a study that combined Hall magnetometry with HFEPR to measure the effects of microwaves on th e magnetization dynamics. The

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193 tunneling at a given reso nant field is enhanced when the microwave frequency corresponds to transitions between states at this resonant field. This resonant enhancement is due to microwave absorption leading to an increase in the la ttice and spin temperatures, and consequently, thermally assisted tunneling. By reducing th e amount of power incident upon the sample through low duty cycle pulsed radiation, we are ab le to reduce the resonant heating and eliminate the non resonant heating of the sample. Comp arison of resonant tunneling for different wait times with and without the presence of extern al microwaves allows us to observe the contribution of microwaves to the tunneling. Finally, we discussed a unique study combining SAWs and HFEPR in order to measure fast rela xation dynamics after the system was pushed out of equilibrium. In addition to being a novel technique, we are able to get time resolved information about the how spins relax in a single en ergy level in this system. From this research we are able to estimate an upper bound of 0. 1-1 ms for the spin lattice relaxation time, T1.

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202 BIOGRAPHICAL SKETCH Jon Lawrence was born in Kalamazoo, Michigan After graduating high school in June of 1997 he enrolled at Grand Valley State University in Allendale, Michigan where he received his bachelors of physics degree in May of 2002. Fo r his senior project he was involved in the synthesis of LaCaMnO and the characterization of its colossal magnetoresistive properties. In August of 2002 he started graduate school at the University of Florida in Gainesville, Florida. In May of 2003 he jo ined the research group of Dr. St ephen Hill. For his career as a graduate student he learned an enormous amou nt about being an experimental physicist and worked on many different projects, including the magnetic resonance spectroscopy studies discussed in this dissertation, as well as the design and construction of a low temperature probe and cryostat. After 4.5 years of hard work, he completed all of the proj ects necessary for his Ph.D. in experimental physics. He re ceived his degree in December of 2007. After graduation he began the ne xt stage of his life, using a ll of the vast experience and knowledge obtained from his physics degree. He promises to only use these powers for good.