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Angle-Dependent High Magnetic Field Microwave Spectroscopy of Low Dimensional Conductors and Superconductors

Permanent Link: http://ufdc.ufl.edu/UFE0012927/00001

Material Information

Title: Angle-Dependent High Magnetic Field Microwave Spectroscopy of Low Dimensional Conductors and Superconductors
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0012927:00001

Permanent Link: http://ufdc.ufl.edu/UFE0012927/00001

Material Information

Title: Angle-Dependent High Magnetic Field Microwave Spectroscopy of Low Dimensional Conductors and Superconductors
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0012927:00001

Full Text











ANGLE-DEPENDENT HIGH MAGNETIC FIELD MICROWAVE
SPECTROSCOPY OF LOW DIMENSIONAL CONDUCTORS AND
SUPERCONDUCTORS
















By

SUSUMU TAKAHASHI
















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


2005


































Copyright 2005

by

Susumu Takahashi

































To my parents, Koh and Teruko Takahashi,

and my family, Ryoko, Kai and Riku















ACKNOWLEDGMENTS

Since the fall of 2001, I have spent a great amount of time working on my

research with many people at the University of Florida (UF) and other places.

Without their assistance, encouragement and guidance, I could not have completed

this dissertation.

First, I would like to thank my advisor, Professor Stephen O. Hill. I have

received numerous benefits by his patient guidance and continuous support over

three and a half years. Steve's enthusiastic discussion has alv--b i encouraged me

to tackle difficult but creative research projects. I was also supported by Steve

to perform many experiments at the National High Magnetic Field Laboratory

(NHMFL), Tallahassee, FL, and to attend conferences in various places. These

experiences are my priceless treasure.

I also would like to express my thanks to other faculty at the University of

Florida. I thank my supervisory committee, Prof. David Tanner, Prof. Peter J.

Hirschfeld, Prof. Mark W. Meisel, and Prof. Daniel R. Talham, for a number of

useful discussion and valuable comments. I also thank Prof. Amlan Biswas for

providing a PCCO sample. I must also thank the technical staff at the University

of Florida. In particular, I thank the machine shop for making a rotating cavity

and giving great -,i:-.- -1 i. us for designing the apparatus.

Many experiments in this thesis were carried out at the NHMFL. I would like

to thank the scientists and staff for supporting our experiments. In particular, I

would like to thank Prof. James Brooks, Dr. Louis Claude Brunel and Dr. Hans

Van Tol for kindly lending equipment for our experiments at the NHMFL.









I also had the great fortune to have discussions with and receive -ii-:.- -1 i ,s

from many conference attendees and visitors. I am especially grateful to Dr.

Phillipe Goy of ABmm, Prof. Toshihito Osada of the University of Tokyo, Prof.

Woun Kang of Ewha Womans University, Prof. Victor M. Yakovenko of the

University of Maryland and Prof. Andrei G. Lebed of the University of Arizona.

I also thank other members of the Hill group and people at UF, Dr. Rachel

Edwards, Dr. John Lee, Dr. Konstantin Petukhov, Jon Lawrence, Norm Anderson,

Tony Wilson, Amalia Betancur-Rodiguez, Saiti Datta, Sung-Su Kim, Dan Ben-

jamin, Emmitt Thompson, Costel Rotundu, Tara Dhakal, N i,.. i: Margankunte,

Hidenori Tashiro, Yoshihiro Irokawa and Stephen Flocks, for kind assistance in

my experiments, and help with my writing and friendship. In particular, I would

like to acknowledge Dr. Alexey Kovalev for useful discussion and assistance with

experiments in the initial stage of my research.

Finally, I would like to thank both my and my wife Ryoko's families for their

constant support, encouragement and love. I would also like to thank Ryoko

and my sons, Kai and Riku, for feeding me, giving me happiness and loving me.

Without them, I would not have even been able to survive Gainesville wildlife.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... .......... iv

LIST OF TABLES ................... .............. ix

LIST OF FIGURES ................... ............. x

ABSTRACT ................... ................. xiii

CHAPTER

1 INTRODUCTION ................... .. ..... .. 1

1.1 Overview of Low Dimensional Systems .............. 1
1.2 Fermi Surfaces of Low Dimensional Conductors ......... 2
1.3 Quasi-one-dimensional and Quasi-two-dimensional Materials 2
1.3.1 The Quasi-one-dimensional Conductor (TM-TSF)2C104 .- 4
1.3.2 The Quasi-two-dimensional Conductor K-(ET)2X . 4
1.4 Instability in Low Dimensional Conductors . . . 6
1.5 Superconductivity in Low Dimensional Materials . ..... 11
1.6 Impurity Effect on the Superconductivity .. . . 12

2 PERIODIC ORBIT RESONANCE . . . . ... 14

2.1 Experimental Techniques to Study Fermi Surfaces . ... 14
2.2 Cyclotron Resonance . . . ....... . 15
2.3 Cyclotron Resonance Involving an Open Fermi Surface: Periodic
Orbit Resonance . . . . . .. 22
2.4 Periodic Orbit Resonance for a Quasi-two-dimensional Fermi Sur-
face ...... ........ . .. .. ............. 27
2.5 POR and Angle-dependent Magnetoresistance Oscillations . 30
2.6 Quantum Effects in the Conductivity . . ..... 37
2.7 Summary .. . . . .. . . .... 41

3 EXPERIMENTAL SETUP .. . . . .. ..... 42

3.1 Overview of Microwave Magneto-optics . . ....... 42
3.2 Experimental Setup ... . . .. . .... 45
3.3 Rotating Cavity .... . . .. . .... 53
3.4 Model of the Resonant Cavity . . . . .... 65
3.5 Positioning Low Dimensional Conductors and Superconductors in
the Cylindrical Cavity . . . . . 69









3.5.1 In-plane Measurements. .................. 69
3.5.2 Interlayer Measurements. ................. ..70
3.5.3 Configuration for Interlayer Measurements Using the Mag-
netic Component of the Microwaves . . ... 71
3.6 Microwave Response of Low dimensional Conductors and Super-
conductors. .................. ........ 72
3.6.1 Skin Depth Regime .................. .... 72
3.6.2 Metallic Depolarization Regime .............. ..77
3.7 Measurement of the C!:i ,i,'. of the Complex Impedance Z . 77
3.7.1 Frequency-lock Method ................ .. 79
3.7.2 Phase-lock Method ................ . .80
3.8 Summary ............... ........... .. 81

4 PERIODIC ORBIT RESONANCES IN QUASI-ONE-DIMENSIONAL
CONDUCTORS ................ ............. 82

4.1 The Quasi-one-dimensional Conductor, (T\ TSF)2C104 . 82
4.2 Semiclassical Description of the Periodic Orbit Resonance and
the Lebed Effect .......... ............ 89
4.3 Observation of the Periodic Orbit Resonances in (T\ iTSF)2C104 93
4.4 Summary .................. ............ 102

5 NON-MAGNETIC IMPURITY EFFECTS ON THE SUPERCONDUC-
TIVITY IN (T\iTSF)2C104 .................. .... 103

5.1 Overview of the Superconductivity in (TMTSF)2C104 ...... .103
5.2 DC (u w 0) Transport Measurements for Different Cooling Rates 106
5.3 Study of the Periodic Orbit Resonance at Different Cooling Rates 109
5.4 All i -i of the Scattering Rate F . . . 110
5.5 Relation Between T, and the Pair Breaking Strength a ..... ..116
5.6 Summary ............... ........... .. 117

6 PERIODIC-ORBIT RESONANCE IN QUASI-TWO-DIMENSIONAL
CONDUCTORS ............. ............. 118

6.1 The Quasi-two-dimensional Conductors K-(ET)2X . ... 118
6.2 Periodic-orbit Resonance in K-(ET)2X . . 121
6.3 Experiments for K-(ET)2Cu(NCS)2 ............. 125
6.4 Experiments for K-(ET)23 .... . .. . 129
6.5 Angle-resolved Mapping of Fermi Velocity: A Proposed Experi-
ment for Nodal Q2D Superconductors ............. ..132
6.6 Summary .................. ............ 139

7 SUMMARY ................... ..... ........ 140









APPENDIX SEMICLASSICAL CALCULATION OF THE ELECTRICAL
CONDUCTIVITY ................... ....... 144

A.1 A Simple Quasi-one-dimensional Model ....... ........ 144
A.2 A General Quasi-one-dimensional Model ...... ....... 147
A.3 A Simple Quasi-two-dimensional Model ....... ........ 149

REFERENCES ...................... ........... 152

BIOGRAPHICAL SKETCH ................... ........ 160















LIST OF TABLES
Table page

3-1 Available magnet systems at UF and the NHMFL. ......... ..48

3-2 Probes used for the cavity perturbation technique. ......... .51

3-3 Resonance parameters for several different cavity modes . .... 61

4-1 Lattice parameters and the AMRO notations for the n-th nearest neigh-
bors. ................... ................ 92

6-1 Unit cell parameters for K-(ET)2X. .................. 119















LIST OF FIGURES
Figure page

1-1 Illustration of the FS by varying the bandwidths tb and t. . 3

1-2 Illustration of the < i -- .1 structure of (T:\ TSF)2C104 and the T\ TSF
molecule ................ ........... 5

1-3 Illustration of the < i 1 structure of K-(ET)2Cu(NCS)2 and the ET
molecule ................ ........... 6

1-4 Illustration of the Peierls instability in a 1D system. . . ... 8

1-5 Illustration of the confinement effect on the trajectories of electrons
in a magnetic field .................. ....... 9

1-6 T-H phase diagram for (TA\iTSF)2C14. ................ 10

2-1 The Fermi surface for the two-dimensional electron system. ..... .. 16

2-2 Real part of a,, as a function of frequency for various values of cwU. 19

2-3 The Fermi surface for the two-dimensional electron system with an
arbitrary direction of the magnetic field. ............ 21

2-4 Oscillatory group velocity v for the Q1D POR. ........... .23

2-5 POR in a-(ET)2KHg(SCN)4 ................ .... 26

2-6 Oscillatory group velocity v for Q2D POR. ............. .28

2-7 AMRO for Q2D conductors. ............. .... 31

2-8 AMRO for Q1D conductors. ............. .... 33

2-9 The Lebed effect in both the dc and ac conductivity. . ... 34

2-10 Ye,"i i ii oscillations. ............... ...... 35

2-11 Numerical calculation of the conductivity for a Q2D FS. ...... ..36

2-12 Landau tube with a magnetic field. .................... .. 38

2-13 DOS of a Q2D conductor in a magnetic field in units of hc. . 39

3-1 Frequency range for the Q2D POR in magnetic fields accessible at
the NHMFL. ................. ... ... ....... 45


lds accessible at
the NHMFL. .... . . . .......... 45









3-2 Overview of the experimental setup. .................... .. 46

3-3 A schematic diagram of the 3He probe. ................ 50

3-4 Vertical temperature distribution in the cryostat for the QD PPMS
7 T and Oxford Instruments 17 T magnets. . . 52

3-5 A schematic diagram of the rotating cavity system. . .... 55

3-6 Photographs of the rotating cavity system. ............. 57

3-7 Schematic diagrams showing various different sample mounting con-
figurations. .................. .. ........ 60

3-8 Angle-dependence of the cavity resonant properties. . .... 63

3-9 A simple description of the resonant cavity. ............. ..67

3-10 Schematic diagram illustrating the various possibilities for exciting
in-plane and interlayer currents in a Q2D plate-like sample. . 70

3-11 Positioning of the sample in the TEO11 mode. ........... ..73

3-12 Typical changes in the amplitude and phase of the microwaves trans-
mitted through the cavity. .................. .... 79

4-1 Electronic properties of the (T:\ITSF)2X. .... . ... 83

4-2 Illustration of i i-1I 1 axes for (T\ TSF)2C104. ............ .86

4-3 The dc AMRO experiment in (T\ITSF)2C1O4 ........... 88

4-4 The oblique real-space crystal lattice. ................ 90

4-5 Resonance conditions and the POR by sweeping the angle and mag-
netic field. . . . . . .. . 94

4-6 Overview of the orientations in the experiments. .......... ..96

4-7 Microwave absorption as a function of the magnetic field. ....... 97

4-8 Angle dependence of the quantity v/Bs... .............. 99

4-9 Angle sweep and field sweep measurements for (T\ITSF)2C104. .. 100

4-10 Summary of the p/q =0, 1 and 1 POR data for sample C. ..... ..101

5-1 Illustration of the C104 anion and the crystal structure of (T\ iTSF)2C104
below TAO . . . . . . .. . 104

5-2 dc transport measurements at different cooling rates. . ... 107









5-3 Summary of the dc transport measurements. . . 108

5-4 Microwave absorption as a function of magnetic field at different rates. 110

5-5 Least-square fit to microwave absorption . . ..... 112

5-6 Cooling rate and temperature rate dependence of the scattering rate. 113

5-7 Comparison between the resistance R,, and the scattering rate F as
a function of the cooling rate. . . . .. .. 115

5-8 T, vs. the scattering rate . . . . .. 116

6-1 Phase diagram for the organic conductors K-(ET)2X. . . 119

6-2 Fermi surface (FS) and trajectories of an electron under a magnetic
field for K-(ET)2Cu(NCS)2. . . . .... .. 120

6-3 Warping on a Q2D FS. . . . . .. . 121

6-4 Numerical calculation of the ac conductivity for a Q2D FS. . 124

6-5 Overview of the orientations in the experiments on K-(ET)2X. . 125

6-6 Experimental data for K-(ET)2Cu(NCS)2. . . . 128

6-7 Angle-dependence of the POR and the SdH oscillations for two kinds
of rotations in K-(ET)2Cu(NCS)2. . . . ....... 129

6-8 Experimental data for c-(ET)23. . . . .. . 130

6-9 Angle-dependence of the POR in c-(ET)213 . . .. 131

6-10 Self-crossing orbits and open trajectories. . . . 134

6-11 Angle-resolved mapping of vp. . . . ..... .. 137

A-l Representation of rotation of a magnetic field relative to a Q1D FS. 145

A-2 Representation of rotation of the magnetic field for a Q2D FS. . 150















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

ANGLE-DEPENDENT HIGH MAGNETIC FIELD MICROWAVE
SPECTROSCOPY OF LOW DIMENSIONAL CONDUCTORS AND
SUPERCONDUCTORS

By

Susumu Takahashi

December 2005

C(! nI: Stephen O. Hill
Major Department: Physics

This dissertation presents studies of angle-dependent high-field microwave

spectroscopy of low dimensional conductors and superconductors. Over the past

20 years, low dimensional conductors and superconductors have been investigated

extensively because of their unusual superconducting, electronic and magnetic

ground states. In order to understand these phenomena, it is important to study

the topology of the Fermi surface (FS). We employ a novel type of cyclotron

resonance to study the FS, the so-called periodic orbit resonance (POR).

In C!i lpter 2, we explain the details of the POR effect using a semiclassical

description. An important aspect of this POR effect is that it is applicable not only

to a quasi-two-dimensional (Q2D) FS, but also to a quasi-one-dimensional (Q1D)

FS.

In C!i ipter 3, our experimental techniques are presented. We outline a rotating

cylindrical cavity, which enables angle-dependent cavity perturbation measurements

in ultra-high-field magnets, and two-axis rotation capabilities in standard high-field

superconducting split-pair magnets.









In C'!i ipters 4 and 5, the results of studies of the Q1D conductor (TA [TSF)2C104

are shown. Using the POR, we determined the Fermi velocity vp and revealed new

information concerning the nature of the so-called Lebed effect in ('!i Ipter 4. In

C'!i lpter 5, we studied the non-magnetic impurity effect and its influence on the

possible spin-triplet superconductivity in (TA[TSF)2CO14.

In C'!i lpter 6, measurements of the POR are performed in the Q2D conductors

K-(ET)2X [X-Cu(NCS)2 and I3]. In X=I3, POR involving the magnetic breakdown

effect was observed for the first time.
















CHAPTER 1
INTRODUCTION

1.1 Overview of Low Dimensional Systems

Electronic band structures give us an idea of the conducting properties of

materials. For the case of an insulator, the allowed energy bands are either filled

or empty and the band gap between the filled and empty bands is large enough

to prevent thermal excitation of electrons to the empty band: thus no current

flows in response to an external field. For the case of a metal, one or more bands

are partially filled. In this case, electrons can move easily in response to an exter-

nal field [1]. For the case of certain kinds of metals, the band structure is highly

anisotropic; i.e., electrons in such metals can move along one direction much

more easily than the other directions. These materials are called low dimensional

systems. In particular, these systems are often classified as either quasi-two-

dimensional (Q2D) or quasi-one-dimensional (Q1D). Recently, the study of these

systems has been attractive because many interesting phenomena have been dis-

covered in them, including unconventional superconductivity, metal-insulator tran-

sitions, antiferromagnetism, spin-density-waves (SDW) and charge-density-waves

(CDW). In order to understand these low dimensional systems, it is important to

study their detailed electronic properties, as well as their superconducting proper-

ties. Although the full band structure gives overall information on the electronic

property, it is often enough to investigate the topology of the Fermi surface (FS) of

the systems because states near the FS dominate the low temperature conducting

properties.









1.2 Fermi Surfaces of Low Dimensional Conductors

For the sake of getting a picture of the FS of a low dimensional system, we

start by considering an isotropic band structure, i.e., the ratio of transfer energies

in a tight binding model t, : ty : tz = 1 : 1 : 1. In general, the shape of the FS

can be complicated, even when the conducting properties are three-dimensional

and free electron-like. However, we here consider a perfectly spherical FS as the

simplest case, as shown in the upper left picture of Fig. 1-1. Starting with this

FS, we change the anisotropy. When the bandwidth along the z-direction becomes

smaller, e.g., t, : ty : tz = 1 : 1 : 1/2, the FS sphere is stretched along the z-

direction. The smaller the z-axis band width, the more the z-direction of the FS is

stretched, as shown in the upper middle figure. Eventually the z-component of the

FS connects with the FS in the next Brillouin zone, and then the FS becomes open

along the z-direction, namely a cylinder-like shape with a small corrugation of the

cylinder, as shown in the upper right figure. In this case, the transfer energy ratio

will be highly anisotropic, e.g., t, : t, : tz = 1 : 1 : 1/100. Such an anisotropic FS

is said to be Q2D. Examples of materials with Q2D FSs are K-(ET)2Cu(NCS)2 and

K-(ET)213 (See O'i p1 6.). Next, we change the band width along the y-direction.

While the band width is reduced, the y-direction of the FS tube is stretched step

by step, as seen in the lower left figure. Eventually, the FS becomes a couple of

plane-like sheets with small corrugations, as shown in the lower right panel of

Fig. 1-1. These corrugations are related to the small transfer energies along the y

and z-directions, e.g., t, : ty : t, = 1 : 1/100 : 1/100, representing a Q1D FS. This is

the case for (T [\TSF)2C104 described in C'i p1 4.

1.3 Quasi-one-dimensional and Quasi-two-dimensional Materials

Here we introduce examples of low dimensional materials which have the FSs

described in the previous section.









1.3.1 The Quasi-one-dimensional Conductor (TMTSF)2C104

The organic metal (TMTSF)2C104 belongs to the family of quasi-one-

dimensional (Q1D) Bechgaard salts [2], having the common formula (T\ TSF)2X.

TMTSF is an abbreviation for tetramethyl-tetraselenafulvalene, and the anion X

is AsF6, C1O4, PF6, ReO4, etc. The (TMITSF)2X compounds are the so-called 2:1

charge transfer salts which transfer one electron from two TMTSF molecules to one

X anion. The < i--I 1 structures of the (TSITSF)2X series are similar. Fig. 1 2(a)

shows the < i,-- 1 structure of (T\ITSF)2C104. As shown to the left of Fig. 1 2(a),

the planar T\ [TSF molecules are stacked along the a-direction. The conducting

properties come from the overlap of r-orbitals on the T\ TSF molecules, as shown

in Fig. 1 2(b). Since the r-orbitals are oriented perpendicular to the T\ TSF

molecule, the molecules couple strongly along the a-direction via the overlap of

partially occupied r-orbitals so that this direction becomes the most conducting

direction. On the other hand, the overlap along the b and c-directions is weak. In

particular, the overlap is extremely weak along the c-direction because the coupling

between T\ TSF molecules is hindered by the insulating CO14 anion sheets, as

shown in the right panel in Fig. 1-2(a). It turns out that the conducting proper-

ties are extremely one-dimensional. In the case of (T\ iTSF)2C104, the transfer

integrals are approximately ta : tb : t = 250 meV : 20 meV : 1 meV [2]. This

common ( i --I 1 structure is the origin of the Q1D conducting properties for all of

the (TMTSF)2X compounds. Thus, (TMTSF)2X, including X C14, PF6 etc., are

good examples for studying Q1D FSs.

1.3.2 The Quasi-two-dimensional Conductor K-(ET)2X

The organic superconductors K-(ET)2X (X = Cu(NCS)2, 13 etc.) belong to the

ET family of charge-transfer salts (CTS), where ET represents bisethylenedithio-

tetrathiafulvalene [(CH2) 22C6Ss (alternatively denoted BEDT-TTF). In contrast

to the planar T\ iTSF molecule (shown in Fig. 1-2), the ET molecule shown in







6

(b)




(a) t,, S



ET molecule



b ET molecule

Figure 1-3. Illustration of the crystal structure of K-(ET)2Cu(NCS)2 and the ET
molecule. (a) Structure of the ET molecule. In contrast to the T\ iTSF
molecule, the ET molecule is not planar. (b) The crystal structure of
K-(ET)2Cu(NCS)2 viewed along the a*-axis.


Fig. 1 3(a) is not exactly planar. This non-planar nature of the ET molecule

complicates the morphology of the ET crystal structures compared with other 2:1

(TMTSF)2X salts. For instance, there are four kinds of iv- I I morphologies in

the case of (ET)213, i.e., the a-, 3-, 0- and K-phases, where a, 3, 0 and K denote

different crystal structures. Fig. 1 3(b) shows the crystal structure viewed along

the a*-axis for the K-phase of (ET)2Cu(NCS)2. For the case of K-(ET)2Cu(NCS)2,

the ET molecules dimerize in a face to face arrangement. The dimer pairs then

pack orthogonally so that the overlap of the r-orbitals of the ET molecules is fairly

isotropic in the be-plane. On the other hand, the overlap along the a-direction is

much smaller because it is prevented by the insulating Cu(NCS)2 anion l .r1-.

It turns out that the conducting properties in K-(ET)2Cu(NCS)2 are highly two-

dimensional, or quasi-two-dimensional (Q2D), i.e., oll/oL ~ 1000 or tll/t

30 [3].

1.4 Instability in Low Dimensional Conductors

According to recent studies of low dimensional systems, the topology of the

high temperature FS contributes to the nature of the ground states significantly.









Examples include the Peierls instabilities, charge-density-wave (CDW) states, spin-

Peierls instabilities, antiferromagnetic (AF) states and spin-density-wave (SDW)

states. In particular, nesting of the FS is important. For example, in the case of

1D systems, the FS consists of a couple of flat sheets, as shown in Fig. 1 4(a). The

instability in a 1D system is related to this shape of the FS, since any points on

one FS can be mapped into the other FS by a single wave vector Qx = 2kF, the

so-called nesting vector. The correlation of electrons on the FS becomes divergently

strong at Qx = 2kg. This is the so-called Kohn anomaly. As a result, the electron-

phonon interaction becomes divergently strong at the nesting vector Q, = 2kF

with decreasing temperature. Therefore, the phonon mode at Q, = 2kF becomes

soft, as shown in Fig. 1 4(b). This soft phonon frequency goes to zero at low

temperature, resulting in a static lattice distortion with Q, = 2kg, the so-called

Peierls distortion. This Peierls transition can be described by a mean field theory,

and the transition temperature Tp is given by a BCS-type gap equation [2]. This

distortion also affects the electronic state. An example for a half-filled 1D system

is shown in Fig. 1 4(c). At high temperatures (T > Tp), there is one electron per

site, and the electron can readily move to other sites, so that the system is metallic.

However, at low temperatures (T < Tp), the system undergoes the Peierls distortion

with Q, = 2kF = 7/2a becoming dimerized. This changes the half-filling to full

filling. As a result, electrons cannot move to other sites, and the system becomes

an insulator. This metal-insulator transition is called the Peierls transition, which

can also happen at any band filling. This transition leads to a modulation of the

charge density, known as a CDW. Furthermore, the nesting also affects other

interactions, e.g., the electron-electron interaction etc. As a result, a similar

transition can happen involving the spin degrees of freedom. This type of transition

is known as the SDW, or AF transition. These transitions are often observed in

Q1D and Q2D organic conductors, e.g., the (T\ilTSF)2X and ET-salts [2].














(a) D
1DFS


kky


/Q = 2kF


High T


itinerant


2D


2 k


metallic state


4-4-4-4-4-4-4-4----~--
a

Low T (< T) no hopping insulating state

--- ------ i~---i--
2a
n = half-filling


Figure 1-4.


Illustration of the Peierls instability in a 1D system. (a) The FS is
represented by a pair of open FS sheets. Any points on one FS can
be mapped into the other FS by a single wavevector Q, = 2kF, the
so-called nesting vector. (b) Kohn anomalies in 1, 2 and 3 dimensional
system. The phonon dispersion is plotted as a function of wavenumber.
The anomaly is seen at Q = 2kF. In the case of 1D, the phonon disper-
sion becomes zero at Q = 2kF. This causes a static lattice distortion,
the so-called Peierls distortion (Reprinted figure with permission from
Kagoshima [4]. Copyright 1981 by the Institute of Pure and Applied
Physics (Japan).). (c) Because of the nesting, the electron-phonon
interaction becomes divergently strong, so that it causes a lattice dis-
tortion, the so-called Peierls distortion. For the case of half-filling,
the system dimerizes below the transition temperature Tp due to the
Peierls distortion. This affects the electronic structure. The distor-
tion results in a fully occupied band, which does not allow electrons
to move to other sites. As a result, the system becomes an insulator.
Thus, the Peierls transition leads to a metal-insulator transition.


















a


Figure 1-5.


9



B =W 4tbb/hwo
oc 1/B






r ^^ 2


W


B Low
B Low B High
W Large W Small

Illustration of the confinement effect on the trajectories of electrons
in a magnetic field. Upon increasing the magnetic field, the transverse
width of the trajectories of electrons decreases. This results in an in-
crease in the one-dimensional properties, i.e., the nesting on the FS
becomes stronger.


In some cases, the nesting property is strongly increased via a magnetic field.

In fact, a reentrance of the SDW phase under a high magnetic field is observed in

the Q1D organic conductor (TMTSF)2X. This phase is the so-called field-induced

SDW (FISDW) state. For (T\ ITSF)2X, the transfer energy along the b-direction,

tb, is significant, tb/t ~ 0.1, so that the motion of electrons is somewhat two-

dimensional. Application of a magnetic field leads to confinement of the 2D

motion. Fig. 1-5 shows trajectories of electrons in a real space. By applying a

stronger magnetic field, the width of the trajectories becomes increasingly smaller.

Eventually, the field confines the width of the trajectory to within one unit cell in

the b-direction. Then the motion of the electrons is effectively one-dimensional.

In the case of (T\ TSF)2X, this dimensional crossover effect is related to the

confinement in the ab-plane, so that the FISDW phase has a minimum critical field

when the field is perpendicular to the ab-plane. In the case of (TMTSF)2C104 [5],

the critical field of the FISDW phase TSDW is 7 T at T = 2 K, as shown in Fig. 1

6. More details of the FISDW state are explained by the standard model [6, 7, 8].




































0 5 10 15 20 25 30
Field (T)


Figure 1-6.


T-H phase diagram for (T:iTSF)2C104. The FISDW phase is ob-
served in high magnetic field, e.g., the critical field BSDW ~ 7 tesla at
T = 2 K. The FISDW effect is caused by the field dependent nesting
(confinement) effect on the ab-plane. The another phase transition was
also proposed at higher magnetic fields using normal and Hall resis-
tance and magnetization measurements, e.g., the critical field of the
second phase ~ 27 tesla at T = 2 K (Reprinted figure with permission
from McKernan et al. [5]. Copyright 1995 by the American Physical
Society.).


7

6

2 5

4


E
2

1

0









1.5 Superconductivity in Low Dimensional Materials

As mentioned, exotic superconductivity has been observed in irn il: low di-

mensional materials, e.g., the HTSC, K-(ET)2X, Sr2RuO4 and (T\ITSF)2X. These

exotica may be represented by anisotropic superconducting energy gaps, spin-triplet

Cooper pairs, non electron-phonon interaction mediated Cooper pairing mecha-

nisms, coexistence of superconductivity and magnetism. For example, (T\ TSF)2X

has recently been considered to be a spin-triplet superconductor since it shows no

7Se Knight shift through the superconducting transition temperature [9], exceeding

the Pauli paramagnetic limit in the upper critical field [10], and non-magnetic

impurity effects [11]. K-(ET)2Cu(NCS)2 is considered to be a d-wave superconduc-

tor because it shows a four-fold symmetry in the magneto-thermal conductivity

tensor [12].

In the case of low dimensional organic superconductors, which are highly

relevant materials to this thesis, the topology of the FS (i.e., nesting) may be

important for generating such anisotropic superconducting energy gaps. Recently,

many theoretical studies have highlighted the importance of such nesting properties

for theories of unconventional superconductors, e.g., low dimensional organic

superconductors [13, 14].

Similar to the anisotropic conducting properties of low dimensional super-

conductors, their superconducting properties are often anisotropic. For the case

of the quasi-two-dimensional (or 1 li, 1, ,) superconductors, the superconducting

anisotropy parameter 7 is represented by the ratio between the interlayer and

in-plane penetration depth or coherence length, i.e., 7 = AI/AI or 7' = /~1_L re-

spectively. In the extreme cases, 7 can be several hundred: e.g., Bi2Sr2CaCu208+y,

7 ~ 50 200 [15]; and K-(ET)2Cu(NCS)2, 7 100 200 [16]. This high degree

of anisotropy is related to a weak Josephson coupling between the superconducting

liv--. The characteristic interlayer Josephson plasma frequency is often in the









range of microwave frequencies. This can be seen as a Josephson plasma resonance

(JPR). We have studied the JPR phenomena for the Q2D organic supercon-

ductor K-(ET)2Cu(NCS)2. The results were reported in Kovalev et al. [17] and

B(: i r,' ,i*, et al. [18].

1.6 Impurity Effect on the Superconductivity

For unconventional superconductors, the non-magnetic impurity effect

is one of the main evidences for unconventional superconductivity. We have

therefore performed an experiment to study the non-magnetic impurity effect in

(TMTSF)2C104. This result is described in C'!h 1 5.

The original work on the impurity effect was developed for magnetic impuri-

ties in conventional superconductors in the 1950's and 1960's. In 1959, Anderson

extended the Bardeen-Cooper-Schreiffer (BCS) theory [19] for very dirty supercon-

ductors, where elastic scattering from non-magnetic impurities is large compared

with the superconducting energy gap [20]. In this theory, Anderson showed that

the Cooper pair with the momentum (k T, -k 1) is immune to non-magnetic scat-

tering. Independently, the impurity effect was discussed by Abrikosov and Gor'kov

in the 1960's (AG theory) [21]. The AG theory predicts a strong suppression and

disappearance of the transition temperature by magnetic impurities, and also pre-

dicts changes of various thermodynamic properties, e.g., the existence of a gapless

regime in the superconducting state.

Impurity scattering destroys the coupling between Cooper pairs. This is

more generally explained in terms of the effect of broken time-reversal symmetry.

For instance, in the case of BCS-type pairs isotropicc gaps with the momentum

(k T, -k 1), the scattering by non-magnetic impurities, which acts on the elec-

tric charges of the Cooper pairs identically, does not destroy the Cooper pairs

because the scattering has little effect on the antisymmetry of the pair (Anderson's

theorem). On the other hand, scattering by magnetic impurities can act on the









spin and ultimately flip one of the spins. As a result, the scattering changes the

pair to a parallel spin alignment (the time-reversal symmetry is broken), and the

Cooper pair is destroyed. Furthermore, extensive studies show that the broken

time-reversal symmetry is not only caused by magnetic impurities, but also in thin

films in parallel magnetic fields, by exchange fields, and proximity effects, etc. [22].

According to the AG theory, the suppression of the superconducting critical

temperature T, is simply related to the scattering time TK of the destructive

perturbation. The function to express the relation is the so-called universal

function given by the following equation.


In( )= +), (1-1)
Tco 2 2 27TkBTC

where Teo = Tc(O) is the superconducting critical temperature without impurities,

(z) = P'(z)/F(z) is the digamma function, and a is the pair-breaking strength,

2a =-, (1-2)
TK

where TK is the scattering time for depairing.

Impurity effects were observed not only in the BCS-type superconductors, but

also in many exotic superconductors. In particular, in the case of the unconven-

tional superconductors, effects were observed even due to non-magnetic impurities.

That is why this non-magnetic impurity effect is considered to be an evidence of

unconventional superconductivity. For example, the spin-triplet superconductor

Sr2RuO4 shows non-magnetic impurity effects [23].

In this chapter, we explained the motivation of the study of low dimensional

materials briefly. In the next chapter, we introduce a periodic orbit resonance effect

to probe the FS. C'i lpter 3 shows our experimental setup. Chapter 4 and 5 explain

the study of the FS and superconductivity for the Q1D material (T i[TSF)2C104.

C'!i pter 6 show the study of the FS for the Q2D materials K-(ET)2X.














CHAPTER 2
PERIODIC ORBIT RESONANCE

2.1 Experimental Techniques to Study Fermi Surfaces

Since the Fermi surface (FS) can explain many aspects of conductors, su-

perconductors and itinerant magnetic systems, many techniques have been

developed to study FS topology. Examples include: angle-resolved photoemis-

sion spectroscopy (ARPES) [24], the de-Haas-van Alphen effect (dHvA) [25], the

Shubnikov-de Haas effect (SdH) [25], angle-dependent magnetoresistance oscilla-

tions (AMRO) [26, 27, 28, 29, 30], cyclotron resonance (CR) [31] and periodic orbit

resonance (POR) [32, 33]. ARPES is now a leading technique to investigate FSs

and has contributed significantly to the investigation of the FS in high temperature

superconductors. Using ARPES, one can measure not only the FS, but also the

band structure. However, since the skin depth of the photon in ARPES is much

shorter than a unit cell, it suffers from the fact that it is a surface probe with a

depth resolution often not more than a unit cell. ARPES is also restricted to mea-

suring in-plane band structures of quasi-two-dimensional (Q2D) systems because

of the resolution of interlayer scattering processes. The SdH and dHvA effects

probe the effective mass, m*, and the quasiparticle lifetime, 7r. However these

quantities are averaged over the FS. The SdH and dHvA effects are also limited to

three-dimensional (3D) and Q2D systems because the technique needs closed orbits

in order to give rise to Landau quantization. Since the SdH and dHvA phenomena

come from quantized Fermi levels under a high magnetic field, as explained in

Sec. 2.6, one can observe these effects only at low temperature, i.e., kBT < hwuc,

where uc is the cyclotron frequency. AMRO can provide additional topographic









information in reciprocal space such as the Fermi wave vectors kF. AMRO is also

applicable in the case of quasi-one-dimensional (Q1D) systems.

CR can be used to determine the averaged m* and r over the FS. Since CR

requires closed orbits, the technique is limited to 3D and Q2D systems. Our new

experimental technique, the so-called POR, is closely related to CR. The difference

between the POR and CR is that the POR can also come from open orbits. Details

are explained in Sec. 2.4. The POR can therefore be observed in Q1D systems.

In the case of Q1D systems, the POR probes the average Fermi velocity vF and

the scattering time T, as shown in C'!h 1 4. In the case of Q2D systems, the POR

probes both the average m* and the k-dependent Fermi velocity VF(k), as well as

r(k), as shown in Chap. 6.

2.2 Cyclotron Resonance

Cyclotron resonance (CR) is known as one of the most useful tools to probe

effective masses experimentally. It was first used in the 1950's to study metallic

elements, e.g., Cu [34, 35, 36], Al [37, 38, 39], Sn [34, 40]. The CR technique is still

being used extensively in some fields .1I-v,, e.g., the two-dimensional electron gas

(2DEG) system in a GaAs/(Ga,As)Al heterojunction [41]. In order to describe the

CR phenomena, one can start by considering semiclassical electron dynamics with

a simple model: a two-dimensional (2D) electron gas, i.e., the band index n is a

constant of the motion and the FS is a complete cylinder, as shown in Fig. 2-1, and

an energy dispersion represented by


E(k) =2* (k + k ), (2-1)


where m* is the effective mass. We now consider a dc magnetic field applied along

the z-axis perpendicular to the conducting plane, i.e., B = (0, 0, B), as shown in

Fig. 2-1. Electrons in the system experience a Lorentz force due to the magnetic

field, and start to change their momentum. The motion of the electrons is given by









the equation of motion for the Lorentz force,


hk -e(v x B), (2-2)


and the definition of the group velocity is


Vg Vk[E(k)]. (2-3)


In the present case, using the energy dispersion in Eq. 2-1, the explicit expressions

corresponding to the above equations are,

eB
m = ky,(t), (24a)
m*
eB
ky kx (t), (2-4b)
m*

k = 0, (2-4c)

and

vX(t) k(t). (2-5a)

VY M h ky M (2-5b)

v, = 0. (2-5c)

Thus, solving the differential equations 2-4 and 2-5, one can see that the resultant

motion is oscillato,;, i.e., kx(t) = kcos(uett+(k)) and ky(t) = ksin(ct+O(k)); the

oscillatory velocities, v, = v cos(ut + 0(k)) and v, = vsin(uct + O(k)); and k, = 0

and v, = 0, where ,c = eB/m* is the characteristic frequency of the oscillatory

motion, the so-called I ;.. u./,..; f,,'.;". ;./ ;; The cyclotron frequency, vc = uc/27r

28 (GHz/T) x 'B, for many materials usually goes into the microwave frequency

range for typical magnetic fields (B < 10 T), since their effective masses are often

in the range of m* ~ O0.1m 10,. where m, is the electron mass. There is no

kinetic energy change, because the motion is caused by the magnetic field. This









oscillatory motion is illustrated in Fig. 2-1. The shape of the constant energy

surfaces is cylindrical because of the 2D nature of the electronic property in the 2D

electron gas. At E = EF, this surface corresponds to the FS. The group velocity,

represented by arrows in Fig. 2-1, is albv-l perpendicular to the FS. As a result,

the group velocity rotates a full 3600 as the electron rotates around the surface, as

shown in Fig. 2-1.

Cyclotron motion affects transport properties. We now calculate the electrical

conductivity resulting from this motion, using a Boltzmann equation within the

relaxation time approximation [31, 33],

2 2 afO k 0 16a)
Vq(ua, B) dk [- ]vp(k, 0) dtvq(k,t) exp( zi)t, (2-6a)
V fd[ Eoo

or an alternative expression is,


,(, B) = dE[ f(E)]N(E) dk2v(k, 0)vq,(, k), (2-6b)

where

Tp(w,k) = dtvp(k,t) exp( iw)t, (2-6c)

where p and q are indices of a cartesian coordinate system, i.e., x, y or z. In

Eq. 2-6c, exp(-iut) is a oscillatory term to give arise the resonance, and exp(1/r)

is a damping term to dephase the oscillatory motion. In the low temperature

limit, O f 6 k kF) or -f(- f( 6(E EF), so that the motion of electrons

affecting the conductivity is restricted to the FS. The conductivity along the x-axis

is then given by the following expression, (See Appendix A.1 for details of the

calculation.)

2eC2 0 17a)
7(wa), B) 2- d2Sv(k, 0) dtvx(k, t) exp( i)t, (2-7a)

2e2v 1 1
V [ (+ ] (2 7b)
V i(w + ) + i(w o) + )


i((a +o +)







19




ar=3


o. =2






oT=0.5


0 1 2 3 4 5


Figure 2-2. Real part of ax as a function of frequency for various values of cur.
The peak in the conductivity represents CR. The resonance condition
is w = wu. For cur >1, the CR is well-pronounced. However, the CR
becomes too weak to observed for Uwr <1. Thus uwr has to be greater
than 1 in order to observe CR.


Thus, the real part of r,, is given by

adc 1 1
Re a1, (, B) dc t + W (2-8)
2 1+(1 t + L')2c2 2 1 )2'

where ard is the dc (w = 0) conductivity and r is the relaxation time of the

electrons on the FS. In the present case, ao, = ayY and az, = 0. The real

part of the conductivity, Re aog, is plotted in Fig. 2-2 as a function of wu/uc.

There is a clear Lorentzian-like peak at the resonance condition, u/wc = 1 (or

wu= wc). This peak in Re ar, is caused by the cyclotron motion. This is ;/. 1.Itron

resonance. Since c = eB/m*, in the present case, wc goes to zero when B 0.

The conductivity at B = 0 is therefore given by


Re a (, B = 0) (29)
1+((2 9)


Eq. 2-9 is of course the ac Drude conductivity which has a peak at w


0 [31].










Thus, the peak corresponding to CR is the same as that of the ac Drude conductiv-

ity. The width and height of the CR peak depend on the parameter, Uwr, as shown

in Fig. 2-2. The peak is well-pronounced if uwr >1. However the peak becomes

smeared out if uwr <1. As with any resonance phenomena, the wur condition is re-

lated to phase memory, or dephasing. The wur < 1, eletrons dephase faster than the

time taken to complete an orbit. Thus, the oscillatory motion is highly damped and

the exponentiol dominates Eq. 2-6c. On the other hand, if uwr > 1, electrons may

execute many orbits before d. pl. '-iw- and the oscillatory term dominates Eq. 2-6c,

resulting in the resonance. Thus, the key to observing CR is an oscillatory 1... H:1

and uwr > 1.

We can now consider the case of a tilted magnetic field, e.g., B = (B sin 0, 0, B cos 0).

(See Fig. 2-3.) The equation of motion and the group velocity are now given by

eB cos O
k1 n* k9(t), (2-10a)
m*

k cos kM(), (2-O1b)
m*
eB sin 0 M
k =es k,(t), (2-10c)
m*

and

,(t)- k(t) (2-11a)
m

Vw(t= -hkw(t), (2-lib)

v, = 0. (2-1 c)

Similarly, the resultant motion therefore follows an oscillatory tr r i* /.. ; kx(t) =

k cos(c2Dt + 0(k)), ky(t) = k sin()2Dt + 0(k)), and kz(t) = k tan 0 cos(c2Dt + Q(k));

the oscillatory velocities, v, = v cos(u2Dt + 0(k)) and v, = vsin(w2Dt + 0(k)); and

v, = 0. The cyclotron frequency is given by

eB cos 0
W2D = (2-12)
m*









which now depends on the angle 0. Again a,, = ayy, similar to Eq. (2-7b), and

a,, = 0 with the resonance condition w = W2D, or,

eB
w -I cos60. (2-13)
m*

2.3 Cyclotron Resonance Involving an Open Fermi Surface: Periodic
Orbit Resonance

In Sec. 2.2, we considered a closed trajectory on a 2D FS to illustrate the CR

effect. However, the resonance phenomenon does not require closed orbit motion,

because the essential ingredient of the resonance is an oscillatory group velocity.

Therefore, the resonance can even be seen in a system with only open trajectories.

This resonance effect is the so-called periodic orbit resonance (POR) [33], or also

the Fermi-H-ai {f..: traversal resonances (FTR) by Ardavan et al. [32]. The POR

was originally predicted by Osada et al. [42] for a quasi-one-dimensional (Q1D)

system. After that, many theoretical works have been performed using different

models, e.g., Hill [33], Blundell et al. [43], Moses and McKenzie [44, 45]. We now

consider the POR for the case of a quasi-one-dimensional (Q1D) FS which consists

of completely open FS sheets. The highly anisotropic energy dispersion can be

written in the form


E(k) = hvp\(I k kF) 2ty cos(kyb) 2t, cos(k,c), (2-14)


where EF > ty and tz, vp is the Fermi velocity, and b and c are lattice constants

along the y and z-directions respectively. t, and tz are the transfer energy associ-

ated with the lattice vectors R, and R,. This energy dispersion describe a Q1D

FS. The 1st term of the energy dispersion is responsible for a flat shape of the

FS, and the 2nd and 3rd term are responsible for the warping along the lattice

vectors. The physical meaning of each transfer energy ty and tz is therefore the

Fourier component of the warping. As an example, the energy anisotropy in the










(a)


Q1D Fermi sur


(b)


23

Fermi velocity

CorIvrugation

z


face B(O)


oscillatory velocity





B(0)


Figure 2-4.


Oscillatory group velocity v, for the Q1D POR. (a) The trajectory
of electrons on the Q1D FS in the case of a tilted magnetic field. The
corrugation is produced by the transfer integral along the z-direction.
The z component of the group velocity becomes oscillatory. This oscil-
latory group velocity causes the resonance in the conductivity za,. The
periodicity of the POR is angle-dependent. (b) Trajectory of electrons
on the Q1D FS in the case of a magnetic field along the z-axis. v, is
not oscillatory in this case, so that no POR is seen.


Q1D organic conductor, (T:\iTSF)2C104 is t : ty : t 250 : 20 : 1 meV [2].

Fig. 2-4 shows such a Q1D FS consisting of a planar sheet which is corrugated due

to the small transfer energy along the least conducting direction. A magnetic field

is now applied parallel to the corrugated plane, as shown in Fig. 2-4(a). A charged

particle will move according to the Lorentz Force. Because the FS is corrugated,

and the direction of the Fermi velocity (indicated by arrows in Fig. 2-4.) is alv-wb

perpendicular to the FS, the motion results in an oscillatory Fermi velocity. This

is the origin of the Q1D POR. The period of the motion can be varied by changing

magnetic field strength, or the angle between the magnetic field and the direction









of the corrugation. For instance, if the magnetic field is applied along the direction

of the corrugation, there is no POR, because there is no oscillatory velocity, as

shown in Fig. 2-4(b).

We now consider the resonance condition for the POR in the case of a Q1D FS

with a magnetic field applied parallel to the FS sheet. The corrugation direction

is assumed along the z-axis. The applied magnetic field is expressed by B =

(0, B sin 0, B cos 0) where 0 is a angle between the magnetic field and the z-axis.

Recalling the equation of motion, Eq. (2-2),


S-2tbbBos() sin[k(t) b] + sin( sin[k, (t)c], (2-15a)

vFy B cos(0)(2-b)
k, = sgn(k,) (215b)

k = -sgn(k,) B sin() (2-15c)

and

v, = sgn(k,)vF, (2-16a)

2t b
v,(t) = 2t sin[k,(t)b], (2-16b)
2t c
v (t) = sin[k (t)c]. (2-16c)

Solving the above equations, the z-component of the trajectory and velocity is

given by
vFeB
k, = -sgn(k)[ sin(O)]t + k,(0), (2-17)

and

2t c vFeBc
v, (t) = sin[k,(0)c sgn(k,) sin(U)t] 2tecsin[k,(0)c sgn(k,)wQlDt],
h h
(2-18)

where WQID = pB, sin(0). Thus, this group velocity is oscillatory, and is charac-

terized by the frequency WQ1D. This frequency is related not only to the magnetic


by the frequency WQ1D. This frequency is related not only to the magnetic









field B and the band parameter (Fermi velocity vp), but also the lattice constant c

which is associated with the corrugation on the FS.

Like the CR, the conductivity is easily calculated. The real part of a,, is given

by,

adc 1 1
Re azz (, B,0) = c[ + 1], (2-19)
2 1 + (u + WQ1D)2 1+ ( Q1 UQD)2

which gives rise to a POR at w = UWQID, i.e., the resonance condition is given by


w BeI sin(eO). (2-20)


Thus, one can measure the Q1D POR at different frequencies w, magnetic fields B

and angles 0. By investigating the position of the POR, one can determine vp from

Eq. 2-20, and one can also determine the scattering time r by analyzing the shape

of the POR.

The FS corrugation pattern of real Q1D conductors can be much more

complicated, particularly for materials with low-symmetry (i ~ I 1 structures,

i.e., monoclinic, triclinic or rhombohedral. As a result, the hopping energy to 2nd

and higher-ordered nearest neighbors becomes considerable. We can see such an

example in the study of the organic conductor a-(ET)2KHg(SCN)4 by Kovalev

et al. [46]. The FS of a-(ET)2KHg(SCN)4, below 6 K, is shown in Fig. 2-5. At

8 K, a-(ET)2KHg(SCN)4 undergoes a phase transition into a charge-density-wave

(CDW) ground state. The FS in the CDW state is characterized by a wave vector

Q. The FS consists of both Q2D and Q1D sections. In the experiments by Kovalev

et al., the Q1D POR was observed by sweeping the magnetic field and the angle

of the magnetic field. The observed POR contain many harmonic resonances, as

seen in Fig. 2-5(b) and (c). The existence of the harmonics implies that the Q1D

section is rather two-dimensional, as can be seen in Fig. 2-5(a). In the case of

a-(ET)2KHg(SCN)4, the Q1D section is not a simple flat sheet. Since the Q1D


mple flat sheet. Since the Q1D


























0 1 2 3 4 5 6
Magnetic field (tesla)


m=2 m=l m=o m=-1

3T


4T


20 40 60 80 100 120 140 160 180 200 220
0 (degrees)


-60 -40 -20 0 20 40 60
0 (degrees)


Figure 2-5.


POR in a-(ET)2KHg(SCN)4. (a) FS in a-(ET)2KHg(SCN)4 below 6
K. a-(ET)2KHg(SCN)4 exhibits a phase transition at 8 K into a CDW
state. This state is characterized by the wave vector Q. As a result,
a-(ET)2KHg(SCN)4 has Q2D and Q1D FS sections. These Q2D and
Q1D sections originate from the high temperature Q2D FS. Thus, the
Q1D FS is strongly warped (Reprinted figure with permission from
Kovalev et al. [46]. Copyright 2002 by the American Physical Society.).
(b) Microwave absorption as a function of the magnetic field. Many
harmonic POR are seen (Reprinted figure with permission from Ko-
valev et al. [46]. Copyright 2002 by the American Physical Society.).
(c) Microwave absorption as a function of the angle of the magnetic
field. The data exhibit many harmonic resonances which imply that
the warping pattern is more two-dimensional (Reprinted figure with
permission from Kovalev et al. [46]. Copyright 2002 by the American
Physical Society.). (d) Summary of the POR in a-(ET)2KHg(SCN)4
(Reprinted figure with permission from Kovalev et al. [46]. Copyright
2002 by the American Physical Society.).









section is reconstructed from a Q2D section belonging to the high temperature FS

(not shown), the Q1D FS sheet still retains a two-dimensional shape, so that it is
strongly corrugated. Such a Q1D FS can be represented by finite higher-ordered

transfer integrals which produce the harmonic resonances. In contrast, the Q1D

conductor (TM TSF)2C104 has a more one-dimensional FS. Experimental results for

the Q1D POR in (TMTSF)2C104 are shown in C'!h 1 4.

2.4 Periodic Orbit Resonance for a Quasi-two-dimensional Fermi
Surface

As shown in the previous section, the POR effect is associated with the

warping of a FS. Therefore, the POR is also observed for a warped 2D FS, i.e., a

quasi-two-dimensional (Q2D) FS. In the case of many two-dimensional conductors,

the conductivity along the z-axis (the least conducting direction), Uaz, is non-zero,

e.g., for K-(ET)2Cu(NCS)2, axx/a t 1000 [3], for Bi2Sr2Ca~_CunOy, aX,/azz

10000 [47], for YBa2Cu30y, a7x/a,, ~ 100 [47] and for Sr2RuO4, a1xx1/az 400-

4000 [48]. These are so-called Q2D conductors. We now show that a,, can also

have a resonance, the so-called periodic orbit resonance (POR). Here we consider a

Q2D electron model, i.e., the energy dispersion is represented by

E(k) 2= (k2 + k2) 2tz cos(kc), (2-21)

where c is the interlayer spacing. The bandwidth in the z-direction, t,, is much

smaller than the Fermi energy, i.e., Ep/tz ~ V/- z/a- ~ 30 for K-(ET)2Cu(NCS)2.

As a result, the shape of the FS is similar to the 2D electron model, but the FS has

a slight corrugation along the z-direction. The magnitude of the small corrugation

of this cylinder corresponds to the bandwidth 4tz along the z-direction. Such a FS

is shown in Fig. 2-6(a). We now consider the motion of electrons on this FS under

a dc magnetic field similar to the case of the 2D electron model. The dc magnetic

field is applied at an arbitrary tilted angle, i.e., B = (B sin,0, B cos0). The












motion is given by the following equations,


X eBcosk(, (2-22a)
m*

Ky eBtc sin 0 eB cos 0
kh (,,-[. (t)c] + k* (t), (2 22b)
h m
eB sin 0
kz k, (t)B (2-22c)
m*
and

vX (t) kh (t), (2-23a)
m

v9(t) -- k(t), (2 23b)
m
2t c
(t) sin[k (t)c]. (2-23c)

Because eBt < $-Bk (except when the field is oriented close to ,'-plane, i.e., 0 ~

90), the equation (2-22b) may be approximated as

eB cos 0
ky k(t), (2-24)
m*

so that the motion can be calculated in the same way as the 2D electron model.

The resultant motion therefore becomes, kx(t) = kcos[2Dt + 0(k)], ky(t)

k sin[wDt + 0(k)], and kz(t) = ktan 0cos[w2Dt + O(k)]; vx = vcos[w2Dt + (k)],

v = vsin[2Dt + Q(k)] and v (t)= 2 sin[kctan 0cos{w2Dt + Q(k)}]. The resonance

frequency is the same as Eq. 2-12. The z-component of the group velocity, vz, is

now non-zero. This gives finite conductivity along the z-direction, a, and to a

resonances. Using the Boltzmann equation, the conductivity, a is given by

Jo(7 tanO) 0 1 1 ,
zz (, B) o tan + ( nt2tan 0)-2 + ,( + 1
1 2r2 +Z _t c n lW2D 272 t (; 2D2
(2-25)

where Jn(x) is the n-th order Bessel function, = kpc, crx and ay are given by

the same expressions as Eq. 2-7b. The conductivity therefore has resonances at









frequencies,
neB
) = nw2D = cos 01. (2-26)
m*

Thus, multiple resonances in ua, are predicted instead of the single resonance in a,,

and ~yy found in the previous discussion for the 2D case. By investigating the POR

at different c, B and 0, one can determine the effective mass, m*. By analyzing

the shape of the POR, one can also determine the scattering time r, since the

half-width of the POR is ~ 2/r.

Although the motion of an electron along the z-direction gives resonances

in the conductivity, a z, with similar resonance conditions, the picture of the

motion is quite different from the motion in the '-plane. Fig. 2-6 shows a typical

trajectory of electrons on the FS. In Fig. 2-6(a), the applied magnetic field is tilted

from the z-axis, so that the trajectory on the FS is also tilted. As shown in the

right panel in Fig. 2-6(a), the group velocity v, becomes periodic because of the

corrugation along the z-direction. This oscillatory v, brings a resonance effect in

the conductivity, za,. On the other hand, when the applied magnetic field is along

the corrugation axis (the z-axis in the present example), the group velocity v, is

not oscillatory. In this case, no POR is observed, as shown in the right panel in

Fig. 2-6(b). In the case of a low-symmetry crystal structure, the corrugation axis

can be different from the z-axis. We see such an example in ('! 11' 6.

2.5 POR and Angle-dependent Magnetoresistance Oscillations

The angle-dependent microwave conductivity measurements we have discussed

here can also be performed via dc measurements. In particular, in the case of

many clean low dimensional conductors, one can observe strong angle-dependent

magnetoresistance oscillations (AMRO) in the dc conductivity (or resistivity) by

rotating a magnetic field of a fixed strength relative to the sample. The AMRO

observed for rotation in different ( i I 11. ..raphic planes in Q2D and Q1D FSs are

named differently. Fig. 2-7 and Fig. 2-8 illustrate each case. Fig. 2-7 explains the




















Yamaji Oscillations in Sr2RuO4


Q2D FS


Figure 2-7.


-90 06 0
8 (degree)


AMRO for Q2D conductors. In the case of a Q2D FS, Yo.i1 .ii oscilla-
tions may be observed by varying the field orientation from the z-axis
to the ';-plane. Here, the conducting plane is the ';-plane. 0 repre-
sents the angle between the z-axis and ';-plane. Q represents the angle
from the x-axis in the I;-plane. The Yei" iii oscillations in Sr2RuO4
are shown. The strongest peak in the resistivity p, is seen at around
0 = 350, and many oscillations are seen at higher angles (Reprinted
figure with permission from Ohmichi et al. [49]. Copyright 1999 by the
American Physical Society.).









AMRO effect for a Q2D FS, so-called Yi,, i ii oscillations. For the observation of

the Ya,, ii oscillations, a fixed magnetic field is rotated between the z-axis and

the 'i-plane. The experimental result of the Yei"" iii oscillations is shown in the

right panel of Fig. 2-7. Sr2RuO4 shows the strongest resistivity peak at around

0 = 35, and many oscillations are also seen at higher angles. On the other hand,

in the case of a Q1D FS, three types of the AMRO effects are named: Lebed (z y

rotation) [26, 27, 51], Danner-Kang-C'l 1d:!:1. (DKC) (z x rotation) [28] and 3rd

angular effects (x y rotation) [29], as shown in Fig. 2-8. In the case of the DKC

effect in (T\iTSF)2C104, the most pronounced peaks are seen at around 0 = 850

and 950, and smaller oscillations are also seen at angles between 0 = 850 and 95.

The strength of the peak depends on the magnetic field strength. The oscillations

become smaller when the magnetic field is weaker. This is because of the lower

product of WJQlDT. In the case of the Lebed effect, many minima are seen. These

angles are the so-called Lebed magic angles. The 3rd angular effect is represented

by two minima in the resistance.

Since the AMRO is nothing more than dc POR, they are easily described by

simply applying u = 0 in the ac conductivity. For example, in the case of the Lebed

effect, one can use Eq. 2-19 with w = 0,


Re ,(w, B,0) = -. (2-27)
t + L;2 T2

Fig. 2-9 plots the inverse conductivity as a function of the angle 0. The plot shows

minima, which correspond to the POR at w / 0, and the AMRO at w = 0. The

positions of the minima shift continuously by changing the frequency. The positions

of the minima at w = 0 represent the directions of the warping. The Lebed effect in

(TMTSF)2C104 is presented in C'!i p1 4.

Next, we consider the case of a Q2D FS (i.e., Yei", iii oscillations). Like

the Q1D AMRO, by applying a = 0 in Eq. 2-25, one obtains the following






















DKC effect in (TMTSF)2C104


?5 /


T2T


20 10 10 2C
,(degree)


Q1DFS

z


x
conducting ax i


Lebed effect in (TMTSF)2PF6
(TMTSF)FPFe T=1.55K P=7.9kbar
S-4 2 c 3
120 } 2

80
c i o b'
; 1 b1.









0.01~ ~ -^1 -------------^-
B

90 60 30 0 30 60 90
Angle (T )

y

B=12T
B=9T




S(TMTSF)2CIO4
" T=1.7K
0.01
90 60 30 0 30 60 90
3rd angular effect in (TMTSF)2C104


Figure 2-8.


AMRO for Q1D conductors. In all cases, Rzz is plotted. In the case
of a Q1D FS, three AMRO effects can be observed: Lebed (y z rota-
tion), DKC (x z rotation) and the 3rd angular effect (x y rotation).
In the case of the DKC effect in (TM iTSF)2C104, the most pronounced
peak is seen around 0 = 850 and 950, and smaller oscillations are also
seen at angles between 0 = 850 and 95. The strength of the peak
depends on the magnetic field (Reprinted figure with permission from
Danner et al. [28]. Copyright 1994 by the American Physical Soci-
ety.). In the case of the Lebed effect, many minima are seen. These
resonance angles are the so-called Lebed magic angles (Reprinted from
Kang et al. [50], Copyright 2003, with permission from Elsevier.). In
the case of the 3rd angular effect, two minima are seen in the resistance
(Reprinted figure with permission from Osada et al. [29]. Copyright
1996 by the American Physical Society.).









1/Gzz


Figure 2-9.


B

0.5 0c


0.25 (0c


/ AMR O \- 0) = 0
-180 -90 0 90 180
0 (degrees)

The Lebed effect in both the dc and ac conductivity. The inverse
conductivity as a function of the angle 0 is simulated using Eq. 2-19.
The inverse conductivity has two minima at w / 0. These minima are
the POR. The inverse conductivity has one minimum at w = 0, the
so-called AMRO.


conductivity,
2J
a, (B, 0) o Jo(y tan 0) + (7tan) (2-28)
S1+ (nUw2272
where w2D u cos 0 and c = eB/m*. Fig. 2-10(a) plots the conductivity given

by Eq. 2-28. Strong oscillations are clearly seen in Fig. 2-10, the so-called Yoii i: ii
oscillations. Just like the POR effect, the amplitude of these oscillations depends

on the product of wc,. When wu-T is high, the oscillations are well-pronounced. On
the other hand, they are smeared when w-r is small. However, the positions of the
maxima and minima of the oscillations are independent of the product wc-. The
position of the dc resonance depends only on 7 = kFc; the figure shows AMRO for
two values of 7 = kpc, i.e., 7 2 and 7 = 3, at wT = 3. Thus one can determine
kFc and, eventually, kF from the Yi,"" iii oscillations.
In Fig. 2-11, we show the frequency dependence of the Ye i iii oscillations.

The calculation for the Boltzmann equation Eq. 2-25 was performed by numerically

























Figure 2-10.


)cj=0.5


(0)c=l
0c1=3


B *4


V y-kFc -2 \'
-60 -30 0 30 60
Angle O(degrees)

Yelin iii oscillations. The conductivity from Eq. 2-28 is shown as a
function of angle between the magnetic field and the least conducting
direction (the z-axis). The plot shows strong oscillations, the so-called
Yeon" i ii oscillations. The oscillations are pronounced due to the large
product wc-. The position of the oscillations depends on 7 = kpc, but
is independent of uwcT.


solving the differential equations in Eq. 2-22 and Eq. 2-23 with the Q2D energy

dispersion E(k) = (k2 + k ) 2tz cos(kzc). The calculation covers v =dc

to 500 GHz and 0 =0 to 90 degrees. Fig. 2-11 shows clearly that peaks in the

conductivity shift to higher angles when the frequency becomes higher. In the case

of ac conductivity measurements, the maxima are called POR.

We introduced the AMRO effect using a semiclassical description in this

section. However the origin of the Lebed effect is still an open question although

many theories have been proposed. The semiclassical description we used in this

chapter is suitable for some theories, but not for others. This difference may be

more explicit when the Lebed effect is considered at dc to microwave frequencies.

Using the POR effect, we recently tested whether the semiclassical description is

applicable to explain the Lebed effect. We will have this discussion in CI '1. 4.


Gzz(B)









2.6 Quantum Effects in the Conductivity

In a quantum mechanical picture, the energy dispersion becomes quantized

under magnetic fields, as does the FS. Such a quantized FS gives rise to important

magnetic quantum oscillation effects in clean compounds and at low temperature,

i.e., hr-1 < hj,,c e.g., the de Haas-van Alphen (dHvA) and Shubnikov-de Haas

(SdH) effects. These can also provide useful information about the FS. Moreover,

both the POR and the SdH phenomena appear in the electronic conductivity,

so that both phenomena can be observed at the same time. In fact, as seen in

C'! '1' 6, we observed both effects in the experiments for K-(ET)2X. In this section,

we introduce magnetic quantum oscillations briefly. We also introduce the angle-

dependence of the frequency of the SdH effect since the angle dependence ilp i,- d an

important role in C'! 1' 6. A more detailed description of the quantum mechanical

effects can be found in many books, e.g., by Shoenberg [25] and Abrikosov [21].

We consider the case of Q2D conductors, i.e., the energy dispersion is modeled

by Eq. 2-21. A magnetic field is now applied along the z-axis. Considering the

gauge transformation, H H' = |p qA 2 + qQ with the vector potential

A = (0, Bx, 0) and = 0, one can get the following energy dispersion,


E,,(k,) = h,(n + ) 2t, cos(kc), (2-29)


where wu(B) = eB/m* is the cyclotron frequency again and n = integer (0, 1, 2...).

The energy dispersion is therefore quantized. Each energy is called a Landau level.

The separation of the energy levels depends on the strength of the magnetic field.

This means that the strength of the magnetic field changes the number of filled

Landau levels below the Fermi energy Ep. Fig. 2-12 shows changes of the Landau

tubes with magnetic field. If B 0, the energy spectrum is, of course, continuous. If

B = finite, the energy spectrum becomes quantized. When n=20, 20 Landau tubes

intersect the FS. Upon increasing B, the number of Landau tubes within the FS









B=O Strength of B












n=oc n=20 n=8 n=4
(continuous)

Figure 2-12. Landau tubes in a magnetic field. If B = 0, the energy spectrum
is continuous. If B = finite, the energy spectrum becomes discrete
(quantized). Upon increasing the magnetic field, the number of Lan-
dau tubes within the FS becomes smaller because the momentum
separation becomes greater.


becomes smaller because the momentum separation kxy becomes greater. In the

quantum mechanical picture, the CR and POR phenomena are represented by a

quantum transition between Landau levels, e.g., n to n+1.

The quantized energy dispersion, Eq. 2-29, brings about radical changes in

the density of states (DOS). Since the Landau levels are discrete, this change leads

to a degeneracy in each Landau level, and the DOS is then also discrete. The

degeneracy is proportional to the strength of the magnetic field. The discrete DOS

in a non-interacting system should be a delta-function, but in a more realistic

system, the DOS has some width due to scattering of carriers. Fig. 2-13 shows a

typical example of the DOS under the magnetic fields in the Q2D system. The

DOS consists of states of each Landau band (n = 0,1, 2...). Each state is given
by 6(E E(n, kz)) (EE(n,k))2+i( 1)2 where is the scattering time. As a

result, each DOS has a finite width. The DOS due to each Landau tube therefore

superimpose upon each other. Moreover the overlap of each DOS depends on 7 so

that the shape of the DOS varies with material, temperature and so on. Because






















EF
iu -h.~ B ---- I Lower B


1/2 3/2 5/2 7/2 9/2
E/hcoc


Figure 2-13


DOS of a Q2D conductor in a magnetic field in units of h',c. The
DOS in a Q2D conductor is given by, N(E) E= n,k,(E E(n, k ))
Zn,k:I (E-(,k7~) (/2)2 The dotted line represents the DOS for
each Landau level. Ti.! solid line is the sum of the dotted lines. The
dashed line represents Fermi energy. When the magnetic field in-
creases, the Fermi ent line moves to left in the figure and vice
versa. Thus, the DOS at the Fermi energy EF is varied by the mag-
netic field. Ti.-, is a the origin of the SdH effect.









of the superposition of states of equally spaced Landau levels, the DOS becomes

oscillatory as a function of the energy. The period of the oscillation of the DOS

depends on the strength of the magnetic field. When a higher magnetic field is

applied, the period becomes greater and vice versa. In Fig. 2-13, the DOS is

represented in units of hu,. When the magnetic field increases, the unit of the

x-axis in Fig. 2-13 becomes larger so that the Fermi energy shifts to left. i.e., the

dashed line representing the Fermi energy shifts to left when the magnetic field

increases, and to right when it decreases. Thus, N(EF) traces the oscillatory

DOS when changing the magnetic field. Recalling Eq. 2-6b, one can see that the

conductivity is proportional to N(EF). This is the origin of the SdH effect. On the

other hand, the POR comes from the oscillatory (k, t). Thus these phenomena

have completely different origins.

As shown in Fig. 2-13, the period of the quantum oscillations is given by

A(EF/hc) = 1. Using the Fermi energy E = h2(kF,1(kz)2 + kF,y(kj)2)/2m* and

the cross sectional area of the FS A = (kFx(kz)2 + kFy(kz)2) = 27 *Ep/h2 at

each k,, the period of the oscillation as a function of the magnetic field can also be

written,
1 he 2we
A( E h (2-30)
B EFm* A
Furthermore, when the magnetic field is tilted, the period is given by


A( h) (2 31)
B EFm* hA cosO0

Thus one can measure the cross sectional area of the FS by studying the magnetic

quantum oscillations. The effective mass can be estimated by studying temperature

dependence and magnetic field dependence of magnetic quantum oscillations. The

book by Shoenberg gives the details [25].


um oscillations. The

book by Shoenberg gives the details [25].







41

2.7 Summary

In this chapter, we introduced the theoretical basis and usage of the POR

phenomena. The physical aspects of the POR are similar to CR and AMRO since

all of them can be explained by the same magneto-transport model. On the other

hand, the SdH and dHvA effects are purely quantum phenomena.















CHAPTER 3
EXPERIMENTAL SETUP

The in i jr results presented in this chapter can be found in the article entitled

Rotating ..:; H:; for high-field 'iu1'.--dependent microwave spectroscopy of low-

dimensional conductors and magnets, S. Takahashi and S. Hill, Review of Scientific

Instruments 76 023114 (2005).

3.1 Overview of Microwave Magneto-optics

In recent years, microwave (millimeter and sub-millimeter wave) technologies,

covering frequencies from 10 GHz to 10 THz (0.33 330 cm 1), have become the

focus of intensive efforts in many fields of research. In engineering and medicine,

THz imaging represents one of the next-generation technologies, enabling non-

destructive materials inspection, chemical composition analysis [52, 53, 54], and

medical diagnoses [52, 53, 55]. In the fundamental sciences, physics, chemistry and

biology, microwave spectroscopy is also very useful for investigating the physical

properties of a material. This is particularly true for the sub-field of condensed

matter physics, where the millimeter and sub-millimeter spectral range can pro-

vide extremely rich information concerning the basic electronic characteristics of

a material [56, 57, 58, 59]. Furthermore, combining microwave techniques and

high magnetic fields (microwave magneto-optics), allows many more possibilities,

including: cyclotron resonance (CR) [32, 33, 36, 39, 43, 46, 60, 61]; electron param-

agnetic resonance (EPR) [62, 63, 64, 65]; antiferromagnetic resonance (AFMR) [1];

Josephson Plasma Resonance (JPR) measurements of li-,. 1t superconduc-

tors [66, 67, 68, 69]; and many others. In each of these examples, the magnetic

field influences the dynamics of electrons at frequencies spanning the microwave











spectral range. Another very important aspect of microwave magneto-optical in-

vestigations is the possibility to study angle dependent effects by controlling the

angle between the sample and the microwave and dc magnetic fields. For instance,

through studies of the angle dependence of CR amplitudes, one can extract detailed

information concerning the Fermi surface (FS) topology of a conductor [46, 61, 70].

Consequently, angle-dependent microwave spectroscopy has been widely used

in recent years to study highly anisotropic magnetic and conducting materials.

Problems which have been addressed using these methods include: high-Tc su-

perconductivity [66], and other low dimensional superconductors, e.g., organic

conductors [32, 33, 43, 46, 61, 68, 69], Sr2RuO4 [60] etc.; the quantum and frac-

tional quantum Hall effects [71, 72]; and low dimensional magnets, including

single-molecule magnets (SMMs) [63, 64, 65, 73].

Unfortunately, the microwave spectral range presents many technical chal-

lenges, particularly when trying to study very tiny (< 1 mm3) single-crystal sam-

ples within the restricted space inside the bore of a large high-field magnet system

(either resistive or superconducting [74].) Problems associated with the propagation

system stem from standing waves and/or losses [75]. Several methods have been

well documented for alleviating some of these issues, including the use of funda-

mental TE and TM\ mode rectangular metallic waveguides, low-loss cylindrical

corrugated HE waveguides [76], quasi-optical propagation systems [76], and in-situ

generation and detection of the microwaves [77]. Standing waves are particularly

problematic in the case of broadband spectroscopies, e.g., time-domain [77] and

fourier transform techniques [78], as well as for frequency sweepable monochromatic

sources [79, 80]. In these instances, the optical properties are usually deduced via

reflectivity or transmission measurements, requiring a large well-defined (i.e., flat)

sample surface area (>A2). For cases in which large samples are not available

(note: A spans from 3.34 cm at 10 GHz to 0.33 mm at 1 THz), resonant techniques









become necessary, e.g., cavity perturbation [56, 57, 58, 62, 75]. This unfortunately

limits measurements to the modes of the cavity. In addition, making absolute

measurements of the optical constants of a sample, as a function of frequency, is

extremely difficult to achieve using cavity perturbation because of its narrow-band

nature [56, 57, 58]. However, the cavity perturbation technique is ideally suited for

fixed-frequency, magnetic resonance (magnetic field-domain) measurements [75].

Furthermore, as we have recently shown, it is possible to make measurements at

many different frequencies by working on higher order modes of the cavity [75]. To

date, measurements in enclosed cylindrical copper cavities have been possible at

frequencies up to 350 GHz (see Takahashi et al. [81]).

The frequency range in the magnetic field-domain technique depends on the

nature of the magnetic resonance. For example, in the case of the Q2D periodic

orbit resonance (POR), the frequency range depends on the effective mass because

the resonance condition for the Q2D POR is given by Eq. 2-26, v, = eB/(27m*)

where v, is the cyclotron frequency, B is the magnetic field strength and m* is

the effective mass. Fig. 3-1 shows the frequency range for the Q2D POR. In the

upper panel, the resonance is shown for different frequencies. In the lower panel,

the frequency ranges are given for different m*. As seen in the figure, the magnetic

field-domain technique works very well in the case of small effective mass. For

instance, in the case of m* = me where me is electron mass, the field range from

0-15 tesla, which is relatively easy to obtain in a laboratory, corresponds to 0-

420 GHz. This wide frequency range is therefore available to increase wc-. The

upper panel in Fig. 3-1 illustrates this point. wc- can be enhanced from 1 to 4

by changing the frequency. However, in the case of a higher effective mass, the

corresponding frequency range is narrower. In this case, one may need to employ

a high magnetic field facility. Fields in the range of 0 45 T at the National High

Magnetic Field Laboratory (NHMFL) in Tallahassee FL, USA corresponds to 0









v=Vc=50 GHz, or=l .1=Vc=100 GHz, or2=2

\ / ,v=Vc=200 GHz, ay=4


500 GHz
(16.7 cm-')
m*-Me

m*=5 me

m*=10 m,


Figure 3-1.


Frequency range for the Q2D POR in magnetic fields accessible at the
National High Magnetic Field Laboratory (NHMFL) in Tallahassee
FL, USA. Since v, = eB/(27rm*), the frequency range depends on
the effective mass. For example, in the case of a small effective mass,
the frequency range is very wide, e.g., the fields in the range of 0 -
45 T correspond to 0 1.26 THz (0 42.08 cm-1) for m* me. For a
higher effective mass, the frequency range is narrower, e.g., the fields
in the range of 0 45 T corresponds to 0 252 GHz (0 8.42 cm-1) for
m* = *,


- 252 GHz (0 8.42 cm-1) for m*

m* = 10me.


".;,. and 0 126 GHz (0 4.21 cm-1) for


3.2 Experimental Setup

Fig. 3-2 shows an overview of our setup for the cavity perturbation technique.

The sample is mounted inside a cylindrical cavity which is positioned in the

magnetic field center. The microwave signal coming from the generator (not shown)

is transmitted through the incident waveguide into cavity and couples to the

sample, and is then returned through a second transmission waveguide. The probe,

which consists of the cavity, waveguides, electronics (thermometer, heater etc.) and

vacuum jacket, is placed inside the cryostat with a small mount of exchange gas.

In the experiment, we study the microwave response of the sample by changing:

the microwave frequency; the strength of the magnetic field; the orientation of the

magnetic field; the temperature etc. This is achieved through a combination of

many instruments.









As a microwave source and detector, we employ a Millimetre Vector Network

Analyzer (\IVNA) with an External Source Association (ESA) option (not shown)

and several Schottky diodes manufactured by AB millimetre [82]. The source

frequency of the MVNA is tunable in the range of F1 = 8-18.5 GHz. By feeding

the source frequency to a Schottky diode, which is a passive non-linear device,

harmonic components of the source (Fmm = N x F1, N = integer) are produced

and transmitted to the waveguide probe. The Schottky diode that act as the source

is called the harmonic generator (HG). The optimized harmonics depend on the

type of Schottky diode. Several Schottky diodes are available [K-band (N = 2,

3, v = 18-40 GHz), V-band (N = 3 and 4, v = 48-72 GHz), W-band (N = 5

and 6, v = 72-110 GHz), and two sets of D-band diodes (N > 6, v > 110 GHz)].

For detection, the microwave signal (Fmm) returning to the MVNA is mixed

with a second microwave signal (F2) at a second Schottky diode, the so-called

harmonic mixer (HM). The beat signal (Fbeat N x F1 N' x F2) and the phase

( ,, = N x k1 N' x Q2) is then sent to a heterodyne vector reciever (VR)

in the MVNA. By choosing appropriate HG and HM so that N = N', and by

locking the phases Q1 = 2, the noise associated with the phase is cancelled, and

a low noise level is achieved. The detection of the signal is performed on the MHz

component in the beat signal, i.e., FMHz N x (F1 F2). The MVNA employs

FMHz = 9.010 !;2S125 or 34.010 !;;2S125 MHz for the vector measurement.

Although the D-band Schottky diode can produce a fairly powerful microwave

signal on harmonics N = 6 and 7, the power diminishes for N > 8. For higher

frequencies (v > 170 GHz), we usually use the ESA option which consists of a

Gunn diode, a directional coupler-harmonic mixer and a multi-harmonic multiplier.

With this option, the multiplier is fed by a more powerful higher frequency

microwave signal (PF1 ~ 30 mW, Fi = 69-82.3 and 82.5-102.7 GHz) from a Gunn

diode. Since the Gunn source frequency is much higher than the internal source in













Table 3-1. Available magnet systems at the University of Florida (UF) and the
NHMFL. The table lists the field geometry and magnet type, the max-
imum available field Bax, the probe length, the available temperature
(T) range, and the outer diameter of the cavity probe. The Quantum
Design (QD) Physical Property Measurement System (PPMS) allows
two-axis rotation.

Magnet Bmax (T) Type length T (kelvin) Probe dia.
45 T (NHMFL) 45 Axial hybrid 1.67 m 1.4- 300 3/4"
33 T (NHMFL) 33 Axial resistive 1.45 m 1.4- 300 3/4"
25 T (NHMFL) 25 Axial resistive 1.6 m 0.5 3001 1"
Oxford Inst. (UF) 17 Axial SC 1.9 m 0.5- 300a 1" (7/8"a)
QD PPMS (UF) 7 Transverse split-coil SC 1.15 m 1.7- 400 1"


the MVNA, one can achieve high frequencies on lower harmonics, thereby enabling

measurements to much higher frequencies with the ESA option. For more details

concerning the MVNA and the ESA option, see Goy and Gross [79] and Mola et

al. [75]. With this setup, we can work in a wide frequency range from 8-700 GHz

(0.27-23.3 cm- ).

Since we wish to investigate the microwave response for a wide variety elec-

tronic and magnetic systems, many magnet systems are needed. Two magnet

systems are available at UF, one produced by Oxford Instruments [83] and one

by Quantum Design [84], and we can also use the magnets at the NHMFL in

Tallahassee, FL, USA [74]. The various magnet systems which are compatible

with the instrumentation described here, are listed in TABLE 3-1. All systems

are standard type magnets with vertical access for measurements (not horizon-

tal access magneto-optical magnets). The standard cryostats designed for these

magnets are all 4He based (either bath or flow cryostats). However the base tem-

perature of the 4He based cryostat is limited, i.e., to roughly 1.4 K. Because of

the demand to work at lower temperatures, we have constructed a simple 3He

refrigerator which is compatible with the 17 tesla Oxford Instruments supercon-

ducting magnet at UF, and the 25 T resistive magnet at the NHMFL, as listed










in TABLE 3-1. A schematic of this refrigerator is shown in Fig. 3-3. The 1.9 m

long waveguide/cavity probe is inserted directly into the 3He space, which is con-

structed from a 7/8" (= 22.2 mm) outer diameter stainless steel tube with a 0.010"

(= 0.25 mm) wall thickness. The lower 254 mm of this tube is double jacketed

with a 1.00" (= 25.4 mm) outer diameter. The volume between the two tubes is

vacuum sealed in order to provide thermal isolation between the 3He liquid and

the surrounding 4He vapor. The 3He condenses by means of heat exchange with

the walls of the 7/8" tube (above the double jacketed region) which is inserted

into the Oxford Instruments 4He flow cryostat operating at its base temperature

of ~ 1.4 K. After condensation of the full charge of 3He (5 liters at STP), sub-

kelvin temperatures are achieved by pumping directly on the liquid by means of

an external sealed rotary pump. The refrigerator operates in single-shot mode,

i.e., the 3He is returned to a room temperature vessel, where it is stored until the

next cooling cycle. A simple gas handling system controls the condensation of 3He

gas, and the subsequent pumping of the gas back to the storage vessel. The 3He

tube and gas handling system is checked for leaks prior to each cool down from

room temperature. Although this design is simple, it has the disadvantage that the

microwave probe comes into direct contact with the 3He vapor, thus potentially

affecting the tuning of the cavity, as well as the phase of the microwaves reaching

the cavity via over 3.8 m of waveguide; such phase fluctuations can cause drifts in

signal intensity due to unavoidable standing waves in the waveguide. However, we

have found that these problems are minimal when operating at the base 3He vapor

pressure (0.15 torr). The temperature of the sample is then controlled by supplying

heat to the copper cavity, which acts as an excellent heat reservoir, i.e., it ensures

good thermal stability. The base temperature of the THe refrigerator is 500 mK and

it provides hold times of up to 2 hours.














Waveguides


jf Pumping Line






Fins




1K Pot



Cavity

Liquid 3He
Figure 3-3. A schematic diagram of the 3He probe used for sub-kelvin experiments
in the Oxford Instruments 17 T superconducting magnet. See main
text for a detailed description of its construction (Reused with permis-
sion from Takahashi et al. [81]. Copyright 2005, American Institute of
Physics.).










Table 3-2. Probes used for the cavity perturbation technique. The coaxial cable,
K- and V-band waveguide probes were built in-house. The corrugated
waveguide (HE mode) probe is produced by Thomas Keating Ltd. [85].
V and R denote vertical and rotating cylindrical cavities respectively. D
and H represents the diameter and height of the cavities respectively.


Probe v range (GHz) Harmonic N Cavity size (D x H)
Coaxial cable 8-18 1 0.70" x i -.I, (V, R)
+ dielectric material (E=15)
K-band (WR-42) waveguide 18-40 2, 3 0.70" x I -.1 (V)
0.52" x 0.52" (V)
V-band (WR-15) waveguide 48-350 3-25 I :I x I :II (V, R)
0. 111 x I I (V)
0.25" x 0.25" (V)
Corrugated waveguide 170-700 15-30 No cavity (flat end plate)


In order to make the best use of the wide frequency range provided by the

MVNA with the ESA option, we have developed several cavity perturbation probes

which have different optimized frequencies. We list the probes in TABLE 3-2. The

coax cable, K- and V-band probes are home-made, and the corrugated probe is

fabricated by Thomas Keating Ltd. [85]. We have also constructed several sizes of

cylindrical cavities for each probe, and rotating cavities for angle-dependent stud-

ies. All cavities are made of copper. The details of the vertical cavity are explained

in our previous paper [75]. The details of the rotating cavity will be introduced in

Sec. 3.3. Probes have been designed separately for each magnet system. For the

sake of good temperature control, the probes are made from a combination of high

and low conductivity materials, i.e., copper waveguide and stainless steel (S.S.)

waveguide. The length of each waveguide section was determined by the temper-

ature profile in the cryostat. Fig. 3-4 shows the temperature profile for the QD

PPMS 7 T and Oxford Instruments 17 T magnet systems. The S.S. waveguides are

used where the temperature change is large, and the copper waveguides are used

for the reminder. See Mola ct al. [75] for further details concerning the microwave

probe design.









3.3 Rotating Cavity

The standard approach for studying angle-dependent effects using the cavity

perturbation technique is to use a split-pair magnet and/or goniometers, and is

widely used in lower frequency commercial EPR instruments, e.g., X-band, K-band

and Q-band [36, 86]. In the case of the split-pair approach, the DC magnetic field

is rotated with respect to a static waveguide/cavity assembly. However, those

approaches are usually limited in terms of the strength of the magnetic fields

obtained. In this section, we outline a method for in-situ rotation of part of a

cylindrical resonator which we developed recently, thus enabling angle-dependent

cavity perturbation measurements in ultra-high-field magnets, and two-axis

rotation capabilities in standard high-field superconducting split-pair magnets.

Details of the rotating cavity have been published in Takahashi et al. [81]. As we

shall outline, the rotation mechanism preserves the cylindrical symmetry of the

measurement, thereby ensuring that the electromagnetic coupling to the microwave

fields does not change upon rotating the sample. This is particularly important

for studies of low dimensional conductors, where sample rotation alone (as in the

case of a goniometer) would lead to unwanted instrumental artifacts associated

with incommensurate symmetries of the sample and cavity. The rotating cavity

described here is compatible with all magnet systems listed in TABLE 3-1.

We note that a rotating cavity has previously been developed by Schrama

et al. [87], at the University of Oxford, also for high-field microwave studies. As

we will demonstrate, the cylindrical geometry offers many advantages over the

rectangular design implemented by the Oxford group. For example, the waveguides

are coupled rigidly to the cylindrical body of the cavity in our design (only the

end-plate rotates), whereas the coupling is varied upon rotation in the Oxford

version, resulting in effective "blind--l.. I i.e., angles where the microwave fields

in the waveguides do not couple to the cavity; in contrast, the cylindrical version









offers full 3600+ rotation. Furthermore, the rigid design offers greater mechanical

stability and, therefore, less microwave leakage from the cavity, resulting in

improved signal-to-noise characteristics [75]. The TEOln cylindrical modes also

offer the advantage that no AC currents flow between the curved surfaces and flat

rotating end-plate of the cylindrical resonator. Consequently, the moving part

of the cavity does not compromise the exceptionally high quality (Q-) factors

associated with these modes. Indeed, Q-factors for the first few TEOln modes

vary from 10, 000 to 25, 000 (at low temperatures), as opposed to just 500 for the

rectangular cavities. This order of magnitude improvement translates into vastly

increased sensitivity, enabling e.g., EPR studies of extremely small single i ,-~I I1

samples. Finally, the cylindrical cavity is machined entirely from copper, and held

rigidly together entirely by screws. This all-copper construction results in negligible

field sensitivity, i.e., the field dependence of the cavity parameters is essentially flat

and, most importantly, the cavities do not contain any paramagnetic impurities

that could give rise to spurious magnetic resonance signals.

The configuration of the rotating cavity is shown in Fig. 3-5. The principal

components consist of the open-ended cylindrical resonator, the cavity end-plate

and worm gear, a worm drive for turning the end-plate, and a wedge which facili-

tates external clamping and un-clamping of the cavity and end-plate. The cavity

assembly is mounted on the under-side of a stage (not shown in Fig. 3-5), and

the end-plate is centered on the axis of the cavity by means of a centering-plate.

The upper part of the wedge is threaded, and passes through a threaded channel

in the stage so that its vertical position can be finely controlled via rotation from

above. Likewise, the worm drive is rotated from above, and accurately aligned

with the worm gear via an unthreaded channel in the stage. Finally, the worm-

drive and centering plate are additionally constrained laterally by means of an

end cap and spindle (see below) which attaches to the under side of the resonator.






















Cavity
End Plateworm Resonator
Wedge
Coupling Hole






Centering-
Plate

End Cap C



Spindle



Figure 3-5. A schematic diagram of the various components that make up the
rotating cavity system. The sample may be placed on the end plate,
which can then be rotated via an externally controlled worm drive.
The wedge is used to clamp and un-clamp the end plate to/from the
main resonator. See main text for a detailed description of the assem-
bly and operation of the rotating cavity (Reused with permission from
Takahashi et al. [81]. Copyright 2005, American Institute of Physics.).









The cavity, end plate, and stage are each machined from copper, thus ensuring

excellent thermal stability of the environment surrounding the sample; the heater

and thermometry are permanently contacted directly to the stage. The remaining

components shown in Fig. 3-5 are made from brass [88].

The internal diameter of the cavity (7.62 mm) is slightly less than the diame-

ter of the end-plate, which is free to rotate within a small recess machined into the

opening of the resonator. On its rear side, the copper end-plate mates with a brass

gear which, in turn, rotates on an axis which is fixed by the centering-plate. As

mentioned above, rotation of the worm gear and end-plate is achieved by turning

the worm drive with the wedge disengaged from the end-plate. During experiments,

a good reproducible contact between the end-plate and the main body of the cavity

is essential for attaining the highest resonance Q-factors. This is achieved by engag-

ing the wedge through a vertical channel in the centering-plate, where it transfers

pressure along the axis of the end-plate. Fig. 3-6 dip-,~'i labeled photographs of

the 1st and 2nd generation rotating cavity assemblies. The 2nd generation version

employs a smaller home-built worm drive, reducing the overall diameter of the

probe to slightly below 3/4" ( 19.1 mm), which enables its use in the highest field

magnets at the NHMFL; the 1st generation probe has an outer diameter of just

under 1".

The wedge and worm gear are driven by stainless steel rods (Diame-

ter = 1/16", or 1.59 mm) which pass through vacuum tight 'o'-ring seals at the

top of the waveguide probe. Small set screws are used to fix the steel control rods

into the worm drive and wedge (Fig. 3-5), and to fix the end-plate within the

worm gear. Rotation of the worm gear is monitored via a simple turn-counting dial

mounted at the top of the probe, having a readout resolution of 1/100th of a turn.

Different worm drive/gear combinations are employ, -1 in the 3/4" and 1" diameter

probes (see Fig. 3-6), with 1/41 and 1/20 gear ratios, respectively. Thus, the angle












(a)


(b)


Figure 3-6.


Photographs of the rotating cavity system; part of the cavity has been
disassembled (end-plate and centering-plate) in order to view the in-
side of the cylindrical resonator. (a) shows the 1st generation rotating
cavity, which fits into a 1" thin-walled stainless steel tube; this cavity is
compatible with the 25 T magnets at the NHMFL and the QD PPMS
and Oxford Instrument magnets at UF (see Table 3-1). (b) shows the
2nd generation rotating cavity, which fits into a 3/4" thin-walled tube;
this cavity is compatible with the highest field (45 T) resistive magnets
at the NHMFL, as well as the 3He probe designed for the Oxford In-
struments magnet at UF (see Fig. 3-7). (Reused with permission from
Takahashi et al. [81]. Copyright 2005, American Institute of Physics.)









resolution on the dial redout corresponding to the actual sample orientation is

approximately 0.090 for the 3/4" probe, and 0.180 for the 1" probe. Although

both probes exhibit considerable backlash (~ 1), this is easily avoided by con-

sistently varying the sample orientation in either a clockwise or counter-clockwise

sense. High resolution EPR measurements on single-molecule magnets (reported in

Takahashi et al. [81]) have confirmed the angle resolution figures stated above.

The stage also performs the task of clamping the V-band waveguides into

position directly above the cavity coupling holes. As with previous cavity designs,

a small channel [0.02" (= 0.51 mm) wide and 0.02" deep] is machined between

the waveguides on the under side of the stage; this channel mates with a similarly

sized ridge located in between the coupling holes on the upper surface of the

cavity housing (see Fig. 3-5). Our previous studies have demonstrated that

this arrangement is extremely effective at minimizing any direct microwave leak

between the incident and transmission waveguides and is, therefore, incorporated

into all of our cavity designs [75]. A direct leak signal can be extremely detrimental

to cavity perturbation measurements, causing a significant reduction in the useful

dynamic range, and to uncontrollable phase and amplitude mixing, as explained in

our previous paper [75].

The internal dimensions of the cylindrical resonator are 7.62 mm x 7.62 mm

(diameter x length). The center frequency (fo) of the TE011 mode of the unloaded

cavity is 51.863 GHz, with the possibility to work on higher-order modes as

well. Using the ESA option, we have been able to conduct measurements up to

350 GHz [89]. While the modes above about 150 GHz are not well characterized,

they do provide many of the advantages of the well-defined lower frequency

modes, e.g., enhanced sensitivity, control over the electromagnetic environment

(i.e., E vs. H field) at the location of the sample, and some immunity to standing

waves. Table 3-3 shows resonance parameters for several unloaded cavity modes









(there are many others which are not listed). The TEOln (n = positive integer)

modes are probably the most important for the rotating cavity design, because

their symmetry is axial. Thus, rotation of the end-plate not only preserves the

cylindrical symmetry, but also ensures that the sample remains in exactly the same

electromagnetic field environment, i.e., upon rotation, the polarization remains in

a fixed geometry relative to the crystal. A sample is typically placed in one of two

positions within the cavity: i) directly on the end-plate; and ii) suspended along

the axis of the cavity by means of a quartz pillar (diameter = 0.75 mm) which is

mounted in a small hole drilled into the center of the end-plate. These geometries

are depicted in Fig. 3-7 for the TE011 mode (see Sec. 3.5 for details), and for

the two dc magnetic field geometries, i.e., axial and transverse. Each geometry

possesses certain advantages for a particular type of experiment. For example, the

quartz pillar configuration is particularly useful for EPR experiments in the axial

high-field magnets [Fig. 3-7(b)] since, for the TEOln modes, the sample sits in a

microwave ac field (H1) which is alv-- transverse to the dc magnetic field (Bo).

We discuss this in more detail in Mola et al. [75] and Takahashi et al. [81].

Another advantage of the TEOln modes is the fact that no microwave cur-

rents flow between the end-plate and the main body of the resonator. Thus, the

Q-factors of these modes are high, and essentially insensitive to the mechanical con-

tact made with the wedge. Table 3-3 lists the key resonance parameters associated

with several modes. For the case of the TE011 mode, the low temperature (~ 2 K)

Q-factor is ~ 21,600, and the contrast between the amplitude on resonance [A(fo)]

and the amplitude far from resonance (leak amplitude, Al) is 31.7dB. These pa-

rameters are essentially the same as the optimum values reported for the fixed

cylindrical geometry in our earlier paper [75], thus confirming the suitability of this

new rotating design for cavity perturbation studies of small single-crystal samples,

both insulating and conducting. We note that the Q-value for the non-cylindrically












(a) End-plate configuration


(b) Quartz pillar configuration


0 rotation


Figure 3-7.


Schematic diagrams showing various different sample mounting con-
figurations for both axial and transverse magnetic field geometries,
including the two-axis rotation capabilities (Reused with permission
from Takahashi et al. [81]. Copyright 2005, American Institute of
Physics.).


Axial
AB


Trans.
B


Axial
AB


Trans.
B










Table 3-3. Resonance parameters for several different cavity modes. The first col-
umn indicates the given mode. The second column lists the resonance
frequencies (fo). The third column lists the Q-factors. The final column
lists the contrast in dB, i.e., A(fo)-A1, where A(fo) is the transmission
amplitude at the resonance frequency, fo, and A, is the leak amplitude
(Reused with permission from Takahashi et al. [81]. Copyright 2005,
American Institute of Physics.).

Mode fo(GHz) Q A(fo)-A1(dB)
TE011 51.863 21,600 31.7
TE212 54.774 3,300 26.5
TE012 62.030 16,600 25.2
TE015 109.035 8,600 19.0


symmetric mode in Table 3-3 (TE212) is almost an order of magnitude lower than

that of the TEO11 mode. As noted above, this is due to the flow of microwave cur-

rents associated with the TE212 mode across the mechanical connection between

the main body of the resonator and the end-plate.

As discussed earlier, even though the end-plate can be rotated, the waveg-

uides are coupled absolutely rigidly to the cavity via the stage. As with earlier

designs [75], the microwave fields in the waveguides are coupled into the resonator

by means of small circular coupling holes which are drilled through the sidewalls

of the cavity. The sizes of these coupling holes [diameter = 0.038" (=0.97 mm), or

~ A/6] have been optimized for the V-band, and the cavity side-wall was milled

down to a thickness of 0.015" (=0.38 mm, or ~ A/15) at the location of these holes.

Once again, these numbers are essentially the same as those reported in Mola et

al. [75]. The key point here is that this coupling never changes during rotation.

Therefore, the full 3600 angle range may be explored, and with excellent mechanical

stability. Fig. 3-8 shows the typical random fluctuations in the cavity resonance

parameters for the TEOln mode during a complete 3600 rotation of the end-plate:

the center frequency varies by no more than 120 kHz (~ 3 ppm, or 5' of the









resonance width); the Q-factor is essentially constant, to within 1 ., and the con-

trast, [A(fo)-A ], fluctuates between 29 dB to 33 dB, corresponding respectively

to leak amplitudes of 3.5'. and 2.-'". of A(fo). Another important consequence

of coupling through the side walls of the cavity involves selection rules for T\ i

cylindrical cavity modes. At the lowest frequencies, only the TE01 mode of the

V-band waveguide propagates. Thus, the microwave H1 field in the waveguide is

polarized parallel to the cavity axis, i.e., it is incompatible with the symmetry asso-

ciated with the H1 patterns of the T\ i modes. Consequently, we do not observe, for

example, the TM111 mode. This offers added benefit, since the TMlln modes are

ordinarily degenerate with the TEOln modes, and steps have to be taken to either

lift these degeneracies, or to suppress the TAi modes all together [75]. Here, we

simply do not couple to these modes at the lowest frequencies. Even at the highest

frequencies, we do not expect coupling to T\ i modes, provided the polarization of

the microwave sources is maintained throughout the waveguide.

The main design challenge in the development of a useful rotating cavity

system concerned the space constraints imposed by the high-field magnets at

the NHMFL; the specifications of each magnet system are listed in Table 3-1. A

prototype configuration was first developed, based on a 1" outer tube diameter.

This prototype was subsequently implemented in both magnet systems at UF,

and in the 25 T, 50 mm bore resistive magnets at the NHMFL. A picture of this

cavity system, which remains in use, is shown in Fig. 3-6(a). The major reason

for the large size of the cavity assembly is the large worm drive diameter (3/8"

9.525 mm), which is determined by the smallest readily available commercial

components [90]; this 1st generation worm drive is made of nylon. A more compact

2nd generation rotating cavity was developed by machining a considerably smaller

custom worm drive (and gear) in-house. This cavity assembly, which is shown in

Fig. 3-6(b), is small enough to fit into a 3/4" diameter thin-walled stainless steel















90 Phase
120 60

150A 30


180 0


210 330

240 300
270


f,=51.86335(GHz) A
3A
A A A A
0 A A
A A
Q=21600



32-
30
21 U


0 50 100 150 200 250 300
Angle(degree)


Figure 3-8.


Angle-dependence of the cavity resonant properties. (a) A polar plot
of the complex signal transmitted through mpe cavity as the frequency
is swept through the TEO11 mode; the polar coordinate represents the
phase of the transmitted signal, while the radial coordinate corresponds
to the linear amplitude. The Lorentzian resonance is observed as a cir-
cle in the complex plane. Points close to the origin correspond to fre-
quencies far from resonance, while the resonance frequency corresponds
to the point on the circle furthest from the origin. The resonance pa-
rameters are obtained from fits (solid line) to the data (open circles)
in the complex plane (see Mola et al. [75] for a detailed explanation of
this procedure). The average center frequency, fo, is 51.863 GHz, and
the Q factor is 21600. (b) Full 3600 angle-dependence of the fluctua-
tions in the resonance parameters for the TEO11 mode: upper panel
- fo; center panel Q; and lower panel difference between the am-
plitude on resonance [A(fo)] and the amplitude far from resonance
(the leak amplitude, Al). These data illustrate the excellent repro-
ducibility of the resonance parameters upon un-clamping, rol ilin :7
and re-clamping the cavity (Reused with permission from Takahashi et
al. [81]. Copyright 2005, American Institute of Physics.).


2'
S:


of Physics.).


2'
S:









tube, thus enabling measurements in the highest field 45 T hybrid magnet at the

NHMFL. Furthermore, this cavity is compatible with the 'He probe constructed

for the 17 T Oxford Instruments superconducting magnet (and the 25 T resistive

magnet at the NHMFL), allowing for experiments at temperatures down to

500 mK. We note that the 1st and 2nd generation cavity assemblies may be

transferred relatively easily from one particular waveguide probe to another,

requiring only that the thermometry be unglued and re-glued to the stage using

GE varnish; all other connections are made with screws. In addition, we have

constructed extra parts for both designs, including several cavities and end-plates.

This enables preparation of a new experiment while an existing experiment is in

progress.

Finally, we discuss the two-axis rotation capabilities made available via a 7 T

Quantum Design (QD) magnet (see Table 3-1). The 7 T transverse QD system is

outfitted with a rotation stage at the neck of the dewar. A collar clamped around

the top of the waveguide probe mounts onto this rotation stage when the probe

is inserted into the PPMS flow-cryostat. The rotator is driven by a computer

controlled stepper motor, with 0.010 angle resolution. The motor control has

the advantage that it can be automated and, therefore, programmed to perform

measurements at many angles over an extended period of time without supervision.

In the 50 250 GHz range, the compact Schottky diodes can be used for microwave

generation and detection. These devices are mounted directly to the probe, and are

linked to the MVNA via flexible coaxial cables (feeding the diodes with a signal in

the 8 18 GHz range). Therefore, the waveguide probe can rotate with the source

and detector rigidly connected, while the vacuum integrity of the flow cryostat

is maintained via two sliding 'o'-ring seals at the top of the dewar. In fact, this

arrangement rotates so smoothly that it is possible to perform fixed-field cavity

perturbation measurements as a function of the field orientation, as has recently









been demonstrated for the organic conductors a-(BEDT-TTF)2KHg(SCN)4 [46]

and (TMTSF)2C104. (See (' 111 4.) We generally use the stepper-motor to control

the polar coordinate, while mechanical control of the cavity end-plate is used to

vary the plane of rotation, i.e., the azimuthal coordinate.

Due to the extremely precise control over both angles, and because of the need

for such precision in recent experiments on single-molecule magnets which exhibit

remarkable sensitivity to the field orientation [73], we have found it necessary

to make two modifications to the 3/4" rotating cavity probe for the purposes of

two-axis rotation experiments. The first involves constraining the cavity assembly

and the waveguides within a 3/4" thin-walled stainless steel tube which is rigidly

connected to the top of the probe. This tube reduces any possible effects caused by

magnetic torque about the probe axis, which could mis-align the cavity relative to

the rotator. The second modification involves attaching a spindle on the under-side

of the end-cap. This spindle locates into a centering ring attached to the bottom of

the PPMS flow-cryostat, thus preventing the waveguide probe from rotating off-axis

(note that the inner diameter of the cryostat is 1.10" as opposed to the 3/4" outer
diameter of the probe).

3.4 Model of the Resonant Cavity

The characteristics of a cavity resonator are affected by many factors,

e.g., shape, dimensions and material of the cavity. In the ideal case in which

the cavity is made by a piece of perfect conducting material and filled with a

lossless dielectric with p and c, the cavity modes are perfectly discrete. The shape

of each resonance is characterized by the delta function, i.e., infinite height and

no width. The resonant frequency for each mode is easily calculated by solving

Maxwell's equations with boundary conditions for the geometry of the cavity. For

the cylindrical cavity we use, the resonance conditions are given by the following











equation [62],

fmnp np -n 2T
m27 27 ( )2 + ( d)2, (3 1)
2w 2wp R d

where m, n and p are integers (n and p >0). R is the radius and d is the height

of the cylindrical cavity. xm is the n-th root of the 1st derivative of the Bessel

function, J((x), and the n-th root of the Bessel function, Jm(x), for the TEmnp

and the TMmnp modes respectively. The electromagnetic field configurations for the

TE mp modes are given below.


H, (r,,z)- pHo J( r)cosm) cos( ), (3-2a)
Wmnpd /l^t R d

np mpwHo Jm(x ) ( 2P
H 7(r, z -)= (~)r sin(mf) cos(- z), (3-2b)
Wmnpd^^/' R d





R d
CU xmn COS() sin( (3 2c)





EmTp(r, ,z) =0. (32f)

In reality, the cavity is not perfect. For instance, the cavity will be made from

several pieces of finitely conducting materials. In particular, the contact between

the various pieces will have resistive, capacitive and inductive losses. The cavity

also need to have coupling holes to connect to the microwave source and detector.

This breaks the symmetry of the cavity modes. All of these factors influence the

characteristics of the cavity resonator. It is too complicated to model a realistic

cavity. Therefore, in order to simplify the model, an equivalent model is often used.

Fig. 3-9(a) shows the equivalent RLC circuit. The impedance of the circuit is given

by [91],
2L + i
Z =R iwL+ = R iwL(1 --), (3-3)
wC w2


2
Z = R iwL + R iL(1 --), (3-3)
wC w2









(b)
(a) 1.0
R L 3

0.5
CT F


0 )o 0

Figure 3-9. A simple description of the resonant cavity. (a) An equivalent RLC
circuit representing the cavity. (b) A Lorentzian resonance produced by
the RLC circuit; wo denotes the resonance frequency; F denotes the full
width at half maximum of the resonance.

where the complex quantity Z is the so-called impedance, the real quantities

R, L and C are the resistance, inductance and capacitance respectively, and

wo = 1/LC. The circuit with the impedance given above has a resonant peak in

the average absorbed power Pave by the resistance R, which is given by,

V R mV2 R2
Pave 2 1 R Vrms rVms2 (3
Z Rw+ (2 w(3 4)
ae12 R l + Lf2 o_ 2)

Thus uo = 27rfo = /t LC is the resonance frequency. In the case of Lwo > wIU U1o ,

Eq. 3-3 is written,

Z -2iL(w o), (3-5a)

where the complex resonant frequency wo is defined as,

F.
= Uo i-, (3-5b)

and F = R/L. Using the above impedance, the power is given by

Pae V2 (R/L2) PO
S(1)2 + (; )2 (11)2 ( )2 (3 )

Fig. 3-9(b) shows the power as a function of frequency. The resonance peak

is clearly seen at the resonant frequency w0. F represents the full width of the









resonance at half-maximum (FWHM) shown in Fig. 3-9(b). The FWHM is often

expressed by using the quality factor (Q), defined by,

wo woL
Q (3-7)
P R

In our setup, the Q-factor for an unloaded cavity can be up to 25000 [75].

When a small (perturbative) specimen is introduced into the cavity, the

characteristics of the resonance change slightly. This may be expressed by a shift

of the resonant frequency from wo to us, and a change of F from Fo to F, (or

the Q-factor from Qo to Q,). The complex impedance is therefore changed from

cj0 to uas. By using Eq. 3-5, the difference in the real and imaginary part of the

complex impedance, which represents the dissipative and dispersive response of the

specimen, is given by,


AZ Z, Zo igoLAJa,
iwo 1 1
SigoL(wu s (7- ))
2 Qs Qo
SigoL(Aw -Ar), (3-8)
2

where we introduced a geometrical factor go which depends on the experimental

geometry (the location and size of the sample in the cavity) [56, 91]. Defining

AZ R + iX (R = sample resistance and X =sample reactance), the sample

response is given by,
AF wo 1 1 R
( = (3-9a)
2 2 Q, Qo goL'

and
X
AU = Us UJo = (3-9b)
go0L









3.5 Positioning Low Dimensional Conductors and Superconductors in
the Cylindrical Cavity

The position of the specimen placed in the cavity is determined by several

factors. These include: the shape and size of the sample; the coupling strength with

the microwaves; favorable polarization of the microwaves for a given experiment;

the choice of whether to use the electric or magnetic component of the microwaves;

the orientation of the sample with respect to a dc magnetic field for magneto-

optics etc... This is particular important when studying anisotropic materials.

In the case of the cavity perturbation technique, the polarization of the induced

currents is often different from the polarization of the microwaves. In this section,

considering the case of Q2D materials with a plate-like shape, we briefly introduce

two cases, in-plane and interlayer measurements. During this discussion, it is

assumed that the least conducting direction of the sample is perpendicular to

sample platelet, and that both skin depths 611 and 6 are smaller than the sample

dimensions (skip depth regime) where 6- and 611 represent skin depth for the least

and the good conducting direction respectively. More details are found in Hill [33].

3.5.1 In-plane Measurements.

In order to measure the in-plane properties, the induced current has to be

polarized along only the in-plane direction, otherwise the interlayer component of

the current dominates the microwave response, as explained in the next subsection.

Fig. 3-10(1) a) illustrates the case of a measurement using the electric component

of the microwaves, with the ac electric field applied in-plane. However, the current

is induced not only along the in-plane direction, but also the interlayer direction.

This is because the current flow is limited to the surface, i.e., within the skin

(penetration) depth 6 (A) of the perimeter of the sample, and this flow is disrupted

in order to screen the electric field within the interior of the sample. Thus, this

configuration is not ideal for in-plane measurements. Another possibility is to










(2) Inter layer measurements


(1) In-plane measurements

a)
-
WI


z


^"-", ^ oi D t fL T.t 11'
H a t
--T ->l~ i~l_,
,. .+-.+++ +.... .
I P'lOil fl, c,

i, __ ,______


Schematic diagram illustrating the various possibilities for exciting
in-plane and interlayer currents in a Q2D plate-like sample. (1) a)
In-plane currents induced by an ac electric field (Reprinted figure with
permission from Hill [33]. Copyright 1997 by the American Physical
Society.). b) In-plane currents induced by an ac magnetic field. We
use the latter configuration for the in-plane measurements (Reprinted
figure with permission from Hill [33]. Copyright 1997 by the American
Physical Society.). (2) Interlayer measurements: a) current induced
by an ac electric field (Reprinted figure with permission from Hill [33].
Copyright 1997 by the American Physical Society.). b) current in-
duced by an ac magnetic field. This latter is the most widely used
configuration (Reprinted figure with permission from Hill [33]. Copy-
right 1997 by the American Physical Society.).


use the magnetic component of the microwaves, as shown in Fig. 3-10(1) b). In

this case, only current parallel to the conducting t-plane is induced. Thus, it is

possible to perform the in-plane measurement with this configuration. However it is

impossible to distinguish between crx and ayy, only the average is measured.

3.5.2 Interlayer Measurements.

This is the most relevant case for our experiments. Since the skin depth or

penetration depth for low dimensional conductors and superconductors is highly

anisotropic, the absorption of microwaves is also highly anisotropic. As a result,

the microwave response can be dominated by the absorption along the interlayer

direction. Let's examine example cases for anisotropic conductors. For the skin

depth regime, the microwave absorption is simply related to the surface area and


b)

Figure 3


Figure 3-10.


L1~ill'r I J Ft









skin (penetration) depth [92], for currents flowing across faces of the sample which

are parallel (||) and perpendicular (1) direction to the in-plane direction, i.e.,

P1- 61a (3-10)
P\\ 611all

where a1 and all are the surface areas parallel to the least and the good conducting

directions, respectively. In the case of the Q1D conductor, (TM [TSF)2C104,

sample iv- I -1 are typically needle shaped so that all ~ 0.4 mm2 and a_ ~ 0.1

mm2. The skin depth is estimated to be 6_ ~ 100 pm and 611 ~ 1 pm at Helium

temperatures at a microwave frequency. As a result, P1/P= 20. In the case of

the Q2D conductor, K-(ET)2Cu(NCS)2, P/IP|i is typically 6-100 at a microwave

frequency. Thus, the energy dissipation P1 along the least conducting direction is

dominant in both cases. One can therefore choose various configurations for the

polarization of the incident microwaves. Here we illustrate two cases which are

likely to maximize the interlayer current. Fig. 3-10(2) a) shows the situation in

which the electric component is applied perpendicular to the in-plane direction.

The current is induced from one face of the sample platelet to the other via

the edge of the sample. Fig. 3-10(2) b) shows magnetic field excitation. The ac

magnetic component is applied along the in-plane direction in this case. The

current is induced over the sample surface again. However, there is no displacement

current in this configuration. In fact, this configuration is the most widely used for

angle-dependent magneto-optical studies of low dimensional materials.

3.5.3 Configuration for Interlayer Measurements Using the Magnetic
Component of the Microwaves

Next we explain actual experimental geometries for angle-dependent magneto-

optical studies of Q2D metals. Configurations are specified for interlayer measure-

ments using the magnetic component of the microwaves.









For the case of the TE011 mode in the cylindrical cavity, which we use most

often, two different examples of the position of the sample in the cavity are shown

in Fig. 3-11. In the case of TEO11 mode, the antinodes of the magnetic component

of the microwaves are located at the center of the cavity, in the middle of the side

wall, and at the halfway point between the center and edge of the end plate, as

shown in Fig. 3-11. Thus, in order to have a strong coupling, we usually place the

sample on a quartz pillar [Fig. 3-11(a)], on the end plate [Fig. 3-11(b)], or on the

side wall (not shown). These configurations are different in terms of the angle-

dependent magneto-optical studies. For example, we can consider the combination

of a vertical magnetic field and a transverse cavity, which is the case for the

rotating cavity with an axial magnet. In the case of the quartz pillar configuration,

as shown in Fig. 3-11(a), the dc magnetic field can be applied from parallel to

perpendicular to the sample platelet while the sample is rotated along the quartz

rod direction using the rotating cavity. On the other hand, in the case of the end

plate configuration, the magnetic field is .i.- li-, along the sample platelet while the

sample is rotated.

3.6 Microwave Response of Low dimensional Conductors and
Superconductors

In this section, we derive an expression for the microwave responses due to

the interlayer properties of the sample. i.e., sample impedance, resistance and

reactance. The problem is mainly to solve Maxwell's equations for E and H with

proper boundary conditions for the sample. There are many papers which discuss

different cases [56, 91, 93, 94]. Here we introduce two important cases, i.e., skin

depth regime and depolarization regime.

3.6.1 Skin Depth Regime

Here we consider good conductors and superconductors, where the skin

(penetration) depth is smaller than the thickness of the sample, are said to be in









the so-called skin depth regime. As discussed in Sec. 3.5, the main contribution

to the absorption comes from the induced currents along the interlayer direction.

We can simplify the problem by considering the electrodynamics of a semi-infinite

medium. The flat conducting sample (the xz-plane) is set at y = 0 so that the

medium for y >0 has the dielectric constant e1, permeability pi and conductivity

aU, and the medium for y <0 is vacuum. The sample surface is placed at an

antinode of the ac magnetic field. We now calculate the magnetic field inside

the sample, assuming the magnetic field is along the x-axis and has harmonic

time dependence, i.e., (Hac)x = i,,. The magnetic field inside the sample is

calculated by Maxwell's equations,


V D p, (3- a)

aH
V x H- --- J, (3-1b)

aB
Vx E + 0, (3- 1c)

V B 0, (3 11d)

and Ohm's law,

J -= E. (3-12)

Using the above equations with p = 0, one obtains the following differential

equation,
a2H OH
V72H- ~atl 2 t- (3 13)

and its solution, with the boundary condition H(y = 0) = Hac, is given by


H(y, t) -/,,. estS, (3-14)

where the complex wavevector is given by


q'= wlli 1- i-91 '/ l-0-i (3 15)
LL)LL









For convenience, the following definition of the complex dielectric constant 7 and

conductivity a are also used frequently:

1
'= 1 + -i+- C e + i2, (3-16a)


and the complex conductivity a is defined as


'e co +-. (3-16b)


Therefore, the real and imaginary parts of the conductivity become,


"1 = W')C2, (3-16c)


and

02 = co O C1). (3-16d)

Additionally, the complex wavevector is also expressed as,

n k
u UN w(- +i-), (3-17)
c c

where V is the complex refractive index, n is the real refractive index and k is the

imaginary refractive index. c is the speed of light c = 1/ Poco.

The induced electric field is calculated easily using Eq. 3-1 b,


E(y, t)= Hoe e y. (318)


Thus one can obtain the impedance of the sample, Z = E/H,

L) 1 i2pl
S. (3 19)
g --- -


Assuming el > co, which is often the case for good conducting materials, we obtain

the well-known expression for the surface impedance,


s = i R + iXs. (3-20)
V71 i172










As discussed in Sec. 3.4, the microwave response is related to Rs and Xs, so that

we can extract both a, and 02 by using the conversions,

RsX,
a, = 2p1w ^R<2, (3-21a)
(R2 + X)2'

and
X2 R2
2 = /1 2 (3-21b)
(R + X2)2
Moreover, rewriting the above equations, one can obtain,


AF oc R +2 (3 22a)



A X, 2 + 2 (3-22b)

where AF = 1/Q- 1/Qo is related to the change of the Q-factor, and Awo= as-Uo

is the change of the cavity resonant frequency.

In the case of superconducting materials, we need to consider the response

of the normal quasi-particles and the Cooper-pairs. This is often treated by using

a general two fluid model, in which a represents the conductivity due to normal

quasiparticles, and (a is the superfluid contribution to the conductivity.


S= 1 2 + 91S ia2S
Tnse2 e2
1i 7i 2 + J6(u) (3-23)
m m*uj

where n, is superfluid density, and the penetration depth A is also used for

nse2/m* = (piwA 2)-1. In a simple case (T ~ 0 and a s-wave superconductor),

at microwave frequencies gssi = 0 so that the complex conductivity often becomes,


a U1 72S w2 (3-24)
piauj\


piauj\









which assumes 92S a1. Thus, the microwave response is given by the following

expression,

AF oc R [1 = p2 2 3, (3-25a)
FL2_ 2a12 2p1

and

Aw oc X, p wA. (3-25b)


3.6.2 Metallic Depolarization Regime

When the skin depth is larger than the thickness of the sample (a), the

electrodynamics are said to be in the so-called metallic depolarization regime

soepresented by qa > 1 [56]. This regime can be understood easily because the

transmitted microwaves induce homogenous currents inside the sample, so that

the microwave loss can be calculated in the quasi-static limit. As a result, the

conductivity is simply related to the sample resistance, namely,


a oc (3-26)
Pi R

Thus, using Eq. 3-9a, one can obtain the following expressions,


AF oc R oc (3-27)
a1


3.7 Measurement of the Change of the Complex Impedance Z

In the experiments discussed here, we focus on investigating relative li.i,

of the sample properties by changing external conditions, e.g., magnetic field,

the orientation of the magnetic field, temperature and frequency. In our setup,

experiments are performed using the following three methods: 1) measuring

the cavity center resonance frequency wo and F (or Q-factor); 2) measuring the

amplitude and phase of the transmitted signal continuously at a fixed frequency,

the so-called fre pii i' v-lock method; 3) and measuring the amplitude and frequency

continuously at a fixed phase, the so-called phase-lock method. Method 1) is the



continuously at a fixed phase, the so-called phase-lock method. Method 1) is the









most straightforward for determining the complex impedance Z, since AF and

Aw are measured directly. However the estimation of AF and Aw is a rather

time-consuming process, which involves sweeping the frequency, subtracting a

background and fitting the cavity resonance. Therefore, we rarely use method 1).

Instead, we mainly use methods 2) and 3), since they allow a much faster data

acquisition rate.

The cavity resonance can be represented by the Lorentzian function seen

in Sec. 3.4. Similar to Eq. 3-6, the amplitude and phase are expressed by the

following equations,

1 1
A(w) = Ao = Ao (3-28a)
(V )2 + -o)2 (2 +( co)2

and,

tan )(co) c- -c o- (3-28b)
( 2Q) ) 2)
where F = o/Q, uo is the cavity resonance frequency, and Q is the quality-factor.

The external conditions are now changed, e.g., by applying a magnetic field. The

changes of the microwave response by the sample are then represented by shifts of

F and the resonance frequency wo, i.e., F -- + AF and uo -i wo + Aw0. Thus the

amplitude and phase of the microwave response also changes,

1
A(w) A(w) = A, (3-29a)
( )2 (a) -w0 Ao)2

and,
c co -A aco
tan (w) -- tan (o) L (3-29b)
( 2
Fig. 3-12 shows a schematic view of these changes. The amplitude at the resonance

frequency becomes smaller because F -- + AF. The change also appeared in the

phase, i.e., the position at (w) = 0 is shifted to at Q(wo + Awo) = 0.









((m)


(b)


0


coo 0 + Am0o


~oo


Figure 3-12.


Typical changes in the amplitude and phase of the microwaves trans-
mitted through the cavity. When the sample properties change by
varying the external conditions, changes appear in the microwave
response of the cavity. As a result, the amplitude and phase are
changed: (a) Typical changes in the amplitude of the microwaves,
represented by AF and Awo; (b) typical changes in the phase of the
microwaves, represented by AF and Awo.


3.7.1 Frequency-lock Method

We perform the frequency-lock using a Phase Matrix/EIP 575B frequency

counter [95]. Replacing w = wo in Eq. 3-29a and Eq. 3-29b, the amplitude and

phase during the freq1' L. .i s-lock are given by


A(wo) A-o a
(F4"'F)2 + (A wo)2


and,

tan 0(wo) = r+Ar
Thus, rewriting the above equations, the microwave response is given by
Thus, rewriting the above equations, the microwave response is given by


2Ao 1
A A(wo) /tan2 o) +


A Ao 1
A(wo) 1 +1
Vtan2 ( o)
When the phase shift is small, tan -+ 0,


AFr^ 2Ao
A(wo)


(3-30a)




(3-30b)


(3-31a)


(3-31b)


(3-32a)


(a) F









and
Ao
Awo Ao ( o). (3-32b)
A( o)
Thus, when the microwave response has small phase shifts, AF depends only on

a change of the amplitude. Eq. 3-32a also shows the change of the amplitude is

linear, i.e., A(wo) = Ao + 6A so [A(o)]-1 = 1/Ao[1 6A/Ao]. The frequency

shift is therefore proportional to (wou). On the other hand, this method can be

problematic when the phase shift is huge, because the frequency becomes off-

resonant, and AF starts to depend on both A(wo) and (awo). Thus the phase and

amplitude become mixed, i.e., dissipation and dispersion are mixed. Empirically,

the frequ-ii'- i,--lock is not appropriate when the phase shift is more than 10 degrees.

However we can correct this problem by performing a vector analysis with Eq. 3-31

or a vector fit [96]. Furthermore we can also avoid the problem using the phase-lock

method.

3.7.2 Phase-lock Method

The phase-lock method involves stabilization of the frequency using the phase

of the microwaves transmitted through the cavity. If the stabilization is perfect,

and the chosen cavity mode is well-separated from the other cavity modes, standing

waves and other sources of df/dv, this method is the most useful. However, the

phase-lock is less stable than the frequency lock so that the signal we obtain by this

method often has a smaller signal-to-noise ratio. Thus, we usually begin by using

the frequency lock. Then we perform the phase-lock measurement if we encounter

problems discussed in the previous subsection. As seen in Fig. 3-12, the phase lock

alh-v-, keeps Q(wo + Aweo)0. Thus w~ o+Awo. Putting = 0 and w =wo+Awo

into Eq. 3-29a, the amplitude becomes,


A(w) Ao (3-33)
F AF









Thus, AF is given by,

S- A r. (3 34)

Therefore, this expression is the same as Eq. 3-32a for the frc'i,-. n' --lock. By

recording the frequency with a frequency counter, we can also measure the fre-

quency shift u + Aow.

3.8 Summary

In this chapter, we explained our experimental techniques. In particular, for

the purpose of angle-dependent high-field microwave spectroscopy, we developed

a rotating cavity which is compatible with several high-field facilities. In the

later part of the chapter, we explained the electrodynamics of low dimensional

conductors and superconductors appropriate to our experimental configuration.















CHAPTER 4
PERIODIC ORBIT RESONANCES IN QUASI-ONE-DIMENSIONAL
CONDUCTORS

The results presented in this chapter can be found in the articles, Periodic-

orbit resonance in the quasi-one-dimensional organic superconductor (T I 1.F)2 C04,

S. Takahashi, S. Hill, S. Takasaki, J. Yamada and H. Anzai, Physical Review B 72

024540 (2005), and Are Lebed's Magic Angles Truly Magic?, S. Takahashi, A.

Betancur-Rodiguez, S. Hill, S. Takasaki, J. Yamada and H. Anzai, submitted to the

ISCOM conference proceedings.

4.1 The Quasi-one-dimensional Conductor, (TMTSF)2C104

The organic metal (TMTSF)2C104 belongs to the family of quasi-one-

dimensional (Q1D) Bechgaard salts [2], having the common formula (T ITSF)2X.

TMTSF is an abbreviation for tetramethyl-tetraselenafulvalene, and the anion X

is AsFe, C104, PF6, ReO4 etc. The (TA TSF)2X series have been the most ex-

tensively studied organic materials. The reason may be the extremely rich phases

for (TA TSF)2X. This includes unconventional metallic states, spin-density-wave

(SDW) states, field-induced spin-density-wave (FISDW) states, possible spin-triplet

superconducting states and so on. Unlike other one-dimensional materials, The

(TMTSF)2X compounds tend to be good conductors at room temperature al-

though the (TA TSF)2X eventually meet the metal-insulator transition at a low

temperatures, except for (TMTSF)2C104, which becomes a superconductor. The

metal-insulator transition in the (TA TSF)2X compounds is seen in the resistivity

of Fig. 4-1(a). This transition is explained by the nesting of the FS. In the case

of the (TA TSF)2X compounds, the transfer energy to the b-direction is finite,

so that the FS is not a perfect flat plane, and the nesting is not as strong as the

































Pressure 5 kbar


Temperature dependence of
resistivity in (TMTSF)2X


Figure 4-1.


Phase diagram for (TMTSF)2X


Electronic properties of the (T\ TSF)2X. (a) Temperature de-
pendence of the resistance in (TMTSF)2X. With the exception of
(T\ iTSF)2C104, all samples undergo a metal-insulator transition
at low temperatures (Reprinted from Bechgaard et al. [97], Copy-
right 1980 with permission from Elsevier). (b) The phase diagram
of (T\ TSF)2X as a function of the pressure. The pressure may be
controlled by replacing the anion X and applying external pressure.
LL denotes a Luttinger liquid regime, 2D (3D) FL denotes a 2- (3-)
dimensional Fermi liquid regime, SDWI represents spin-density-wave
insulating states and, SC is a superconducting state. Varying pres-
sure, high temperature and ground states are changed. For example,
(T\ TSF)2PF6 exhibits a SDWI ground state at ambient pressure.
However, under 6 kbar, the ground state of (T\ITSF)2PF6 becomes
superconducting (See J6rome [98], Moser et al. [99] and Dressel [100]
for the details of the phase diagram.).









1D system as we saw in Sec. 1.4. This imperfectly nested FS therefore shifts the

instability to a lower temperature. In the case of the (T\ TSF)2X compounds, this

nesting causes the SDW instability, so that the ground state is the SDW insulating

state. Moreover, the anion X affects the nature of the SDW instability, since it

pl a role in determining the spacing of the T\ TSF molecules. By replacing the

anion X, one can therefore change the spacing, and then control the coupling of the

TMTSF molecules along the b and c-directions, i.e., one can change the transfer

energies tb and tc. This change may be small compared with the Fermi energy EF,

but it is significant enough to shift the critical temperature TMI for the SDW state.

This is actually seen in Fig. 4-1(a). By replacing the anion X BF4 to PF6, TMI

shifts from 80 to 15 K. This may be understood in terms of a change of the dimen-

sionality, and can be probed by detailed investigation of the FS. More importantly,

by applying pressure, one can control the dimensionality more systematically. For

instance, Fig 4-1(b) shows a proposed phase diagram of (T\ TSF)2X as a function

of the pressure. The pressure may be controlled by replacing the anion X and ap-

plying external pressure. Varying pressure, high temperature and ground states are

changed. For instance, The TMI in (T\ TSF)2PF6 decreases rapidly under pressure.

Above 8 kbar, the SDW insulating phase disappears, and then the superconducting

phases appears as a ground state. In fact, this was the first superconductivity

discovered in the organic systems in 1979 [101].

Among the Bechgaard salts, (T\ iTSF)2C104 is the only material which

has a superconducting state under an ambient pressure. (T-\TSF)2C104 has a

structural phase transition of the C104 anions at 24 K. This due to the ordering

of the non-centrosymmetric tetrahedral C104 anion, so-called anion ordering. The

anion ordering is extremely sensitive to the cooling process of the crystal. X-ray

scattering studies reveal that a superlattice structure with Q = (0, 1/2, 0) is

formed at the anion ordering temperature (TAO = 24 K)[102, 103]. This occurs









when (T\[TSF)2C104 is cooled slowly at around 24 K (< 0.1 K/min). Then

(TMTSF)2CO14 has a metallic and superconducting state in the low temperature

regime, the so-called relaxed state. On the other hand, when (TMTSF)2CI04

is cooled rapidly (> 50 K/min), the anion is disordered. This is the so-called

quenched state. We have investigated the cooling process dependence of the

electronic properties of (TA\ TSF)2C104, especially in terms of its influence on the

superconductivity. This investigation is introduced in C'i p 5.

The < i--r .1 structure of (TMTSF)2C104 shown in Fig. 1-2 is triclinic. The

lattice parameters in the relaxed state are a = 7.083 A, b = 15.334 A, c = 13.182 A,

a = 84.400, 3 = 87.620 and 7 69.000. In general, the shape of the (T:\ TSF)2C104

single < il I 1 is needle-like, as shown in Fig. 4-2(a). The most conducting direction

is the needle direction (the a-axis). For convenience, the b', c' and c* axes are often

emploi-l 1 The b' (c') axis is defined as the projection of the b- (c-) axis onto the

plane perpendicular to the a-axis, i.e., b' = b sin 7 and c' = csin3. The c*-direction

is defined as the direction perpendicular to the ab-plane, i.e., c* = c'sina'.

Similarly, The b*-direction is defined as the direction perpendicular to the ac-plane,

i.e., b* = b'sin a' where a' is the angle between b' and c'. a* is also often used for

the angle between b* and c*. These angles a* and a' are simply expressed by

cos 3 cos 7 cos a
a* = 7 -' cos -1 [ ].os (4-1)
sin 3 sin 7

(TMTSF)2C104 is a 3/4-filled system similar to other (TA\JTSF)2X. The Fermi

surface estimated by band calculations reveals that the FS consists of four sheets,

as shown in Fig 4-2(b), because of the superlattice structure caused by the anion

ordering. The planar FS has a small warping due to small transfer energies along

the b' and c'-directions.

As shown in Sec. 1.4, (TAUTSF)2C1O4 has a FISDW phase in high magnetic

field. Since this FISDW phase is caused by the enhancement of the nesting