Quantum transitions in antiferromagnets and liquid helium-3

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QUANTUM TRANSITIONS IN ANTIFERROMAGNETS
AND LIQUID HELIUM-3







By

BRIAN C. WATSON


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


2000















ACKNOWLEDGEMENTS


I would like to thank my thesis advisor, Professor Mark Meisel, for his guidance

and encouragement over the past three and one-half years. I gratefully acknowledge the

members of my supervisory committee, Professors Art Hebard, Kevin Ingersent, Yasu

Takano, and Dan Talham. In addition, I am grateful to Fred Sharifi for sharing his

knowledge and experience with me. There are several additional people that have

contributed to this thesis, and I am indepted to Dr. Naoto Masuhara for his enlightening

physics conversations and Dr. Jian-sheng Xia for his technical acumen. I am also grateful

to Drs. Stephen Nagler and Garrett Granroth for their assistance with the neutron

diffraction experiments. Garret Granroth also deserves thanks for teaching me the

laboratory basics during the semester that we both worked together. Stephen Nagler has

also contributed to this thesis by writing portions of the MATLAB fitting routines. Every

member of the Department of Physics Instrument Shop has been extremely helpful. I

would especially like to thank Bill Malphurs for his attention to detail and for noticing

when my instrument designs were geometrically impossible. Dr. Valeri Kotov deserves

thanks for his guidance during my foray into theoretical physics. I am grateful to Larry

Frederick and Larry Phelps in the Department of Physics Electronics Shop for their

support. Once again, I acknowledge invaluable input from Professor Dan Talham and the

members of his research group, including Gail Fanucci, and Jonathan Woodward for








operating the EPR spectrometer, as well as Melissa Petruska, Renal Backov, and Debbie

Jensen for synthesis of the antiferromagnetic materials studied in this dissertation. The

assistance from Dr. Donovan Hall during experiments at the National High Magnetic Field

Laboratory was invaluable. I would also like to thank Professors Gary Ihas and Dwight

Adams for loaning equipment and for their help during experiments at Microkelvin

Laboratory.

















TABLE OF CONTENTS




ACKNOWLEDGEMENTS . . ......... .

ABSTRACT . . . ..............


CHAPTERS
1 INTRODUCTION . . . . . . . . .
1.1 BPCB . . . . ....................
1.2 MCCL . . . . ................... .
1.3 Zero Sound Attenuation in Normal Liquid 3He . ....... .
1.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3He-B


2 EXPERIMENTAL TECHNIQUES
2.1 SQUID Magnetometer . .
2.2 Vibrating Sample Magnetometer *
2.3 AC Susceptibility . ...... .
2.4 Tunnel Diode Oscillator . .
2.5 Conductivity . ....... .
2.6 Neutron Scattering ..... .
2.7 Nuclear Magnetic Resonance
2.8 Electron Spin Resonance . .
2.9 Pulsed FT Acoustic Spectroscopy

3 THEORETICAL TECHNIQUES
3.1 Exact Diagonalization .
3.2 The XXZ Model . .... .


9
* 9
* . 10
. . 12
. . 16
. . 19
. . 24
* . 26
* . 28
. . 36
. . 36


4 STRUCTURE AND CHARACTERIZATION OF A NOVEL
MAGNETIC SPIN LADDER MATERIAL ...... .
4.1 The Structure and Synthesis of BPCB ...... .
4.2 Low Field Susceptibility Measurements . .....
4.3 Low Field Magnetization Measurements ..... .
4.4 High Field Magnetization Measurements ..... .
4.5 Universal Scaling . . . . . .
4.6 Neutron Scattering . . ............ .


. . . 61
. . . 63
. . . 67
. . . 82
. . . 92
. . . 113
. . . 119


* 1
2
4
5
7








5 MAGNETIC STUDY OF A POSSIBLE ALTERNATING
CHAIN MATERIAL . . . .................. .
5.1 Structure and Synthesis of MCCL . . ............ .
5.2 Electron Paramagnetic Resonance . . ............ .
5.3 Low Field Susceptibility Measurements . ..........
5.4 High Field Magnetization Measurements . . .......... .

6 ZERO SOUND ATTENUATION NEAR THE QUANTUM
LIMIT IN NORMAL LIQUID sE CLOSE TO THE SUPERFLUID
TRANSITION . . . . . . . .
6.1 Experimental Details . . .
6.2 Zero Sound . . . . . . .
6.3 First Sound . . . .
6.4 Error Analysis and Final Results . . ............. .

7 DIRECT MEASUREMENT OF THE ENERGY GAP OF
SUPERFLUID 3HE-B IN THE LOW TEMPERATURE LIMIT . .
7.1 Details of the FT Spectroscopy Technique . . ..........
7.2 Thermometry Issues . . . ................. .
7.3 Edge Effects . .. . . . . . . ...
7.4 Temperature Dependence . . . ............... .
7.5 Pressure Dependence . . . ................. .
7.6 Error Analysis . . . . ................... .
7.7 Absolute Attenuation . . . ................. .


8 SUMMARY AND FUTURE DIRECTIONS . . .
8.1 BPCB . . . . . . .
8.1.1 Summary . . . ................. .
8.1.2 Future Directions . . .............. .
8.2 M CCL . . . . . . . .
8.2.1 Summary . . . ................. .
8.2.2 Future Directions . . .............. .
8.3 Zero Sound Attenuation in 3He . . ............ .
8.3.1 Summary . . . ................. .
8.3.2 Future Directions . . .............. .
8.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3He-B -
8.4.1 Summary . . . ................. .
8.4.2 Future Directions . . . ..............
8.5 Concluding Remarks . . . ................ .


S 245
S 245
S 245
* 246
S 246
S 246
S 248
S 248
* 248
S 249
250
S 250
S 251
S 252


APPENDICES
A LOW TEMPERATURE PROBE DRAWINGS
B COIL FORMER DRAWINGS . ......
C PRESSURE CLAMP DRAWINGS . ...... .
D NMR PROBE DRAWINGS . . . .


127
128
132
139
155



162
165
177
187
198


212
216
219
221
225
231
241
242








E POLYCARBONATE SAMPLE SPACE DRAWINGS . ....... . 333
F APPLESCRIPT ROUTINES . . . ................ 338
G ORIGIN SCRIPTS . . . ..................... 344
H MATLAB FITTING PROGRAMS . . .. ............. 347
I DATA SET PARAMETERS FOR THE LANDAU
LIMIT EXPERIMENT . . . . . . . ... 350

LIST OF REFERENCES . . . . .................... 354

BIOGRAPHICAL SKETCH . . . .................. 362















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


QUANTUM TRANSITIONS IN ANTIFERROMAGNETS
AND LIQUID HELIUM-3

By

Brian C. Watson

December 2000



Chairman: Mark W. Meisel
Major Department: Physics


Effects arising from quantum mechanics are increasingly common in new devices

and applications. Two different, but related, topics, low-dimensional antiferromagnets and

liquid 3He, have been studied to obtain a deeper understanding of the quantum mechanical

properties that govern these systems. Low dimensional magnetism provides a means of

investigating new quantum phenomena arising from magnetic interactions. Superfluid and

normal liquid 3He exist in a very pure form and therefore allow severe tests of theoretical

descriptions. More specifically, the magnetic properties of bis(piperidinium)

tetrabromocuprate(II), (C5H12N)2CuBr4, otherwise known as BPCB, and

catena(dimethylammonium-bis(j12-chloro)-chlorocuprate), (CH3)2NH2CuC13, otherwise

known as MCCL, have been measured and are reported herein. Theoretically predicted








scaling behavior has been observed, for the first time, in BPCB. In superfluid 3He, the

pair-breaking edge has been measured at low temperature, thereby allowing for a

measurement of 2A. These data indicate that the energy gap at low pressure is

significantly less than predicted by BCS theory. Finally, subtle effects due to the

attenuation of zero sound in normal liquid 3He have been measured. Evidence for the

quantum correction to zero sound attenuation, predicted by Landau over 40 years ago, is

presented herein.














CHAPTER 1
INTRODUCTION


Quantum mechanical properties of various systems are increasingly important both

fundamentally and technologically as new materials and devices are being generated at the

boundary between the classical and quantum worlds. This dissertation addresses the

quantum mechanical properties of two apparently disparate systems, gapped

antiferromagnets and 3He. In both cases, however, the quantum mechanical nature of

these systems is apparent in their macroscopic properties. In fact, these two systems are

models for the verification of various quantum mechanical predictions.

The dissertation is arranged as follows. Chapters 2 and 3 detail the nine

experimental and two main theoretical techniques that were used to collect and analyze the

data presented herein. Chapter 4 reports the experimental results concerning the gapped

antiferromagnetic material bis(piperidinium)tetrabromocuprate(II), (Cs5Hn2N)2CuBr4,

otherwise referred to as BPCB, and Chapter 5 discusses the alternating chain material

catena(dimethylammonium-bis(jt2-chloro)-chlorocuprate), (CH3)2NH2CuCI3, otherwise

referred to as MCCL. These chapters include magnetization and neutron diffraction data

from experiments at the National High Magnetic Field Laboratory (NHMFL) and Oak

Ridge National Laboratory (ORNL), respectively. In addition, electron paramagnetic

resonance (EPR) measurements performed by Professor Talham's research group in the

Department of Chemistry at the University of Florida are included. The liquid 3He studies








are presented in Chapters 6 and 7 which discuss the low temperature acoustic experiments

performed in the University of Florida Microkelvin Laboratory. The first 3He experiment

is an absolute measurement of zero sound attenuation in 3He above the superfluid

transition temperature. The second 3He experiment uses Fourier transform techniques and

is a measurement of the 2A pair breaking energy in superfluid 3He-B. The final chapter

summarizes the experimental results and lists possible future experiments.



1.1 BPCB

The objective of this work is to better understand quantum phase transitions in

antiferromagnets. Low dimensional, gapped, insulating, antiferromagnetic materials are

ideal candidate systems for the experimental realization of quantum phase transitions.

These transitions are defined as phase transitions that occur in the low temperature limit

(T 0), where quantum fluctuations have energies larger than thermal fluctuations

(hco > kBT), and are driven by a change in some aspect of the system other than

temperature. A current review of quantum phase transitions is given by Sondhi et al. [1].

When thermal and quantum fluctuations are equally important (hwo kBT), the state of the

system is referred to as being in a quantum critical regime. Quantum critical behavior is

important in two dimensional antiferromagnets, and the behavior of charge and spin

density waves in the quantum critical regime of two dimensional doped antiferromagnets is

observed to play a role in high Tc superconductivity [2].

To better understand the quantum critical behavior in two dimensional materials,

we begin by studying quasi-two dimensional systems. The logical intermediate step

between two dimensional planes and one dimensional chains are ladder materials. The








long range order that occurs in a two dimensional plane of spins can be approximated by

ladders of increasing width [3]. Ladders are formed by two or more one dimensional

chains arranged in a ladder geometry with electronic spins at the vertices of the ladder

interacting along the rungs of the ladder with exchange J1 and along the legs of the ladder

with exchange J1. In order to further the analogy between the two dimensional cuprate

high Tc superconductors and quasi-two dimensional ladders, we choose to study ladder

systems with Cu2 S = 1/2 spins. Ladders with an even number of legs are expected to

have a gap to magnetic excitations otherwise referred to as a spin gap, A [3]. A spin gap

can be measured indirectly in nuclear magnetic resonance experiments or directly in

neutron scattering experiments. In addition, a spin gap will manifest itself in

magnetization studies at low temperature (T --* 0) as a critical field, Hci, below which the

magnetization is zero. Recent studies have revealed a connection between the spin gapped

state in ladder materials and superconductivity [4-6].

Until now, the best experimental realization of a 2-leg ladder was thought to be the

material Cu2(l,4-diazacycloheptane)2Ch4, otherwise known as Cu(Hp)Cl [7-14].

However, the low temperature properties of Cu(Hp)Cl have been recently debated [15-

19]. Although quantum critical behavior has been preliminarily identified in Cu(Hp)Cl

near H6, this assertion is based on the use of scaling parameters derived by fitting the data

rather than the ones predicted theoretically. Clearly, additional physical systems are

necessary to test theoretical predictions of 2-leg S = 1/2 ladders including quantum critical

behavior.

Chapter 4 in this dissertation describes the investigation of the gapped

antiferromagnetic S = 1/2 ladder material bis(piperidinium)tetrabromocuprate(II),








(C5H12N)2CuBr4, otherwise referred to as BPCB. In 1990, the room temperature crystal

structure of BPCB was determined, in an x-ray scattering study by Patyal et al. [20], to

resemble a 2-leg ladder. This crystal structure has been recently verified in neutron

diffraction experiments. In addition, magnetization and EPR measurements have

elucidated details of the magnetic exchange. Finally, evidence for quantum critical

behavior in this material is presented.



1.2 MCCL

The simplest antiferromagnetic low dimensional materials are electronic spins

arranged in one dimensional chains with a single exchange constant between spins, J. An

exact solution of the isotropic S = 1/2 one dimensional chain was provided by Bethe [21]

in 1931 for the isotropic nearest neighbor case. In 1983, Haldane [22] predicted a gap in

the spin excitation spectrum or spin gap for isotropic integer spin chains. The Haldane gap

for both S = 1 [23,24] and S = 2 [25] systems has been experimentally observed. Spin

gaps may also occur in half integer spin chains if the exchange between spins alternates

between two values, Jd and J2, where, to leading order, 1J, J2 = A/kB.

Chapter 5 presents the results concerning the alternating chain material

catena(dimethylammonium bis(pt2-chloro)-chlorocuprate), (CH3)2NH2CuCl3, otherwise

referred to as MCCL. The room temperature crystal structure of MCCL was determined

in 1965 [26] to consist of S = 1/2 Cu2+ spins arranged in isolated zig-zag chains with

adjacent chains separated by (CH3)2NH2 groups. The distance between spins alternates

between two values and the bond angle between spins is approximately 90 degrees.

Consequently, the magnetic structure is expected to be an antiferromagnetic alternating








chain with the exchange constant alternating between the values J\ and .J2. Preliminary

neutron diffraction work at ORNL has verified the crystal structure. In addition, a

structural transition has been observed at approximately 250 K and the possibility exists

for a second structural transition occurring between 11 and 50 K. A description of the

magnetic exchange is obtained by analyzing the results of magnetization and EPR

experiments.



1.3 Zero Sound Attenuation in 3He

In 1956, Landau advanced a theory based on the properties of normal Fermi

liquids, and this description is commonly referred to as Fermi Liquid Theory [27,28]. In

the 1960's, it was realized that 'He at low temperatures was a model system for

verification of this theory. At this same time, the experimental apparatus needed to study

3He below 100 mK became available. Since then, this theory has afforded an extremely

accurate description of the properties of 3He. Landau Fermi Liquid Theory describes a

perfect Fermi gas, where the interactions between atoms are added as a perturbation.

These interactions are included by considering elementary excitations with effective mass,

m*, which are termed quasiparticles. There are two primary modes of sound propagation

in 3He depending on the time between quasiparticle collisions, r oc lIT2, and the angular

frequency of the sound, ca At high temperatures (an << 1), quasiparticle collisions

provide the restoring force and the sound propagation is termed hydrodynamic or first

sound. Consequently, the viscosity and therefore the attenuation of first sound decrease

roughly with the square of temperature. At low temperatures (an-r >> 1), quasiparticle

collisions can no longer provide the necessary restoring force to propagate hydrodynamic








sound. Instead, sound is transmitted, through quasiparticle interactions, as a collective

mode by an oscillatory deformation of the Fermi sphere and is referred to as collisionless

or zero sound. The attenuation in the zero sound regime increases as the square of

temperature since the relaxation rate of this collective mode increases due to quasiparticle

collisions. At temperatures well below the Fermi energy (T << TF), and above the

superfluid transition temperature, (T> Tc), the attenuation of both first and zero sound are

well described by Landau Fermi Liquid Theory.

In the zero sound regime, the attenuation is dominated by scattering within a

continuous band of quasiparticle energies near the Fermi energy, AE = EF + kBT. At high

frequencies (kBT << h o << kBTF) collisions will scatter quasiparticles to unoccupied

energy levels greater than ksT away from the Fermi energy. This quantum scattering

produces a second term in the attenuation, and the attenuation of zero sound may be

written as




ao(o,T,P) =a(P)T2[h1 1+. (1.1)




Because the second term is effectively temperature independent, determination of this term

requires measurement of the absolute attenuation in the zero sound regime.

Several attempts have been made to verify this second term [29-3 1], and the most

recent effort was reported by Granroth et al. [32]. In the latest experiment, the

temperature and pressure were held fixed while the frequency, f= d2ni, was swept from 8

to 50 MHz. To provide absolute attenuation, the received signals were calibrated against








the attenuation in the first sound regime. The result of this measurement was that the

frequency dependence of the quantum term was a factor of 5.6 + 1.2 greater than the

prediction. However, the frequency range was limited by the polyvinylindene flouride

(PVDF) transducers that were used so that fm. 50 MHz. By extending the experiment

to higher frequencies, it should be possible to more accurately determine the quantum

term. However, using the first sound regime as a means of calibration places an important

restriction on the highest useful frequencies. For example, this type of calibration was not

possible in most other reports [29-31].

The objective of this work is to measure the absolute zero sound attenuation in

He as a function of frequency. In this experiment, relatively low-Q, crystal LiNbO3

transducers were used to extend the frequency range to approximately fmax -110 MHz.

Again, absolute calibration of the attenuation was determined using measurements in the

first sound regime. For both the zero and first sound data, the temperature was held fixed

while received signals were averaged at several discrete frequencies. Chapter 6 contains a

complete description of the results and the analysis at the pressures of 1 and 5 bars.



1.4 Measurement of the 2A Pair Breaking Energy in Superfluid 3He-B

The pairing energy of Cooper pairs in the superfluid, 2A, can been estimated in the

limit of weak coupling using BCS theory [33,34] as 3.5 kBTc, where Tc is the transition

temperature from the normal to the superconducting state. Deviations from BCS theory

have been introduced by Serene and Rainer [35] who used quasi-classical techniques to

incorporate strong coupling corrections in a treatment known as weak coupling plus

(WCP) theory. One of the first attempts to measure 2A(7) was performed by Adenwalla








et al. [36] in 1989 who worked at T/Tc > 0.6 and between 2 and 28 bars. In 1990,

Movshovich, Kim, and Lee [37] measured the 2A pairing energy over a range of pressures

(6.0 to 29.6 bars) and temperatures (0.3 < T/Tc < 0.5). However, in both cases,

experimental limitations required that the results were either dependent on a particular

temperature scale or involved extrapolation to zero magnetic field.

The measurement of the pairing energy in superfluid 3He-B using a novel acoustic

Fourier transform technique [38,39] is described in Chapter 7. Both the temperature and

pressure dependence of the 2A pair breaking energy are included. In addition,

comparisons are made with the existing BCS and WCP plus theory as well as the results

from previous experiments.














CHAPTER 2
EXPERIMENTAL TECHNIQUES


In this chapter, the experimental techniques employed to study both 3He and the

antiferromagnetic materials are discussed. The first three Sections, 2.1 through 2.3,

describe magnetic susceptibility measurements using a SQUID magnetometer, a vibrating

sample magnetometer (VSM), and AC mutual inductance techniques. The vibrating

sample magnetometer research was conducted at the National High Magnetic Field

Laboratory (NHMFL) in Tallahassee, FL. The next two Sections, 2.4 through 2.5, discuss

tunnel diode oscillator (TDO) and conductivity measurements, respectively. Section 2.6

describes neutron diffraction measurements that were carried out at Oak Ridge National

Lab (ORNL), Oak Ridge, TN. Section 2.7 outlines the design of a nuclear magnetic

resonance (NMR) probe as well as details of the spectrometer and superconducting

magnet. Section 2.8 describes electron spin resonance (ESR) measurements which were

performed by Dr. Talham's research group in the Department of Chemistry at the

University of Florida. Two 3He acoustic spectroscopy experiments are described, and

both were conducted in the University of Florida Microkelvin Laboratory. Section 2.9

highlights only the general experimental approach of both 3He experiments while leaving

the details of each experiment to the relevant chapters.








2.1 SQUID Magnetometer

The SQUID magnetometer used in our magnetization experiments (model

MPMS-5S) was from Quantum Design, Inc., San Diego, CA. The system is composed of

a computer and two cabinets. The first cabinet houses the electronics and the second

cabinet contains the liquid He dewar. The SQUID communicates with the computer over

the IEEE-488 general purpose interface bus (GPIB), and control of the measurement

system is accomplished using software provided by Quantum Design. The temperature

and magnetic field can be varied automatically via the computer software. The

temperature is controlled by two heaters and the flow of cold He gas, and the useful range

of operation is from 1.7 K to 300 K with an estimated error of less than 0.5% [40]. The

temperature can be lowered from 4.5 K to 1.7 K by applying a vacuum over a small liquid

He reservoir. For practical purposes, the minimum temperature is 1.8 K and typically

2.0 K was used to decrease the measurement time. The superconducting magnet provides

reversible field operation over +/- 5.0 T.

In this commercial device, the measurement is accomplished using a rf SQUID. A

SQUID consists of a superconducting ring with a weak link or Josephson junction. The

electrons in the ring form Cooper pairs, which must be described by a single wave

function. The phase of the electron wave function on either side of the boundary is

equivalent. Therefore, the flux through the loop is quantized and must be an integer of the

flux quantum, h / 2e. A screening current will increase in the ring to enforce this criteria

until each integer flux quantum is reached. Similarly, the voltage across the boundary will

oscillate with a period of one flux quantum. Theoretically, the SQUID can measure

magnetic flux with a resolution less than 1 flux quantum. However, practical design








considerations make it impossible to measure the flux directly; e.g. the SQUID must only

detect the flux due to the sample and not from the magnet. In a rf SQUID, the

superconducting ring is shielded from the magnet and connected to the pick-up coils with

an isolation transformer. A rf signal is applied to an electromagnet so that the flux

through the ring oscillates. A DC bias is also applied, using a feedback loop, so that the

voltage across the link remains at the single flux quanta condition. This DC bias is

proportional to the signal from the pick-up coils and therefore the magnetization from the

sample.

The samples are mounted on the end of long stainless steel rods and lowered into

the sample space. The magnetization of the sample is measured by moving the sample

through the pick-up coils using a microstepping controller. The pick-up coils have been

wound so that the voltage in the coils is proportional to the second derivative of the

magnetization. The computer reads the voltage output as a function of position and

compares it to a theoretical curve using a linear regression technique. This theoretical

curve depends slightly on the geometry of the sample. The standard curve assumes a

cylindrical sample. For all our experiments, 48 position steps were used over a 4.0 cm

scan length.

The output of the SQUID is given in units of "emu", which is an abbreviation for

"electromagnetic units" but it is not really an actual unit. The manner in which "emu" is

used to output the data has led to some confusion. In cgs units, the "emu" is equivalent to

cm3 or erg/G2 [41,42]. Accordingly, the units of molar volume susceptibility can be

derived from Curie's Law and may be written in unit form as









nN 2, ( erg )2
S_ nNAg2P 2PB gauss erg emu cm3
V 3kBTV ks (erg/K)T(K)cm3 =gausscm3 cm3 cm3' (2.1)



where n is the number of moles, g is the dimensionless Lande g factor, and NA =

Avagadro's number. The units of magnetization can be obtained from a straightforward

calculation of total spin as



M = nNAMB =nNAPB( erg ) = emuG = cm3G. (2.2)
gauss



2.2 Vibrating Sample Magnetometer

High field (0 < H < 30 T) magnetization experiments were performed at the

National High Magnetic Field Laboratory (NHMFL), Tallahassee. These measurements

used a 30 T, 33 mm bore resistive magnet and a vibrating sample magnetometer (VSM).

The general setup of the VSM is shown in Fig. 2.1. Powder and single crystal samples

(m ; 100 mg) were packed into gelcaps and held in place at the end of a fiberglass sample

rod with Kapton tape. The sample rod screws into the VSM head and is locked in place.

To position the sample in the field center, the height of the VSM head is adjusted until the

VSM signal is at a maximum. The VSM uses a pair of counter wound pick-up coils (3500

turns/each, AWG 50). The sample is vibrated at 82 Hz in the center of the pick-up coils

to generate a signal. This signal from the VSM is sent through a 19 pin breakout box and

then to a Lakeshore model 7300 VSM controller. The VSM controller
















VSM head

V valve

top plate


Gas handing
valm


He level


platUom


heated
- ~lSmsm


Figure 2.1: Overview of the VSM setup [43].








does not have an IEEE interface and, therefore, a Keithley 2000 multimeter reads the

EMU monitor on the VSM controller and communicates with the computer over the

GPIB bus. Absolute signal calibration was not necessary during our measurements,

because we were able to reach saturation magnetization. In addition, at saturation, we

were also able to measure and subtract a small linear correction with negative slope that

corresponds to the diamagnetic contribution from the gelcap as well as the diamagnetism

from the sample. The VSM has a resolution of 103 emu and a maximum signal of-104

emu. The largest sample signal was at least an order of magnitude below this limit at 30

T, so the VSM pick-up coil response remained in the linear regime. The sample signal

was greater than the minimum signal resolution of 103 emu at a magnetic field of

approximately 1 T.

Temperature control was achieved by varying the pumping speed on either a 4He

or 3He bath. A heater was not used in our experiments. The resistance values of a

calibrated cemrnox thermometer were measured and converted to temperature using a

Conductus LTC-20 Temperature Controller. The cemrnox resistor is calibrated only down

to a temperature of 2 K. In addition, the cernox resistor has a field dependence that must

be corrected using the results of Brandt et al. [44]. This thermometer is placed in a

location directly adjoining the sample space (see Fig. 2.2). The sample space as well as

the middle layer surrounding the sample space is filled with a small amount of 4He gas.

When the temperature of the 3He bath falls below the 4He lambda transition temperature,

some of the gas in the surrounding sample space becomes superfluid and the thermal

connection with the bath is made. For this reason, above a temperature of 2.0 K, where









77 K



I 1 I




i sample
rod
rod 150 cm
gelcap

4 He gas


3 He bath
cernox
4He L thermometer
superfluid
0 - 0.64 cm




Figure 2.2: A sketch of the thermometer setup and thermal conduction mechanisms below
the 4He lambda transition temperature in the NHMFL vibrating sample magnetometer.




the thermometer is calibrated, the thermometer values were used directly. Below this

value, the temperature is estimated from the He pressure.

In our experiment, the lowest 3He bath temperature was 0.58 K. The actual

sample temperature was warmer than this temperature. The main mechanism for thermal

conduction inside the sample space was a small amount of 4He gas. At such a low

temperature, an error in the temperature of 0.1 K becomes very important. By estimating

the heat leak, we can determine the worst possible error in temperature from









QH =Qc ,and (2.3)



A AT AT
Area -K= Area-- A K (2.4)
AL radius



where QH and Qc are the rate of heat transfer into and out of the sample space and K is

the thermal conductivity of the 4He gas. The area and radius characterize the inside of the

AT
0.64 cm diameter tube. The factor is determined by considering the dewar geometry.
AL

The liquid N2 bath temperature of 77 K is 150 cm above the sample. If we assume that

the temperature gradient is a constant of 0.5 K/cm, then we arrive at a AT of 0.2 K. This

calculation is obviously an overestimate; however it gives us a basis for determining the

maximum possible error. Consequently, the lowest temperature in our experiment,

originally reported as 0.58 K, was estimated to be 0.7 + 0.1 K.



2.3 AC Susceptibility

The AC susceptibility measurement system is a standard mutual inductance

technique that consists of a dewar, probe, and electronics. Detailed drawings of the probe

are listed in Appendix A. Computer control of the instruments was accomplished using

Labview and a GPIB interface. Five instruments were employed: a Picowatt AVS-47

Resistance Bridge, HP3457a digital multimeter, HP6632 power supply, and two PAR

124A lock-in amplifiers. A schematic of the susceptibility setup is shown in Fig. 2.3. The

signal from the secondary coil is split into inputs A and B of both lock-in amplifiers which





















PAR 124A Lock-in

D + 90'
Signal
RefOut A B Out


4.5 kQ
Resistor box


PAR 124A Lock-in

Signal' .
Ref In A B Out

S HP6632

Output


HP3457a


Input 0 Input 1


Dewar


Sample


leatet-


SThe"rm!

primary secondary


Lakeshore carbon
glass resistor
CGR-1-1000 /


*I "= =
* 9 3 ,



* l:


Figure 2.3: (A) Schematic diagram of the mutual inductance circuit used to measure AC
susceptibility. (B) Overview of the copper sample plate indicating the position of the
thermometer, heater, and susceptibility coil.


Voltage
AVS-47
Current


S Twisted
pair


Heater








are operated in 'A-B' mode. The inputs are filtered so that high frequency components

(f > 1 kHz) are attenuated. At the start of the measurement, usually at the lowest

temperature, the lock-in amplifiers are adjusted so that the signal from one is a maximum

and the phase difference between them is w 90 degrees. The adjustable reference output

provides the primary excitation voltage (typically VREF = 5 Vp-p). Because the resistance

of the primary coil is small (50 Q), a 4.5 kM resistance box is placed in series with the

primary so that the reference output behaves as a constant current source.

The heater consisted of approximately 100 turns of manganin wire on a copper

core, which was bolted to the copper sample support (Fig. 2.3 B). The resistance of the

heater was approximately 50 C. Five watts of heater power was sufficient to bring the

temperature from 4.2 K to 77 K. The Labview software controlled the heater to achieve

an input drift rate, typically 0.2 K/minute. The temperature could be lowered from 4.2 K

down to 1.7 K using the 1K pot. Temperature control in this range was achieved either by

controlling the 1K pot pressure manually using the pumping valve or by opening the valve

all the way and letting the computer control the temperature using the heater.

The temperature was determined by measuring the resistance of a CGR-1-1000

Lakeshore Cryotronics carbon glass resistor and converting to temperature. The resistor

was wrapped in copper shim stock and bolted to the sample plate. The resistor wires were

attached to the cold plate using GE varnish. The resistance was read using the AVS-47

resistance bridge on the 2 kW range using a 1 mV excitation at 15 Hz.

The susceptibility coil consists of two coil former, a primary (insert) and a

secondary (outer). The coil former were manufactured from phenolic rod and the

drawings are shown in Appendix B. The primary has 300 turns on two layers using








copper AWG 40 wire. The secondary consists of two counter wound coils separated by a

small gap with 1500 turns/side on 40 layers using the same wire. The insert is a tight fit

into the secondary and the position of the insert is adjusted so that the signal from the

secondary coil is a minimum. The sample is placed on one side of the insert so that any

magnetic signal from the sample will unbalance the coil. The susceptibility coil was

attached to a 16 pin connector using GE varnish and plugged into the sample plate. The

sample plate is suspended below the 1K pot using two stainless steel tubes.



2.4 Tunnel Diode Oscillator

The tunnel diode oscillator (TDO) technique uses a tank circuit that is sensitive to

the inductance of a coil where the sample is located. The sample is placed inside the coil.

As the inductance of the sample changes, the resonant frequency of the circuit also

changes. In the case of a conducting sample, this inductance change is related to the skin

depth. The TDO technique is particularly sensitive to superconducting transitions. For a

discussion of the superconducting applications, please see the dissertation of Philippe

Signore [45].

The circuit used for the TDO experiment is shown in Fig. 2.4. The tunnel diode

(Germanium Power Devices, model number BD-6) is the active element in the inductance

circuit. This device has a negative I-V curve when biased at the correct voltage and

therefore behaves analogously to a negative resistor. A 5 pF capacitor is added in parallel

with the tunnel diode to stabilize the oscillation. In series with the tunnel diode are the

wire capacitance and the coil inductance. The wire capacitance is on the order of 300 pF.


























GPIB


Computer


GPIB


Dewar


Battery voltage
(0 1.5 Volts)


HP 5385A
S Frequency
Counter


T



Trontech
RF Amplifier


Twisted pair


Coaxial
cable
i-
Wire cap
(500 pF)


BD-6 TD
5 pF


+- .
15V
HP E3630A
DC Power Supply


Coil Sample


Figure 2.4: Schematic diagram of the tunnel diode oscillator setup.


AVS-47
V I


liP 6632


-- +


Lakeshore
carbon glass
resistor
CGR- I- 1000








Additional capacitance can be added to change the resonant frequency which is typically

10 MHz.

The coil is wound on a phenolic former fabricated in the University of Florida

Instrument Shop. The drawings for various designs are listed in Appendix B. For 100

turns of AWG 40 copper wire, we expect a coil inductance of -5 gH. Power is supplied

by two 1.5 V Mercury batteries at room temperature. The voltage bias is adjusted until

the circuit begins to oscillate. It is useful to watch the signal output on an oscilloscope as

the voltage bias is adjusted. The voltage where the signal appears most sinusoidal usually

corresponds to the maximum voltage output. However, because this point usually occurs

near the end of the voltage range for stable oscillation, it is not practical to balance the

circuit at this point. It is more useful to balance near the center of the voltage bias curve

so that changes in the sample inductance, due to a change in temperature, do not push the

circuit out of its oscillation condition. After the DC component is removed, the signal

voltage (-50 mVp.p) is increased (-1 Vp-p) by a Trontech W500K rf amplifier. A

HP5385A frequency counter reads the frequency and communicates with the computer

using GPIB bus. Temperature control is identical to the method discussed for the AC

susceptibility technique (Section 2.3).

A major complication of this method is that the properties of the tunnel diode

circuit are temperature dependent. Even without a sample, the resonant frequency of the

circuit will change due to a change in the wire resistance, thermal contraction of the

sample coil, and most importantly the thermal dependence of the tunnel diode. From

4.2 K to 77 K, this change is typically on the order of a few MHz. Frequency transitions

related to the sample are at least an order of magnitude less, and a temperature dependent








background must be subtracted. One possible way to decrease the background

contribution is to mount only the tunnel diode on the underside of the 1K pot. The leads

must be attached to the copper using GE varnish. As long as the 1K pot needle valve is

open, the temperature of the 1K pot will remain stable and no background will need to be

subtracted. However, at temperatures above 20 K, this method will boil-off a

considerable amount of liquid He from the bath. To avoid this limitation, the wires

connecting the tunnel diode to the circuit should have low thermal conductivity, such as a

CuNi alloy. The increased resistance of the wires will lower the output signal slightly.

It has been observed that the transition temperature of high Tc superconductors

[46] and hole doped ladder materials [47] can be increased by the application of pressure.

Two pressure cells have been constructed to study the effects of pressure on the

superconducting transition temperature using the previously described tunnel diode

technique. Both pressure clamps are based on the design of J.D. Thompson [48], which is

an improvement over a previous design [49]. The second design is approximately twice as

large as the first. The clamp bodies and sample cells are made from beryllium copper and

Teflon, respectively, at the University of Florida Instrument Shop. The tungsten pushrods

and wafers are made of tungsten carbide by Carbide Specialties, Waltham, MA. The

larger cell is shown in Fig. 2.5. Complete drawings for this cell are given in Appendix C.

The sample coil (- 1 mm diameter for the smaller cell, 100 turns, 4 layers of AWG

40 wire) is contained inside the Teflon sample cell. A second coil of the same size is

usually included and contains lead wire for the pressure calibration since Tc(P) for








0 0
0

Oi


LIII







rn


cz


Figure 2.5: Overall drawing of beryllium copper pressure clamp.


lead is well known [50]. The wires exit through the top of the Teflon cell and are secured
to the brass lid with 2850 epoxy. The cell is filled with isopentane liquid to distribute the
pressure. A retaining ring is placed on the bottom of the Teflon cell to prevent the cell
from rupturing outward. The entire pressure clamp is placed in a press, and the tungsten
carbide pushrods on either side of the cell transmit the pressure. As the pushrods are
compressed, the top of the beryllium copper clamp (the side without the wires) is
tightened. By using the surface area of the Teflon cell and the pressure applied it is
possible to calculate the actual cell pressure.








2.5 Conductivity


As part of a collaboration with Dr. Talham's research group (Department of

Chemistry, University of Florida), several attempts were made to measure the conductivity

of Langmuir Blodgett (LB) films deposited onto glass slides. These films are typically a

few hundred angstroms thick and are expected to be semiconducting. Since the resistivity

of these films was expected to be in the 10 Ma-cm range, masks were engineered to

optimize the conductivity measurements. The size and dimensions of the masks used are

shown in Fig. 2.6. The masks were cut from stainless steel sheets using an electric

discharge machine (EDM) in the Department of Physics Instrument Shop.

The transport measurements were performed in the following manner. First, a

mask was placed onto the LB film, the bottom of the glass slide was glued to the metal
"puck" using rubber cement, and the whole assembly was placed in the evaporation

chamber. During the evaporation, the sample is inverted. To prevent the mask from

falling, the mask is held in place by gluing the sides of the mask (which hang over the

sample) to additional glass slide pieces using rubber cement. The evaporation chamber

was placed under vacuum using a combination mechanical/diffusion pump system. When

the pressure reached 1 x 10.8 Torr (after approximately three hours), the evaporation

would begin. The current source was set to 180 A, giving a gold evaporation rate of

2 A/s. Typically, 300 A of gold were deposited. After removal from the evaporation

chamber, the glass slide was attached to a G 10 support using rubber cement. Using silver

paint, gold wires were attached as current and voltage leads to the evaporated gold.

These wires were also glued to flat copper contacts on each edge for strain relief The

circuit diagram for the four-probe technique is shown in Fig. 2.7. Measurements were

done at 19 Hz and used an initial excitation voltage of 200 pV. If the result was infinite

resistance, the excitation voltage was increased to 20 mV. In spite of considerable effort,

no reasonable resistance ( < 100 MQ) was obtained. The samples were expected to be











0.010cm


0.010cm


010
cm











..








005cm


- 0.20 cm


. . ... .. .



E
LC)












E
i :. :.. .












0 0:
S." .' '"

,. . .


i'.". ,,:" :: :. u

, ," :,,, : :; ' : 1
** ... 0


..': '*...* '.'.: 0'
.. L O
C)..-


Figure 2.6: Stainless steel mask designs used for gold evaporation onto Langmuir
Blodgett films.







semiconducting, so warming the samples should have increased the carrier density and


hence the conductivity. Each sample was placed under a lamp to increase the temperature


up to 100 C above ambient. Efforts were also made to improve gold contact with the


conducting layer of the LB sample through gold "scarring". To rule out the possibility


that the contact resistance was unusually large, the contact resistance was measured using


the mask in Fig. 2.6 (A). In every case, the contact resistance was on the order of 0.5 n,


and the sample resistance was infinite.


E
0 "
0
0 : "

.- ,.



t .' '*'


o0.

3(












G10 Support copper shim stock


gold super LR-700
gold glue
wire g Resistance Bridge
Silver V I
Paint LB Film
glass slide
evaporated
gold








(A) (B)


Figure 2.7: Circuit diagram for four-probe AC measurements on LB film samples. The
figure in (A) is a magnified view of the LB film while (B) is an overview of the entire
experimental setup.



2.6 Neutron Scattering

Neutron scattering experiments were performed at the Oak Ridge National

Laboratory High Flux Isotope Reactor (HFIR). When the reactor is operating at full

power (85 MW), it produces a large thermal neutron flux of 1.5 x 10'5 (neutrons/cm2sec).

This flux is accessed by four 10 cm diameter beam tubes that extend horizontally from the

midpoint of the reactor core. The neutron flux passes through a sapphire filter to limit the

amount of fast neutrons. At each of the access points, there is a triple axis spectrometer,

labeled HB-1 through HB-4. All of the experiments listed in this dissertation utilized the








HB-3 spectrometer (see Fig. 2.8). The typical monochromatic neutron flux (resolution

1 meV) after collimation is 3 x 107 (neutrons/cm2 sec) [51]. The angle between the

sample and the incident and reflected beams can be changed independently. Changing the

20M monochromator angle can continuously vary the incident energy. After interacting

with the sample, the neutron beam is defracted by the analyzer which is usually the same

material as the monochromator. It is also possible to operate without an analyzer, which

essentially allows all final neutron energies. After the analyzer, a 3He detector registers

the neutron flux.

Both q scans at integrated final energy and AE scans at fixed q were performed.

For both types of scans, the monochromator was pyrolitic graphite (PG)[002] with a fixed

incident energy of 14.7 meV or 30.5 meV. The collimation was 20' at positions C2 and

C3, before and after the sample. A collimation of 60' represents the open beam which has

dimensions of 5 cm x 3.75 cm. A PG[002] filter was also used to remove second order

reflections. For the constant AE scans at fixed q, a PG[002] analyzer was used to select a

fixed final energy. With the collimation, the resolution was typically 1 meV. The BPCB

sample was a single deuterated crystal with the approximate dimensions of 10 mm x 7 mm

x 2 mm, while the MCCL sample was approximately 1 g of deuterated powder. The

specimen was attached to a sample support which also served as a thermal anchor.

Temperature control was accomplished by varying the pumping speed on an in-house 4He

cryogenic system.












HB-3 Spectrometer
(


Collimator (C1)-.




Collimator (C2)


Shutter

Sapphire Filter


Monochromator


Collimator (C3)

Analyzer Crystal
Collimator (C4)
3He Detector


Figure 2.8: Overhead view of HB-3 beamline at the HFIR facility at ORNL [51].



2.7 Nuclear Magnetic Resonance

The nuclear magnetic resonance system can be divided into four main sections:

finger dewar, probe, spectrometer, and superconducting magnet. The drawings for the

dewar, probe, and magnet are included in Appendix D. The finger dewar was purchased

from Kadel Engineering, Danville, IN, and was designed specifically for our Oxford

magnet. The overall length of the body is 49 inches and the tail portion is 33 inches long.

It contains a liquid N2 shield with a 5.7 liter capacity designed to have a 2.5 day hold time

and a liquid He capacity of 35 liters with a 13.2 day hold time. A stainless steel 0-ring








flange bolts to the top and necks down to a LF flange (A&N, part number

LF100-400-SB). Six copper baffles are suspended below this flange along with three 4-40

stainless steel support rods. There are seven quick connects (A&N, part number QF16-

075) arranged in circle 45 degrees apart for access to the He bath.

The NMR probe was designed to be a versatile platform to study protons as well

as other nuclei. It is based loosely on a design by A.P. Reyes and co-workers [52]. The

main additional objective of this design was to allow the ability to tune the NMR

capacitors from room temperature while the probe vacuum can was cold. In addition, the

probe should allow for stable temperature control between 1.5 K and 300 K. The

combination of these goals presents another difficulty, i.e. 4He gas exists as single atoms

and therefore does not contain any molecular vibration modes. Therefore, He has a low

ionization energy with a minimum at -1 x 10-3 Torr [53], which is incidentally the typical

vacuum pressure produced by a mechanical pump. At high power (P > 100 W), current

will arc between the terminals of each capacitor. To overcome this difficulty, the He

vacuum pressure must be higher than 1 Torr or lower than 1 x 10-6 Torr. For the 1K pot

to be effective, the lower pressure must be chosen. Although this pressure is easily

reached using a turbo or diffusion pump, the lack of any exchange gas means that the

sample must be thermally anchored by other means. We have chosen two ways to solve

the capacitor arcing dilemma. First, the sample can be contained in a separate

polycarbonate sample space (see Appendix E). Second, the capacitors can be contained in

their own vacuum cans and pumped independently. The flexibility in the design allows us

to switch between the two methods or use a combination of them. In the following

paragraphs, I will discuss the notable features of the NMR probe design.








Machining of the probe parts as well as the probe assembly took place at the

Department of Physics Instrument Shop. The top of the probe is built on a LF flange that

mates to the stainless steel adapter on the top of the dewar (Fig. 2.9). The electronic

connections are made through the brass box at the top of the probe. The main pumping

line is also connected to this box with a quick connect flange. There are five stainless steel

rods that connect the room temperature flange to the vacuum can: the main pumping line,

two capacitor pumping lines, the 1K pot pumping line, and the needle valve control rod.

The horizontal position of the 1K pot pumping line is shifted near the top of the baffles,

using a brass adapter, to prevent a direct line of sight from room temperature. The 1K pot

pumping line also contains its own set of internal baffles. At the junction of the pumping

line and the 1K pot, the design has been carefully chosen to prevent superfluid 4He from

climbing the pumping line walls. The 1K pot is located near the bottom of the vacuum can

and filled from a 0.050 inch OD capillary connected to the needle valve.

The capacitor pumping lines also accommodate the capacitor adjustment rods.

These capacitor control lines are fabricated from a combination of 1/8 inch diameter

stainless steel and phenolic rod. Each rod is fitted with triangular phenolic spacers so that

it remains in the center of its pumping line and ends with a stainless steel screwdriver tip

designed to fit the variable capacitor. The phenolic portion of the rod is located at the

bottom to prevent any increase in capacitance. Each control rod must be spring loaded

because the height of the capacitor changes upon rotation. A double 0-ring seal at the top

of each rod allows adjustment of the capacitors without significant increase in vacuum

pressure at 1 x 10-8 Torr.









* a a a a C
* 0
* 0
* a a a a a


Baffle


3/8Cajun-_


Needle Valve
Control Rod






Main Pumping Lin(


Double 0-ring
Seal


LF Flange

1K Pot Pumping Line

Capacitor Adjustment
Rods

Baffle


Sample
Support^


tl r ,
Sample i
Space
L j


1K Pot



I .69 inches


Capacitor
Can


(A) (B)

Figure 2.9: Detail of the NMR probe top (A) and bottom (B).






The capacitors are 40 pF non-magnetic trimmer capacitors from Voltronics,

Denville, NJ (part number NMTM38GEK). The position of the capacitors has been

chosen so that the distance from the NMR sample coil is minimized. The frequency range,

over which the NMR circuit can be tuned, is limited by this length. Each capacitor is

screwed into the bottom of its pumping line using a Vespel spacer to prevent any electrical

connection with the stainless steel can. As mentioned earlier, each capacitor can be sealed

in its own individual vacuum can using soft solder.








The vacuum can is 28 inches long so that the vacuum can top flange is higher than

the dewar neck. The vacuum seal is achieved using indium wire. The distance between

the vacuum can outer wall and the dewar inner wall is only 0.055 inches. During a magnet

quench or some other unforeseen event, the He below the probe would not be able to

escape easily. A hard Styrofoam piece must be bolted to the vacuum can bottom to

exclude any He liquid. The vacuum can contains its own set of six copper baffles to limit

the radiation heat leak from the pumping lines. The CuNi twisted pair for the

thermometer and heater is thermally connected to the top of the vacuum can and the 1K

pot. The NMR signal is transmitted using semi-rigid coax which is thermally connected at

those same points using hermetic feedthroughs (Johnson, part number 142-000-003). The

use of hermetic feedthroughs insures that the inner conductor is also thermally anchored.

Care has been taken at these points to insure that thermal contraction of the semi-rigid

coax is permitted. The polycarbonate sample space has a separate 0.025 inch outer

diameter capillary that runs to the top of the probe. The gas in this capillary is also

thermally anchored using copper bobbins at the top of the vacuum can and the 1K pot.

The spectrometer is a commercial instrument designed by Tecmag Inc., Houston,

TX. The spectrometer can be divided into five main components: NMRKIT II, Libra,

PTS 500, American Microwave Technology (AMT) Power Amplifier, and G4 PowerMac.

A continuous wave rf signal is sent from the PTS-500 frequency generator to the

NMRKIT. The NMRKIT mixes the signal with an intermediate frequency of 10.7 MHz

heterodynee detection) and uses only the lower side band. The signal is amplified by the

AMT Power Amplifier for a maximum pulse power of 400 W. It is important to note that

the AMT is limited to a maximum frequency of 300 MHz. The in-phase and quadrature








components are recorded by the Libra with a time resolution of 100 ns. The software

allows the returned signal to be viewed in either time or frequency space and to perform

data manipulation. The phase of the transmitter pulse is cycled to decrease the signal due

to coherent noise. Due to the phase cycling, all sequences must contain a multiple of four

pulses. The computer communicates with the Libra using a set of control lines connected

through a ribbon cable to an internal Tecmag card. The computer is connected to the

NMRKIT II using a Mini-Din serial cable. Since the G4 PowerMac does not have a serial

port (the spectrometer was originally designed for an older PowerMac version), a

Keyspan (Richmond, CA) adapter card was purchased to meet that need. Tecmag has

provided software that allows pulse sequences to be created. In addition, Applescripts

[54] can be written to access the spectrometer software and perform automated data

acquisition.

A schematic of the pulsed NMR setup is shown in Fig. 2.10. The crossed diodes

(Hewlett Packard Shottky, part number 1N734) have been chosen with a 2 ns response

time to insure proper operation up to 300 MHz. The purpose of the crossed diodes in

series after the power amplifier are to prevent unwanted noise before or after the pulse.

Use of a directional coupler instead of a "magic tee" is preferred for pulsed applications.

A directional coupler concentrates the losses to the input side, which can be overcome

with additional power, while a magic tee divides the losses between the input and output

[55]. The NMR circuit is balanced to 50 Q prior to connection with the directional

coupler using a HP8712 Network Analyzer. The cable length between the directional




















PTS 500


I Main output


Various
Synthin interconnects
TECMAG
NMRKIT


Probe in


TECMAG
Libra


AMT
Power
Amp


Rfin


Rfout



Shottky 1N734
crossed diodes
in series


Crossed diodes
in parallel


Mini-Circuits
15542
Directional Coupler


Semi-rigid
SMA


Dewar


Lakeshore
CX-1030-SD
Cemrox
resistor


balanced to 50 ohms
using the network analyzer
(includes cables)


Sample


Figure 2.10: Schematic diagram of the pulsed NMR setup.


u

Serial'
Line


Rfout


Miteq
AU- 114
pre-amp


GPIB


TS-530A


AVS-47


Twisted
pair


I Kohm
I1/4W
Metal film
chip resistor


Ribbon ,jtlc








coupler and the pre-amp should be X/4 (speed of signal 2c/3, where c = speed of light).

The impedance of a k/4 transmission line obeys the following relationship:



ZINP *ZOU =50 Q (2.5)



When the pulse reaches the directional coupler, ZOUT = 0, due to the crossed

diodes. The effective input impedance will be infinite along the pre-amp side and 50 Q

along the NMR circuit side. Consequently, all of the power will be directed toward the

NMR circuit. After return from the NMR coil, ZouTr = 50 Q, and the pulse will see a 50 K2

pre-amp circuit impedance.

Temperature control is achieved using the Picowatt system AVS-47 resistance

bridge and TS-530A temperature controller. The G4 PowerMac controls these

instruments using the GPIB bus and Labview software. Thermometer resistance set points

are sent to the temperature controller, which adjusts the heater power to achieve those set

points according to the time constants set by the software. The temperature controller is

limited to 1 W of heater power which is sufficient for most applications. The thermometer

is a Lakeshore CX-1030-SD Cernox resistor calibrated from 1.4 K to 100 K. The heater

is a 1 kQ, 1/4 W, metal film chip resistor. Both the heater and thermometer are attached

to the polycarbonate sample space with Emerson & Cuming 2850 epoxy.

The Oxford superconducting magnet has an 88 mm bore diameter and a maximum

center field of 9 T located 570 mm below the top flange. When at the maximum field of

9 T at the center of the solenoid, the field at the top flange is -0.06 T. There are four

shim coils to improve the field homogeneity. This magnetic field variation is less than








6 ppm in a 10 mm diameter spherical volume at the maximum field. The main magnet and

the shim coils are energized using Oxford power supplies. The power supplies

communicate with the G4 PowerMac computer over the GPIB bus. It is important to

shunt or "dump" the shim coils when charging the magnet to full field. Failure to perform

this step will result in current being trapped in the shim coils, and this excess field may

cause a quench.



2.8 Electron Spin Resonance

The electron spin resonance (ESR) measurements were carried out by Dr.

Talham's research group at the Department of Chemistry at the University of Florida. The

ESR spectrometer is a commercial Brueker X-band (9 GHz) spectrometer. Temperature

control down to 4 K was achieved with an Oxford ESR 900 Flow Cryostat. Further

details are available by reviewing the dissertation of Garrett Granroth [56]. The ESR

measurements discussed in this dissertation were made on both powder and single crystal

samples. Typical ESR spectra consisted of a single broad line with a width of

approximately 500 G. Exact calibration of the Lande g factor for each material was made

by comparison with the center frequency of the free radical standard DPPH.



2.9 Pulsed FT Acoustic Spectroscopy

Pulsed Fourier Transform acoustic spectroscopy experiments on liquid 3'He were

conducted at the Microkelvin Laboratory at the University of Florida. The sample cell

was placed in a Ag tower mounted on a Cu plate attached to the top of the Cu

demagnetization stage of Cryostat No. 2. Further details of this cryostat design are








described by Xu et al. [57]. A Ag powder heat exchanger in the Ag tower provides

thermal contact for cooling the liquid sample. Miniature coaxial cables with a

superconducting core and a CuNi braid were used between the tower and the 1K pot.

Stainless steel semi-rigid coaxial cables were used from the 1K pot to the room

temperature connectors. Above 40 mIK, the temperature was measured using a calibrated

ruthenium oxide (RuO2, Dale RC-550) resistor which has approximately a 500 Q room

temperature resistance [58]. The value of this Ru02 thermometer was measured using a

Picowatt AC Resistance Bridge (Linear Research). A heater mounted on the nuclear

stage supplied up to 1 jLW of thermal power. From 40 mK to 1 mK, the temperature was

measured using a 3He melting curve thermometer. Below nominally 3 mnK, a Pt NMR

thermometer (PLM-3, Instruments of Technology, Finland) was used and calibrated

against the 3He melting curve [59]. The pressure was determined using a strain gauge

mounted next to the tower. Details of the Tecmag commercial spectrometer were

discussed in the previous section.

A cross sectional view of the sample cell, nominally a cylinder with a radius ofR =

0.3175 + 0.0010 cm, is shown in Fig. (2.11). In order to obtain a large frequency

bandwidth, low-Q, coaxial LiNbO3 transducers were used and were separated by a 3.22 +

0.01 mm MACOR spacer and held in place with BeCu springs. The spacing was measured

before the experiment and verified in situ by measuring the time delay between successive

zero sound reflected pulses. The transducers had a fundamental resonant frequency of

21 MHz and were operated in four frequency windows: 8-12, 16-25, 60-70, and

105-111 MHz. Figure 2.12 shows the power reflection coefficient vs. frequency for the









Ag Cell Body

^ ^ --,,.- BeCu Spring
Upper Signal B S'rin
Cable
CuNi Shield -- ..


- L F -- Upper Crystal

\-- MACOR spacer

Lower Crystal -CuNi Shield


$ Lower Signal
$ Cable





Figure 2.11: Cross section of the 3He acoustic cell.


0.40

0.35

0.30

0.25

0.20

0.15

0.10

104


106 108 110 112
Frequency (MHz)


114 116


Figure 2.12: Frequency response functions of the 5'h harmonic for two LiNbO3
transducers at 0.3 K [39].








5th harmonic of each LiNbO3 transducer at 0.3 K [39]. The overlap of the transmitting

and receiving transducer bandwidths determines the useful frequency range at each

harmonic window. For the case shown in Fig. 2.12, this operational range is

approximately 3 MHz centered at 108 MHz. Similar results were obtained for the other

frequency windows. Due to the highly structured resonance peaks in the frequency

response function of each transducer, the resulting signal is structured.

A schematic circuit diagram of the pulsed FT acoustic technique is shown in

Fig. 2.13. The strain gauge and the 3He melting curve thermometer (MCT) were both

monitored using a capacitance bridge (General Radio Company, type 1615-A) and the

output was sent to a PAR 124A lock-in. The balance of each lock-in was determined by

measuring the function output voltage using a HP34401A multimeter. The multimeters

communicated with the G4 PowerMac over the GPIB bus. The pulse output of the

Tecmag spectrometer was sent to an inline attenuator before reaching the connection

points on the dewar. Although the pulse output of the spectrometer can be adjusted, it is

easier to consistently set the voltage level of the input pulse using an attenuator. Because

the output level of the NMRKIT II was controlled manually by a knob, for repeatability it

was set to the maximum of 13 dBm at all times. The attenuator was set to either -10 or

-20 dB. The received signals from the sample cell were amplified approximately 20 dB by

a Miteq AU-1114 preamp. The Pt NMR thermometer signal was monitored using a

TDS-430A digital oscilloscope. Each Pt NMR signal trace was sent to the PowerMac

over the GPIB bus and recorded for later analysis. Labview software was used to

communicate with the GPIB instruments. However, the spectrometer was operated using






















GPIB cable t)
0-


PAR 124A
Lock-in Funcon
A Out


Detector

GRC
Capacitance
Bridge


Unknown


PAR 124A
Lock-in F-uncti
A out


i HP34401A
SInput



HP34401A


Detector

GRC
Capacitance
Bridge


Unknown


PTS 500


NMRKIT II
Mai
oupWt Probe in


Attenuator


Various
interconnects


Miteq I DC power
AU-1114 power


--- PreampF


Dewar


-. PLM-3 NMR
Thermometer


Pt wire
NMR
therm


Figure 2.13: Schematic diagram of the pulse FT acoustic 3He spectroscopy technique.


Libra


TDS-430A


MCT


Strain
gauge








commercial software provided by Tecmag. The synchronization between Labview and the

Tecmag software was accomplished using Applescript routines (Appendix F). Typically

the spectrometer data at each frequency averaged the results of 128 pulses with a 4 s wait

step between each pulse. Each transmitter pulse was 0.4 uts (Fourier transform

spectroscopy) or 4 ts (amplitude/time of flight acoustic spectroscopy), depending on the

experiment, and the pulse power at the sample cell was estimated to be approximately

-20 dBm. The waiting time between different frequencies was at least 8 minutes. The

real and imaginary components were separately digitized at 10 M samples/s for a 2048

samples. Spurious signals, resulting from electrical crosstalk, appeared in the first

microsecond of data. Before taking the FT, this region of the data was blanked (set to

zero). In addition, for some experiments, echoes were eliminated by blanking to avoid

adding spurious structure in the frequency spectrum. This blanking and subsequent FT

were accomplished using Origin scripts (Appendix G). Calibration of the MCT involved

several steps. In the first step, the 3He MCT pressure was changed using a standard

zeolite absorption He pressure bomb (i.e. "dipstick") and the resulting pressure was

measured using a Digiquartz transducer. Figure 2.14 shows the result of a capacitance vs.

pressure calibration of the melting curve thermometer using this method. This step should

be accomplished at a relatively warm temperature (-150 mK) to decrease the amount of

time needed to reach equilibrium when changing pressures. The calibration of the strain

gauge was accomplished using a similar technique. Once the capacitance vs. pressure

calibration is complete, a predetermined temperature vs. pressure curve is required. The

experiments in this dissertation used the 3He melting curve of Wenhai Ni (Fig. 2.15 [59]),

which is consistent with the Greywall scale [60].












32 xpernmenial roinis
Polynomial Fit

L- 31
0.
0)O/
C,
M 30

CL,
o 29


28 -

28 29 30 31 32 33 34 35

Pressure (bars)

Figure 2.14: The capacitance vs. pressure of the MCT at 150 mK. The solid line is a fifth
order polynomial fit to the data.





By combining the capacitance vs. pressure calibration with the pressure vs.

temperature relationship, we can convert capacitance into temperature. However, it is still

necessary to obtain an absolute calibration of the temperature curve using fixed

temperature points. In this experiment, TA and TN, were used for this purpose. These

fixed points, in the 3He phase diagram, can be easily identified as changes in slope when

slowly (50 gK/hr) sweeping temperature (see Figs. 2.16 A and B).

The absolute calibration results in the y-axis of the temperature vs. capacitance

curve being adjusted by a constant value to match the fixed points. During the two 3He

experiments listed in this dissertation, the vertical adjustment was typically 0.5 gK. The


















34 Solid Phase
:N AB A ^
33 -

C,)
32









1 10 100
T(mK)



Figure 2.15: Melting curve of 3He as determined by W. Ni [59]. The superfluid 3He
ordering transitions, A (2.505 mK), AB (1.948 mK), and the solid ordering transition N
(0.934 mK), are marked with arrows.










31.760 i- i- '' i-'


31.755 A


31.750
CU

31.745
C.
CU
I 31.7450


TA = 31.745 + 0.002 pF
31.735


31.730 ----------
13000 14000 15000 16000 17000
Time (s)

31.790 '

< B
31.788 g
LL

S31.786


M 31.784 TN = 31.7873 + 0.0005 pF ".
o "-,'%.

31.782


31.780 ,-,-, ,-
4000 5000 6000 7000
Time (s)

Figure 2.16: Identification of the capacitance value of the temperatures, TA and TN, from
plots of the MCT capacitance vs. time while slowly warming. Solid lines have been added
as guides to the eye. Both figures are different sections of the same data set.








final temperature vs. capacitance relationship is shown in Fig. 2.17. The solid line is a fit

to a 7th order polynomial. This polynomial was used to convert the MCT capacitance

values into temperature above TN.

The Pt NMR thermometer was calibrated using the MCT by sweeping the

temperature from 3 mK to 0.5 mK and recording both the MCT and Pt NMR

thermometer. There was a wait of at least 5 minutes between each Pt NMR pulse so that

the nuclear spins could relax. The MCT temperature was recorded before and after each

Pt NMR trace and averaged. The entire temperature sweep would take approximately

12 hours. The temperature scale was checked by comparing Tc(P) [61] with the

temperature where there was a dramatic crossover from high to low attenuation in the

zero sound signal. A graph of MCT temperature vs. Pt NMR integrated signal is shown in

Fig. 2.18. The digitization rate of the TDS-430A oscilloscope was faster than the PLM-3,

and therefore each Pt NMR trace was read by the oscilloscope and integrated after the

experiment by the Labview software. The solid line represents a fit to the data from

TN (0.934 mK) to 1.5 mK using



A
MT~c (2.6)
Mp, +B



where TMCT is the MCT temperature, Mpt is the Pt NMR integrated signal, and A and B are

fitting parameters. The temperature range for the fit was chosen so that the Pt NMR

integrated signal was at least a factor of 10 above the noise. The 3He melting curve

determined by W. Ni and co-workers [59] did not extend much above 322 mK (the








minimum in the melting curve). At this temperature, a separate calibration was used and

was based on measurements by Grilly et al. [62]. The resultant temperature calibration

from the Grilly et al. scale was adjusted by a constant to match the value of the 3He

melting curve at the minimum given by Ni et al. [59].


31.70 31.72 31.74 31.76
Capacitance (pF)


31.78


Figure 2.17: The temperature vs. capacitance curve generated by combining the
capacitance vs. pressure relationship in Fig. 2.14 with the 3He melting curve in Fig. 2.15.
The curve has been adjusted by a constant to match the fixed temperature points, TA and
TN. The solid line is 7th order polynomial fit.





47











i -
3.0


E 2.4

U)

,-- 1.8

E 1.2
(D


0 0.6


0.0 6I-I, I I
0 5 10 15 20 25

Pt NMR Integrated Signal (a.u.)



Figure 2.18: The MCT temperature vs. Pt NMR integrated signal. The solid line is a fit
to Eq. 2.6 from TN to 1.5 mK.















CHAPTER 3
THEORETICAL TECHNIQUES


In this chapter, two theoretical models are presented that were used to investigate

the magnetic behavior of the low dimensional magnetic materials reported in this

dissertation. In each section, the relevant theory as well as details of the software are

discussed. In addition, the advantages and restrictions of each model are considered. The

first section discusses the exact diagonalization method, which is essentially the calculation

of the partition function for a cluster of spins. Although, this approach, in principle, is an

exact calculation, the number of spins included with this technique is limited by the

computing power. This restriction places limits on the temperature range over which

these systems can be accurately modeled. With respect to the Hamiltonian, this method is

flexible and can be applied to any cluster of interacting spins with only minor changes to

the software. The second section considers the mapping of the ladder Hamiltonian onto

the XXZ model. Although the XXZ model is exactly solvable, the mapping approximates

the magnetic behavior since it only includes the low energy states of the ladder

Hamiltonian. Nevertheless, this method can model the magnetic behavior of ladders at

temperatures significantly below the thermal energy represented by the magnetic

interactions.








3.1 Exact Diagonalization

The partition function for an arbitrary system of discrete states is written as




Z = exp(- E' (3.1)




where E, are the energy states of the system and kB is the Boltzmann constant. For a

cluster of spins, the magnetization can be calculated in a straightforward manner once the

partition function is known; i.e.




g93-S, Sexp E,
M =E (3.2)
.exp -E '




where S, represents the total spin of each state. A cluster of two Ising S = 1/2 spins that

interact with a single exchange is the simplest cluster to model. This system can be

represented by a Hamiltonian, where J is the magnetic exchange, given by



Asng = J S, S,, -g uH S,. (3.3)
1=1 j=1,2



In Eq. 3.3, the symbol g represents the Lande g factor and /a is the Bohr magneton. The

first and second terms represent the interaction between spins and the interaction with the








magnetic field, respectively. Using Eqs. 3.2 and 3.3, we arrive at the magnetization for n

moles of spins that are arranged as interacting Ising pairs; namely




M= (nNlt B exp(g kT p BH (3.4)I
2 xp B+exp B + 2exp-d
S. kT2kT)



where NA is Avagadro's number. It is easy to see that this equation will behave properly in

the limit of high temperature or field.

We can use a similar method to calculate the magnetization for a system of

Heisenberg spins. Again, for simplicity, we consider a system with only two interacting

spins and a single exchange constant J. The Hamiltonian resembles Eq. 3.3, except that

the spin operator is now a vector,



is = J S, S,+i, g9B SH g (3.5)
1=1 i=1,2


In the Ising case, the spin basis states were also the eigenvectors of the spin operator. For

the Heisenberg case, this is not true. We explicitly write the spin basis eigenvectors that

represent the electron wave functions with either spin up, a), or spin down, 10). The

spin operator S, can be divided into its components S, = Si' + Si + Sz. The components

are also operators that act on the spin basis function according to the rules given in

Table 3.1.









Table 3.1: The spin operators acting on the spin basis functions.

SIcX) SJB)
& Y 12I) Y210)


/210) -210)


We apply the Hamiltonian, Eq. 3.5, onto the basis function for two spins to obtain
a matrix representation for the energy states of that system. Omitting the field interaction
term, this matrix may be written as:


(aal
(ap I
(/iaj
(9P I
(fta\


Iaa) la/ )
J/4 0


0
0
0


-J/4
J/2


I\a)
0
J/2
-J/4


ifp)
0
0
0
J/4


=0.


(3.6)


The matrix is blocked according to the total spin value, S = 1, S = 0, and S = -1. Each
matrix corresponding to a particular spin can be diagonalized individually. For the case of
only two spins, it would be just as easy to diagonalize the entire matrix at once. However,
for large clusters of spins, a great reduction in the necessary computing power is achieved
by diagonalizing each total spin matrix individually. After determining the eigenvalues, the







field interaction term may be included. For two spins, we obtain the eigenvalues: -3J/4,

J/4, J/4 gpsH, and J/4 + guBH, which correspond to the familiar singlet and triplet states

for two interacting S = 1/2 spins. The triplet states are degenerate in zero magnetic field,

and we can display this graphically using the diagram sketched in Fig. 3.1.



E

S = M s = 0
MH=





S~l^ -- ---- M^=O
S=0 -M = 0

Ms = .1


Figure 3.1: A graphical depiction of the energy eigenvalues for a system of two S = 1/2
spins in a magnetic field showing the singlet and triplet states.


Once we have the eigenvalues, it is trivial to plug them into Eq. 3.2 to obtain the

magnetization. Again, for practical purposes, we have assumed n moles of spins arranged

as dimer pairs, and have



[ -n g rexp -g.H
2 nNA9gUB k+T (3.7)
2 exp+PIr exp -g-rexp 1
L k T )k T YkkTT











In the low field limit, Eq. 3.7 becomes the Bleaney-Blowers equation [42] ; which can be

written as


nNAg2 1 (3.8)
A3kBT l+ exp(Jl/k)
3



At high temperatures, Eqs. 3.7 and 3.8 become the S = 1/2 Curie law, namely


nN 2 22 SSI
S Ag B- _nNS+l (3.9)
4kBT 3kBT



Figure 3.2 shows a graph of molar magnetic susceptibility vs. temperature in a

magnetic field of 0.1 T produced using Eq. 3.7 with an exchange constant of J = 12 K.

Below the peak temperature of approximately 7 K, there is an exponential decrease in

susceptibility due to singlet formation. The peak in the curve corresponds to the thermal

energy (gap) needed to form triplets. This type of curve is common to low dimensional

gapped magnetic systems.

Using this approach, we could have calculated other thermodynamic quantities as

well. For instance, using the partition function, Z, and the energy eigenvalues, E,, for two

Heisenberg spins, we write the entropy [63] of this system as:








1 ( E + I exp2( E2 )(
o'=-e k 'k+to +--exp ---= +lnZ +
Z ep kT ln Z \ kBT)[kBT


(3.10)


Sp- E T In )ep( E E4 )( _E4T nZ
Z kBT kT ,Z Bt, kTk


Figure 3.3 shows the entropy vs. temperature for a system of two Heisenberg spins that

interact with an exchange constant J in a magnetic field of 0.1 T. In the high temperature

limit, the entropy approaches ln(number of energy states) = ln(4), and in the low

temperature limit, it approaches zero.


25

20
E0
E 15
E
E 10

5


0 10 20 30 40 50 60 70
T(K)
Figure 3.2: The molar magnetic susceptibility vs. temperature in a magnetic field of 0.1 T
for dimer pairs of Heisenberg spins that interact with an exchange constant of J = 12 K.
The curve was produced using Eq. 3.7.










1.50 ,-' '' ,

1.25

1.00
()
G)
> 0.75
0
L-
"C 0.50

0.25
J=12K
0.00
I I I I p I p p I p ,
0 10 20 30 40
T(K)

Figure 3.3: The entropy vs. temperature in a magnetic field of 0.1 T for a dimer pair of
Heisenberg spins that interact with an exchange constant of J = 12 K. The curve was
produced using Eq. 3.10.





Calculating magnetization using the partition function is only trivial when the

number of interacting spins, and hence the number of energy states, is small. This exact

diagonalization method was used by Robert Weller [64] in 1980 to calculate the

susceptibility for larger clusters (N > 12) of S = 1/2 magnetic spins. This method relies on

mathematics that have long been understood, however, it was not a viable alternative until

cheap computing power was available. Although this method can be easily scaled up to

larger systems, the corresponding matrix size increases as the factorial of the number of


spins. For N spins, the maximum matrix size of N or N choose N12, is such that
w e N1 canobe mn/ ip y ta
wN/2

when N = 12, the matrix size is 924 x 924. A matrix of this size can be manipulated by a








desktop computer in a few minutes. For 20 spins, the maximum matrix size 184,756 x

184,756 and the computation quickly becomes impossible. At higher spin values, e.g. S =

1, the matrix size increases even faster. However, we have not utilized all possible

symmetries of the problem. By considering geometric symmetry of the spins, we can

reduce the problem computationally by several orders of magnitude. This method is

referred to as the Lanczos algorithm [65,66]. Using this approach, the magnetization for a

system of as many as 30 spins can be calculated. I did not use this method, and so I do

not discuss the details here.

Since the importance of the boundaries of a model system decrease with increasing

system size, it is important to use as many spins as possible. In addition, a small number

of spins can accurately describe a low dimensional material only as long as the correlation

length does not exceed the total length of the system. At T = 0, for quasi-two dimensional

systems, such as spin ladders, the correlation length can become infinitely long. For these

reasons, this method will only give accurate results for temperatures that are the same

order of magnitude as the exchange constants or higher, T > J. If the system size is too

small, this method introduces erroneous plateaus in the magnetization curves as the

temperature is lowered below the exchange constants.

In this dissertation, the spins were arranged in either a ladder or alternating chain

geometry. However, any arrangement consisting of interacting spins with exchange

constants, J1, J2, ... JN, could have been used. Both the ladder and alternating chain model

systems used 12 spins which were arranged in a ring to help alleviate the boundary

problem. The programs were written in MATLAB and produced theoretical curves using

the exact diagonalization method described above. I am grateful to Steve Nagler (ORNL)








for his assistance in writing these programs, which have been included in Appendix H.

Fitting the data involved three steps. First, the experimental curves were fit using a high

order polynomial ( > 5). Second, the software would generate multiple curves over a

preset parameter space. Each curve would be compared to the polynomial and the

difference between the polynomial curve and the theoretical curve would be recorded as a

chi2 value. The chi2 value is the sum of the square of the difference between each

theoretical point and the polynomial curve. Finally, when the program was finished

generating curves, the chi2 values would be searched to obtain the lowest value, hopefully

corresponding to the best fit. It was beneficial to generate a curve using those final

parameters to ensure that the theoretical curve matched the data. A typical parameter

search generates approximately 300 curves and takes approximately 18 hours. It is

possible to increase the efficiency of the process by allowing the software to choose the

next parameters instead of blindly searching the whole parameter space. This procedure is

described in the dissertation by Robert Weller [64]. However, this technique was

abandoned as it tended to find local minima in the parameter space.



3.2 The XXZ Model

During my investigation of low dimensional materials, it became necessary to

produce low temperature (T << the lowest ladder exchange constant, e.g. J1 ; 4 K)

magnetization curves for spins arranged in a ladder geometry. At the lowest experimental

temperature of 0.7 K, there exists a feature in the data at half the saturation magnetization,

Ms /2, that could not be modeled using the exact diagonalization method. As discussed in

the previous section, the exact diagonalization procedure is increasingly inaccurate as the








temperature is lowered below the exchange constants. In addition, the exact

diagonalization method also introduced erroneous plateaus in the theoretical curves that

resemble the feature at Ms /2. Therefore, another method was required to study the

magnetization of ladder materials at low temperature.

Chaboussant et al. [14] have previously created low temperature magnetization

curves for the ladder-like material, Cu(Hp)Cl, by mapping the ladder Hamiltonian onto the

XXZ model which was initially solved by H. Bethe [21]. The thermodynamics of the

XXZ model have been completely described by Takahashi and Suzuki [67]. I begin with

the ladder Hamiltonian including the field interaction term



N12 N-2 N
Ladder =J -22- '92, +Jl 1, S,+2 + g91B S, (3.11)
t=1 i=1 -=1



We can consider only the restricted Hilbert space composed of a singlet S = 0, ms = 0)

and the lowest energy triplet IS = 1,ms = -1) on each rung. These are the two lowest

energy states (see Fig. 3.1) and therefore the most populated. This approximation is valid

since we are interested in the critical region where the magnetic field is on the order of J.

We can rewrite the effective Hamiltonian on this restricted Hilbert space as



N/2 "r+1 + SrYl + Z. rZ)+Hff Sz, (3.12)
r=1 2 r=l


where the effective field is given by








Heff =Ji + gBH' (3.13)




and Sr now represents the total spin of rung r. It should be noticed that this Hamiltonian

is completely symmetric around Heff = 0. Hence, any quantities computed from this

Hamiltonian will also be symmetric around this point. This Hamiltonian (Eq. 3.12) can be

identified as the effective S = 1/2 XXZ model. The thermodynamics of the XXZ model

have been reduced to a set of non-linear differential equations by Takahashi and Suzuki

[67]; such that



In r(x) = -3-3 Js(x) + s(x) ln(l + u(x)) (3.14)
kBT


u(x) = 2B(x) cosh 3-BH-ff +K 2(x) (3.15)


and ln t(x) = s(x) In(1 + (x)), (3.16)



1 I
wheres(x)=-sechI-x, is the convolution product and q(x), u(x), and K(x) are
4 '2

parameters in the model. These equations must be solved iteratively from a known

solution for each value of the temperature and magnetic field. In this case, the known

solution was rq(x) = 3 and Kc(x) =2 for J11 = 0 and Heff = 0. The convolution products

are calculated as discrete integrals using 200 points. Since, the hyperbolic secant function

x
and hence, s(x), decays quickly, was used instead of x in the argument to increase the
10








resolution. The convolution product must therefore be divided by 10 as well. Typically

10 iterations were sufficient to reach equilibrium with a 200 point resolution. Once a

stable solution is reached, the free energy per spin can be calculated using


F J
F 11-kBTInc(O). (3.17)
N 2



dF
The magnetization is proportional to M --dH The curves produced must be


normalized so that the maximum overall magnetization is 1. Curves generated this way

using J11 = 0 were compared to the exact dimer results, Eq. 3.7, to ensure that the method

was correct. It should be noted that there are three typographical mistakes in the

treatment ofChabbousant et al. [14]: a sign error and a missing s(x) factor in Eq. 3.14 (or

Eq. 30 as listed in the Chaboussant et al. paper) and a factor of 2 difference in Eq. 3.17

(or Eq. 32 as listed in the Chaboussant et al. paper). These integral equations were solved

using MATLAB software. For reference, these programs are included in Appendix H.














CHAPTER 4
STRUCTURE AND CHARACTERIZATION OF A NOVEL MAGNETIC SPIN
LADDER MATERIAL


Magnetic spin ladders are a class of low dimensional materials with structural and

physical properties between those of 1D chains and 2D planes. In a spin ladder, the

vertices possesses unpaired spins that interact along the legs via J11 and along the rungs via

J/, but are isolated from equivalent sites on adjacent ladders, i.e. interladder J' << J\1, J.

Recently, a considerable amount of attention has been given to the theoretical and

experimental investigation of spin ladder systems as a result of the observation that the

microscopic mechanisms in these systems may be related to the ones governing high

temperature superconductivity [2,6]. The phase diagram of the antiferromagnetic spin

ladder in the presence of a magnetic field is particularly interesting. At T = 0 with no

external applied field, the ground state is a gapped, disordered quantum spin liquid. At a

field Hcl, there is a transition to a gapless Luttinger liquid phase, with a further transition

at Hc2 to a fully polarized state. Both Hci and Hc2 are quantum critical points [1]. Near

Hci, the magnetization has been predicted to obey a universal scaling function [68]. Using

a symmetry argument, this universal scaling can also be shown to be valid at Hc2. Until

now, this behavior has not been observed experimentally.

A number of solid state materials have been proposed as examples of spin ladder

systems, and an extensive set of experiments have been performed on the compound










Cu2(1,4-diazacycloheptane)Cl4, Cu2(C5Hi2N2)2C4, referred to as Cu(Hp)Cl [7]. The

initial work identified this material as a two-leg S = 1/2 spin ladder [7-14]. Although

quantum critical behavior has been preliminarily identified in this system near Hci, this

assertion is based on the use of scaling parameters identified from the experimental data

rather than the ones predicted theoretically [13,14]. Furthermore, more recent work has

debated the appropriate classification of the low temperature properties [15-19]. Clearly,

additional physical systems are necessary to experimentally test the predictions of the

various theoretical treatments of two-leg S = 1/2 spin ladders.

Herein, we report evidence that identifies bis(piperidinium)tetrabromocuprate(II),

(CsHi2N)2CuBr4 [20,69], hereafter referred to as BPCB, as a two-leg S = 1/2 ladder that

exists in the strong coupling limit, JI/J, > 1. High-field, low-temperature magnetization,

M(H < 30 T, T > 0.7 K), data of single crystals and powder samples have been fit to

obtain J = 13.3 K, J\ = 3.8 K, and A 9.5 K, i.e. at the lowest temperatures finite

magnetization appears at Hci = 6.6 T and saturation is achieved at Hc2 = 14.6 T. An

unambiguous inflection point in the magnetization, M(H,T = 0.7 K), and its derivative,

dM/dH, is observed at half the saturation magnetization, Ms/2. This behavior has not been

detected in Cu(Hp)Cl [8-10]. The Ms/2 feature cannot be explained by the presence of

additional exchange interactions, e.g. diagonal frustration JF, but is well described by an

effective XXZ chain, onto which the original spin ladder model (for strong coupling) can

be mapped in the gapless regime HcI < H < Hc2. After determining Hci and with no

additional adjustable parameters, the magnetization data are observed to obey a universal










scaling function [68]. This observation further supports our identification of BPCB as a

two-leg S = 1/2 Heisenberg spin ladder with J'<< J .

This chapter is divided into six sections. In the first section of this chapter, I will

discuss the structure and synthesis of BPCB. The second and third sections report the

results of low field susceptibility and magnetization measurements, respectively. The

fourth section presents the high-field magnetization work performed at the National High

Magnetic Field Laboratory, while section five details the universal scaling behavior of

BPCB. Results from the neutron scattering experiments, performed at Oak Ridge

National Laboratory, are provided in section six.



4.1 Structure and Synthesis of BPCB

The crystal structure of BPCB has been determined to be monoclinic with stacked

pairs of S = 1/2 CuE ions forming magnetic dimer units [20]. The CuBr4-2 tetrahedra are

co-crystallized along with the organic piperidinium cations so that the crystal structure

resembles a two-leg ladder, Fig. 4.1. The rungs of the ladder are formed along the c*-axis

(the c*-axis makes an angle of 23.4 with the a-c plane and the projection of the c*-axis in

the a-c plane makes an angle of 19.8 with the c-axis) by adjacent flattened CuBr4-2

tetrahedra related by a center of inversion. The ladder extends along the a-axis with

6.934 A between Cu2' spins on the same rung and 8.597 A between rungs. The three

dimensional crystal structure of BPCB, including the organic cations, viewed along the

E -axis is shown in Fig. 4.2. The atomic positions have been taken from the x-ray






































Figure 4.1: A schematic diagram of the crystal structure of BPCB viewed down the [010]
axis as determined by Patyal et al. [20]. The magnetic exchange between S = 1/2 Cu2+
spins is mediated by non-bonding Br-Br contact. The two primary exchange models
considered were a ladder model, with parameters J and J\\, and alternating chain model,
with parameters J1 and J2. In the ladder model it is possible to include a frustration
exchange, J.a





















0 a


SCu ON

O Br Q C

0 H
Figure 4.2: The crystal structure of BPCB viewed along the c -axis. The ladder direction
is along the a-axis. The c*-axis, rung direction, makes an angle of 23.4 with the a-c plane
and the projection of the c*-axis in the a-c plane makes an angle of 19.8 with the c-axis.
The solid lines indicate the unit cell.










scattering data of Patyal et al. [20] and verified in the neutron scattering studies (see

Section 4.6). The hydrogen positions have been calculated using symmetry arguments.

The ladder structure is viewed edgewise (dark spheres) in Fig. 4.2, and it is apparent that

the rungs of the ladder extend out of the a-c plane. Adjacent ladders are separated by

12.380 A along the c-axis and 8.613 A along the b-axis. Although the b-axis separation is

approximately the same as the rung separation, it is unlikely that the organic cations

provide significant superexchange between ladders along the b-axis and hence the

magnetic exchange between ladders is expected to be small (J' << J11). The magnetic

exchange, Ji, between Cu2 spins on the same rung is mediated by the orbital overlap of

Br ions on adjacent Cu sites. The exchange between the legs of the ladder, J1, is also

mediated by somewhat longer non-bonding (Br Br) contacts and possibly augmented

by hydrogen bonds to the organic cations. A diagonal exchange, JF, is possible, although

it should be weak (JF<< J ), and since the diagonal distances (9.918 A vs. 12.066 A) are

not equal, only one JF was considered in our analysis.

Shiny, black crystals of BPCB were prepared by slow evaporation of solvent from

a methanol solution of [(pipdH)Br] and CuBr2, and milling of the smallest crystals was

used to produce the powder samples. The stochiometry was verified using carbon-

hydrogen-nitrogen analysis [70]. In addition, deuterated single crystal and powder

samples were produced and used in neutron scattering studies performed at the High Flux

Isotope Reactor at Oak Ridge National Laboratory. The protonated BPCB material has a

molecular weight of 583.49 g/mol and a density of 2.07 g/cm3.










The previous study by Patyal et al. [20] reported the Lande g factor along all three

single crystal axes for BPCB as g(a-axis) -= 2.063, g(b-axis) = 2.188, and g(c-axis) =

2.148. ESR measurements at a frequency of 9.272 GHz were performed on a powder

sample of BPCB at room temperature and on a single crystal sample along the c-axis from

20 to 300 K. The room temperature results were completely consistent with the

previously reported data, i.e. g(powder) = 2.13 [20] and g(c-axis) = 2.148. At all

temperatures, the EPR signal consisted of a single broad line approximately 500 G wide.

Figure 4.3 is a sample derivative trace, d//dH of the EPR signal intensity at 75 K. By

plotting the area under the EPR intensity curve I(H) as a function of temperature, we

obtain the graph shown in Fig. 4.4. This graph closely resembles the susceptibility curve

of BPCB. The Lande g factor measured along the c-axis decreases monotonically from

2.148 to 2.141 from 300 K to 20 K as shown in Fig. 4.5. This magnitude of change in the

Lande g factor will not adversely affect the quality of the magnetization fits, which

assumed g to be the temperature independent value of 2.148 along the c-axis.



4.2 Low Field Susceptibility Measurements

Although the crystal structure of BPCB resembles a ladder, other possible

exchange pathways can produce similar results from macroscopic measurements [71,72].

Initially, an additional magnetic exchange model, i.e. alternating chain, was considered

during the analysis of the magnetization data. Figure 4.1 shows the two primary exchange

pathways considered, i.e. an alternating chain with exchange constants J\ and J2, and a

ladder with exchange constants JL and J\. The Hamiltonians for N spins that interact with

















20

15

10

5

*-' 0
C6


-5

-10

-15

-20

2000


2500 3000 3500 4000
Field (G)


Figure 4.3: The first derivative of the EPR signal intensity, dI/dH,
single crystal (m = 18.6 mg) at a frequency of 9.272 MHz and 75 K.


30
CO

S25
0)

n 20

0
uJ

15

CL
cQ.
E 10
<
"o
0)

r.-5
0)
0)
CE


0 50 100


150
T (K)


4500




vs. field for a BPCB


200 250


Figure 4.4: The integrated EPR signal intensity vs. temperature for a BPCB single crystal
(m = 18.6 mg) at a frequency of 9.272 MHz.


I I I I I '

0 H IIH c-axis

0
0
0
0
o
0
0
0
000000

0
.O
0 0
0
00



I 0; 0* o ~ 0 100o












2.150


2.148


2.146


2.144


2.142 -


2.140 1 I I 1 I
0 50 100 150
T(K)


200 250 300


Figure 4.5: The Lande g factor along the c-axis of a single crystal sample of BPCB
(m =18.6 mg), determined by the EPR line center frequency at 9.272 MHz, vs.
temperature. The room temperature value ofg agrees with the value reported earlier [20]
(g = 2.148.). The temperature dependence of the Lande g factor is most likely due to the
thermal contraction of the lattice.


either a ladder or alternating chain exchange can be written as


N12 N-2 N
0,dr = JL j 2,-, *s,2 + J1 g, -S,+2 + gBL-fH
1=1 i=1 1=1


N12 NI2-1 N
* .n = JI g2,-, .2, +J2 2-S2'g21,+ +gu S, .Hi ,
I=] i=1 i=1


(4.1)


(4.2)


respectively.










Low field (H < 5 T) magnetic measurements were performed using a Quantum

Design SQUID Magnetometer. The low field, 0.1 T, magnetic susceptibility, X, of a BPCB

powder sample (m = 166.7 mg) is shown as a function of temperature in Fig. 4.6. The

general shape of the curve is typical of low dimensional magnetic systems, and more

specifically, it possesses a rounded peak at approximately 8 K and an exponential

dependence below the peak temperature. No evidence of long range order was observed

at the minimum temperature of 2 K. A temperature independent diamagnetic contribution

of Xdiam = -2.84 x 10-4 emu/mol was subtracted from the data in Fig. 4.6. The

diamagnetic contribution is the sum of the core diamagnetism, estimated from Pascal's

constants to be -2.64 x 104 emu/mol, and the background contribution of the sample

holder. For all of the susceptibility data in this chapter, a diamagnetic contribution has

been subtracted from the data and although no Curie impurity term was subtracted, in

some cases a S = 1/2 Curie contribution was included in the fit. This Curie contribution is

typically -2 % of the total number of S = 1/2 spins. In Fig. 4.6, the susceptibility data

have been fit (solid line) using a high temperature expansion by Weihong et al. [11] based

on the ladder Hamiltonian, Eq. 4.1, providing the exchange constants of J = 13.1 + 0.2 K

and JH = 4.1 + 0.3 K. The first 14 terms of the expansion (up to fourth order in J/T)

were used for the fitting procedure. These same data were also fit (Fig. 4.7) using the

method of Chiara et al. [7], which assumes the alternating chain Hamiltonian, Eq. 4.2,

providing the exchange constants of J1 = 13.74 + 0.03 K and J2 = 5.31 + 0.04 K. The

fitting method of Chiara et al. [7] includes the data below the peak in the susceptibility and

consequently is more accurate than the high temperature series expansion method of










Weihong et al. [11]. However, although there are differences in the values of the two sets

of exchange parameters, both cases provide physically plausible results. Therefore, using

only the low field X(T) data, we are unable to distinguish between the ladder and

alternating chain model. Similar results are obtained for BPCB single crystal samples.

The magnetic susceptibility vs. temperature for BPCB single crystal (m = 46.9 mg) is

shown in Fig. 4.8. The sample was zero field cooled to 2 K and then measured in a field

of 0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam =

-3.16 x 10-4 emu/mol has been subtracted. Incidentally, the diamagnetic contribution for

the single crystal samples is larger because more diamagnetic support material was used to

ensure proper crystal alignment during the measurement. The solid line represents the

best fit using a high temperature series expansion by Weihong et al. [11] with the

parameters J1 = 12.9 + 0.3 K and J11 = 3.8 0.3 K. Figure 4.9 shows this same data fit

using the method ofChiara et al. [7] with the results J, = 13.66 + 0.14 K and J2 = 5.57 +

0.12 K. Analogous to the ladder and alternating chain fits of the powder susceptibility

data, both fitting methods generate plausible results. In addition, although the exchange

constants from the single crystal and powder samples do not quite agree within

uncertainty, the fitting results appear to be self consistent for both methods. The fitting

results for powder and single crystal samples along all three axes have been summarized in

Tables 4.1 and 4.2.

The choice of 0.1 T as the applied field in the susceptibility measurements was not

arbitrary. Figure 4.10 shows the molar magnetic susceptibility for BPCB single crystal

with H 11 a-axis and applied fields of 1, 2, 3, 4, and 5 T. At high temperatures (T > A/kB),




















25 1 1 1 1 1 1 1 1-
0 O Experimental Data
S -- Ladder Fit:
20
SJQ j =13.1 + 0.2 K
0 J1= 4.1+0.3K

E
-15 -8 ^sImpurity Conc. =1.2 0.2 %-
-0
-0
o 10 :0
0
-0
-0
50
o BPCB Powder
H H=0.1 T
0 1 1 I
0 20 40 60 80 100
T(K)

Figure 4.6: The molar magnetic susceptibility vs. temperature for BPCB powder
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of
0.1 T. A small constant diamagnetic contribution of Xdiam = -2.84 x 10-4 emu/mol has
been subtracted. The solid line represents the best fit using a high temperature series
expansion by Weihong et aL. [11] with parameters J = 13.1 + 0.2 K and JI = 4.1 + 0.3 K.





















25 1 -1-1-1-1
0 Experimental Data
Alternating Chain Fit:
20 J, = 13.74 0.03 K
I J2 = 5.31 0.04 K
E 15 Impurity Conc. = 0.9 0.1 %

E




5 BPCB Powder
H0 =0.1 T
0 I- --------------i -~
0 20 40 60 80 100
T(K)

Figure 4.7: The molar magnetic susceptibility vs. temperature for BPCB powder
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of
0.1 T. A small constant diamagnetic contribution of Xdiam = -2.84 x 10-4 emu/mol has
been subtracted. The solid line represents the best fit using the method of Chiara et al. [7]
with parameters J, = 13.74 0.03 K and J2 = 5.31 0.04 K.





















2 5 1 I-I I 1 I 11 1 1
0 Experimental Data
-- Ladder Fit using:
20 = J =12.9+0.3K
',J= 3.8+ 0.3K






5 BPCB Single Crystal

~- H=0.1 T||Ia-axis I
0 15I Impuit Ioc 3.I .
E




?20 40 60 80 10010
5
BPCB Single Crystal
H=0.1 TiI1 a-axis

20 40 60 80 100
T(K)

Figure 4.8: The molar magnetic susceptibility vs. temperature for a BPCB single crystal
(m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of
0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam =
-3.16 x 10-4 emu/mol has been subtracted. The solid line represents the best fit using a
high temperature series expansion by Weihong et al. [11] with parameters J =
12.9 +0.3 K and J, = 3.8 + 0.3 K.




















25 1 1 1 i- |i'-
0 Experimental Data
S--- Alternating Chain Fit:
20 % J,=13.660.14K
557= 5.57 0.12 K
15 Impurity Conc. = 3.3 0.4 %
E
(D
10



BPCB Single Crystal
H =0.1 T jj a-axis
0 1 1 1 1 I
0 20 40 60 80 100
T(K)

Figure 4.9: The molar magnetic susceptibility vs. temperature for a BPCB single crystal
(m 46.9 mg). The sample was zero field cooled to 2 K and then measured in a field of
0.1 T. A small constant diamagnetic contribution of Xdiam = -3.16 x 10-4 emu/mol has
been subtracted. The solid line represents the best fit using the method of Chiara et al. [7]
with parameters J, = 13.66 + 0.14 Kand ,/2 5.57 + 0.12 K.




















25 .. I '


20

0
E 15

E ____--1 Tesla
10 o 2 Tesla
SaA 3 Tesla
5 v 4 Tesla
o 5 Tesla
H a-axis
0 i I I I a
0 5 10 15 20
T(K)

Figure 4.10: The molar magnetic susceptibility vs. temperature for a BPCB single crystal
(m = 14.2 mg). The sample was zero field cooled to 2 K and then measured in the fields
of 1, 2, 3, 4 and 5 T. A small constant diamagnetic contribution of-4.06 x 104 emu/mol
has been subtracted. The data collapse onto a single curve at high temperatures
(T> A/kB) indicating the approximately constant susceptibility at high temperature.










Table 4.1: The alternating chain parameters, J, and J2, determined from fitting the
susceptibility vs. temperature data using the method of Chiara et al. [7].

mass (mg) J, (K) J2 (K) Impurity Conc. (%)
powder 166.7 13.74 + 0.03 5.31 + 0.04 0.9 + 0.1
a-axis 46.9 13.66+0.14 5.57+0.12 3.3+0.4
b-axis 13.6 12.76 0.10 5.18 + 0.10 4.8 + 0.5
c-axis 24.4 13.65+0.10 6.05+ 0.10 3.8+0.4


Table 4.2: The ladder parameters, J1 and J determined from fitting the susceptibility vs.
temperature data using the high temperature expansion by Weihong et al. [11].

mass (mg) Jj_ (K) J11 (K) Impurity Conc. (%)
powder 166.7 13.1 + 0.2 4.1 + 0.3 1.2 0.2
a-axis 46.9 12.9 0.3 3.8 + 0.3 3.3 0.4
b-axis 13.6 13.4 0.3 3.7 + 0.2 7.5 1.0
c-axis 24.4 13.3+0.4 3.8+0.5 3.0 1.0


all of the susceptibility data collapse onto a single curve demonstrating the approximately

constant susceptibility. However, below the peak, the susceptibility curves begin to

deviate. In addition, the peak temperature decreases with increasing field. At fields above

the gap, A/gJIB 6.8 T, the peak in the susceptibility curve should disappear entirely.

Although, a larger applied field would increase the signal to noise ratio of our

measurements, we would measure the field and temperature dependence of the sample

simultaneously, thus complicating our analysis.

The inverse susceptibility as a function of temperature for BPCB powder

(m= 166.7 mg) is shown in Fig. 4.11, and similarly, the inverse susceptibility vs.

temperature for a BPCB single crystal with H 11 a-axis (m = 46.9 mg) is shown in









Fig. 4.12. At temperatures above the spin gap, T >> A 8 K, the inverse molar

susceptibility should be linear with temperature. The slope of this line can be determined

by inverting the S = 1/2 Curie law,



I =(T +e) 4kB (4.3)
x NAg2J '2



where NA is Avagadro's number. The value of 0 is somewhat more difficult to calculate.

Johnston et al. [41] have written a high temperature series expansion, by inverting a

susceptibility expansion from Weihong et al. [11], for the inverse susceptibility in terms of

J1 and J11 containing 42 non-zero terms. The first four terms of that series can be written

as



1 B 1+(2J,, +J)2+(2J2 +j 2)- +(2J3+J13 +'"., (4.4)
Z NAg2PB2 2 2 3


1
where x = -. By comparing Eqs. 4.3 and 4.4, we can write the Curie temperature, 0,
2T

to second order in T as



Er(2J,+J) (2J,'+2 J ) (2Ji 3 + 3) (4.5)
e= 4+... (4.5)
4 8T 24T2





















0.8 1 1 1 1 1 1 1 1 1|I
o Experimental Data
0.7 Linear Fit (100 K to 300 K)
0 slope = 2.313 0.002 (mol/K emu)
0. E )=4.9 0.3 K

E 0.5

o 0.4 0
0
E0.3 0

T 0.2
-0.1 : BPCB Powder
0.1 =0.1T
0.0 1= 1
0 50 100 150 200 250 300
T(K)

Figure 4.11: The inverse molar magnetic susceptibility vs. temperature for BPCB powder
(m = 166.7 mg). The sample was zero field cooled to 2 K and then measured in a field of
0.1 T. A small constant diamagnetic contribution of Xdia.m = -2.84 x 10-4 emu/mol has
been subtracted. The solid line is a linear fit over the temperature range from 100 K to
300 K giving a slope of 2.313 + 0.002 mol/(emu K) and 0 = 4.9 + 0.3 K.




















0.8 1
o Experimental Data
0.7 Linear Fit (100 Kto 300 K) -
slope = 2.50 0.01 (mol/emu K)
0.6 E=5.81.2K
a
E 0.5
a)
Vo 0.4
0
E 0.3 -
-0.2
0 FBPCB Single Crystal
01 H = 0.1 T 11 a-axis -
0.0 1 1 1 1 1
0 50 100 150 200 250 300
T(K)

Figure 4.12: The inverse molar magnetic susceptibility vs. temperature for a BPCB single
crystal (m = 46.9 mg). The sample was zero field cooled to 2 K and then measured in a
field of 0.1 T parallel to the a-axis. A small constant diamagnetic contribution of Xdiam =
-3.16 x 10-4 emu/mol has been subtracted. The solid line is a linear fit over the
temperature range from 100 K to 300 K with a slope of 2.50 + 0.01 mol/(emu K) and 0 =
5.8 +1.2 K.










After plugging in nominal values for our exchange constants and comparing the magnitude

of each term, it is clear that we only need to consider the first term as long as our linear fit

begins above approximately 100 K. The solid lines in Figs. 4.11 and 4.12 represent linear

fits over the temperature region from 100 K to 300 K. The slope and Curie constant for

these two samples, as well as BPCB single crystal specimens along the b and c-axis, are

listed in Table 4.3 and compared to the theoretical result from Eq. 4.5. The exchange

constants, J and J, used in Eq. 4.5 have been taken from Table 4.2.



Table 4.3: The slope of the inverse molar susceptibility, l/X, vs. temperature and the
Curie temperature, E, determined from a linear fit to the data from 100 K to 300 K. The
theoretical slope, 4kB/g2 B2NA (Eq. 4.3) and Curie constant, (J + 2J,,)/4 (Eq. 4.5), are
included for comparison. The parameters used in the calculation of the Curie constant
were taken from Table 4.2.

slope 4kBg2,UB2NA 9 (K) (J,1 + 2JI1)/4 (K)
(mol/K emu) (mol/K emu)
powder 2.312 + 0.002 2.35 + 0.02 4.9 0.3 5.3 0.3
a-axis 2.50 0.01 2.51 0.02 5.8 + 1.2 5.1 0.4
b-axis 2.24 + 0.02 2.23 0.02 3.9 + 1.1 5.2 0.3
c-axis 2.29 + 0.02 2.31 0.02 4.6 + 1.6 5.2 + 0.6


The molar magnetic susceptibility multiplied by temperature vs. temperature for

BPCB powder (m = 166.7 mg) is shown in Fig. 4.13. The solid line is the theoretical high

temperature Curie value for XT of 0.425 (emu K)/mol. At temperatures above

approximately 100 K, XT approaches a horizontal line with the Curie value, indicating

paramagnetic behavior. Below 100 K, the value of XT decreases when lowering the











temperature, indicating antiferromagnetic behavior. If we had not subtracted the

appropriate diamagnetic contribution, XT at high temperatures would have a non-zero

slope.


0.5


0.4

0
E 0.3

:3
E
S0.2

0.I-1
0.1


0 50 100 150 200 250 300
T (K)

Figure 4.13: The molar magnetic susceptibility multiplied by temperature vs. temperature
for BPCB powder (m = 166.7 mg). The sample was zero field cooled to 2 K and then
measured in a field of 0.1 T. A small constant diamagnetic contribution of
Xdiam = -2.84 x 10-4 emu/mol has been subtracted. The solid line is the theoretical Curie
value of 0.425 (emu K)/mol.




4.3 Low Field Magnetization Measurements

The magnetization measurements were performed with a commercial SQUID

magnetometer, which can apply a maximum field of 5.0 T. This limitation is particularly

unfortunate in the case of BPCB, since the spin gap, expected from the susceptibility

measurements, is approximately 7 T. The spin gap is calculated to first order as










A/kB = JL-J11 (4.6)



for the ladder exchange or



A/kB = Ji J2 (4.7)



when considering an alternating chain model [41]. For each measurement listed, the

samples were zero field cooled from 300 K. The overall form of the low field

magnetization measurements can be understood by examining the behavior of two

Heisenberg S = 1/2 spins with a single exchange constant J. The molar magnetization of

such a system can be calculated using Eq. 3.7. At low temperatures (T << A/kB), the

magnetization will remain zero until the gap field is reached (H = A/g/iB) and then

afterwards have a positive first derivative. At high temperatures (T > H and T >> J),

Eq. 3.7 becomes approximately linear with applied field. Between these two temperature

extremes, the magnetization will have a small positive first derivative (compared to the

paramagnetic result of dM(H)/dH < NAg2JUB2/4kBT) and a positive second derivative. The

molar magnetization vs. field for BPCB powder (m = 166.7 mg) at a temperature of 2 K is

shown in Fig. 4.14. The solid line is a fit using Eq. 3.7 with an exchange constant ofJ =

12.6 + 0.1 K. The general shape of the curve matches the data commendably considering

the simplicity of the model. This agreement is an indication that, regardless of which

magnetic model is correct, BPCB exists in the strongly coupled limit, i.e. JJ/J\ >> or

JI/J2 >>l.










To facilitate fitting the magnetization data more accurately, we used the exact

diagonalization technique discussed in Chapter 3. For all of the fits that are discussed,

unless otherwise noted, the calculations used 12 spins arranged in a ring. The molar

magnetization as a function of field for a BPCB single crystal (m = 166.7 mg) at the

temperatures of 2, 5, and 8 K is shown in Fig 4.15. The solid line at 2 K represents the

best fit using the 12 spin exact diagonalization procedure and an alternating chain

Hamiltonian, Eq. 4.2. The magnetization curves at 5 K and 8 K were produced using the

same best fit exchange parameters derived from the 2 K data, J1 = 13.20 + 0.05 K and J2 =

5.20 + 0.05 K. The experimental curve at 2 K is reproduced extremely well by this fitting

technique. However, at higher temperatures, using the same exchange constants, the

theoretical and experimental curves begin to deviate. The same fitting procedure can be

applied to the data using the ladder Hamiltonian, Eq. 4.1. Figure 4.16 shows the same

data fit using the exact diagonalization with a ladder Hamiltonian. The ladder best fit

exchange parameters are JL = 12.75 + 0.05 K and J = 3.80 0.05 K. Contrary to the

case for the alternating chain Hamiltonian, the higher temperature experimental and

theoretical magnetization curves agree using the same exchange constants at higher

temperatures. This agreement suggests that the data may be more accurately modeled

using the ladder Hamiltonian.

There are two reasons why the error in fitting the low field magnetization

measurements is relatively large compared to the error in fitting the susceptibility

measurements. First, a small discrepancy in the mass or temperature measurement will





















400 1 1i i i 1i
BPCB Powder
T=2K K
300
30 0 Experimental Points
- Dimer Fit
( J=12.6+0.1K
S200
E


100



0
0 1 2 3 4 5
H(T)

Figure 4.14: The molar magnetization vs. field for BPCB powder (m = 166.7 mg) at a
temperature of 2 K. The solid line represents the best fit to Eq. 3.7, the molar
magnetization for pairs of S = 1/2 Heisenberg spins with a single exchange constant of J
12.6 + 0.1 K.
















1000 --. ,
0 T=2K
A T=5K
800 V T=8K
--- Exact Diagonalization ,
E 0 J= 13.20 K
"-600 -ir .9600 J2= 5.20 K ,

E
01 400


200

SH II a-axis "
0
0 1 2 3 4 5
H(T)

Figure 4.15: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg)
with H I a-axis at the temperatures of 2, 5, and 8 K. The solid lines are produced using
the best fit parameters, J, = 13.20 0.05 K and J2 = 5.20 + 0.05 K, determined from the
2 K data using the 12 spin exact diagonalization procedure and an alternating chain
Hamiltonian. At higher temperatures, using the same exchange constants, the theoretical
and experimental curves begin to deviate.

















1000 i 1 1 1 1--- x
0 T=2K
A T=5K
800 V T=8K
-" Exact Diagonalization
0
E 00 J= 12.75 K
(.9 Jll= 3.80 K

E
S 400


200

LeIH a-axis -
01
0 1 2 3 4 5
H(T)

Figure 4.16: The molar magnetization vs. field for a BPCB single crystal (m = 166.7 mg)
with H II a-axis at the temperatures of 2, 5, and 8 K. The solid lines represent the best fit
to the 2 K data, using the 12 spin exact diagonalization procedure and a ladder
Hamiltonian. The ladder best fit exchange parameters are J, = 12.75 + 0.05 K and J: =
3.80 + 0.05 K. The higher temperature experimental and theoretical magnetization curves
agree using the same exchange constants.










result in a large difference in the best fit exchange constants. Experimentally, it is easier to

hold the magnetic field constant than the temperature. Second, because we did not reach

the saturation magnetization, or even the critical field Hci, absolute calibration of the mass

or the critical fields, Hci and Hc2, is not possible. Determination of the exchange

constants from the low field magnetization data relies on the absolute magnetization

values. On the other hand, the susceptibility data contains a maximum with a unique

temperature dependence that is sensitive to the values of the exchange constants. Figure

4.17 shows the 2 K magnetization data from the previous two figures. The solid and

dotted lines represent the exact diagonalization fits extended to 20 T using the alternating

chain and ladder Hamiltonians, respectively. At a temperature of 2 K, it should be

possible to distinguish between the two models by continuing the magnetization

measurements to high field. By lowering the temperature to 1 K, this difference will

become more pronounced (see Fig. 4.18). Figure 4.19 shows the magnetization data for

Cu(Hp)Cl at 0.42 K [14]. The solid line represents the best fit using the exact

diagonalization procedure and an alternating chain Hamiltonian with exchange constants

J1 = 13.20 0.05 K and J2 = 2.3 0.05 K. The first derivative of the data and theoretical

prediction are provided in the inset. The asymmetry of the curve is obvious from Fig. 4.19

and is a result of the asymmetry in the magnetic exchange (see Fig. 4.19), i.e. J,11 J2.

These results suggest that Cu(Hp)Cl, which has been considered a two-leg ladder material

[14], is better described by an alternating chain model.



















6000 . I i . 1 i . .
0 Experimental Data
-- Exact Diagonalization using /
5000 Alt. Chain Hamiltonian /
J1= 13.20K //
75 4000 J2 = 520 K
E -- Exact Diagonalization using //
6 Ladder Hamiltonian //
J 3000 J=12.75 K
E J = 3.80 K /
Q^/
2000

1000 H II a-axis
rT=2K
0 I . . I . . I -, ,
0 5 10 15 20
Field (T)

Figure 4.17: The molar magnetization vs. field for a BPCB single crystal vs. field from 0
to 5 T. The solid and dotted lines represent the exact diagonalization fits extended to 20
T using the alternating chain and ladder Hamiltonians, respectively. At a temperature of
2 K, it should be possible to distinguish between the two models by continuing the
magnetization measurements to high field.
























* Exac
Alt. C
J = 1
J2= 5
Exac
Ladd
J= 1
JII=


1000 [-


t Diagonalization using /
hain Hamiltonian /
13.20 K /
5.20 K /
t Diagonalization using /
er Hamiltonian //
12.75 K //
3.80 K




H II a-axis -
T=1 K


5 10 15
Field (T)


Figure 4.18: The solid and dotted lines represent the exact diagonalization fits from the
previous graph (Fig. 4.17) calculated at a temperature of 1 K. At this temperature, the
difference between the curves becomes more pronounced. At a field of approximately
10.6 T, there appears to be an inflection in the predicted magnetization using the ladder
Hamiltonian.


6000


5000


5 4000
E
0
, 3000
E

2000



















1.0 %- U nP_,J,'L
T=0.42 K
So Data
0.8 Alt. Chain Model 0.25 .............

JI = 12.85 K 020 -
S0.6 = 4.35K K f
1 0,15

0.4 0.10 o

00. .05
0.2
0.00
0 5 10 15 20
0.0 -H (T)

0 5 10 15 20 25
H(T)

Figure 4.19: The data from Fig. 7 in Reference 14 have been digitized. After
interpolating to 200 equally spaced points, the 1st derivative was taken using 13 point
smoothing (open circles in the inset). The theoretical curve was created with the exact
diagonalization procedure using 12 spins arranged in a ring and an alternating chain
Hamiltonian (solid lines). The exchange constants of J, = 12.85 K and J2 = 4.35 K were
determined by varying the exchange constants to minimize the square of the distance
between the theoretical and experimental curves. The theoretical curve consisted of 3000
points. A 250 point adjacent averaging procedure was applied to the first derivative of the
theoretical curve (inset). A value of 2 was assumed for the Lande g factor.










4.4 High Field Magnetization Measurements

The high-field, H < 30 T, magnetization, M, of a BPCB powder sample (m =

208.2 mg) normalized to its saturation value, Ms, is shown as a function of field and

temperature in Fig. 4.20. Since the saturation magnetization was reached on our studies,

we were able to measure and subtract a small, temperature-independent contribution

(Xdiam a -2.84 x 10-4 emu/mol), which is the same value obtained in the low field work

(Section 4.2), by performing a linear fit to the data above 20 T. The data were acquired

while ramping the field in both directions, and no hysteresis was observed. Although

approximately 3000 points were acquired at each temperature, the data traces are limited

to 150 points for clarity. The lines are fits using the 12 spin exact diagonalization and an

alternating chain Hamiltonian, Eq. 4.2. The best fit exchange constants, which are listed in

Table 4.4, have a systematic temperature dependence with J\ increasing and J2 decreasing

with increasing temperature. The theoretical curves adequately reproduce the

magnetization data at the two highest temperatures of 3.31 K and 4.47 K. However, at

the temperature of 1.75 K, the exact diagonalization curve deviates significantly from the

data at Hci =6.6 T and HC2 =14.6 T. Data were also taken at 0.7 K; however, the exact

diagonalization technique fails to produce a reasonable curve for the reasons discussed in

Chapter 3, and consequently, that theoretical curve is not shown. It should be noted that

the exchange constants, J\ 13 K and J2 7.0 K, do not match the exchange constants

obtained from the susceptibility data, .1\ 13.7 K and J2 5.5 K. Similar results are

obtained for magnetization measurements of single crystal samples. The high field, H <

30 T, magnetization of a single crystal sample (m = 18.9 mg) with H 11 a-axis is shown in




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