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I

This is to certify that the
dissertation entitled
INVESTIGATION OF ELECTRON SPIN RESONANCE
IN MULTILAYERED CUMN/CU SPIN GLASSES
ABOVE THE FREEZING TEMPERATURE

presented by

Diandra Laia Leslie-Pelecky

has been accepted towards fulfillment
of the requirements for

Ph.D. degreein PhYSiCS

 

    

' Major professor

Date 09111/91

MSU is an Affirmative Action/Equal Opportunity Institution 0-12771

 

 

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MSU Is An Affirmative ActiorVEqual Opportunity Institution

ammo-91

[\WSTIGAHC
CUE/CC S

 

INVESTIGATION OF ELECTRON SPIN RESONANCE IN MULTILAYERED
CUMN/CU SPIN GLASSES ABOVE THE FREEZING TEMPERATURE

By

Diandra Lain Loslie-Pelecky

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment for the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1991

 

[\WES'UGAF

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ABSTRACT

INVESTIGATION OF ELECTRON SPIN RESONANCE IN MULTILAYERED
CUMN/CU SPIN GLASSES ABOVE THE FREEZING TEMPERATURE

By

Diandra Laia Leslie-Pelecky

Interpretation of the Electron Spin Resonance linewidth, AH and resonance peak
position, Ho, above the freezing temperature, T,, is difficult due to simultaneous physical
processes and even a question as to whether the linewidth is a probe of the long-range
’ordered’ spin glass phase, or a measure of short range correlations. We have measured
AH(T) and Ham in multilayered Cu,,,,Mn, spin glasses with x = 7% and 11%, and spin
glass layer thicknesses, W”, from 1 nm to 1000 nm at 9 GHz. Finite size and
dimensionality effects on T, have previously been observed in these multilayers.

AH(T) is shown to have the same general form for all layer thicknesses, with
samples having W30 5 3 run more accurately described by a corresponding form
appmpriate for T, = 0. Systematic variations in all parameters are seen as a function
of layer thickness. The separation of effects due to the depression of T, and effects on
the linewidth itself is necessary in order to correctly understand the physical origin of the
trends.

The observed behavior of AHCI') is explained in terms of a correlated cluster
Picture. In the correlated cluster model, the existence of spin correlations at

tCmperatures significantly above T, leads to a distribution of cluster sizes and relaxation

 

 

tees. with the 1
Sisters become

along wit.“ a D3

.-

resonance line.

gL255e5 above T,

behavior of the if
phase. in uhich

Efiffleratures be

“Mature of t,

times, with the linewidth dominated by configurations with faster relaxation times. As
clusters become increasingly correlated, the internal fields generated by the clusters,
along with a DM anisotropy field, inhomogeneously broadens the exchange narrowed
resonance line. This interpretation suggests that the observation of resonance in spin
glasses above T, is a characteristic of an intermediate phase between the spin glass and
paramagnet phases, and not a measurement of the spin glass phase per se. The effect
of decreasing layer thickness on the formation and growth of clusters, and the resulting
behavior of the linewidth is qualitatively discussed. The possibility of a ’Griffiths-like’
phase, in which regions of the sample are condensed into the spin glass phase at
temperatures between the bulk freezing temperature and the measured freezing

temperature of the sample, is considered.

C

St 1}
“I‘(: 1
R'ic’.’

(3f I}

"Someday, after we have mastered the winds and the
waves, the tides and gravity, we will harness for God the
energies of love and then, for the second time in the history
of the world, man will have discovered fire. "

-Teilhard de Chardin-

This thesis is dedicated to my family:
my mother and father,
my brothers Theodore and Steven,
and most of all,
my husband Richard.

iv

IWOUId like
he injtal idea for
the fine points of t

members of my cc

"y?“é

Me ofideas. qua
rte adsice. inspira‘.
interact with these
I owe a dc};
Hudson, Stephanie
for doing their job
Life as a gl
encouragement I n

Eidaate students I

most. who was a

I
I
I
Iwould alstj
i9 '

MLSSlstance in II

International for SI I

a! i
love
Ofmvh
. U ”
~.L,' 5b
ships

 

ACKNOWLEDGEMENTS

I would like to thank Dr. Jerry Cowen for financial support and for suggesting
the initial idea for this series of experiments, and Dr Jack Bass who tirelessly pursued
the fine points of characterizing multilayered spin glasses. I would like to thank the
members of my committee, especially Dr. Phillip Duxbury who has been an unending
source of ideas, questions, advice and knowledge, and Dr. Norman Birge, who has given
me advice, inspiration, and his time. I am privileged to have had the opportunity to
interact with these people.

I owe a debt of gratitude to Peggy Andrews, Tom Palazzolo, Jim Muns, Tom
Hudson, Stephanie Holland, Jean Strahan, Barry Tigner, Bob Raines and Diane Clark
for doing their jobs so well.

Life as a graduate student was made much easier by the moral support and
encouragement I received from Mark Holtz, Rich Kennan, Viv Shull, and the cast of
graduate students I have had the opportunity to work with. Special thanks goes to Reza
Loloee, who was an enormous help both as CMP Specialist and as friend.

I would also like to acknowledge Jim Anderson of Research Specialties, Inc, for
his assistance in the renovation and upgrading of the ESR system, and Rockwell
International for support through a Rockwell Fellowship.

Finally, I could not have completed this task without the unending patience and
love of my husband Richard, who made me understand the importance of the small
things.

LIST OF TABL
LIST OF FIGL'.

CHARTER ON]
INTROE
THE EX
THEORI

57
{T7
gHXI—VHJQ

_J._J

TABLE OF CONTENTS

LIST OF TABLES ...................................... x
LIST OF FIGURES ..................................... xii
CHAPTER ONE: INTRODUCTION ........................... 1
INTRODUCTION: WHAT IS A SPIN GLASS? ............... 1

THE EXPERIMENTAL SIGNATURES OF SPIN GLASSES ........ 2
THEORETICAL PROPERTIES .......................... 7
Frustration and Disorder .......................... 9

Metallic Spin Glasses ............................ 10

The RKKY Interaction ........................... ll

Anisotropy .................................. 14

SUMMARY ...................................... 16
CHAPTER TWO: INTRODUCTION TO ESR ..................... 17
INTRODUCTION .................................. 17
BASIC THEORY OF ESR ............................. 17
Classical Equations of Motion ....................... l8

Bloch Equations ............................... 22
EXPERIMENTAL DETECTION OF THE RESONANCE ......... 24
SUMMARY ...................................... 26
CHAPTER THREE: SPIN GLASS THEORIES .................... 27
INTRODUCTION .................................. 27
PHASE TRANSITIONS AND FINITE SIZE EFFECTS ........... 28
Critical Slowing Down ........................... 30

Finite Size Effects on Critical Exponents ................ 33

MEAN FIELD THEORY .............................. 35

The Edwards-Anderson Model ...................... 35

The Sherrington-Kirkpatrick Solution .................. 37

Replica Symmetry Breaking ........................ 39
Ultrametricity ................................. 41

AT and GT Lines .............................. 43

Dynamics Within the Mean Field Model ................ 45

THE DROPLET MODEL ............................. 47

The Behavior of Thin Films Within the Droplet Model ....... 50

The Effect of Magnetic Fields ...................... 52

vi

THE GRll: El
SI'MMAR‘.

CHAPTER FOL'R.
SPIN GLA‘
INTRODU
MULTILA i

Finch

Free

Ottl
Fini'
Elie I

MI‘LTlLA‘

TWO-DIM}

THE C [R ii

EH3

CHINE}; FIVE;
ISTRODL'
Gen!

ESR IN 5p
ESP

ESIi

Rel;

Res

The

Lin

 

Dir
Mc

0t

SUMMARY ...................................... 56
CHAPTER FOUR: REVIEW OF PREVIOUS STUDIES OF MULTILAYERED

SPIN GLASSES ................................... 58

INTRODUCTION .................................. 58

MULTILAYERED METALLIC SPIN GLASSES ............... 59

Finite Size and Dimensionality Effects on T, .............. 60

Frequency Dependence of the Freezing Temperature ......... 62

Other Measurements on Multilayered Spin Glasses .......... 64

Finite Size Effects in Metallic Films ................... 66

Effect of Magnetic Field on T, in Metallic Multilayers ........ 66

MULTILAYERED SEMICONDUCTING SPIN GLASSES ......... 68

TWO-DIMENSIONAL SPIN GLASSES ..................... 69
THE CURRENT PICTURE OF DIMENSIONALITY AND FINITE SIZE

EFFECTS IN MULTILAYERED SPIN GLASSES .......... 69

CHAPTER FIVE: ESR IN SPIN GLASSES ...................... 71

INTRODUCTION .................................. 71

General Behavior .............................. 71

ESR IN SPIN GLASSES: THEORY ...................... 72

ESR of Localized Moments in Metallic Hosts ............. 73

ESR Equations for Transition Ion-Metal Systems ........... 73

Relaxation Rates ............................... 75

Residual Width and Thermal Broadening ................ 80

The Effect of Magnetic Ordering on Linewidth ............ 82

Lineshape, Correlation Functions and the Relaxation Rate ..... 85

Dimensionality Effects on ESR Linewidth and Lineshape ...... 91

EXPERIMENTAL RESULTS ........................... 93

Metallic Spin Glasses ............................ 93

CuMn ................................. 93

AgMn ................................. 95

Other Spin Glasses ............................ 100

Eu,Sr,,,S .............................. 100

Concentrated Metallic Spin Glasses .............. 101

GdAl ................................ 102

Misc. Spin Glasses ........................ 102

Ferromagnets, Antiferromagnets and Low Dimensionality ..... 103

ESR in l and 2 Dimensional Materials ................ 106

THEORIES OF ESR IN SPIN GLASSES ................... 108

Mean Field ESR Theory ......................... 109

The Correlated Cluster Model ...................... 112

Comparison of the Models ........................ 115

SUMMARY ..................................... 116

vii

MOTIVATIO.‘
THES If

CHAPTER SIX: EXI
INTRODL'CTI
Sample
SAMPLE FAE
ngm
Spuuefi
CHARACTER
XeRay
SherOst
Resistis
SQUID
ELECTRON S
ESR.S£

Skin 1).
BaCkgr.
Measur
SUMMARY ,

CHAPTER SEVEN;
ANALYSIS c
UNE\VIDTH

Filling

Depart

C0ncei

LAYER THK
QUaiit;

Remix

EITCCIS

DCPCni

Depart

IIS of
G -
MPH?
Zero 1

Field 2

MOTIVATION FOR THE EXPERIMENT DESCRIBED IN THIS

THESIS ................................... 1 17
CHAPTER SIX: EXPERIMENTAL METHODS .................. 121
INTRODUCTION ................................. 121
Sample Naming Convention ....................... 121
SAMPLE FABRICATION ............................ 122
Target Production ............................. 122
Sputtering Apparatus ........................... 124
CHARACTERIZATION ............................. 126
X-Ray Studies ............................... 128
Microscopy ................................. 130
Resistivity .................................. 131
SQUID Measurements .......................... 132
ELECTRON SPIN RESONANCE (ESR) TECHNIQUES ......... 132
ESR Sample Preparation ......................... 135

Skin Depth Effect ............................. 135
Background Signal ............................. 139
Measurement Protocol .......................... 140
SUMMARY ..................................... 143
CHAPTER SEVEN: DATA AND ANALYSIS ................... 144
SUMMARY OF SAMPLES ANALYZED .................. 144
ANALYSIS OF THE FREEZING TEMPERATURE ............ 149
LINEWIDTH ANALYSIS ............................ 155
Fitting Protocol .............................. 159
Departure from the Power Law Behavior Near T, .......... 15 9
Concentration Dependence ........................ 160
LAYER THICKNESS DEPENDENCE .................... 168
Qualitative Notes on the Raw Data ................... 191
Reproducibility ............................... 191
Effects Due to the Layered Structure ................. 192
Dependence of the Position of the Minimum Linewidth on WSG . 194
Departure From Power Law Behavior ................. 194

Fits of AHCT) as a Function of WSO .................. 200
General Behavior of the Fits ....................... 201
INTERPRETATION OF THE MEANING OF T, IN EQUATION 7-4 . 210
Zero Temperature (2D) Scaling Forms ................ 211

Field and Frequency Effects on T,(WSG) ............... 228

Effect of the Depression of T, on Equation 7-4 ........... 233
COMPARISON WITH THEORY ........................ 246
Comparison with the Mean Field Picture of ESR .......... 247
Comparison with the Correlated Cluster Model ........... 249
Summary of the Comparison with Theory .............. 252

viii

INTERPRETA

CLrST
LINESHIFI A
SL'MMARY .

CHAPTER EIGHT: .
SUMMARY A
SUGGESTION

APPENDIX A: SOL'

APPENDIX B: C RIT

INTERPRETATION OF AH(T) IN THE CONTEXT OF ACTIVATED

CLUSTERS ................................. 253

LINESHIFT ANALYSIS ............................. 257
SUMMARY ..................................... 259
CHAPTER EIGHT: CONCLUSIONS AND SUGGESTIONS ........... 260
SUMMARY AND CONCLUSIONS ...................... 260
SUGGESTIONS FOR FUTURE EXPERIMENTS .............. 269
APPENDIX A: SOLUTION TO THE BLOCH EQUATIONS .......... 274
APPENDIX B: CRITICAL EXPONENT S IN SPIN GLASSES .......... 278

Table 3-1:
Table 3.2:
Table 5-1:
Table 7-1:
Table 7-2;
Table 7-3;
Table 7-4;

EDIE 7-5;

‘ 1 Concentra:

: Dcl‘t‘ndene

Summary
Phases Re
Parameter

concent
C onftgurat
Configura:
Configura:
Coniigurar
Parameters
Concentrat,

Minimu

 

Table 3-1:
Table 3-2:
Table 5-1:

Table 7-1:
Table 7-2:
Table 7-3:
Table 7-4:
Table 7-5:
Table 7-6:
Table 7-7:

Table 7-8:
Table 7-9:

Table 7-10:
Table 7-11:
Table 7-12:

Table 7-13:
Table 7-14:
Table 7-15:
Table 7-16:
Table 7-17:
Table 7-18:
Table 7-19:

Table 7-20:
Table 7-21:

Table 7—22:

Table 7-23:

LIST OF TABLES

Summary of Critical Exponents .......................
Phases Resulting From the Sherrington-Kirkpatrick Analysis
Parameters describing the dependence of AH(T) on Mn and Sb
concentration in Ag,,,,Mn,,Sb, from Mozurkewich, et. al.‘°5 ......
Configurations of Run 243 - Cu,,,,,Mn,,ll .................
Configurations of Run 185 - Cu0_93Mn0_07 .................
Configurations of Run 245 - Cu,,_93Mnom .................
Configurations of Additional Samples Run ................
Parameters from Finite Size Scaling Fits of T, .............
Concentration Dependence of AHCT) in Bulk Spin Glasses ......
Concentration Dependence of the Positions and Values of the
Minimum Linewidth for Cuo_89Mno_,, ..................
Dependence of t‘ on concentration
Characteristics of the Minimum Linewidth for Run 243
Temperature at which AH(T) Departs From a Power Law for Run
243 ' C“o.39Mno.rr
Temperatures at which AHCT) Deviates from a Power Law for Run
185 - CuomMnom/Cu ...........................
Temperature at which AHCI) Diverges from a power law for Run
245 - CuomMnom/Cu ...........................
Fits to A+BT+Ct"‘ for Run 243 - Cuo,89Mn,,,u/Cu ..........
Fits to A+BT+Ct“‘ for Run 185 - Cu0_93Mn0m/Cu ..........
Fits to A+BT+Ct"‘ for Run 245 — C%.93Mno_m/Cu ..........
Fits to A+BT+CT" for Run 243 - Cu0_89Mn,,,,,/Cu .........
Fits to A+BT+C"I“" for Run 185 - Cu0_93Mn0_m/Cu .........
Fits to A+BT+C'T‘“ for Run 245 - Cuo,93Mnom/Cu .........
Predicted Dependence of T, on Magnetic Field for
Cuo,,9Mno,,,/Cu ...............................
Extrapolation of T, to 9.3 GHz for Cu0,89Mno,,,/Cu ..........
Parameters A and C after removal of T,(WSG) dependence of
Equation 7-4 for Run 243 - Cu,,_89Mn0_,,/Cu
Parameters A and C after removal of T,(WSG) dependence of
Equation 7—4 for Run 185 - Cu0_93Mn0_m/Cu
Parameters A and C after removal of T,(WSG) dependence of
Equation 7-4 for Run 245 - Cuo_93Mn0m/ Cu

0000000000000

0000000000000

31
38

97
145
146
147
148
150
161

163
168

198

199
202
203
204
213
214
215

230
232

   
  
 

Tahiti-14'. Fits of R
Tahiti-25: Fits to A
Table'I-Zh: Fits to IT
TableB-l: Definitior
TableB-Z: Summar}
TableB-i'. Summary
Table B-4: Values of

 

Table 7-24: Fits of Run 243 - Cuo,89Mno,u - to Equation 7-11 ........... 255

Table 7-25: Fits to A" +BT+C" exp(D T“) for Run 243 - Cu,,,,,,,Mn,,_ll ..... 256
Table 7-26: Fits to (H('I‘)-Ho(oo)) ~ t'3 for Run 243 - Cu0,39Mno.u ........ 258
Table B—1: Definitions of Some Common Critical Exponents ........... 279
Table B-2: Summary of Relationships Between Critical Exponents ........ 280
Table B-3: Summary of Static Exponents for Spin Glasses ............. 281
Table B-4: Values of the Dynamic Exponents for Spin Glasses .......... 284

xi

Titirel-l: Typicali
Tiiure l-Z: Free enc
Figure 1-3: A Frustr':
Figure 14: The Ritl
Figure 2-1: Position

below .
Figure 2-2 (a) Abso.

(b) thei
Figure 3-1: An exam

Figure 3-2: Topoloog
Figure 3-3:

   
   
  
 

 

Ref70)
Figure 5-1: -
FigureS-Z: .
TigureS-i: .
metallic
Block or
Backgrou
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Figure 6.1.
@m6&
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Fiture 7.2-

Fiture 7.3;

Figure 74 T
Figme 7-5:

173».
“El-TE 7-6:

Figure 1-1:
Figure 1-2:
Figure 1-3:
Figure 1—4:
Figure 2-1:

Figure 2—2
Figure 3-1:
Figure 3—2:
Figure 3-3:
Figure 5-1:
Figure 5-2:
Figure 5-3:
Figure 6-1:
Figure 6—2:
Figure 7—1:
Figure 7-2:
Figure 7-3:

Figure 7-4:
Figure 7-5:

Figure 7-6:
Figure 7-7:

Figure 7-8:
Figure 7-9:

Figure 7-10:
Figure 7-11:

Figure 7-12:
Figure 7-13:

LIST OF FIGURES

Typical Susceptibility of a 1000 nm Cuo.93Mn,,m Sputtered Film 3
Free energy surfaces for (a) a spin glass and (b) a ferromagnet 8
A Frustrated Plaquette of Spins ...................... 9
The RKKY interaction as a function of distance between atoms . . . 13
Position of the effective magnetic field (a) above, (b) at, and (c)

below resonance as viewed in the rotating reference frame ..... 21

(a) Absorptive and dispersive parts of the Lorentzian lineshape and
(b) their derivatives ............................. 25

xii

An example illustrating finite size effects on the specific heat . 34
Topology of ultrametric states ........................ 42
Temperature - dimension phase diagram for Ising spin glasses (after

Ref 70) .................................... 55
Available relaxation paths for transition ion-metal systems ...... 76
Allowed relaxation paths for the correlated cluster model ...... 113
Measurement regimes of techniques used in the investigation of

metallic multilayers in the time—magnetic field plane ........ 119
Block diagram of the ESR system .................... 134
Background due to dopants from the silicon wafers .......... 141
Depression of T,(Ws,,) as a function of Wso for Run 243 -

Cu,,,,,,,1\/In,,,u ................................ 151
Depression of T,(Ws,,) as a function of W30 for Run 185 -

CU093Mnom ................................. 152
Depression of T,(WSG) as a function of Wso for Run 245 -

Cu,,.9,_.,Mn,,_,,7 ................................. 153
T,(WSG)/T,(00) as a function of Wso for all concentrations ..... 154
Representative lineshapes for a) a 1 nm Cu,,,,.,Mn,,_11 and b) a 500 nm

CuMn,,,,,,Mn,,_11 sample .......................... 157
Contributions from the real and imaginary parts of the susceptibility

and the background in fitting a CuoggMnoJ, 500 nm sample . . . . 158
Dependence of AH(T) on concentration for bulk samples ...... 162
Dependence of the Residual Width, A, on Mn Concentration . . . . 164
Dependence of the Thermal Broadening, B, on Mn Concentration . 165

Dependence of the Divergence Strength, C, on Mn Concentration 166

Concentration Dependence of the Position and Value of the

Minimum Linewidth in Bulk Samples ................ 167

AH(T) and H,,(T) for Sample 243-12 - Cu,,_,,,1VIn,,11 - 1000 nm . 169

AH(T) and H,,(T) for Sample 243-11 - Cu,,_,,,,Mn,,_ll - 500 nm . . . 170

7"
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Figure 7-14:
Figure 7—15:
Figure 7-16:
Figure 7-17:
Figure 7-18:
Figure 7-19:
Figure 7—20:
Figure 7-21:
Figure 7-22:
Figure 7-23:
Figure 7-24:
Figure 7-25:
Figure 7-26:
Figure 7-27:
Figure 7-28:
Figure 7-29:
Figure 7-30:
Figure 7-31:
Figure 7-32:
Figure 7-33:
Figure 7-34:

Figure 7-35:
Figure 7-36:
Figure 7-37:
Figure 7-38:
Figure 7-39:
Figure 7-40:
Figure 741:
Figure 7-42:
Figure 7-43:

Figure 7-44:
Figure 7-45:

Figure 7-46:
Figure 7-47:

Figure 7-48:
Figure 7-49:

AHCI') and H,,(T) for Sample 243-8 - Cuo,89Mno.,, - 50 nm 171
AHCT) and HO(T) for Sample 243-7 - Cuo,,9Mno,u - 30 nm 172
AHCI‘) and HOCT) for Sample 243-6 - Cumano,ll - 10 nm . . . 173
AH(T) and H,,(T) for Sample 243—4 - Cu,,,,,9Mn,,.11 - 7 nm ..... 174
AHCI‘) and HOCI') for Sample 243-2 - Cuoanm, - 3 nm ..... 175
AHCI') and H,,('T) for Sample 243—1 - Cumanm, - 1 nm ..... 176
AH(T) and Ham for Sample 185-1 - Cu,mMnom - 1000 nm . . . 177
AHCI') and Ham for Sample 185-5 - CuoflMnom - 100 nm . . . . 178
AH(T) and Ham for Sample 185-2 - Cu,,.,,,Mno_,,7 - 50 nm 179
AH(T) and H0(T') for Sample 185-4 - CuomMnom - 30 nm 180
AH(‘T) and HO(T) for Sample 185-3 — CuomMnom - 7 nm ..... 181
AH(T) and H0(T) for Sample 185-10 - Cuo_9,Mnom - 5 nm 182
AHCT) and H,,(T) for Sample 185-6 - Cuo_9,Mno_m - 3 nm ..... 183
AH(T) and Ho(T) for Sample 245-1 - CuomMnom - 30 nm 184
AH(T) and H,,(T) for Sample 245-3 - Cuo.9,Mn,,,,,, - 10 nm 185
AH(T) and H,,(T) for Sample 245-4 - Cuog,Mnom - 9 nm ..... 186
AH(T) and H,,(T) for Sample 245-6 - Cuo_93Mn,,_m - 7 nm ..... 187
AH(T‘) and HOCT) for Sample 245-7 - Cuo_93Mnom - 6 nm ..... 188
AH(T) and H0(T) for Sample 245-8 - Cuo.93Mnom - 5 nm ..... 189
AH(T) and H,,(T) for Sample 245-12 - CuomMnom - 3 nm 190
Comparison of Two Separate Data Runs on Sample 243-2

- Cuaanm, ................................ 193

Comparison Between a Cu,,,,Mnom Film with W30 = 500 nm and a

500/30/1 CuomMnQu/Cu Sample ................... 195
Pit of AH(T) for Sample 243-l2 - Cu,,,,,,Mn,,,ll - 500 nm ...... 205
Fit of AH(‘T) for Sample 243-1 - Cuongnm, - 1 nm ....... 206
Comparison of AH(T) for the 243 Series - Cu,,,,,,,Mn,,.ll ....... 207
Excess Linewidth, AH(T)°", for the 243 Series - Cuo_,9Mno,,, . . . 208
Excess Linewidth, AH(T)°", for the 245 Series - Cu0_93Mno_m . . . 209
Variation of the Residual Width, A, with Wso for Run 243 -
Cu0_89Mno_,, ................................. 2 16
Variation of the Thermal Broadening, B, with W50 for Run 243 -
CUQggMno." ................................ 217
Variation of the Divergence Strength, C, with W8C, for Run 243 -
CuaggMnm, ................................ 218
Variation of x with W30 for run 243 - Cu,,_,,,,Mn,,_,1 ......... 219
Variation of the Residual Width, A, with Wso for run 185 —

Cuo.93Mnom ................................ 220
Variation of the Thermal Broadening, B, with Wso for run 185 -
Cu0_93Mnom ................................ 221
Variation of the Divergence Strength, C, with Wso for run 185 —
Cuoanom ................................ 222
Variation of x with Wso for run 185 - CuomMnom ......... 223
Variation of the Residual Width, A, with W80 for run 245 -
Cuo_93Mno_m ................................. 224

xiii

Figure 7—51:
Figure 7-53:
Figure 7-53:
Figure 7-54:
Figure 755:
Figure 7-56:
Figure 757;
Figure 7-58:

Figure 7-59;

Fig-II? 7-60;

Figure 8-1:

I Variatit

C u; e331
Variatit
cut-95M
Variatit‘
Extrapo
Field a:
Compar
for Rur
Compa'
for R3"
Comps:
for Rs."
Compar
Run IS
Compar
Run 2;
COmpar
Run 24

PIOPOSCt;

hiuhil-

ht:

 

Figure 7-50: Variation of the Thermal Broadening, B, with WSO for run 245-

Cuo.”Mno_m ................................ 225
Figure 7—51: Variation of the Divergence Strength, C, with WSO for run 245 -

CUoflgMnom ................................ 226
Figure 7-52: Variation of x with W30 for run 245 - Cuo_93Mn,,m ......... 227
Figure 7-53: Extrapolation of T,(Ww) at 9.3 GHz for Cu0,89Mno,,,. ...... 231
Figure 7-54: Field and Frequency Effects on T,(Wso) for Cuo.,9Mno,,, ...... 234
Figure 7-55: Comparison of the Residual Width from Equation 74 and 7-7

for Run 243 - CuoggMnoJ, ....................... 240
Figure 7—56: Comparison of Divergence Strength From Equation 7—4 and 7-7

for Run 243 - Cu039MnOJ, ........................ 241
Figure 7—57: Comparison of the Residual Width from Equations 7-4 and 7-7

for Run 185 - CuomMnom ........................ 242
Figure 7-5 8: Comparison of the Divergence Strength from Equations 7—4 and 7-7 -

Run 185 - Cuo_93Mno_m .......................... 243
Figure 7—59: Comparison of Residual Width from Equations 7-4 and 7-7 -

Run 245 - Cuoanom .......................... 244
Figure 7-60: Comparison of Divergence Strengths from Equation 74 and 7—7 -

Run 245 - Cuo_93Mnom .......................... 245
Figure 8-1: Proposed Temperature - Layer Thickness Phase Diagram for

Multilayered Spin Glasses ........................ 268

xiv

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I

'Spin
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CHAPTER ONE: INTRODUCTION

INTRODUCTION; WHAT E A SPIN GLASS?

"Spin Glasses are magnetic systems in which the interactions

between the magnetic moments are ’in conflict’ with each other, due to

some frozen-in structural disorder. Thus no type of conventional long-

range order (of ferromagnetic or antiferromagnetic type) can be

established. Nevertheless these systems exhibit a ’freezing transition’ to

a state with a new kind of ’order’ in which the spins are aligned in

random directions."1

This definition, from a review article by Binder and Young, captures the essential
nature of spin glasses: although we know a spin glass when we see one, we have a hard
time explaining why it is a spin glass. Considerable theoretical and experimental effort
has gone into understanding the novel properties which appear in such a wide variety of
materials. Most of the effort has focused on the behavior of spin glasses in the spin glass
phase proper; this thesis investigates the behavior of spin glasses as the transition is
approached from above the freezing temperature using Electron Spin Resonance. The
development of nanoscale fabrication and characterization techniques has prompted recent
experiments on the behavior of the freezing temperature as a function of sample size.
Measurement of the Electron Spin Resonance linewidth and position in bulk and
multilayered samples has shed additional light on both the nature of the spin glass
transition and the characteristics of multilayered spin glasses.

The organization of this thesis is as follows; Chapter One provides an
introduction to the experimental properties of spin glasses, and the theoretical concepts

necessary to explain these properties. Chapter Two explains the basics of Electron Spin

1

Resonance (ESR). 3'
glasses. and the expt
rifh chapter is divide
localized moments in
inspin glasses. Havi
with the motivation I
detailed in Chapter

concltsions and 51::

W

Efifimure depem

2
Resonance (ESR), and Chapters Three and Four outline the major theories of spin

glasses, and the experimental progress in the study of multilayered spin glasses. The
fifth chapter is divided into two sections; the first section outlines the theory of ESR of
localized moments in metals, and the second section summarizes the current state of ESR
in spin glasses. Having laid the necessary conceptual foundation, Chapter Five concludes
with the motivation of the work presented in this thesis. The experimental technique is
detailed in Chapter Six, the data and analysis are presented in Chapter Seven, and

conclusions and suggestions for further experiments are found in Chapter Eight.

THE EXPERINIENTAL SIGNAIUE QF SPIN QLASSfi

The primary - and most easily observed - signature of a spin glass is a cusp in the
temperature dependent susceptibility, x(T), accompanied by hysteretic behavior at
temperatures below the cusp. Throughout this thesis, I will refer to the temperature of
the cusp as the freezing temperature, T,, although it is also commonly referred to in the
literature as the glass temperature. This cusp was first measured by Cannella and
Mydosh2 in the frequency dependent susceptibility of AuFe, but is seen in a wide variety
of materials including dilute and concentrated metallic alloys, insulators and magnetic
semiconductors‘*3". A typical plot of x(T) for a CunnMnom 1000 nm sputtered film
is shown in Figure 1-1. The solid symbols correspond to measurements made after the
sample has been cooled with no applied magnetic field, and the Open symbols represent
the behavior of the susceptibility when the sample is cooled in a magnetic field of 100 G.

A measuring field of 100 G was used for both curves. Above T,, the behavior is the

 

 

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same for both the field cooled (FC) and the zero field cooled (ZFC) curves, but below

the cusp, the field cooled curve remains roughly constant, while the ZFC curve returns
toward zero. The cusp in the susceptibility is reminiscent of the behavior observed in
ferro- and antiferro- magnets, which are known to undergo phase transitions, so the
initial assumption was that this cusp represented a phase transition. Measurements such
as the onset of remanence and irreversibility in the magnetization, and sharp features in
Mossbauer, nonlinear susceptibility and anomalous Hall effect experiments also indicate
a phase transition at T,, yet corresponding behavior has not been observed in other
thermodynamic quantities. The magnetic specific heat, resistivity and thermopower show
only broad features at T,, leaving Open the question of whether or not a true phase
transition occurs.

Specific heat measurements“ in CuMn and AuFe show that the entropy is about
70% of the maximum entropy above T,, indicating that some short range magnetic order
exists above the transition temperature. Along with thermopower and resistivity7
experiments, these measurements indicate that a spin glass above T, can not be treated
as a simple paramagnet. Additional evidence for this picture is found in diffuse neutron
scattering8 in AuFe at zero magnetic field, which shows two quasielastic lines with
different linewidths. One of these lines is interpreted as corresponding to fast relaxation
processes of single spins, while the other line is attributed to the behavior of correlated
pairs, triples, etc. As the temperature decreases, the second line grows at the expense
of the first, indicating a gradual growth of short range ordering.

The strongest evidence for a true phase transition is the divergence of the

nonlinear susceptibility, as first measured in AgMn by Monod and Bouchiat9 and later

by 153,10, and 1m 1

amended in powers t

where 1,, is the r.

 

 

 

susceptibility could

acta'acteristic expor

5
by Levy”, and Levy and Ogielski“. If the temperature dependent susceptibility is

expanded in powers of the magnetic field, we find

= 10(1') - 11mm + Orderm‘) 1-1

ml:

where x“, is the nonlinear susceptibility. Bouchiat12 found that the nonlinear
susceptibility could be fit to a power law in the reduced temperature (t= (T-Tf)/T,) with

a characteristic exponent, 7

 

The extraction of an exponent from data appears to depend heavily on the region of
temperatures and magnetic fields considered in the scaling analysis. Bouchiat’s
measurements indicate that restriction of the temperature range to a narrow region around
Tf results in a well defined value of 7 = 2.2 :1; 0.2. Alternate analyses of the nonlinear
susceptibility have been suggested” .

Dynamic measurements of the susceptibility have shown“1 that the relaxation
time spectrum is very broad, even far above T,. As the temperature is lowered, the
width of the spectrum dramatically increases, so that time scales are of macroscopic
order at and below Tr. The shortest relaxation times, 1m are thought to be on the order
of 1010 to 1013 seconds. The average time scale, T,,, and the minimum time scale, rm,

are consistent with an Arrhenius-type law,

where 13 is a propor
therm‘ily activated p
maximum relaxation
the above picture. 2'

Vogel-Fulcher law.

he \‘ogel-Fulcher

Amhenius law, but t
correspond to the ffo

Above '1‘
Cmital 51

f) the

 

Ong dOWn

 

(E
1: = to expkfi] 1-3
3

 

where To is a proportionality constant and E,“ is an activation energy which measures
thermally activated processes involving temperature independent barriers; however, the
maximum relaxation time, rm, does not obey an Arrhenius law, and is inconsistent with
the above picture. rm displays a strong temperature variation, and is described by a

Vogel-Fulcher law,

1' = ‘0 exp (3%) 1’4
The Vogel—Fulcher law describes the frequency dependence of Tf better than the
Arrhenius law, but the variable To is generally treated as a parameter and does not
correspond to the freezing temperature.

Above T,, the behavior of the relaxation times may be described in terms of a

critical slowing down, with the maximum relaxation time following

rm = to (T _ Tf)“" 1-5
where v is the critical exponent of the correlation length (5 o: t"), and z is the dynamical
critical exponent (1' o: E").

If the spin glass transition is not a true phase transition, the cusp in the

susceptibility may represent a ’falling out of equilibrium’. Because the time scales of the

reisxation are macro
probe has a charact-
scale longer than t?
equilibrium behavior
toincrease with deer
udue to nonequilibt
the magnitude of the I
flout the ability of 2'
field.

Below T,, the
relaxation time eff K.
forms dépending on I

AS this thesit
ap’r‘w'OaChlng T I

f! e X p\
missed

THE

The conflictlr.

 

7

relaxation are macroscopically long, the system is never completely in equilibrium. Any
probe has a characteristic time scale, such that, when the system is relaxing at a time
scale longer than that of the measurement, the probe is incapable of measuring
equilibrium behavior. Experimentally, the position of the freezing temperature is found
to increase with decreasing measuring times, supporting the interpretation that the cusp
is due to non—equilibrium effects. The freezing temperature is also found to depend on
the magnitude of the magnetic field used in the measurement, which brings up questions
about the ability of a system to support spin glass order in the presence of a magnetic
field.

Below T,, the primary experimental spin glass signatures are irreversibility and
relaxation time effects. The temperature dependent susceptibility has very different
forms depending on whether or not the sample was cooled in a field, as seen in Figure
1-1. As this thesis is mainly concerned with the temperature regime above and
approaching T,, experimental investigations of the spin glass phase per se will not be

reviewed.

Ifl EQRETIQAL PRQPERTIES

The conflicting experimental evidence surrounding the nature of the spin glass
phase and transition complicates theoretical analysis. Calculations are further hampered
by the lack of a well-understood Hamiltonian for the system, no obvious choice of order
parameter, a complicated and non-periodic ground state, and the analytical difficulties

inherent in dealing with random systems.

8

The free energy surface of a spin glass may be schematically thought of as a
multi-valleyed structure, with a large number of local minima, as shown in Figure l-2a.

The free energy surface of a ferromagnet is shown in Figure 1-2b for comparison.

 

 

 

 

 

Order Parameter Order Parameter

 

 

(a) (b)

 

 

Figure 1-2: Free energy surfaces for (a) a spin glass and (b) a ferromagnet

The complicated free energy of a spin glass leads to a large number of states
which, although different microscopically, are characterized by the same macroscopic
measurable quantities. These degenerate states are separated by free energy barriers
which grow as the temperature decreases. The question of whether there is a single (to
within field symmetry) ground state, or a large number of ground states is still
controversial""‘. The answer may in some part be dependent on the vector character

(i.e. Ising or Heisenberg)“'17 and range of magnetic interactions in the material.

Fmstration at

A mic:
steels) and th.
dsorder and h
Lectures a r2

positioning of

 

 

 

FruStrat
Wishing. A

Frustration and Disorder

A microscopic spin glass theory which is to explain the nonperiodic ground
state(s) and the complicated free energy surface must have two fundamental elements:
disorder and frustration. Disorder refers to the lack of periodicity of the spins, which
introduces a random element to the interactions. Disorder may be a result of random

positioning of magnetic ions on a crystalline lattice or amorphous crystal structure.

 

 

 

 

 

 

Figure 1-3: A Fmstrated Plaquette of Spins

Frustration is the inability of a system to simultaneously satisfy all of its ordering

constraints. A simple example of a frustrated system is a plaquette of three Ising spins

on the vertices of 2
between A and C a:
Band C is antiferros
their preferred orie
responsible for the r

This combine
can not be removed
both frustration and t
Teal materials. The
“on material to mat:

in dilute metallic Spi

Metallic Spin Glassl

 

Dilute metalli

Sllbsu'tuun i

10

on the vertices of an oblique triangle, as shown in Figure 1-3. If the interactions
between A and C and between A and B are ferromagnetic, but the interaction between
B and C is antiferromagnetic, we can not have a situation in which all three spins are in
their preferred orientations. This prevents the onset of long range order and is
responsible for the many valleyed free energy.

This combination of frustration and disorder results in a ’serious’1 disorder which
can not be removed by a mathematical transformation. Spin glass theorists agree that
both frustration and disorder must be present in any spin glass theory which is to explain
real materials. The physical mechanisms responsible for frustration and disorder vary
from material to material, so the specific interactions which result in spin glass behavior

in dilute metallic spin glasses will be considered next.

Metallic Spin Glasses

Dilute metallic alloys are the canonical spin glasses and are created by randomly
substituting a small number of transition metal ions - usually Mn, but sometimes Fe - in
the lattice of a host noble metal (Cu, Au, Ag or Pt). The interactions are long-ranged,
and the spins behave in a Heisenberg-like manner. Metallic spin glasses have been
extensively investigated due to the ease of preparation and the range of concentrations

over which spin glass behavior is observed.

lire RKKY Interat

The Rudcr.’
explained the long rl
conduction electrons
in Cu interact. Hunt

orlentation of the mo

31”: be described by

‘k

 

1 l
The RKKY Interaction

The Ruderman-Kittel-Kasura-Yosida‘8'19'20 (RKKY) interaction originally
explained the long range interaction of nuclear spins via hyperfine coupling to a sea of
conduction electrons. This interaction can also explain how widely separated Mn atoms
in Cu interact. Hund’s rules determine the magnitude of the local Mn moment, with the
orientation of the moment determined by the exchange interaction, J(R,,). The interaction
can be described by a Heisenberg Hamiltonian,

H = -z w 2 ~53 1-6

it}

The exchange polarizes the spins of the conduction electrons near the Mn ion. The Pauli
exclusion principle causes the conduction electrons to respond with a characteristic
wavelength, x, = 1r/ kp, where kF is the Fermi wavevector. The resulting spin
polarization of the conduction electrons is oscillatory and long ranged. The sign of the
interaction between the Mn ions depends on whether the second Mn ion is at the crest
or the trough of the spin polarization wave created by the first Mn ion, so that both
ferromagnetic and antiferromagnetic interactions are present. As the distance between
the Mn ions increases, the strength of the interaction decreases. The derivation of the

RKKY interaction may be found in a number of texts”. The result is

Jug) = ~

where A is the meat.
:efl. Ru is the distar
host metal. The exp
The RKKY interactit

mean free path effec

JU

 

12

 

he, 5 .
J(Ry) = - [131'] [LT—h“ makfkv)'2krRu°°s(2kFRu) (ii! 1-7
I“ «3 (ZkFRyy

where A is the mean free path of the conduction electrons, 303 is the volume of the unit
cell, Rt,- is the distance between Mn ions i and j, and kP is the Fermi wave vector of the
host metal. The exponential term accounts for the finite mean free path of the electron.
The RKKY interaction is frequently written in terms of the asymptotic limit, neglecting

mean free path effects

 

cos(2k,Rv + am]

J( )=Jo
R” [ (kFRyi’

Figure 1-4 shows the RKKY interaction as a function of the distance between the
moments (R,,-). The random placement of the ions provides disorder and the presence of
the RKKY interaction is responsible for both ferro- and antiferro- magnetic interactions,
resulting in frustration.

In theory, we expect an equal number of positive and negative interactions, which
would yield a simple Curie Law paramagnetic susceptibility in the region above Tf.
Morgownik and Mydosh22 carefully investigated the high temperature susceptibility in
an attempt to corroborate the expression for the RKKY interaction in metallic spin
glasses. They found that deviations from ideal random mixing occurred in such a way
as to increase the probability of ferromagnetic interactions in CuMn, AuMn, AuFe and

PtMn. Their analysis showed that even small deviations from random mixing make big

differences it

andt'erromagr

F—

 

Figure 1-4;

13

differences in the interactions. In CuMn, the Mn nearest neighbor interaction is

antiferromagnetic, while the next nearest neighbor interaction is ferromagnetic.

J
M 'j

Figure 1-4: The RKKY interaction as a fitnction of distance between atoms

 

 

 

 

 

 

The Mn-Mn distance23 is maximized by preferential occupation of next nearest neighbor
sites by Mn, so that the predominant interactions are ferromagnetic.

Many theoretical calculations attempt to predict the behavior of T, in metallic spin
glasses from the form of the RKKY interaction, and especially to explain the observed
sublinear dependence of Tf on concentration. Larsen” has presented a mean field

calculation in which the freezing temperature is given by

where the square to
over neighbor spins

using a quenched u:
greater than the axe
ion. Lexy and Z"
corrections to the H
byafactor of more

besix times more ll‘
more iimportant tha'
aSymptotit: form of 1

than 6 nm in CuMn

there will always be

corrections to the 1.. l

.lnhotropy

AnisotrODies i
tonstraints on the int‘
so
.sems, which thee
dinen'

sro ' ‘

ns, exhibit

 

14
kr, ~ 2]: ‘12]: [10,)? 1-9

where the square root represents the local energy scale of spin j (with the sum running
over neighbor spins). Larsen has derived a sublinear dependence of T, on concentration
using a quenched uniform model in which only those moments which are at a distance
greater than the average nearest neighbor distance are included in the sum for a given
ion. Levy and Zhang”36 have demonstrated that failure to include preasymptotic
corrections to the l/R’ behavior overestimates T, by a factor of two for Cuo.”Mno.m, and
by a factor of more than three for cuogsMnQos. They find preasymptotic corrections to
be six times more important than mean free path effects in CuWMnM6 and eight times
more important than mean free path effects for Ago-“Mug“. This means that the
asymptotic form of the RKKY interaction can not be applied to pairs separated by less
than 6 nm in CuMn and 3.5 nm in AgMn” ‘53 In concentrations larger than 100 ppm,
there will always be neighbors within this limit, demonstrating the importance of the

corrections to the UK3 behavior.
Anisotropy

Anisotropies increase the frustration in a system by placing additional ordering
constraints on the individual spins. Some authors?7 contend that nominally Heisenberg
systems, which theoretically are not expected to exhibit a phase transition in three

dimensions, exhibit phase transitions only because of the presence of anisotropy.

Computer simulati
anisotropy is not fur

question.

 

 

 

Although dip

|

of these anisotropies
most significant at
(Dll) anisotropy. ‘
scattering from a th
isastrong spin orb

is at the origin. ant

15

Computer simulations28 and experimental measurements” offer evidence that
anisotropy is not fundamental to the ordering phenomenon. This remains an unresolved
question.

Although dipolar anisotropies are present in metallic spin glasses, the magnitudes
of these anisotropies are too small to be responsible for ordering in CuMn or AuFe. The
most significant anisotropy in metallic spin glasses is the) Dzyaloshinskii-Moriyawm
(DM) anisotropy, which described the interaction between two spins via spin—orbit
scattering from a third atom. This anisotropy is especially pronounced if the third atom
is a strong spin orbit scatterer (i.e. Pd, Fe, or Co impurities in CuMn). If the impurity
is at the origin, and the two spins are at R1 and R2, the DM anisotropy has the form

SinlkF(Rr + R2 + (Rz'Rr))] + 450M
[1+ CkF(R1+ R2 + (R2 - R1))]

 

Hm: = (‘10)!)

(El-fl.) (Exit) .(sgxsg)
R1319: (RI-R2)k13’

1-10

 

Because of the coupling of the spin cross product to the spatial cross product, this
anisotropy is unidirectional. Typically, JDM is approximately 1/10 of the magnitude of
the exchange coefficient of the RKKY interaction. The DM anisotropy is most important

below T,, where the anisotropy acts to maintain the frozen direction of the spins.

 

In this (
briefly review er
transition has b
tme equilibriun

prepenies - has

16

In this Chapter, some of the experimental properties of spin glasses have been
briefly reviewed. Experimental evidence for and against the existence of a true phase
transition has been discussed, and an alternative explanation - that a failure to achieve
true equilibrium during the course of the measurement is responsible for the observed
properties - has been suggested. The theoretical concepts necessary to a discussion of
spin glasses have been introduced, with particular attention paid to the properties of

metallic spin glasses.

[\TRODL'C T

Chapter
general theory
Willis is discus
Where these b

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CHAPTER TWO

INTRODUCTION TO ESR

INTRQDQQTIQE

Chapter Two focuses on the basic theory of Electron Spin Resonance (ESR). The
general theory of ESR in non-interacting and interacting spin systems far from critical
points is discussed. The material in this chapter provides a foundation for Chapter Five,
where these basic formulas will be modified to describe ESR in the much more

complicated spin glass system.

BASI THEORY F R

As an idealized case, consider a system of spins, each with angular momentum

S. Each spin has a moment, it, given by,

ii = ”g “'3 g 2'1
where the #3 is the Bohr magneton,
ch
= _ 2-2
“B 2mc

17

and g is the
positive. and

defined as

“hen
he occupatio
statistical meci
0i ireqUClle' h

microwave en.

18
and g is the ’ g factor’. The charge of the electron, e, will always be taken to be

positive, and the minus sign accounted for explicitly. The gyromagnetic ratio, 7, is

defined as

3 “a 2-3

When a magnetic field is applied, (ZS +1) Zeeman levels result for each spin.
The occupation of these levels by an ensemble of spins is described by the appropriate
statistical mechanics, and results in a net magnetic moment. An applied microwave field
of frequency by induces transitions between Zeeman levels. Resonance occurs when the

microwave energy is equal to the energy difference between two Zeeman levels.

hv = guDH 2-4

Classical Equations of Motion

We can treat the spin as a charged, classical ’top’ with angular momentum its,
and moment u=-7fiS. The motion of the charged top in a magnetic field will be

governed by the classical equation

t—tw

who"
the moment 2
LgrnOT P780

AILtO

calculfilic’“ le

there the HF:

which is the s

the microwas

he Same resr

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In exp

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19

When H is a constant vector (Ho = Hok), the resulting motion is precession of
the moment at a fixed angle with respect to Ho, with frequency w, = 71-1,. This is called
Larmor precession, and the frequency w, is the Larmor frequency.

Although the above description is entirely classical, a quantum mechanical

calculation leads to the same results. We start from the Heisenberg equation of motion,

. d

where the Hamiltonian is the Zeeman interaction. Calculation of the commutator leads

to

d _.
Tit-(flop) = y]? x <p'op> 2-7

which is the same as Equation 2-5. The details of extending the quantum case to include
the microwave field may be found in a number of references32'33'3‘. As we obtain
the same results classically as quantum mechanically - even in treating the presence of
the microwave field - the more intuitive classical picture can be used in discussing these
effects.

In experiments, we apply a static field, Ho, and a smaller, time dependent field,
H,, at right angles to the applied field. The time dependent field is usually written as

the properly rotating part of an oscillating field,

The counter-rotatir‘
Equation 2-8 is any

and may be negleci

We can trar;
. . I
attest. This reter

equation of motion

mmhmkmi

ttqtiency of the alte

 

20

H, = H,cos(m)i + Hlsin(wt)f 2-8

The counter-rotating part of this field is obtained by letting w -> -w. If the field of
Equation 2-8 is anywhere near resonance, the counter-rotating part is far from resonance

and may be neglected. The presence of H, modifies Equation 2—5

5% = -a x y (H, + rim) 2-9

We can transform this equation to a reference frame in which both H, and H, are
at rest. This reference frame rotates about the z direction with frequency wk. The

equation of motion in the rotating frame is

I .
(gt) = -,-, x v ,7” 2-10

where the prime denotes the derivative taken with respect to quantities in the rotating

frame, and

= 5( 0-2} . 11,: 2-11

In this reference frame, the moment precesses about H4, in a cone of fixed angle.
The motion of the moment is periodic, with frequency 7H,". At resonance, the

frequency of the alternating field is equal to the Larmor frequency; equivalently, we see

21

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that this implies th
rotating reference f
zcomponent. as s2"
[1,, is negative. as
when H,” = H“ 2.:
WW. H, is along

In a quantt.
quantum levels. T-
from an energy 5,3:

energy correspondi
Bloch Bouations

Up to this p

being transferred 0 I

wOilld ha\e a rESOI
Mag [lanSltiong b
with

Some net abso

in I
39 S, or to the latt‘
ton .
.. e
TESonance 1,,

pie
”oleOeicant

B

l
OCh DOSilet

hit,
AS.‘X Ex

 

22
that this implies that H,,, = H,. Figures 2-1 a,b, and c illustrate the behavior in the

rotating reference frame. If H,, is above resonance (greater than wl'y), H,,, has a positive
2 component, as shown in Figure 2- la. If H(, is below resonance, the 2 component of
H,,, is negative, as shown in Figure 2-lc. Figure 2-lb shows the system at resonance,
when Heff = H,, and (H,,-wl'y) is zero. In all three figures, H,,, is shown by the shaded
arrow. H, is along the x-axis and H,, - w/‘y-iS in the x direction.

In a quantum description, a spin flip corresponds to a transition between two
quantum levels. The absorption of energy from the microwaves enables the spin to flip
from an energy state corresponding to being aligned with the field to a state with a higher

energy corresponding to a position antialigned with the field.

Bloch Equations

Up to this point, we have considered the spins as perfect oscillators, with energy
being transferred only between the spins and the magnetic fields. This ideal system
would have a resonance line with no width. In a real system, spins are constantly
making transitions both from lower energy states to higher energy states, and back again,
with some net absorption. In order to de-excite, a spin must transfer energy to other
spins, or to the lattice. Both processes take a finite amount of time, which results in a
finite resonance linewidth. Felix Bloch” introduced the idea of relaxation times
phenomenologically into the equations of motion to account for these processes.

Bloch posited that the magnetization, if pushed out of equilibrium, will tend to

relax exponentially to its thermal equilibrium value, M0. The 2 component of M grows

as individual morn,
moments to give u;
relaxation time. T;
Interactions

I

applied field. FOIL
locally. so that e.;
magnetization is re
components. and is
The Bloch l

rotating reference 2

 

23

as individual moments making up M flip to relax Mz toward Mo, which requires the
moments to give up energy to the lattice. This process is characterized by the spin-lattice
relaxation time, T,.

Interactions between spins can destroy the magnetization perpendicular to the
applied field. For example, the dipolar interaction can randomly modify the value of H,,,
locally, so that each spin precesses at a slightly different rate. The perpendicular
magnetization is relaxed by destroying the coherence in the individual perpendicular spin
components, and is characterized by the spin-spin relaxation time, T2.

The Bloch Equations, which describe the motion of interacting spins are (in the

rotating reference frame):

dM Mo-M

 

 

 

d; = wanna): + T1 2 2-12a
dM M
f = "Y (fiwaLy + T: 242"

The details of the solution to the Bloch equations can be found in Appendix A.
The spin—lattice relaxation time is usually much longer than the spin-spin relaxation time,
and the rotating field, H, is taken to be much smaller than the applied field, H,,. The
susceptibility will be complex, as the magnetization has components both in and out of
phase with the rotating field. The imaginary part of the susceptibility (X") describes the
absorptive behavior of the sample, while the real part (X') describes the dispersion. In

metals, the complex surface impedance results in the absorbed power being proportional

to a mixture of ti

I' .
lIIIEWIdm A H i
’ s 5

IS r -
p‘opOnlOna] ,0

24

to a mixture of the real and imaginary parts. The susceptibility from the Bloch equations

is

 

 

- T
xi = growl “t “’3 ]
1+(m-ma) T2
// 1 l
x = —x,(w,T,) 2-14
2 (1+(0-00)2 1,2]

The functions which result from the solution of the Bloch equations and their
derivatives are shown in Figure 2-3a and b; they are Lorentzian and result from the
assumption of exponential relaxation. Other types of lines are observed experimentally,
and these lineshapes may be derived by assuming other forms of the relaxation. The
linewidth, AH, is usually measured as the peak-to—peak value of the first derivative, and

is proportional to the inverse of the relaxation time, T2.

EXPERIMENTAL DE TI N F THE RES NAN E

The simplest experimental set up consists of a microwave source, which sends
microwaves down a transmission line and into a resonance cavity. Using various
microwave guides, the reflected (or transmitted) microwave energy is sent to a detector
and converted into a current. This current is proportional to the power reflected (or
transmitted), and is smaller than the incident power (P,) by an amount equal to the power

absorbed by the sample (P,). As the magnetic field is swept, (P,—P,) vs. field is plotted,

\
a» , : (0)4650)

F

3

3;

i

g \

l

“Etna 2.2

“(’3qu

T. i

A 5......
- A315 53 van a

T:

 

 

 

 

25

 

Absorptive and Diaperslve Lineshape:
l l l l l

 

 

 

Pm Absorbed (Arte. Unite)

 

 

 

are/an “:5. Units)

 

 

 

 

 

 

 

 

Figure 2-2: (a) Absorptive and dispersive parts of the Lorentzian lineshape and (1)) their
derivatives

and can ills
interest lwi
order to in?

description.

 

motion of a
and Spin-lat
1Equations

Spins. A pr

Eli en.

26

and can then be analyzed to extract parameters which relate to physical quantities of
interest (which will be explained in Chapter 5). This basic set up can be modified in
order to improve the detection of the signal, but the essentials are all present in this brief

description.

SUMMARY

The basic elements of electron spin resonance have been presented in terms of the
motion of a classical moment in a field. Relaxation times, which result from spin-spin
and spin-lattice interactions have been phenomenologically considered within the Bloch
Equations. Chapter Five will extend this simple model to the case of strongly interacting
spins. A preliminary description of the experimental detection of the resonance has been

given.

 

Spin
PTWt‘duraI t
kincluded.
questions ab
SC-LIOOIS of j

Eduard s- A r

low lying er
droplet t
arising fI'Om

In or
0mm [he SIC
lite formalis
Chapter brief

CHAPTER THREE

SPIN GLASS THEORIES

INTRQDQQTIQN

Spin glasses pose a challenging problem for theorists. In addition to the
procedural complications inherent in treating disordered systems, frustration must also
be included. In addition, the wide variety of materials displaying spin glass behavior and
questions about ergodicity and measuring time must be considered. There are two major
schools of spin glass theory; the first developed from the mean field theory of the
Edwards-Anderson and Sherrington-Kirkpatrick models, and is referred to as the
hierarchical or mean field model. The second is a scaling theory which expresses the
low lying excitations of a spin glass in terms of critical exponents, and is referred to as
the droplet theory. This section will summarize the development of and important results
arising from these two schools of thought.

In order to understand spin glass theory, and especially the question of whether
or not the spin glass ordering is a true phase transition, the idea of phase transitions and
the formalism used in their description must be understood. The first section of this
chapter briefly reviews phase transitions and scaling formalism. The concept of ’slowing
down’ near a critical point will also be reviewed. The second part of this chapter focuses
on developing a conceptual understanding of the two major spin glass theories. While

the hierarchical model is mathematically rigorous, the upper critical dimension - the

27

IOWESI dlIlli
torea‘ systi
predictions
experiment

A v
identifying
Ect‘lIOtl S;
glass phage
themes dl
et‘lpi'timent;
CIOplet m0

be,
riferenCes‘6

lllSCOnrjnun‘

get

28

lowest dimension at which mean field exponents are valid - is 6, which makes application
to real systems questionable. While less theoretically rigorous, the droplet theory makes
predictions which distinguish between two and three dimensions and can be
experimentally tested.

A word of caution: most of the theoretical effort to this point has focused on
identifying the nature of the spin glass transition and of the resulting phase. The
Electron Spin Resonance experiments discussed in this thesis investigate how the spin
glass phase is approached from above the transition temperature. A lot of spin glass
theories dismiss this regime as being ’paramagnetic’ and do not address the
experimentally observed ordering effects above T, discussed in Chapter One. Neither the
droplet model nor the hierarchical model has specifically addressed Electron Spin

Resonance experiments.

PHASE TRANSITION AND FINITE IZE E T

Details of the study of phase transformations may be found in many
referencesm7. Mathematically, a phase transition is indicated by a singularity or
discontinuity in some derivative of the free energy. Experimentally, this is indicated by
the divergence of some measurable quantity (i.e. the magnetic susceptibility and specific
heat in ferromagnets). The point at which the transition occurs is called the critical
point, which separates the two phases.

A phase transition corresponds to the spontaneous breaking of some symmetry,

so that the two phases must be described by two different functions which can not be

ana’yticz
symmetr
order pa
above tl
therrnodj
the order
non-zero
“lli be 5:
P:

“thh a ,
EPPTOachi
.‘tial re,-
[hemSelye
Elle rise
discussed
Th
applod't‘he
amlical e
as 3,, Cr.
Blimprmic

lemma“)

29

analytically continued across the critical point. In the ferromagnet example, a rotational
symmetry exists above the Curie temperature, and is broken when the spins align. The
order parameter, is (generally) a thermodynamically measurable quantity which is zero
above the critical point and non-zero below and is introduced to describe the
thermodynamics of the phase with the broken symmetry. In the ferromagnet example,
the order parameter is the magnetization, which is zero in the paramagnetic phase and
non-zero in the ferromagnetic phase. Anticipating future results, the order parameter
will be symbolized by q.

Fluctuations measure the range of correlation functions; i.e., the distance over
which a disturbance - like a spin wave - ’remembers’ its initial state. As the system
approaches a phase transition, fluctuations in the order parameters extend over large
spatial regions and persist for very long times. Mathematically, the fluctuations manifest
themselves as singularities. The time dependent properties of the critical fluctuations
give rise to the dynamics via the fluctuation - dissipation theorem, which will be
discussed later in the context of ESR theory.

The divergent behavior of a thermodynamic quantity as the critical point is
approached is described by scaling, and the nature of the singularity is characterized by
a critical exponent. The mathematical definition indicates that the behavior of a function
as the critical point is approached is given in terms of a limit, so that f(x) is
asymptotically proportional to x‘. This asymptotic limit may not exist, but the

accompanying formalism has been shown to describe experiment very well. We say that

where Ax «

Crir
ariety of r
they belong

relaxed to n

0r the com

Table 3-1 s

Critica| Slt

3O

lim lnf(x)
x ~ Ax " - — = A 3-1
f() ( ) Ax,” [mum
where Ax ~ (x-xc) and x, is the critical point.

Critical exponents have proven to be exceptionally useful. In many cases, a large
variety of materials can be characterized by the same set of exponents, so we say that
they belong to the same universality class. Many of the thermodynamic quantities are

related to the reduced temperature, t, which is defined by
t = T- Tc
TC

or the correlation length, 5, which measures the range of a pair correlation function.

3.2

 

Table 3-1 summarizes and defines some of the common critical exponents.

Critical Slowing Down

As the critical point is approached, a system will take increasingly longer times
to reach the equilibrium state due to the presence of fluctuations. The description of this
process is based on the thermodynamics of irreversible processes, and is called the
thermodynamic or conventional theory of critical slowing down”. Physically, critical
fluctuations of various sizes (with the average size being equal to the correlation length)
are always excited. These excitations can then split up into smaller excitations, merge

with other excitations, or dissipate into random heat, etc, with the available dissipation

Syn

\-\,.\.,\.\..\-\\

 

31

Table 3-1: Summary of Critical Exponents

 

   

i ' iii-V'"t.‘ii’-D¢fll“ifi°n  {'.53-:35: - 

 

Specific Heat

 

Order Parameter

 

Susceptibility

 

Correlation Length

£~ t"

 

Equation of State

H ~ M8 (t=O)

 

 

 

Relaxation Time

 

r~£z

 

 

paths deter
used to do
approach t

omens

Where z is
critical poir
Of an ordc
mfléureme

reElation l

NOE that o

exDO“em.

32

paths determining the relaxation time of the system. Critical slowing down has been
used to describe the behavior of ordered ferromagnets and antiferromagnets as they
approach their ordering temperatures-”’9'“. Dynamic critical scalin g predicts that the
characteristic relaxation time of a system, 1, goes like
_ ’3'
1: ~ [-13] 3—3

7}

where z is the dynamic exponent, and u is the correlation length exponent. As the
critical point is approached, the relaxation time will become infinite, indicating the onset
of an ordered state. Care must be taken in relating this equation to experiments, as
measurements usually measure an average relaxation time, and not a characteristic
relaxation time. The average relaxation time is related to an average exponent, zm by

_. -z,,v
{av or (:1, J 3-4
Tr

Note that the scaling laws are written in terms of the characteristic relaxation time. For
spin glasses, Ogielski‘“ has shown that 2,, = z - B/v where B is the order parameter

exponent.

finite Si;

therefore.
M am
l'Olei‘ c
are aliou
An exam
3‘1. TV]:
line shcm

T

33
Finite Size Effects on Critical Exponents

A phase transition can occur only when the thermodynamic limit (Nam) is taken;
therefore, a finite system will display a rounded cusp with a finite height, instead of a
peak, and the position of the cusp will be displaced from the position of the infinite
volume case. The simplest example is that of a system in which all dimensions but one
are allowed to become infinite. The size of the finite dimension will be denoted by L.
An example of the behavior due to the finite size on the specific heat is shown in Figure
3-1. The dashed line shows the specific heat in the infinite volume limit, and the solid
line shows the results for a finite system.

The difference between the position of the peak in the finite case and the position
in the infinite case is measured by e, the fractional shift.

T. -T,,.(N>
T

C

€(N) = 3-5

We can also define a temperature, T’, at which the first deviations from the infinite limit
occur (indicated by the arrow in Figure 3-1), so that we may further define a fractional
rounding, 6(N), where

T.(N) -Tn(N) 3'6
T

C

 

5(N) =

 

 

 

 

 

 

FlEuro 3.];

3161mm]-
EYOUnd Slate
energy is ll

Heighbws at

(limeogmwi

34

 

Specific Heat

 

 

 

Tea.) T,,(oo)

 

 

 

Figure 3-1: An example illustrating finite size efi’ects on the specific heat

The magnitude of the critical temperature is a measure of the overall strength of
the interactions. In a naive mean field picture, k,,Tc is directly proportional to the total
ground state energy. As an example, for a finite nearest neighbor Ising model, the total
energy is that of the infinite system minus lost coupling energy due to the lack of

neighbors at the surface.

N151; « E(N) - dJN — JN" M

where N" is the number of boundary spins, J is the exchange, and d is the

dimensionality. The shift will obey

35

where A is called the shift exponent. The correlation length, 5, which is a measure of

the range of the correlation function, diverges with the reduced temperature as

5(7) == t" 3'9

When the correlation length for a finite sample approaches sample size, finite size effects

should be observed. A and v are (in general, but not always”) related by

1 = 1 3-10
V

NIEAN FIELD THEIRY

The Edwards-Anderson Model

The Edwards-Anderson (EA) model”, was the first to apply standard mean field
theory to the spin glass problem. This model considers classical spins placed on the
vertices of a cubic lattice, with only nearest neighbor interactions. An equal likelihood
of ferromagnetic and antiferromagnetic interactions is assumed, with the interaction

strengths given by a Gaussian distribution. This model successfully predicts a cusp in

in my

5133.7 m:
ukhfl

Edua'ds

A»
“linen

Ckof:

Shag

If] 113an
C

fimah

36

the susceptibility, but also predicts a cusp in the specific heat, which is not
experimentally observed.

Solving the mean field problem for spin glasses requires a quenched average, in
which the log of the free energy instead of the free energy itself must be averaged.
Edwards and Anderson used the ’replica trick’, in which the partition function, Z, is

written

Z"~l+nan 3'11

so that the quenched average, denoted by an overbar, can be written

 

‘7' lim [Z"-l] 3-12
n-vO

n

Use of the replica trick allows the calculation of the quenched average.
The EA order parameter is the spin correlation function q“, which correlates a

spin at time tl to its value at some later time, t2;

qEA = <§r(tr) ' §t(t2)> 3-13

In using the replica trick, n copies of the original system have been introduced. In

replica language, qEA can be interpreted as the correlation between a single spin, in two

difrerem rep?

definition 0ft

“here QM C I.

The Sherringr

37

different replicas. If the Greek indices (1 and 3 represent two replicas, an alternate

definition of q is:

4“ = (ff.§;p> “#35 3‘14
where q“ defined this way is called the overlap.
The Sherrington-Kirkpatrick Solution

Sherrington and Kirkpatrick (8K)”44 extended the EA model to the infinite
range case, in which interactions are allowed between all spins. The replica trick is used
to evaluate the quenched average, which allows calculation of the free energy and
thermodynamically measurable quantities. In the evaluation of the average over the
disorder, the replicas become correlated. Instead of one mean field parameter, as in the
EA model, the SK model introduces two parameters - qua and ma, where the greek
indices identify replicas. SK assumed that all replicas must be equal, so that these
parameters should display no dependence on the replica index. This assumption, called
replica symmetry, takes qafi = q (V (13) and ma = m (V 0:). These parameters are
defined by

ml = < 2 <Sl>>d = $2 "'1 3-15
1 N r

where m
thermal ;

' .
ran-go, 1
*¥§§A‘uint

\.

38

q = (’11?! 213332), = 2‘: m? 3-16

.1.
N l
where this q is the same as the Edwards-Anderson order parameter, < > indicates an

thermal average, and < > 4 indicates an average over disorder. The different phases

resulting from this solution are summarized in Table 3—2.

Table 3-2: Phases Resulting From the Sherfington-Kirkpatrick Analysis

 

 

 

 

q m A pr... 1 r
0 0 Paramagnet
96 0 0 Spin Glass
:6 0 rt 0 Ferromagnet

r

 

 

 

 

 

There are some difficulties with this solution. For T _<_ Tr, the spin glass phase
does not represent a minimum of the free energy — the paramagnetic phase has a lower
free energy. Even more troubling, at low temperatures, the entropy is negative, with
S(T=O) = -0. 16. Theorists have questioned the use of the replica trick, the order of the
analytic continuation of the replica trick and the thermodynamic limit, and the validity

of a mean field approach in general. Thouless, Anderson and Palmer (TAP)45 showed

that. at high
trick. but of
among other
assum tion 0

broken led tc

Replica Sym

39

that, at high temperatures, the same solution can be obtained without using the replica
trick, but obtained different results below T,. This led de Almeida and Thouless“,
among others, to the conclusion that the problem with the SK solution was the
assumption of replica symmetry. The crucial realization that replica symmetry must be

broken led to the current conceptual picture described by the hierarchical model.

Replica Symmetry Breaking

The replica symmetry breaking (RSB) solution of the mean field problem is rather
detailed, and only a brief outline will be given for the purpose of establishing a
conceptual spin glass picture. Excellent reviews of the details are available in other
sources".

Imagine that the qua can be arranged in a matrix, with the or index running along
the horizontal and the 6 index along the vertical. The diagonals of this matrix will be
zeros. Measurable quantities will be related to the trace of this matrix, and the trace of
the product of this matrix with other matrices. a and B both run from 0 to n, where n
is the number of replicas and eventually must be taken to O to recover the free energy.

In the first iteration of symmetry breaking“, Parisi suggested dividing the matrix

into blocks, so that the diagonal blocks form one set, and the off diagonals form another.

Elected:
El.

blocks a

M

”3.53311

Ur

40

Elements in the diagonal blocks are given the value q,, and those in the off-diagonal

blocks are given the value (10, as illustrated below:

Oq,q,qllqoqoqoqo
qloqlqllqoqoqoqo
qquOqllqoqoqoqo

4141410 Iqoqoqoqo

_

40 40 40
40 go go
go go 40

3-17

*—

40 90 40 901—0? @721:—

40 I 41 O 41 ql
40 I 41 q] 0 ql
qo I q, q, q, 0

Parisi then suggests49 that this procedure be repeated to infinite order. Each time, the

diagonal matrices further are subdivided, so that, on the second iteration, the matrix

shown in Equation 3-17 becomes

Oqzlqlqllqo qo qoqo
420 lqlqllqo qo qoqo

(11 (11 l 0
q, «11 l 9;
71-0— (Io-710
40 40 40
qo qo qo
40 co qo

42' 4., 4., <1qu
0 I40 40 qoqo
Fro—l WEI-5:9,—
qol 420 lqrqr

qolqlqll 042
qolqlqllqzo

3-18

Monte Carlo simulations“ of spin glasses predict an internal energy at zero

temperature of U(O) = -O.76 :1; 0.01. The replica symmetric solution of the SK model

results i
gites St

W) =

CZETEV I
5.

l'ltramt

>1

41
results in S(0) = -0. l6, U(0) = -0.798. The first iteration of replica symmetry breaking

gives 8(0) = - 0.01 , and U(O) = -0.765, and the second iteration gives 8(0) = -0.03 and
U(0) = -0.7636, indicating that the replica symmetry breaking solution converges
rapidly. In the limit of infinite iterations, the order parameter, qua becomes a function.
The free energy then becomes a functional of q(x), and can be maximized with respect
to this continuous variable. Various approximations can be made to calculate the free

energy for different temperature and field regimes.

Ultrametricity

Mezard, et. al.’°, and Mezard and Virasoro"1 related the hierarchical
organization of states to the calculation of statistical quantities. They find that, given any
three states, at least two pair of them will have the same overlap with probability one.
This is the characteristic of an ultrametric space: any three points in space will form an
isosceles triangle, with the two equal angles larger than or equal to the third one. For
any value of q, the states are organized into non-overlapping clusters, such that any pair
of states inside the cluster has an overlap greater than q. Each of these clusters can be
further decomposed into smaller clusters which group together states with overlap greater
than some q’, where q’ > q. This procedure can be repeated an infinite number of
times, indicating that the set of pure states obeys a hierarchical ordering. The

organization of hierarchical space is schematically pictured in Figure 3-2

42

 

 

Cluster

 

 

Configuration

 

 

 

Figure 3-2: Topology of ultrametric states

In this figure, the highest level will correspond to a cluster. If we define a pure
state as a state which cannot be broken down into a combination of other states, the three
vertices shown correspond to pure states, with q1 = q“. q2 will be equal to l and
correspond to different configurations within a pure state. To calculate the magnetization
at a given site, an average is taken over a large representative set of configurations
contained in the cluster.

The overlap between two states depends only on their closest common ancestor.
The higher up you have to go on the tree to find a common ancestor, the smaller the
overlap. All the pure states with a common ancestor share a common property; there
exists a partition of the N sites into disjoint macroscopic cells such that the average

magnetization of each of these states in every cell is the same, within some given

. 1“ :
Tfifltuit
o 3: ~
«1111.75 a

tilt: the

AT and

43

resolution. This has enabled the development of a rather complicated procedure for
taking averages using the hierarchical tree for the purpose of computing the free energy
(and thermodynamic quantities calculable from the free energy ) at any temperature and

field.

AT and GT Lines

The mean field picture has stimulated some exotic calculations based on the
hierarchical structure of states, but has also resulted in some more conventional and
experimentally testable predictions. The question of what happens to a spin glass when
a magnetic field is applied has been addressed by both theoreticians and experimentalists.

Within mean field theory, two lines in the magnetic field - temperature plane
describe the effect of the field on the spin glass transition temperature. De Almeida and
Thouless‘“S (AT) predicted that the dependence of the freezing temperature on field, for

the Ising mean field model, would be described by

 

Tr" § 3 19
HAT(T) «x T ‘
f

Above this line, the free energy is smooth with a single valley as found in the free
energy of a paramagnet, while below this line, the free energy becomes the complicated
free energy surface of a spin glass. The time the system takes to transit between any two

valleys is macroscopic, so we see irreversible behavior associated with the onset of this

state. The A

the paramagn
For H

the componcr
Gabay and T

ShOU‘rd (X‘Cu r

The longitudir
Claimed that tl
replica Sl‘mmc
pan to “Perth
freezing of [he
be thought 0f
i”filerstbte b:
irreversibility

existence of a (

44

state. The AT line marks the transition between the reversible behavior associated with
the paramagnetic phase and the irreversible behavior due to the spin glass phase.

For Heisenberg spins, the presence of a field selects a preferred direction, so that
the components of the spin longitudinal and transverse to the field will freeze differently.
Gabay and Toulouse’2 (GT) predicted that the freezing of the transverse components

should occur at the GT line, which is defined by

 

T-T-l
HGT(T) «( fr ] 2 3-20
f

The longitudinal components will freeze at the AT line. Gabay and Toulouse originally
claimed that the region between the GT and AT lines represented the region where the
replica symmetric solution is correct, but this has been shown to be incorrect”, due in
part to experimental indications that the longitudinal components will be affected by the
freezing of the transverse components. For Heisenberg spin glasses, the GT line may
be thought of as the line marking the transition from reversible behavior to weakly
irreversible behavior, and the AT line as indicating the transition from weak
irreversibility to strong irreversibility. Little experimental evidence exists for the
existence of a GT line in real materials. Anisotropy which couples the longitudinal and
transverse degrees of freedom (like the DM) has been predicted to be responsible for a
crossover from Ising-like behavior in low fields to Heisenberg-like behavior in high
fields. This has been experimentally observed by Fert and Levy“ and predicted within

mean field theory by Kotliar and Sompolinsky”.

Dinamlcs “'ilh

Dynamic
hierarchical app
below Tf. and “
introduces dynar

The SK
miticially introd
were used in the
the autocorrelatit
doun with a dec.

Sompolin
equations for the
Which favors [hp
calculation GSche
cll'iation of moth
which is the Tegp'

 

of the system is

 

 

45
Dynamics Within the Mean Field Model

Dynamics in the mean field model can be broken into two approaches. The
hierarchical approach uses the mathematics of ultrametricity to predict the dynamics
below T,, and will shed no light on the present investigation. The second approach
introduces dynamics into the SK model and can be used above T,, as well as below.

The SK model has no natural intrinsic dynamics, so the dynamics must be
artificially introduced, usually within an equation of motion approach. Glauber dynamics
were used in the SK paper to predict that the susceptibility should diverge as t", and that
the autocorrelation function (C(t) = < S(t) S(t+1) >) should experience critical slowing
down with a decay rate pr0portional to t”.

Sompolinsky and Zippelius"5 used a self consistent formulation of the Langevin
equations for the relaxation of soft spin (S is a continuous variable in a symmetric well
which favors the values 1 1) fluctuations in the Edwards-Anderson model. Their
calculation eschews the use of replicas, relying on linear response theory to generate an
equation of motion for the spin variable in terms of a propagator (Green’s Function),
which is the response function of the spins to the field. Above T,, the low frequency
spin fluctuations will be characterized by a generalized damping function. The relaxation
of the system is predicted to diverge at a rate t‘. Their results above T, are in agreement
with the results from the SK model and calculations on generalized short-ranged mean

field spin glasses. The relaxation time may be found from

there C(t) is ‘

46

=_1_.. _ 3-21
I 1_C_[[C(t) C]dt

where C(t) is the spin spin correlation function. The average relaxation time is given by

1:” = fP(t’) t’dr’ 3'22
o

where P(r’) is a probability distribution of relaxation times.

Using dynamical scaling at zero field, Sommers and Fischer-‘7 predict zv = 2.
Use of the mean field value of v = 0.5 predicts z = 4. The average relaxation time
diverges with z,,,v = l, predicting that 2,, = 2. When a field is applied, the exponent
zvaries from4atH = 0,T =T,toS.0angoestoinfinityanthoO. Ina
Heisenberg spin glass, Fischer” finds that z keeps its universal value of 4 in non-zero
field along the entire GT line. Dynamic critical exponents for spin glasses are
summarized in Appendix B.

Experimentally, the hierarchical picture has been used to predict the decay of
remanent magnetization and ageing phenomena below T,. Most of these experiments
focus on cooling the system through the freezing temperature in a constant magnetic field
to some final measurement temperature. As the cooling takes place, the number of states
increases rapidly as a result of the formation of the multivalleyed free energy; the spin
glass is ’caught’ in a large number of metastable states. The number and properties of

the states have been calculated within the hierarchical model. Various phenomena can

be examined by

.. t'ore making t:

 

after the meast;
predicts that t'tel
it takes for the 3
field. As the

temperature. rte
change the mea
are concerned

hierarchical mc

W

47

be examined by varying the magnitude of the cooling field, the amount of time elapsed
before making the measurement, the length of time the cooling field is allowed to persist
after the measurement temperature has been reached, etc. The hierarchical model
predicts that field cooling decreases the barrier heights between states, so that the time
it takes for the system to change from one state to another should depend on the applied
field. As the amount of corrugation in the free energy increases with decreasing
temperature, new metastable energy minima are constantly appearing, which should
change the measurable quantities of the system. The experiments presented in this thesis
are concerned with the regime above and nearing Tf, so the predictions of the

hierarchical model below Tf will not be further considered.

THE DROPLET MODEL

The droplet model was first proposed by McMillan59’6", and Bray and
Moore61’62. Fisher and Huse“""“ (FH) have formalized and extended this model,
making predictions which differentiate between two dimensional and three dimensional
behavior. Fisher‘56 has noted that, contrary to conventional dynamic scaling, where one
assumes that the relaxation time of a system with fluctuations on scale L goes like I},
the nature of disordered systems implies that the natural log of relevant parameters and
not the parameters themselves are the proper scaling variables. In ordered systems, the
singularity in the free energy which causes a critical point is due to competition between
energy and entropy. The contribution of entropy to the free energy is on the order of

T, but since the characteristic scale of the free energy is also of order T, there will in

effect he no
behavior of
ertelg.V Mm
base region
is very wide
factor. the d:
The l

piaced Ising
represent the
describe the n
presented is 01
coherent collet
group of spins
exci'ations will
short spatial et
5ehavior, will :
W8. The t
thermodynamic
The FH ,
ft"Toniagnet, 0 i:

and argue the

 

48

effect be no barriers, and no activated behavior. Barriers will exist in systems where the
behavior of the free energy is caused by competition between two different types of
energy with at least one energy being random. Since every quantity on the scale L will
have region to region fluctuations due to randomness, the distribution of barrier heights
is very wide. If the time for surmounting a barrier is proportional to the Boltzmann
factor, the distribution of barrier heights should result in activated scaling.

The Fisher-Huse model starts with an Edwards-Anderson lattice of randomly
placed Ising spins with short range interactions. In the same way that spin waves
represent the low energy excitations for ferromagnets, FH developed an ansatz to
describe the nature of the low lying excitations in spin glasses. The conceptual picture
presented is of a collection of non-overlapping droplets. A droplet is a minimum energy
coherent collection of spins with one phase, separated by a domain wall from another
group of spins which differ by a universal spin flip. The majority of the low energy
excitations will be high frequency excitations involving small groups of spins, and having
short spatial extent. The low frequency excitations, which will dominate the long time
behavior, will involve the coherent flipping of a droplet containing a large number of
spins. The thermodynamic properties of the system are determined by how
thermodynamic quantities scale with droplet size.

The FH Ansatz is that the free energy of a droplet of size L scales as L’. For a
ferromagnet, 0 is equal to d-l, but in a spin glass, frustration will reduce the exponent,

and FH argue that

The distributit
median. and 5.

If 0 <
some length 3i
will prevent 3p.

and the Param.

e s51 3.23

The distribution of droplet free energies will have a width which is comparable to its
median, and scales like L‘.

If 0 < 0, large, low energy excitations at T = 0 will be numerous, so that, on
some length scale, there will be a large density of quasi-independent excitations which
will prevent spin glass ordering. In this regime, the transition temperature must be zero,

and the paramagnetic correlation length, 5 +, is given by

1

5* ,, (lTJI—fi 3-24

which allows us to identify it = (NO).

If 0 is positive, then very few of the large scale droplets will be thermally
activated at low temperatures. The distribution of droplet widths is very broad, so there
will always be some droplets which will be active at any temperature, and these droplets
will dominate the equilibrium measurements. Low lying excitations will occur on all
length scales greater than the correlation length.

Relaxation of the low lying excitations is hindered by the presence of energy
barriers. The height of barriers to relaxation of a droplet of length scale L are assumed
to scale like L“, for L s E and where d-l 2 ill 20. The relaxation of these modes is
proportional to the Boltzmann factor. The characteristic relaxation time of the whole

system, 1, will be due to those droplets at scale E; at scales greater than 5, the effects

of comp). red'u
The relii‘lauon

be t‘oufid in a 2

forT > Tr: am
The Behavior 0

The Fish
glass systems it
relaxation time (
than the correlat
the measurement

or the cusp bein':

 

 

 

Decreasin

behavior. 0
T {C

 

50
of entropy reduce the barriers, and the regions of size E will flip roughly independently.

The relaxation time of the full system (assuming a zero transition temperature as would

be found in a 2D system) satisfies;

Int ~ 3-25

 

Tl ’Vz":

for T > T,, and where subscripts on the exponents denote their dimensionality.

The Behavior of Thin Films Within the Droplet Model

The Fisher-Huse theory has been modified‘53 for application to multilayered spin
glass systems with spin glass layer thicknesses W50. On time scales shorter than the
relaxation time of the system, only fluctuations of the system on length scales shorter
than the correlation length will be measured. A cusp in the susceptibility results when
the measurement time, tn, becomes shorter than the equilibration time, with the position

of the cusp being given by

7',(:_) fl W“,

T

C

326

 

 

(., * '1'263)]-(l + v2’2)

 

Int

Decreasing Wso results in a crossover between two and three dimensional

behavior. For temperatures far below T,, the relaxation time at scale s is

which results i

hme,

This PTEdiction
W50 13’ ger thar

Near T,.

0

ll'tl

~7r

is? size scalii

51

(*3 ” '2 V293)
WSG

TO * ’2":)

Int .. 3-27

 

which results in a measured freezing temperature which depends on the measurement

time,

1,0,.)

T8

~

 

 

l
w'3 * W’3 7'7; 3-23
ln(t")

 

This prediction assumes thin layers, TAWSG) much less than the bulk value T,(oo), and
Wm larger than the average separation between Mn impurities.
Near T,, the relaxation time of the film will be governed by conventional dynamic

finite size scaling,

, ,, W33 3-29

where h is the three dimensional dynamic critical exponent. (In the case that dynamic
critical scaling doesn’t apply, we can simply write 1(WSG,Tg) as a general function of
W3C, and the true freezing temperature, T,,). The relaxation time of the system must be
less than the measuring time, or the system will fall out of equilibrium before the
correlation length reaches the size of the film. The relaxation time of the system is given

by

or

In [her

by

The Effect- Of 1‘

 

 

52

 

 

ln(1: t“) ~ (t’e Wm)" ‘ ”2%": 3-30a
or

t ~ t"'"’ exp[ tvam + emu) W32“ + Gama] 3.30],

In the critical region, the shift of the freezing temperature from T, will be given
by
_ '7 [(‘Vs '* V2'2°3)"3]-1
T‘ T’a‘") ~ W” 3 In t" 3—31
T8 W8”;

 

 

 

 

which introduces logarithmic corrections to the measurement of v3.
The Effect of Magnetic Fields

For three dimensional droplet models in an applied field, three phases are
predicted; the first is a paramagnetic phase, as for the Ising case. The second phase
occurs when the symmetry in the direction of the field is broken, but the rotational
symmetry about the field still exists, in analogy to the GT line in mean field theory. The
third phase is when symmetry is broken in all directions. Only two ground states are
expected to exist.

In the Fisher-Huse model, the presence of a small applied field will suppress the

freezing transition. The relaxation times will grow, but not diverge, as Tf is approached.

The broad dist
temperature TA
time scales lon,i

is predicted to l

Which is only a
enough, The l
1“ magnetic fie
U‘kaTJ. the

ailllreciable tht

W

Griffithsr

n .
Onanalyttc fUn C

i
43mpf-‘ralllf‘e 0f l

that

 

of the PUre

 

ill
aCTOScOpiC re 3.

r k
elumen: Mm l
t

 

53

The broad distribution of relaxation times will result in freezing characterized by a
temperature T,(H,tm), below which the majority of the relaxation phenomena occur on

time scales longer than the measuring time. For small fields, the freezing temperature

is predicted to be

 

_ 2 «146
T8 2(HJ...) ~ Hy + 5 (1n("’to))“y * B) 3-32

8

 

which is only weakly dependent on the measuring time if the measuring time is small
enough. The logarithmic factor is a correction to the de Almeida-Thouless prediction.
In magnetic fields where the Zeeman energy is comparable to the exchange energy
(J ~ kBTg), the collective aspects of spin glass freezing should not be seen, and

appreciable hysteretic phenomena will occur only at very low temperatures.

THE GRIFFITHS PHA E

Griffiths67 showed that the free energy of a dilute Ising ferromagnet is a
nonanalytic function of the external magnetic field for all temperatures below the critical
temperature of the corresponding non-dilute system. When the temperature is between
that of the pure system and the ordering temperature of the dilute system, there exist
macroscopic regions of the sample which are ordered. Static measurements, which

represent average quantities, are incapable of detecting these regions.

PI".
ordering 1
thus dorn
ground st
lnfiequen
the size r
nmemm

J'Y,‘
Linalfi ma

anonetp

terromag

54

Physically, large ’rare’ regions which are characteristic of systems with higher
ordering temperatures will relax more slowly than the average of the system, and will
thus dominate the long time dynamics. These large clusters will lock into one of two
ground states (related by a global spin reversal) and will flip into the opposite state
infrequently. Regions of a particular size will dominate on a particular time scale, with
the size of the dominant regions increasing with time. The Griffiths temperature, To,
is the temperature of the corresponding non-dilute phase, which is higher than that of the
dilute magnet; T, 5 To.

In the Griffiths phase (T, < T < To), relaxation of these clusters will result in
a nonexponential form for the spin autocorrelation function, C(t). For a random Ising

ferromagnet“;

J. -
co) ~ eXPI-A(1nt)"‘] 3 33

and for a random Heisenberg ferromagnet69

1 3-34
C(t) ~ exp(-Bt 2)

Randeria, Sethna and Palmer"0 have considered the possibility of a Griffiths
phase in spin glasses. In a d-dimensional short range Ising spin glass, they find the
correlation function to be bounded below by the expression in Equation 3-34 in some
temperature range above the spin glass transition. The bound disappears to all orders in

perturbations about high temperature and dimensions, which is perhaps why there is no

evidence

 

55

evidence for this phase in mean field theory. They suggest the phase diagram shown in

 

 

 

T

 

 

 

Figure 3—3: Temperature - dimension phase diagram for Ising spin glasses (afier Ref 70)

The authors emphasize that the lower bound is strictly for an ideal system; in a
real system, sample inhomogeneities would enhance clustering, and the bound would be
swamped. The experimental observations of ordering above T; described in Chapter One
support the existence of an intermediate phase between the spin glass ordering and
paramagnetic phases.

Theoretical existence of this phase is found in Ogielsld’s‘" 1‘3 Monte Carlo

simulations of the dynamic behavior of short range three dimensional Ising spin glasses

regime

of tires

5630 as

(I,
:3
‘3...

“.3

56

in a magnetic field. Three distinct regimes are noted; Above the Curie point of the
nonrandom ferromagnetic Ising model, (Tc = 3.8 Tf), paramagnetic behavior is observed.
As the temperature is lowered, a different type of behavior is seen between 3. 8 Tf and
1.5 T,, in which short range correlations cause a change in the form of the decay of the
relaxation functions. The spin-glass correlation length and the correlation times in this
regime are very small compared to those in the spin glass phase, as is the rate of increase
of these quantities with decreasing temperature. From 1.5 T, to T,, critical behavior is
seen as the spin glass phase is approached. Ogielski finds z,,,v = 7.2 i 0.1 from these

simulations.

UMTVIARY

The hierarchical! mean field and droplet models present two very different pictures
of spin glasses. The biggest difference between them for our purposes is the
dimensionality of the models. The hierarchical model is a mean field model with an
upper critical dimension of 6. The droplet model considers experimentally measurable
differences between two and three dimensional behavior. The droplet model has been
extensively used in previous analyses of multilayered spin glasses.

The other major difference between the models is the effect of an applied
magnetic field. Within mean field theory, the AT and GT lines predict the dependence
of the freezing temperature on magnetic field. Experimentalists, as we shall see in the
next chapter, have claimed measurement of the AT/ GT line(s) in fields of up to

15,000 G. The droplet picture predicts a similar dependence of the observed Tf (to

ll tlll
measur-

ther

of the

JHCEfi

bfhay

aPPTO;

57

within a logarithmic dependence on measuring time), but with the caveat that this
measurement represents a falling out of equilibrium and not a phase transition. In the
Fisher-Huse formalism, the presence of any finite magnetic field will prevent a spin glass
transition.

The novelty of the spin glass phase and questions about the nature of the transition
have resulted in a lack of theoretical investigation of the temperature region above Tf.
The formation of clusters and the onset of short range order at temperatures far above
Tf is well documented experimentally and indicates that labeling this region as simply
’paramagnetic’ is incorrect. The idea of a ’Griffiths phase’, in which macroscopic
regions of a sample become correlated at temperatures above the ordering temperature
of the sample as a whole, has been suggested to explain the behavior in this regime. No
experimental evidence of a Griffiths phase in spin glasses has been reported.

This brief foray into spin glass theory may appear to have shown only that neither
of these theories will be of any use in explaining results above T,. A conceptual
understanding of these pictures is important in that any theory developed to explain the
behavior above Tf must be consistent with one of these interpretations as T; is

approached.

rro_
Ex}
(/'
K3

lTestsi
frequ,

of mi

dlmEr

CHAPTER FOUR
REVIEW OF PREVIOUS STUDIES OF

MULTILAYERED SPIN GLASSES

INTRODUCTION

This chapter summarizes the current understanding of multilayered metallic spin
glasses. Extensive measurements of Tf in multilayers and subsequent analysis in terms
of both finite size and dimensionality effects are reviewed, and the picture of the
transition from three to two dimensions is established. The importance of measuring
frequency and field will be investigated for relevancy to the ESR study. Measurements
of multilayered semiconducting spin glasses and a two dimensional Ising spin glass will
be compared with results from metallic multilayers. The final section of this chapter will
summarize the current understanding of finite size and dimensionality effects in
multilayered metallic spin glasses.

An important question in the study of spin glasses is the value of the lower critical
dimension (LCD). The LCD is the dimension below which a phase transition can only
occur at zero temperature. Computer simulations28 indicate that the LCD for long range
RKKY spin glasses should be between two and three. Advances in thin film fabrication
and SQUID technology have made possible the creation and measurement of spin glass
samples on a nanometer scale. In long range ordered magnets, like ferromagnets, finite

size effects are not evident until sample size is on the order of monolayers, due in part

58

to the

systen

much

59

to the strength of the spin-spin interactions, and the non—random nature of the spin
system. In spin glasses, the complex interactions cause observable finite size effects at
much larger layer thicknesses. Further reduction of the size of the sample can reduce
the dimensionality of the material. These properties make thin film spin glass structures
ideal for investigation of the LCD and finite size/dimensionality effects.

As the spin glass thickness decreases, the corresponding magnetic signal becomes
too small to measure with standard susceptometers. This necessitates the fabrication of
multilayers - structures which separate the spin glass layers with buffer layers of
nonmagnetic material of sufficient thickness to prevent coupling between different spin

glass layers.

MULTILAYERED METALLI PIN LASSES

Kenning, et. al. ,71'72 fabricated the first multilayered CuMn structures to
experimentally determine the lower critical dimension (LCD). Details of the fabrication
and characterization of these samples will be given in Chapter 6, as the same techniques
are used in this thesis. Samples were fabricated with spin glass layer thicknesses, W30,
ranging from 1000 nm to 2 nm, with Mn concentrations of 4, 7, l4 and 21 atomic
percent. Interlayers of Cu and Si were studied, with experimental results indicating that
30 nm of Cu and 7 nm of Si are sufficient to prevent interlayer coupling. Measurements
were made in a SQUID susceptometer in magnetic fields of 100 - 200 G and measuring

times of 300-400 seconds. The freezing temperature from these quasi-static

llifii

Pin

the
the

TA

057$

bestJ

60

measurements is extracted from the position of the cusp in the temperature dependent

susceptibility.

Finite Size and Dimensionality Effects on T'

Both finite size and dimensionality effects have been observed in the behavior of
the freezing temperature as a function of layer thickness. Finite size scaling predicts that

the freezing temperature of a sample with spin glass layer thickness Wso (represented by

T,(WSG)) is given by

Tf(oo) - T/(WSG) ~ W82 4.1
1,0»)

 

where x is the inverse of the correlation length exponent, u. Although data over the
entire range of layer thicknesses can be fit to Equation 4-1, finite size scaling should be
valid only in a small region about the bulk value of T,. Fitting the data from all
concentrations to a single exponent results in a value of A = 0.7 :l; 0.05, which implies
a value of u = 1.4 i 0.1. By plotting log “(Wm/)Txoofl vs. longSG], two regimes
are observed, separated at about 20 nm. (Figure 7-4 shows an example of such a plot.)
Based on this observation, and the frequency dependent data which will be discussed in
the next section, Kenning, et. al. chose to use finite size scaling for samples with Wso
2 20 nm, resulting in a value of v = 1.1 :t 0.3. This value can be compared with
other measurements of 1: shown in Appendix B. If the correlation length is defined to

be the sample thickness at which size effects are first noticeable, Kenning, et. al. find

50m

"T
P.
'19

intl

Poi..-
. 'J.

atn

61
E = 100 nm. The normalized plot of TKWSG)/T,(oo) vs. layer thickness shows that

concentrations from 4 at. % to 14 at. % all exhibit the same behavior, indicating that
larger concentrations do not display markedly increased ferromagnetic interactions.

One prediction of finite size scaling - a rounding of the divergence - is not seen
in these samples. Sputtered samples may be sufficiently strained so that the cusp is
already broadened beyond the point where rounding might be seen. Alternatively, the
rounding may not be observed because the feature in the susceptibility is a cusp and not
a true divergence.

Below the 20 nm length scale, the data can be analyzed using the Fisher-Huse
prediction for the freezing temperature for small layer thicknesses as given by Equation

3-31. We can write this as

T/(Wso) ., W4 4-2

 

where a = (¢3+¢2u203)/(1+v2¢2) and the subscripts on the exponents indicate
dimensionality. Fitting to 2 5 W30 5 20 nm, a value of a = 0.5 i 0.1 is obtained.
An extension of this study to thinner samples and including data for AgMn73 finds a
revised value of a = 0.8, which is in agreement with the requirement from the Fisher-
Huse theory that a 5 ll! 5 2. Stubi, et. al.74 have shown that similar behavior is seen
in AgMn/Ag, AgMn/Cu and CuMn/Ag as is seen in CuMn/Cu, suggesting that these

structures belong to the same universality class.

 

Freq

iiiat 1h

62
Frequency Dependence of the Freezing Temperature

At high temperatures, spin glass susceptibilities obey a Curie law. At some
temperature (as high as 5 T,,”), deviations from this behavior appear as the relaxation
times begin to increase. Below this temperature, the susceptibility is frequency
dependent, and the magnetization exhibits irreversibility and waiting time effects. The
’static’ or equilibrium susceptibility still follows the Curie law, but longer and longer
times are needed to experimentally measure this behavior. The ability to measure the
susceptibility over a wide range of frequencies provides a powerful means of comparison
with theory. We have seen in Chapter 3 that standard phase transition dynamics predict
a critical slowing down of the relaxation times, so that the maximum relaxation time goes

like

 

_ "ZV
, ”of; Te) 4—3

where T, is the freezing temperature measured at frequency w = 1/ 1m, and T: is the
equilibrium freezing temperature. The Fisher-Huse thermally activated dynamics predicts
that the frequency dependence of the freezing temperature should behave like

_ 4v
111(2) .. (Tf :8) 4-4

to T,

ln tl
Fret
seco
l'nit

oil

Semi
Zl’ V:

of t:-

63
In the case of two dimensional behavior, Equation 4-4 is modified by setting T, = 0.

Frequency dependent susceptibility measurements with measuring times from 10“ to 10‘
seconds have been made on multilayered CuMn/Cu samples by the group at Uppsala
University”. Zero-Field Cooled (ZFC) measurements were made at measuring fields
of 10 G. As the observation time decreases, (increasing frequency), the location of the
cusp moves towards higher temperatures and broadens in all layer thicknesses; however,
the magnitude of these effects in thick (W$0 = 1000 nm) samples are only about 20%
of the those seen in the thin (WSG = 3 nm) sample. The frequency dependence can be

typified by the quantity k;

”I 4.5

k = .1. __
T, d (logmt)
For the 1000 nm film, k has a value of 1/200 at t = l s, in agreement with
measurements of bulk (not sputtered) CuMn films. In the 3 nm film, k was found to be
1/40. This is in qualitative agreement with Monte Carlo simulations on 2D short-range
Ising spin glasses"5 which found k to be approximately 1/20, and 3D simulations",
which predict a value of k = 1/60.
If the dynamic behavior of the films is analyzed in terms of Equation 4-3, the
1000 nm film is fit with 2:: = 9 :1; l, and T, = 66 i 0.2 K, where To is 1013”"
seconds. Fitting the 3 nm data to this expression, however, results in a value of
2» = 19, which suggests that standard critical slowing down is not a good description

of the dynamics of thin spin glass layers.

64

Fitting the data from the 3 nm sample to Equation 4-4 with a zero T, results in
a value of 11w = 1.6 i 0.2, with To = 10‘3“” seconds. The value of l+¢v is in rough
agreement with the value of 2 found from Monte Carlo simulations75'7". The data may
also be fit to Equation 4-4 with a finite transition temperature. For 0 < T, < 10 K, W
is between 1.1 and 1.6.

These measurements have been extended to samples with WSO from 2 nm to
1000 nm. For W80 5 4 nm, fitting to the activated dynamics with zero freezing
temperature yields a consistent value of W = 1.6 i 0.1. If this fit is continued to
thicker samples, the exponent grows anomalously. Samples with WSG s 50 nm can be
fit to Equation 4-4 with the same exponent of (it: = 1.6 -l_- 0.1. These measurements
have been important in confirming the change in the dimensional character of

multilayered spin glasses as the spin glass layer thickness is reduced.

Other Measurements on Multilayered Spin Glasses

Gavrin, et. al.", have investigated sputtered multilayers of CumMnQog/A1203.
The choice of A1203 as the interlayer material is designed to eliminate concerns about
interfacial alloying, transmission of the RKKY interaction across layers, and to simplify
characterization. WSG ranged from 4 nm to 125 nm, with the A1203 layer having a
constant thickness of 7.5 nm. Sample composition was confirmed by microscopy and
x-ray analysis.

The values of T, from susceptibility measurements were fit to Equation 4-1, with

an exponent of A = 0.64 i 0.07 (v = 1.6) describing the data for all thicknesses. DC

 

measure:
from 10v
vs. thick
8.8 nmt
suscepti‘:
for all s

Values c

65
measurements of nonlinear susceptibility, in which the linear susceptibility is extrapolated

from low field measurements, show distinct differences in the scaling behavior for thin
vs. thick samples. The crossover exponent, 6, is found to be about 7 in thin (6 nm and
8.8 nm) samples, and 4.4 in the bulk. The value of u obtained from the nonlinear
susceptibility measurements is 3.4 in thin samples, and 1.6 in the bulk film. B as 0.9
for all samples. The value of n is 0.5 in thin samples and 0.11 in thick samples. These
values can be compared to values in Appendix B.

Measurement of the nonlinear susceptibility in CuMn/Cu multilayers has been
made by the Uppsala group79 using an ac technique at low fields, and a dc technique
at high fields. The measurements were made on a 2 nm Cuo_87Mno_,3/Cu sputtered
sample, and critical exponents different from those of Gavrin,et. al. were reported. The
extracted exponents were: 7 = 9.3, v = 4.5, d = 0 and n = 0, some of which are quite
different from those found by Gavrin, et. al. Our experience with dc nonlinear
measurements has indicated that the extrapolative determination of the linear
susceptibility is not straightforward, and we would expect the more sensitive ac

measurements of the Uppsala group to be more accurate.

Finite Size Effects in Metallic Films

The anomalous Hall effect, which is a measure of the magnetization of the
impurity spins, has been studied in flash evaporated Auoer,ml and Au0,9,Feo_0, films
varying from 40 nm to 4 nmm‘. The strong spin-orbit coupling in AuFe results in

an anomalous component in the Hall resistivity when an external magnetic field is

less

Elle

66

applied. T, is determined from the maximum in the temperature dependent Hall
resistivity. The Hall measurements are made by zero field cooling the sample, then
applying a measuring field between 100 G and 500 G. As the sample thickness
decreases, the freezing temperature decreases, with larger measuring fields broadening
the cusp. The freezing temperatures are fit to Equation 4-1 ,with an exponent of
v = 1.6 i 0.4. A deviation from finite size scaling was seen in films with thicknesses

less than 10 nm, but was not analyzed in terms of possible dimensionality effects.

Effect of Magnetic Field on T, in Metallic Multilayers

Arguments have been made for a DM anisotropy induced crossover from Ising
to Heisenberg-like behavior in bulk spin glasses as the magnetic field strength is changed.
Experimental measurement of the nonlinear susceptibility82 in bulk AgMn and CuMn
has been analyzed using anisotropy as a scaling parameter. Two sets of critical
exponents result, one of which is ascribed to ’Ising like’ behavior and results from low
field measurements, and the other of which is labeled ’Heisenberg’ behavior and is
observed in high fields. Torque measurements made by Campbell, et. al.83 on
AgomMno026 indicate that, for fields between 1 kG and 20 kG, the value of T,(H) does
not change by more than 10%. Similar results are cited for CuMn.

Kenning, et. al.“4 have studied the effect of magnetic field on the freezing
temperature in Cuo,”Mnom/Cu multilayers with Wso from 2 nm to 1000 nm in fields

from 2 G to 15000 G. The behavior in low field is described” by

67

 

 

1 2
2 - sun” 3 4-6
T0!) = 1(0) 1 — (El—)3 __
f f 4 k,7‘,(0)
and high field behavior by
you = 13(0) — 47

8

 

1 - ((n+2><n+1))%[ 8%” ]%

 

1:, no)

where, in both expressions, n = S(S+1), and T,(0) and T,(0) are the Ising and
Heisenberg zero field freezing temperatures. Kenning, et. al. find the behavior for all
W80 to be qualitatively described by these equations, although the prefactor of the Ising
expression differs from the experimental measurement by an order of magnitude. The
depression of T, is more pronounced in the Ising regime (from 2 G to about 500 G) than
in the Heisenberg regime (above 500 G). No dimensionality or finite size corrections

to this behavior are noted, and the crossover field appears to be the same for all W30.

MULTILAYERED SEMIQQNDUQTINQ SPIN GLASSE

Awschalom, et. al.“5 studied the dilute magnetic semiconductor Cd,_,,Mn,,Te, with
x = 0.069 to 0.20. Multilayered structures with CdTe interlayers were grown by
Molecular Beam Epitaxy (MBE) with spin glass layer thicknesses from 8.6 nm to
1.8 nm. X-ray diffraction and low temperature photoluminescence confirmed the layered
structure and concentration. The ac magnetic susceptibility was measured at 97 Hz and

a variety of temperatures. Contrary to the effects seen in metallic multilayers, decreasing

 

.. l‘ l.l Ill

1

h

A‘l

I.

68
W80 resulted in a broadening of the susceptibility peak, but no shift to lower

temperatures. The observation of rounding may be due to the low strain in MBE grown
samples, which may explain why this phenomena is not seen in sputtered films.

The interactions in CdMnTe are short ranged, with the nearest neighbor
interaction 5 times that of the next nearest neighbor exchange. This means that there are
a number of isolated single spins and also small clusters of spins which preferentially
interact with each other and not with surrounding spins. At low temperatures, the
paramagnetic clusters result in a l/T divergence of the susceptibility. The rounded
maximum indicates the onset of short range spin glass ordering, and the broadness is due
to the blocking of the establishment of long range order. This is interpreted as proof that
a 2D structure is incapable of supporting spin glass order.

Hysteretic effects were seen in the thinnest samples with the highest
concentration, with the warming susceptibility having a greater magnitude than the
cooling measurements. The hysteretic effects are due to the trapping of the system in
a metastable state; when the system is cooling, the establishment of short range order
creates preferential states. As the temperature continues to decrease, the energy barriers
increase, and the system is trapped. In the lowest concentration (x= 0.07), a 1/T
behavior is seen for all thicknesses, which indicates that when the system consists

primarily of small isolated clusters of spins, dimensionality is unimportant.

D

 

B

actit

if; "
77+

69
TW -DIMENSI NAL P LA

Dekker, et. al.87 have investigated the behavior of the two dimensional Ising spin
glass Rb2Cu,-,,CoxF4, with x = 0.22. AC susceptibility measurements over 15 decades
of time show frequency dependent behavior beginning as high as 2 T,, with an extremely
broad distribution of relaxation times. Fits to Equation 4-3 result in a value of zu = 15.
The activated dynamics of Fisher-Huse given by equation 4-4 with T, = 0, as
appropriate to a true two dimensional system, find (in; = 2.2 i 0.2 and
To = (2il) x 10'”. Use of the same equation with a finite T, resulted in a worse fit.
Ageing phenomena seen in this material88 are found to be qualitatively different than

those seen in 3D.

THE CURRENT PICTURE QF DINIENSIQNALITY AND FINITE SIZE EFFEQIS
IN MULTILAYERED SPIN GLASSES

Based on the static and the dynamic measurements, we have the following picture
of dimensionality crossover and finite size effects in multilayered spin glasses. For
20 nm 5 W86 5 1000 nm, deviations from bulk behavior in the static and dynamic
measurement of T, can be analyzed in terms of finite size effects. The correlation length
exponent, v, has a value of 1.1 i 0.3 when fit to samples in this thickness regime.

For thin samples, the behavior of the freezing temperature is described by the
activated scaling of Fisher-Huse, as given by Equation 4-4 and its zero freezing

temperature modification. Frequency dependent susceptibility measurements over 8

 

decades in ti r
for No dime

rind a crosso

 

 

 

 

The tr
1
decrease its

demonstrate t

 

70

decades in time show that T, = 0 for W30 5 5 nm, and provide convincing arguments
for two dimensional behavior. In the intermediate regime (5 nm 5 W86 5 20 nm), we
find a crossover behavior described by Equation 4-4 with a finite T,.

The freezing temperature is observed to increase with increasing frequency, and
decrease with increasing magnetic field; the former effect has been shown to

demonstrate dimensionality dependence, while the latter has not.

 

 

This c

   

ESR gluen in

summarizes t

 

ordered matel
0i ierro- and
lith'lllel to S]
magnels is c
EllOFt‘d to l

(Writ n

CHAPTER FIVE

ESR IN SPIN GLASSES

NTRODUCTION

This chapter is divided into three parts: the first part extends the basic theory of
SR given in Chapter 2 to the ESR of magnetic moments in metals. The second part
immarizes the experimental situation of ESR in spin glasses and other magnetically
rdered materials. Of special interest are results of linewidth and peak position studies
f ferro- and antiferro- magnets, as many of these materials behave in an analogous
tanner to spin glasses. The ESR of one and two dimensional ferro- and antiferro-
tagnets is discussed. The third section looks at theoretical explanations specifically
.ilored to ESR in spin glasses. Having provided the necessary background, Chapter

ive concludes with the motivation and goals of this experiment.

eneral Behavior

In spin glasses, the linewidth as a function of temperature, AH(T), has two trends,

)th of which are also seen in other materials. At high temperatures, the linewidth is

near in temperature and described by

71

 

E

l-

Whit

72

[AHU'HM = A + BT 5-la

As the freezing temperature is approached from above, the linewidth diverges as

a power law in the reduced temperature, t, with a characteristic exponent, x

T

[AH(T)]“ = c (Iii) 5-1b
f

The standard equation used in the analysis of ESR data above T, is therefore

 

T- r -s
AH(T) =A + BT +0[ T f] 5-2
f

ESR IN SPIN GLASSES: THEOR,_Y

This section begins with a theoretical explanation of the linear behavior common
to localized moments in metallic hosts, and is followed by a summary of the approaches
which have been used to explain the behavior near the ordering temperature in

ferromagnets, antiferromagnets, and spin glasses.

ESR

tag .
lmou

Oilsei

no
I I

{he Cor

73
ESR of Localized Moments in Metallic Hosts

The major applications of ESR in metals are; 1) determination of spectroscopic
states of ions in various hosts, 2) determination of relaxation times, 3) information about
the exchange interaction between localized moments, and 4) studying long and short
range magnetic order. The two measurable quantities we will be interested in are the
resonance peak position, H0 (related to the inverse of the effective g value) and the
linewidth, AH. Measurement of the g value provides information about the spectroscopic
state of the ion and the energy scale of the resonance, and measurement of the linewidth
provides information about relaxation phenomena.

The first experimental report of the ESR of localized moments in metallic hosts
was made by Owen, et. al.89 in 1956 in Moanom, where M is Cu, Ag or Mg. Mn“
impurities in non-metallic hosts show six separate resonance lines. Owen, et. al.
observed a much wider single line, with no fine or hyperfine structure, centered around
g = 2. As the temperature was reduced, the linewidth broadened, and the position of
the resonance peak shifted to lower fields. At the time, spin glasses had not been
identified as such, and the divergent behavior was explained in terms of

antiferromagnetic interactions.

ESR Equations for Transition Ion-Metal Systems

In the ESR of transition ions in metals, two spin components must be considered;

the conduction electrons, which are denoted here by a subscript ’e’, and the ions, denoted

 

alt?

l‘he

331d

74

by the subscript ’i’ . Consider first an assembly of conduction electrons and ions in the
approximation of immediate relaxation and far from any critical point. Taking all ions
to be equivalent (and all electrons to be equivalent), a simple relation exists between the
average magnetization and the effective field, so that we can write the static

magnetization of the ions and of the conduction electrons per ion (or per electron) as

Hi 5 xtHo = X?(Ho + All?) 5-3a
H, 5 1,17. = X201 + Afir) 5-3b

where A is the molecular field coefficient, given by

A = ____l2 (31(Rq) 5-4

2
8; 8. Fe

and the susceptibilities of the ions and electrons are given by

0 S 3+1
x, = (any 4,7,3 S-Sa

x2 - gamma) 5-5b

where p(E,,) is the density of states at the Fermi surface.

Re

 

75
Relaxation Rates

The relaxation processes in the ESR of localized moments in metals are governed
by the presence of three distinct components; the lattice, the ions and the conduction
electrons. In Chapter Two, we considered only one type of spin which could interact
with other spins of its own type and with the lattice. In the ESR of transition ions in
metals, there are four relaxation processes. The conduction electrons and the localized
spins can each relax to the lattice, with characteristic relaxation times T,,, and TiL
respectively. In addition, the conduction electrons and the localized moments can cross-
relax (with times T,, and T,,). T,,, the Korringa relaxation time, is a measure of how
quickly the spins of the localized moments relax to the conduction electrons. T,,, the
Overhauser rate is a measure of the reverse process. The Korringa and Overhauser rates
can be calculated by using a rate equation approach.

1 4 1t 2
7}, , (pl)

1 8
— = —rtcS 5+1 12 5-6b
T“ 3, ( )p
where J is the exchange coefficient and c is the concentration of magnetic ions.

Figure 5-1 schematically illustrates the relaxatidn paths available to these systems.

T,, and Tci are the relaxation times between the ion and the conduction electrons, and TcL

76

and TiL are the relaxation times between the spins and the lattice. TiL is very long

compared to the other times.

 

 
     
 

Conduction
Moments Electrons

 

 

 

1:4
g r '1 m r '1

iL eL

 

 

 

LATTICE

 

 

 

Figure 5-1: Available relaxation paths for transition ion-metal systems

We can identify three regimes: in the first two regimes, we can treat the
electrons as if they were in static thermal equilibrium. If the electrons relax rapidly to
the lattice (isothermal approximation), the linewidth is determined by the ion-electron
relaxation rate, l/Tg. In the adiabatic regime, the electron magnetization doesn’t

respond rapidly enough to follow the rf field, so that the resonance width is determined

3y the relative values of T,,, and T,,. In these two regions, we can replace the conduction

77

electron spin and number operators by their expectation values. This leaves us with an

equation of motion for the ionic magnetization.

as?
7?! = 3,11,57,41? + A”) = 81% (1 +Axe)M,xH 5.7
which introduces an effective g factor,
5-8

s = g..(1 + Ax.) = 3.0 + Ax? + A’xi’x.)

The first two terms are temperature and concentration independent, while the last
is proportional to concentration and inversely proportional to temperature. The g-shift
due to the second term is called the Knight shift and is observed for many metals with
Eu“, Gd“, and non—S state ions. In the adiabatic regime, both the first and second

order (in A) g-shifts are seen, while in the isothermal regime, the second order shift is

not observed. The linewidth is given by

AH~ =—1—+—l—1——6-]EA+BT 5-9
T,, T,, T

_1_
Tel?
where A is the linewidth at T = 0, called the residual linewidth, and B is the thermal
broadening coefficient. In theory, l/TiL is the residual (T = 0) linewidth, although in
practice, there may be other contributions to this quantity. The presence of the (l-0/T)

term is due to assumption of a Curie-Weiss behavior of the susceptibility. Note that this

expression results in a concentration independent value for B. If we cannot treat the

 

electrons
cannot In
of the SM
Pn
maintain e
directly 1c
relaxation
Of'spins [C
to me lat-f
Can 1mg"
bffu'een l
20 {he 10‘»
become .
“Gene
in gene

will be

W e\

78

electrons as being in thermal equilibrium, then T6L becomes important. If the electrons
cannot transfer energy to the lattice fast enough, the system is ’bottlenecked’ , and neither
of the shifts in equation 5-8 are observed.

Physically, we can explain the bottleneck as follows: the impurity spins want to
maintain equilibrium with the lattice, and there are two routes to do this; relaxation
directly to the lattice, and relaxation via the conduction electrons. We expect that the
relaxation of the spins directly to the lattice would be slow compared to the relaxation
of spins to the conduction electrons. If the conduction electrons can transfer their energy
to the lattice fast enough, the linewidth will be dominated by how fast the localized spins
can transfer energy to the conduction electrons (l/Tk). If the relative relaxation rates
between the spin systems are such that the conduction electrons can transfer energy back
to the localized moments faster than they can transfer energy to the lattice, energy can
become trapped between the conduction electrons and the moments, which results in a
bottleneck. The linewidth will be due to both sets of spins, as they are locked together.
In general, a bottlenecked system will not display a g-shift, so that the observed g value
will be close to that of the conduction electrons.

lHasegawa90 modified the Bloch equations to include the cross relaxation between

the elements in the system, resulting in the Bloch-Hasegawa equations.

d”. _ 1 1 g. 1 540
7 - gills”: X (H + AM.) ’ ("T—+3.2) 55?: + g-T—dbfi‘ a

 

Var
isothermal

large. we :

 

Where

and

79

 

am?
. = mm, x (mm) - _1- + _L m, + film, slob
dt T,, Ta. 3‘ T“

Various approximations are made to solve this system of equations. In the
isothermal regime, (1/Td) is very large and the electrons relax quickly. If (gi/g,) is very
large, we recover the adiabatic regime. In the strongly bottlenecked limit, gi ~ go and

T11. 22 TcL, with the rates of relaxation to the lattice both small, the equation of motion

may be written

 

d A? +1?
_(___.___.2 .. guamnmx g - 1501.11.19 5-11
dt T
where
g = ggxg + 81x] 5_12
x; X}
and
_1_,o .. LXO‘
1 _ I", ‘ Ta ‘ 5-13
Td x: + xi

In general, the host susceptibility is much smaller than that of the magnetic ions

(x," > x,"), the ion g factor is equal to the effective electron g-factor, and we find that

the effective relaxation rate (and hence the linewidth) goes like

 

where B is
to Equatio
but is diff
mine. it

systems. s

account it

line 3th

ReSidUa

Wm;

311101

1*.6 C;

80

+

l._1_ _1- .1. _
T Til. Tel. TIL Tel. ie

:2er

+ T“ [7:] a A+BT 5-14

where B is inversely proportional to the concentration of magnetic ions. This is similar
to Equation 5-9, which describes the behavior in the adiabatic and isothermal regimes,
but is different in that the bottlenecked expression is concentration dependent. The
g value, indicated by the position of the resonance peak, should be constant. In real
systems, some combination of bottlenecked and isothermal/ adiabatic behavior is expected.

In more detailed calculations, the Bloch-Hasegawa equations can be modified to
account for the motion of the conduction electrons throughout the sample, hyperfine, and

fine structure splitting.
Residual Width and Thermal Broadening

The nonzero value of the residual width at low concentrations is attributed to local
internal field effects (such as the presence of anisotropy) and not to relaxation effects,
although these effects are included in the ion-lattice relaxation rate. From Equation 5-14,

we can write the residual width in field units as

 

illd tilt” the

there bK
Ex
hmptrtie'
.lithOtigt
no addir
{Mattie
Tm of
ailgnec‘
bTOade

theta

Meg

is f(
hunt

hfi‘e

81

A = 7‘ 5-15a
g “3 Tu.

 

and the thermal broadening coefficient may be written

BT = — be 5—15b

where bK = (1/T,J/T is the unbottlenecked Korringa rate given by Equation 5-6a.

Experimental studies91 correlate the residual width to local strains and structural
properties. Barnes91 suggests that the behavior is due to internal field distributions.
Although the RKKY interaction commutes with the total magnetization and should cause
no additional linewidth, the part of the RKKY interaction transverse to the field is
frequency dependent. This suggests that, if the appropriate limits hold, the transverse
part of the RKKY interaction cannot follow the rf field. The spins are locked into an
aligned configuration, producing an internal field, which results in inhomogeneous
broadening. For S-ion states, random strains should have a fairly negligible effect, since
there is almost no angular momentum in the ground state. Dipole-dipole broadening may
affect the residual width, but this quantity is small in the Mn alloys.

For AgMn with concentrations on the order of a thousand ppm, the residual width
is found to decrease with increasing concentration for concentrations higher than a few
hundred ppm. Concentration and frequency dependence has not been tested in detail,

because magnetic ordering phenomena obscure the linear behavior. When a third

 

 

impurity (S

residual wit

explained th

there B is
‘thich tan ‘

the talue o

the Hits

L t
Wham

‘
9th C

82
impurity (Si or Al) is added to CuMn (Mn concentration on the order of 0.1%), the

residual width is roughly proportional to the concentration of the additional impurity.
In general, the residual width would be expected to be positive. Stewart92

explained that A is related to the ”true" residual width, A0, via

A=Ao-—Be 5-16

where B is the same thermal broadening coefficient, and 0 is the Weiss temperature,
which can be measured from high temperature susceptibility measurements. If 0 > 0,

the value of A measured by ESR experiments will be negative.

The Effect of Magnetic Ordering on Linewidth

Line broadening or narrowing is the result of interactions between spins. Specific
mechanisms for the spin glass problem will be discussed in the last section of this
chapter. Linewidths may be broadened homogeneously, or inhomogeneously.
Inhomogeneous broadening is a result of the existence of different Larmor frequencies
throughout the sample. In effect, each of the spins sees a different field, and has a
slightly different resonance frequency. This results in a distribution of individual lines
merged into one overall line or envelope, but with no change in the lifetimes of the
states. Inhomogeneous broadening may be due to experimental artifacts, such as poor

homogeneity of the magnetic fields across the sample, or unresolved fine or hyperfine

 

structure.
or irregula
The

change the

S) motion;
Interaction
trpolar in

in

War

83

structure. The linewidth may also be inhomogeneously broadened due to inhomogeneity
or irregularities in the sample.

The width of a homogeneously broadened line is entirely due to processes which
change the lifetime of the states between which the transition takes place. This may be
due to; l) dipolar interaction between like spins, 2) spin-lattice interaction, 3) interactions
with the radiation field, 4) diffusion of excitations throughout the sample, or
5) motionally narrowing fluctuations in the local field, such those due to as exchange
interactions. Two of these interactions will be important to the problem at hand: the
dipolar interaction, and the phenomenon of exchange narrowing.

In the case of dipole-dipole interaction between spins of different types, random
local fields are set up at each spin site. The interaction between a pair of spins depends
on the distance between them and their orientations. The dipolar field displaces the
resonance line from that of a single ion by an amount which depends on the strength of
the interaction. The greater the number of pairs with a given interaction strength, the
greater the net intensity of the interaction and the further the shift. As we look at more
distant ions, the interaction will diminish in size as the separation between the spins
increases, but there are more spins with which to pair at larger distances. The resonance

lines which are displaced a small amount from their single ion position are due to strong
interactions with a small number of other spins, and so the intensity of these lines is
greater. Lines which are displaced by a large amount are a result of interaction with a

larger number of spins at a greater distances, and will be less intense. The line profile

84
should then reflect the distribution of the dipolar fields, with a width on the order of
traff-

If the interaction is between like spins, the field is homogeneously broadened.
The precessing components of one magnetic dipole set up an oscillatory field at another
dipole, which (because all the spins have the same Larmor frequency) is at the precise
frequency to cause resonance transitions, and vice-versa. These transitions are equivalent
to the exchange of quanta between the spins, and gives an additional broadening which
is homogeneous in nature, as the lifetime of the individual ion in a given quantum state
is shortened.

The other important mechanism is the exchange interaction, which we can think
about simply by considering a two level-two spin system. For antiferromagnets, the
antiparallel state is lowest in energy; however, the one up, one down state (I t l)) is not
an eigenstate of the exchange Hamiltonian. The eigenstate is a linear combination of
l t l) with | l t), which results in the system oscillating between the two states at the
exchange frequency, J/h. When the exchange energy is large compared to other
energies, the orientation of the spins is being changed at a rate of order J/h, which
causes a fluctuation in the local dipolar field at a similar rate. The fluctuations cause the
dipolar broadening to be less effective. This phenomena results is a line which is

narrowed near the center and broadened at the wings, and is called exchange narrowing.

85
Lineshape, Correlation Functions and the Relaxation Rate

The formalism used to describe the behavior of ESR lines in the presence of spin-
spin interactions near a critical point has been developed by Kubo, Mori, Kawasaki and
Tomita93'94’95’96'97. We are interested in the linear response of a system (the spins)
to a perturbation (the magnetic field).

The fluctuation-dissipation theorem links a spectral density function with a time
correlation function; alternatively, it may be thought of as describing the relationship

between the response of a driven system and the correlations in its equilibrium

fluctuations.

J(w) = f<A'(r)B(0)> em dt 5-17

where J is the spectral density distribution, and <A(t) B(O)> is a time correlation
function which measures the fluctuations of the system in response to the applied driving
force. In the case of ESR, the spectral density distribution is the power absorbed as a
function of frequency (or field) - which is the lineshape and will be denoted by I(w). The
response function will be the correlation function of the random forces which result from
spin-spin interactions.

The Bloch equations with an rf field are the same as the equation of motion for

a damped oscillator. The oscillator has natural frequency we, which is modified by the

86

driving force in a random way, with this randomness broadening the lineshape. The
frequency of the oscillator may be written

w(t) = too + wl(t) 5-18

where w,(t) represents the random fluctuations in frequency, with a time average of zero.

In the ESR experiment, w(t) = 7H(t), where H(t) contains the static and rotating rf

fields. The equation of motion of the magnetization is

5-19

 

’ ' (0M
= to)
dt x

We are interested in an ensemble average of M,. Assuming that we can replace the time

averages with ensemble averages, we can integrate Equation 5-19.

I
<M,(r)> = Mac) cw < exp(if01(t') dt’)> 5-20
o
¢(t), the relaxation function of the oscillator, is
t
¢(t) = exp [if (910’) dt’] 5-21
0

so that we can write

87

<M.<r>M.ror> = IM.(0) 12 e""'°‘<r>(r) 5-22

which links the expectation of the magnetization to the relaxation function. By the

fluctuation-dissipation theorem, Equation 5-17, we can write the resonance absorption

spectrum, 1(w-w0) as

1(0) - ”0) = fifemwwomdt 5-23

where this expression has been normalized to 1. This equation represents an absorption
centered at (.00 which is broadened by the random modulation wl. We can define two
characteristic parameters of the broadening; The amplitude of modulation, A, is given
by the average of the square of the modulation process, < (0,2 > . Using the central limit

theorem, we can write the modulation in terms of a Gaussian,

A2 = f(w-wo)21(m-wo)dm 5-24
The correlation time, 1,, is defined
1:, = film) d: 5.25
o

where the correlation function of the modulation, II/(t), is given by

88

W) = 31-2-(w,(r) w,(t+r)) 5-26

¢(t) measures how much the modulation at time t+1' is influenced by the value of the
modulation at some earlier time t; if Mt) is 0, there is no correlation. r, corresponds to
the time after which the correlation function is reduced to less than one half the initial
value -- this tells us how fast the effects of the perturbation vanish after the perturbation
is removed.

If r, is large compared with l/A (slow modulation), the intensity distribution will
directly reflect the distribution of the modulation. The width of the lineshape will be
about A. If r, is small (fast modulation), a disturbance will not last for a significant
time, so that fluctuations are smoothed out, and the resonance line will be sharper around
the center. In the limit Are a O, the line will be Lorentzian, and the half-width will be
on the order of A27, (which is less than the corresponding linewidth for the Gaussian).
T > 1, for most times, so the Lorentzian behavior will be most significant near t = 0,
and the line will be Gaussian in the wings.

In the ESR spectra of magnetic solids, the slow dipolar interactions broaden the
lines, resulting in a Gaussian shape, while the exchange interactions - if large enough -
narrow the line. The speeds of the respective modulations are about 108 - 109 Hz for the
dipolar interaction, and an order of magnitude faster for the exchange interaction, so that
the exchange narrowing condition is fulfilled in most cases.

In the limit of rapid exchange modulation;

89

<exp[if w1(t’)d:/] > ~ exp[-f(t - 1:) (0)1(1) 091(0) )dt] 5-27
0 o
and the Kubo-Tomita formula results:

5-28

 

W) .. exp[-f<w.(r> w,(0)>dr
0

We recognize the brackets as being proportional to the correlation function of the
modulation. In this limit, the relaxation function decays exponentially;

¢(t) ~ exp(-t/I‘), where I‘ is a complex value related to the linewidth and lineshift.

Including all of the constants,

r = l [exp“‘°°‘ «91(1) w,(0)> dt 5-29
X o

where X is the static susceptibility. The linewidth is given by the real part of I‘, and 1le
times the imaginary part gives the shift in the resonance position. This equation may
also be derived from a Green’s function solution of the equation of motion, where l‘ is
the self energy. In general, Equation 5-29 will apply if

I I
((01 (m1 (0)) = (1H (r>.M.(0)1 [M-(0).H (0)1) “0
<M.(0)M_(0)>

90

where H’ is a perturbation Hamiltonian, and the time dependence of the perturbation
Hamiltonian is accounted for by an interaction representation approach. The important
point about this equation is not its complexity, but rather that it contains a four spin
correlation function, as does the relaxation function of the modulation.

In exchange narrowed dipolar broadening, the strength of the exchange is greater
than the dipolar strength, so we can treat the Hamiltonian, H, as the unperturbed
Hamiltonian, HZeeman + Hacmc, plus a perturbation H’ = Hmm. We can redefine the
relaxation function to remove the faster Zeeman and exchange frequencies, and focus on

the slower dipolar effect. Evaluating the correlation function for the dipolar case,

Mt) = '12 22 p5! m“ 35:11:) 5-31
1h M an

where M is the change in the total Zeeman quantum number (M=O,:l, or $2), the
F01”) the angular dependence from the dipolar functions, and the Si M(er) is a four spin

time correlation function.

55;: = <S.-..(t)5,,(t)S,,,(0)s,,(0)> 5-32

where the greek indices are 2, + or -, and the i,j,k,l label the spin sites. This function
measures the probability that, at time 1, the spin value on site i is a, and the spin value
on site j is 6, given that the spins on sites k and 1 had values 5 and 'y at time = O. This
four spin correlation function must be decoupled in order to find expressions which can

be compared with the experimental measurements. Given the number of combinations

91

of spins which must be considered, this is not a simple calculation, and usually must be
tailored to the system in question. In cases away from the critical point, the four-spin
correlation function can be decoupled into products of two spin or one spin correlation
functions. This is the starting point of the analysis which will be used in the latter part

of this chapter.

Dimensionality Effects on ESR Linewidth and Lineshape

The magnetic dimensionality of a sample is not necessarily the same as the spatial
dimensionality. Magnetic dimensionality will depend on the lattice structure, but also
on the strengths and the ranges of the interactions. The characteristic time for decay of
the spin correlation function, 1,0, in three dimensions, is proportional to h/J . Propagation
of disturbances in one dimension is much more difficult than in three dimensions, as
there are fewer relaxation pathways available. In lower dimensions, long time behavior
of the relaxation function will be dominant. Experimentally, the decay of ¢(t) is
determined to be Gaussian for short times, but diffusive (approximately t“ ) for long
times. The dimensionality dependence is a result of the diffusive nature of the motion
of the disturbances.

The expression for the characteristic time contains an integral over the correlation
function, with the correlation function diverging like t"2 for 1D and like ln(t) in 2D. One
of the experimentally observed effects is a significantly different lineshape, which falls
somewhere between Gaussian and Lorentzian. The shape of one dimensional lines is

close to Lorentzian in the center, but decreases more rapidly. The slow decay of the

92

correlation in a one dimensional system does not allow the system to reach the fast
exchange regime typical of Lorentzian lines, but leaves the system in an intermediate
state. Short time phenomena are generally dependent on the number of nearest neighbors
coupled by the interaction, while the long time phenomena are most dependent on the
availability of paths for the disturbance. The one dimensional lineshape is predicted to
be the Fourier transform of exp(-t3’2), and the angular and temperature dependences
should also be different from that of the three dimensional prediction.

Observation of the different lineshapes and angular dependence of lower
dimensionality materials is not straightforward. The differences are much greater in one
dimensional materials than in two, with most two dimensional materials having a
lineshape indistinguishable from a Lorentzian. The appearance of a Lorentzian lineshape
in one and two dimensional materials is due to the fact that only a very weak interchain
or interplanar coupling is needed to create paths for propagation of a disturbance, thus
simulating three dimensional behavior”. The same reasons are responsible for the onset
of divergent behavior seen in one and two dimensional materials which should not
support magnetic order. In many cases, the deviations from Lorentzian behavior are so
small that they can only be seen in single-crystal precision experiments specifically

geared to sensitive observation of the lineshape.

EXPERINIENTAL EEULTS

This section focuses on the experimental investigation of spin glasses in the

regime above and approaching T,, and presents the two models used to understand these

93

results. The same general behavior of the linewidth and shift is seen in a wide variety
of spin glasses; we will see that many ferro- and antiferro- magnets share the same
characteristics as spin glasses. Representative data from long range ordered magnets, as
well as data from low dimensional magnets will be reviewed for its pertinence to the

present problem.

Metallic Spin Glasses

Cub/Ln

The first ESR investigations of spin glasses were performed by Owen”, and
Griffiths99 on CuMn. The anomalous behavior of the linewidth and shift were discussed
in the beginning of this chapter.

Salamon and Hermann100 found the same behavior - a divergent linewidth and
a shift of the resonance to lower fields - in Cuo_—,5Mn0_25, attributing the divergence of the
linewidth to the critical nature of the dynamics of the spin glass phase in support of a
phase transition at T,. Critical slowing down of the relaxation time as T, is approached
lessens the effectiveness of the exchange narrowing of the dipolar broadening causes the
linewidth to diverge, as is also seen in insulating antiferromagnets“.

Starting from the Equation 5-29, the random forces are roughly approximated by
writing <w,(t),w,(0)> = of, the dipolar frequency, for times between 0 and 76"”.

The relaxation rate and the fractional g-shift are given by

94

2
w t
4 , Ag = _l_ (I): 1: 5-33

1
T x e 2x

 

or, relating these quantities to the linewidth and resonance shift in field units;

AH = i— ofite (H - Ho) = immite 5-34
893 8'43

where x is the static susceptibility, did is the dipolar frequency = gzaazlhv and v is the
volume per spin. 1, is the characteristic spin-spin relaxation time, and is equal to the
inverse of the exchange frequency, «1,. The expression for the line shift in Equation 5-34
is a low frequency approximation. We can combine the equations in 5-33 to find an

expression for the relaxation rate in terms of the shift and the linewidth.

i .__ _S_ 2 5-35
re

By plotting the relaxation rate as a function of temperature, Salamon showed that
the effective relaxation rate goes like the t". A more detailed analysis of data for
CuosnMan103 found the same behavior with exponent -l.3. This is compared to a
mflan-field calculation of Kinzel and Fischer‘o‘, which predicts that the relaxation rate
Should diverge as t‘2. The fit of the relaxation rate did not take the contribution from the
linear behavior at high temperatures into account, which may have influenced the value

Obtained for the exponent. Deviations from the power law behavior are seen near T,.

95

Both the shift and the excess width are related to the four spin correlation

function, so we can derive a relationship between them.

 

.. [Hf ”(8 ”’18 5-36

where H(g =2) is the value of the resonance peak position corresponding to a
g-value of 2, and E is predicted to be twice the exponent governing the linewidth
divergence. Experimentally, this relationship is difficult to test, because of the small
temperature range over which both Ho significantly differs from the high temperature

value and the power law behavior still holds.

Most of the ESR work in metallic spin glasses above Tf has been performed on
Ag,,XMn,, in part due to the lack of chemical clustering and the ease of sample
preparation. Mozurkewich, et. al. "’5 proposed that the dominant mechanism for
broadening in metallic spin glasses is the DM anisotropy and not the dipolar interaction.
To test this theory, a spin orbit scatterer, Sb, at a concentration y (in at. %) was added
to some of the AgMn samples, so that the study considered Ag,_an,Sby with x from
.026 to .10 and y from 0.06 to 0.57. Measurements were made over a frequency range
from 1 to 9 GHz. These measurements and the subsequent analysis have the most

relevance to our experiment, so they will be reported in more detail.

96
AH(T) in both the AgMn and the AgMnSb samples exhibited the behavior

characterized by Equation 5-2, with the parameters obtained from fitting shown in Table
5-1. The residual width, A, increases linearly with y and decreases linearly with x. The
thermal broadening, B, increases linearly with increasing Sb concentration, with the rate
of increase largest in the smaller Mn concentrations. This behavior is due to the
lessening of the bottleneck due to the introduction of another relaxation path.

The divergence strength, C, increases linearly with increasing Sb concentration,
with the strong dependence on Sb concentration precluding dipole-dipole interaction as
the major source of line breadth near T,. Calculations using a DM anisotropy106 as
the primary broadening mechanism predict a linear dependence of the excess linewidth
on both x and y. The linear dependence on y is observed, but a weaker than linear
dependence on x is seen, which is attributed to the same effects responsible for the
sublinear dependence of Tf on x. The possibility of chemical clustering at the higher
concentrations is also noted.

The values of the exponent it all lie between 1.2 and 1.8, with the values for the
undoped AgMn being roughly concentration independent and ranging from 1.2 to 1.4.

The addition of Sb tends to increase the exponent from 1.3 at small concentrations to 1.8

97

Table 5 -1 .' Parameters describing the dependence of AH (T) on Mn and
Sb concentration in Ag ,_,Mn,Sb, from Mozurkewich, et. al.”.

 

‘ y<at%>

1:.;:B<crr<>j:

 "

:i1:..c.<c>:;z

 

2.48

21

1.4

 

 

0.06

2.75

24

1.3

 

0.17

9.79

3.15

24

1.6

 

 

0.22

9.65

3.29

30

1.5

 

0.46

9.10

16

4.35

43

1.8

 

2.6

0.57

8.67

32

4.60

48

l.8

 

5.8

19.28

—43

2.53

34

1.2

 

7.8

24.56

2.67

37

1.4

 

10.3

 

 

31.43

 

 

2.68

 

40

 

1.4

 

 

98

at the largest concentrations. This is attributed to depression of the range dependence
of the exchange coupling by the spin orbit scatterers. Departure from power law
behavior is observed as the freezing temperature is approached, with the departure
occurring at higher reduced temperatures for higher frequencies and lower
concentrations.

At high temperatures, the resonance peak position is found at g z 2. As TeTf,
the line shifts to lower fields, with the shift being much stronger for lower frequencies
than for higher ones. The shift cannot be properly described as a pure g-value shift, nor
as a simple, frequency independent internal field. The magnitude of the low frequency
shift is insensitive to Mn concentration (at any given reduced temperature), but shifts
more strongly with increasing Sb concentration. The behavior of the shift with Sb
concentration is correlated to the behavior of the linewidth with Sb concentration,
indicating that they arise from the same mechanism. From equation 5-34, at low
frequencies, the resonance peak position should diverge as the reduced temperature to
a power twice that of the linewidth divergence. As already mentioned, this is
experimentally difficult to verify, because the resonance peak position is different from
the high temperature value only over a small temperature region before the line becomes
too broad to accurately measure. The strong Sb dependence of the data reported by
Mozurkewich,et. al. , indicates that the dominant linewidth mechanism is an exchange
narrowed anisotropy, and not the previously predicted exchange narrowed dipolar
broadening.

Wu, et. al. “’7 have measured frequency dependence of the linewidth and shift

in Ag,,,Mn, (x=2.6 at.%) at frequencies from 1 and 10 GHz. The critical part of the

99

spin-spin relaxation rate is calculated using the DM anisotropy instead of the dipolar
interaction as the broadening mechanism. Critical scaling in both field and frequency are
(separately) applied, and critical exponents extracted. For frequency scaling, the

linewidth is written

(1311)“ = Ct" GUM) where 1: = tot‘" 5-37

and uz is found to be 2.5 -__l- 0.3. Field scaling results in an expression for the excess

linewidth

m _
(AH)"‘ = Ct"“G(Hr) where 1: = Tot 2 5 38

which yields a value of (B + 7)/2 = 2.5 i 0.3. The crossover exponent,
<15 = B + "y = 5.0 i 0.5. These values can be compared with other measured and
theoretically calculated values shown in Appendix B.

An interesting sidenote to this analysis is that one freezing temperature is used in
the scaling for all frequencies, which the authors claim determines a unique (to about
10%) T,, good for all fields and frequencies. In this field regime, the Gabay-Toulouse
line predicts a constant value of T,, so Wu, et.al. interpret Tf as the GT transition
temperature.

Baberschke, et. al.108 and Mahdjour, et. al. “’9 noted that the magnetization,
and not the temperature is the variable occurring in the Bloch-Hasegawa equations, and

plotted the linewidth as a function of susceptibility, where the susceptibility is measured

100
at a magnetic field equal to the resonance peak position. As Tf is approached, the

magnetization becomes highly nonlinear, departing from its usual Curie—Weiss
temperature dependence. This departure is larger for larger applied magnetic field
(corresponding to larger ESR frequencies), which they cite as the reason why ’frequency
dependence’ is observed. By plotting AH(T) vs. x, all frequency dependence is removed,
which indicates that this is a field and not a time window effect. The excess part of the
linewidth can be analyzed in terms of a power law in x,/Hz, where X. is the nonlinear
susceptibility, and the scaling holds for all frequencies and for reduced temperatures

down to T,. A similar analysis has been applied to (LaGd)A12”°r‘“.

Other Spin Glam

EULSILXE

EuxSrHS is a Heisenberg system, with cubic NaCl structure. The compound can,
depending on concentration, be paramagnetic, ferromagnetic, or a spin glass. The
nearest neighbor interaction is ferromagnetic, and the next nearest neighbor is
antiferromagnetic, with long range magnetic dipolar interactions the sole source of
broadening. The non-metallic nature of this system precludes the bottlenecked behavior;
the linewidth is temperature independent until magnetic correlations start to cause
broadeningm. The same type of frequency dependence (a weakening of the divergence
with increasing frequency) is observed”. Dynamic scaling has been used to explain

the frequency dependence of the linewidth”.

101

In concentrated metallic spin glasses, direct exchange, and not the long range
RKKY interaction dominates. Hou, et. al. “5"“ analyzed this behavior in FegNiano,
and FewN 17°F”, finding that these materials behaved similarly to the materials mentioned
above.

Park, et. al.117’m, studied amorphous Fe,Niw,P,4B, and sputtered Mn..,B,2
films. Contrary to results found in dilute metallic spin glasses, the linewidth does not
diverge strongly. As T, is approached, the linewidth becomes very broad, but remains
measurable through the freezing temperature. As the temperature is lowered below T,,
the linewidth decreases and finally disappears. The divergent behavior above T, is fit to

a phenomenological expression,

= stern-Ir

where n is either 1 or 0. Frequency effects similar to those seen in the other spin glasses
are seen, but there appears to be a ’resonant frequency’ , for which the rise in linewidth
is faster than either higher or lower frequencies. This resonant frequency is about 3 GHz
for the Fe compound, and 4 GHz for the Mn compound. In contrast to the dilute
metallic spin glasses, the ESR linewidth shows no history dependence on the cooling field
for fields up to at least 17 k6. In the Fe series, increasing the iron concentration

decreases the strength of the divergence and increases the breadth of the minimum in

102
AH(T). Bhagat, et. al. ”9 have shown that Equation 5-39 can be used to fit data from

both concentrated and dilute metallic spin glasses.

Malozemoff and Jamet120 studied amorphous Gdo.37Alo,3 films sputtered onto
Kapton. The linewidth is fit to a power law with exponent between 1.5 and 2.
Frequency dependent measurements reveal a flattening of the divergence with increasing
frequency. Exchange narrowing is predicted to account for the linewidth at temperatures
above T,; near the freezing transition, where a departure from the power law behavior
is observed, a local field model based on the presence of ferromagnetic clusters is

suggested.

Misc, Spin gilasses

l. ‘2‘, found divergent behavior in the insulating spinel solid

Fiorani, et. a
solution CdIn2,,,Cr,S4 for x between 0.6 and 1.7. The spin glass mechanism is
preferential occupation of the octahedral sites of the spinel structure by the Cr, which
gives rise to F and AF bonds. As with the concentrated metallic spin glasses, the
linewidth measurements are insensitive to the magnetic field in which the sample was
cooled. The broadening of the linewidth and the shift in the resonance peak position are

attributed to random internal fields. The value of the linewidth saturates near T,, then

decreases, with an accompanying deviation from a Lorentzian shape well above T,. In

103

the region near T,, the lineshape is Gaussian in the center and very narrowed in the
wings. Similar results have been seen in CukCr2,Sn2,2,S4‘22.

Aronson, Salamon and Hauser123 studied the concentrated amorphous spin glass
MnSi, in which the spin relaxation is dominated by random anisotropic fields, with
linewidth and shift results identical to the concentrated metallic spin glasses.

Koche, et. al.124 studied the divergence of the linewidth in pseudobrookite (Fez
TiOS) in a percolation context, modeling the sample as one ’infinite’ cluster of spins
coexisting with smaller finite clusters. The low energy excitations of the infinite cluster
are treated as spin waves, with the finite clusters providing relaxation channels. The DM
interaction links the finite clusters and the infinite cluster. The linewidth can be
satisfactorily fit to an exponential form,

(AH)"‘ = A exp[-l) 5-40

T;

Ferromagnets, Antiferromagnets and Low Dimensionality

The divergence of the ESR linewidth is noted for many -- but not al --
ferromagnets and antiferromagnets. Taylor and Coles‘” identified a series of
intermetallic compounds, including both ferromagnetic and antiferromagnetic materials,
which display the same temperature dependence of the linewidth and the resonance peak
position as seen in the spin glasses. The materials are atomically well ordered, and

include dilute alloys like GdZnn, as well as compounds like GdAl2 and szEu.

104

Measurements were made on powdered samples at 9 GHz. At high temperatures, these
materials follow an A+BT behavior, and at some temperature, denoted by T,,, deviation
from the linear behavior in the form of an diverging linewidth is noted.

In the ferromagnetic compounds, T,, is between 1.2 and 1.5 T,, for all compounds
except Cu6Gd, which broadens at about 6Tc. In the antiferromagnets, broadening is also
seen at 1.5 - 2 Tc in most compounds, with the exception of: Cu4Gd (2Tc), EuAl4(4Tc),
GdZn12 (5Tc), and GdZZn,—,( 1011). The high temperatures at which broadening occurs is
related to the presence of short range order. The short range order is induced by
anisotropy in the exchange interaction; large enough anisotropies can effectively reduce
the dimensionality of the material.

ZnnMn has a highly anisotropic crystal structure; along the c axis, Mn atoms are
5.1 A apart, with only a single Zn atom between them. Adjacent chains of Mn atoms
are separated by at least 6.6 A, with several Zn atoms in between the Mn. Susceptibility
measurements indicate that chains of Mn along the c axis order first, at a temperature of
about 120 K. This intrachain ordering is ’quasi one dimensional’ and therefore weak,
but strong enough to sustain short range order. Full 3-d ordering is not seen until about
23 Km. ESR measurements show that the g value of the material is equal to 2 at high
temperature, begins to increase at 120 K, and reaches a value of 3 at 4 K. The
broadening at temperatures far above T6 is linked to the onset of the quasi-l-dimensional
(or short range) order. One possible source for a large amount of short range order
occurs when the RKKY interaction favors spin ordering which cannot be maintained due

to the strict periodicity of the lattice. This frustration would result in a fairly extended

105

temperature range over which the free energy would be minimized by the existence of
short range order.

In the short range ordered regime, Kawasaki“ used the formalism outlined in the
first part of this chapter and calculated that the linewidths in antiferromagnets could be

described by

AH(T) a (T—TN)""' 5-41

with an exponent of P = 5/3. Experimental values of P range from 0.5 to 1.6,
depending on the material. Seehra and Castner127 measured AH(T) for Man, a
uniaxial antiferromagnet, and found P= 1.17. Burgiel, et. all” found the same
behavior, but with exponent 3/8. Zimmerman, et. al.129 measured P = 0.58 in the
antiferromagnet CuCl2 2HZO. Measurements of MnO and MnS‘m”, found P = 3/4
for both materials.

Seehra”2 measured the frequency dependence of the excess linewidth in MnF2
near the critical temperature. The frequency dependence starts at a reduced temperature
of about 0.15, with higher frequencies producing narrower linewidths. Frequency
dependence is also noted in MnSm. All of the measurements of P cited above, except
Seehra and Caster’s were performed at 35 GHz, which may explain the low values of the
exponents.

As in spin glasses, anisotropy appears to play a very important role in the
presence of a divergence. Although the linewidth of MnF2 - a strongly anisotropic

antiferromagnet - diverges, the cubic antiferromagnets RanF3, KMnF3, and KMn:

106

MgF3 show a slight narrowing as TN is approached from above. The role of anisotropy
in the divergent nature of the ESR linewidth was theoretically investigated in
antiferromagnets by Huber‘o‘, and Seehram, in much the same manner as
Mozurkewich, et. al.‘°5 did with AgMn. These theories will be discussed in the last

section of this chapter.
ESR in 1 and 2 Dimensional Materials

Because of the importance of long time relaxation behavior in 1 and 2 dimensional
systems, the dependence of linewidth on temperature should be different. In one
dimension, for exchange narrowed dipolar broadening, the expression corresponding to

Equation 5-34 is

% ~ (032/3 (1.)1/3 5-42

In the canonical linear chain antiferromagnets TMMC ((CH3)4NMnCl3)‘3", and
CsMnCl32HZO‘35, the same divergent behavior of the linewidth is observed; however,
the line broadening begins at much higher temperatures than the resonance shift and the
ordering behavior is dependent on the direction of the sample with respect to the field.
As the temperature decreases, short range order begins to grow, so that small clusters
are formed. These clusters randomly provide local fields, which results in a shift of the

resonance line. Although a one dimensional compound can not support order, the small

107

clusters are ordered enough to approximate the behavior seen in the three dimensional
ordered compounds. 1

Other ESR experiments with TMMC have substituted diamagnetic or
paramagnetic impurities, which tend to block or slow down the rate of spin correlations.
Linear chain ferromagnets, such as CHAC [(C,H,,NH3)CuCl3] and CHAB
[(C6H,,NH3)CuBr3] also display the g-shift seen in the antiferromagnets; the linewidth
results were not reported“. The lineshift move towards higher fields for the hard
axes, and towards lower fields for the easy axis.

In magnetically 2-D manganese bromide and manganese chloride salts, AH(T)
also obeys a straight line at high temperatures (due to coupling of the spin system and
the phonon system through the DM anisotropy, and a divergent part which follows
Equation 5-2'37. The exponent is observed to have a value on the order of 2, and a
different angular dependence of the linewidth than three dimensional compounds.

Richards and Salamon138 studied the behavior of the two dimensional
antiferromagnet KzMnF4. The problem is treated within the theory of two dimensional
exchange narrowed dipolar interactions, and the linewidth may be calculated by assuming
a Gaussian form for the short time behavior of the relaxation function, and a diffusive
behavior for long times. Qualitatively, the calculated linewidths predict the same
temperature dependence as that observed. The numerical agreement is not ideal, but
given that the theory has no adjustable parameters, these results are very good evidence
that the exchange narrowed dipolar broadened formalism can distinguish two dimensional

behavior from three dimensional behavior.

108

The antiferromagnet KzMncMng4 displays crossover behavior from one to two
dimensions, depending on the Mn concentration. As the Mn sites are diluted with Mg,
the paths for spin diffusion becomes blocked, resulting in one dimensional behavior
indicated by a change in the lineshape. The crossover to one dimension can be
determined within the context of a percolation problem. As the amount of Mn increases,
the lineshape becomes increasingly Lorentzian although, even at c= 1, the line is
significantly non-Lorentzian. The behavior of AH(T) is dependent on the angle between
the normal to the 2D plane and the applied field, with the dependence being more
pronounced for lower concentrations. Although the data was not fit to a theoretical form,
the raw data seems to indicate that the exponent takes on one value for all of the samples
obeying two dimensional behavior, and gradually increases as the Mg concentration
increases.

Walsh, et. al. ‘39 found the same behavior in szMncMghF.” a dilute two
dimensional antiferromagnet which is closely related to the 2D Ising spin glass
Rb2Cu,_xCoxF4. These effects were explained in the context of percolation theory, which
predicts that the linewidth will diverge like T4" in the 2D case. The data do show a
divergence in the samples which are two dimensional, and a much weaker divergence in

the ’one dimensional’ samples, but the predicted power law behavior is not observed.

THEORIES OF ESR 1N SPIN GLASSES

Now that we have looked at the general theory of ESR in localized metallic

moments, and data from spin glasses, ferromagnets and antiferromagnets, we turn to the

109
specific theories which have been advanced to explain the behavior of the ESR linewidth

in spin glasses. There are two general approaches. The first theory can be characterized
as a mean field approach, which uses the Mon-Kawasaki treatment of ESR near critical
points. The second approach is a cluster model in which the dominant processes above
T, are the formation of regions of correlated spins. These two theories have a subtle, but
very important difference. The mean field theory assumes that the behavior of the ESR
linewidth is a divergence associated with the phase transition into the spin glass phase.
The correlated cluster model implies that, at temperatures above T,,, but below T,, the
behavior of the ESR linewidth in spin glasses is actually a measure of a paramagnetic

phenomena.

Mean Field ESR Theory

Huber studied the behavior of the linewidth near critical points in
ferromagnets‘”, antiferromagnets”, and AgMn”1 using the Mon-Kawasaki
treatment. The Hamiltonian is taken to include the RKKY exchange interaction and the
DM anisotropy is the broadening mechanism. The Random Phase Approximation used
to decouple the four-spin correlation function emphasizes two spin correlation functions,
which is analogous to considering that single spin flip events are most important. In
ferromagnets, the linewidth is predicted to diverge as x3". In antiferromagnets
(specifically Man), dipolar anisotropy is introduced, with the linewidth is predicted to
diverge as ('T/x)£5’2. Using the relationships between the susceptibility and the

correlation length, and between the correlation length and the reduced temperature, he

110
finds that the linewidth should diverge like (T-TN)“, where x is between 1.2 and 1.6.

This lower limit is determined by using the mean field relationship between E and t, and
the upper limit by using the experimentally measured relationship 5 = (f-TN)'2’3. The
accuracy of both expressions decreases as the temperature gets farther from T,,. Critical
slowing down is suggested as the reason for the divergence. The predictions are valid
in the temperature range from TN to 2 TN. Theoretical predictions of the angular
dependence of the linewidth within the framework of this theory have been
experimentally verified‘”.

The broadening of the linewidth as a function of temperature and the eventual
departure from the predicted power law behavior can be qualitatively considered as
follows: When the exchange energy is much larger than the anisotropy and Zeeman
energies, the usual picture of exchange narrowing applies. As the Neel temperature is
approached, the mean free path of an excitation increases due to increasing correlations
between the spins. Cluster formation decreases the frequency of the exchange
modulation, resulting in broadening. At some temperature, the modulation frequency
becomes comparable to the anisotropy frequency or the Larmor frequency, and the
exchange narrowing picture fails. Below the temperature where the exchange frequency
is on the same order as the Larmor frequency, the measuring frequency becomes
important, so that increasing measuring frequency results in a deviation from power law
behavior at a higher reduced temperature.

[.105

Based on the measurements of Mozurkewich, et. a , Levy, Morgan-Pond and

Raghavan‘o" used the Mod-Kawasaki/Kubo-Tomita formalism with a DM anisotropy as

be

111

the broadening mechanism. Configurational averages over the random distributions of
the exchange, DM and dipolar couplings must be made for application to spin glasses.

The four spin correlation function is expanded by expressing it in the basis in
which the RKKY interaction is diagonal. The decoupling is exact for an infinite range
model, but emphasizes single spin flips as being the most important contributions to the
line. A mean field expression for the two spin correlation functions is assumed for the
resulting decoupled expression. The assumptions cause the model to break down near
T,. Morgan-Pond142 estimates that the corrections to the mean field approximations
become important below t = 1/4 in AgMn.

Both the divergence of the linewidth and the shift of the resonance peak position
to lower fields are predicted from this theory, but these quantities are not presented in
a closed analytic form. Their quantitative predictions for AH(T) as a function of
temperature agree remarkably well with the low frequency data, especially considering
that no adjustable parameters are used. Higher frequency data does not match as well
in the region where it differs from the low frequency data, due to the approximations
used in the decoupling procedure. This analysis has been successfully applied to
EuSrS‘43r‘“; it is simpler in this case because the only broadening mechanism is the
dipolar interaction. Agreement with low frequency experiment has been extraordinary,

considering that there are no adjustable parameters.

l 12
The Correlated Cluster Model

Experimental evidence indicates that correlations begin at temperatures far above
T,. Barnes145 adapted and extended the Bloch-Hasewaga treatment of ESR of localized
moments in metals to treat spin glasses. At high temperatures, correlations between
spins can be neglected, and the behavior of the linewidth is explained by the Bloch-
Hasegawa equations. As T, is approached, correlations will become increasingly
important as pair, triples, etc. form. Clusters will form when the correlation between
the spins in the cluster excwds some critical value of J. The Boltzmann factor weighting
causes these clusters to essentially drop out of the thermodynamics of the system.

We now have a four component system consisting of the lattice, the conduction
electrons, correlated clusters, and the uncorrelated ions. A schematic picture of the
system is shown in Figure 5-2.

This results in the introduction of a myriad of additional relaxation times. Tcc and
To, represent the relaxation time of a spin in a correlated cluster to a free spin and vice-
versa. We now have three Bloch-Hasegawa equations; one for the conduction electrons,
one for the correlated clusters, and one for the uncorrelated ions.

We have to consider the conditions under which the correlated spins and the
uncorrelated spins will have a common resonance. The solution for the case of common
resonance requires that an appreciable number of correlated spins are present, which is
satisfied in a temperature regime with an upper limit of a few times T,. The same range

also applies to the bottlenecked case.

113

 

T11
Localized -
Moments 1
T. -

     
   
  
    

     
  
 
   

Correlated
Clusters

 
    

  

T-l

00

 

 

 

 

 

 

 

if Conduction g
g Electrons g
g— l T g r"
? § cL
l LATTICE l

 

 

 

 

Figure 5-2: Allowed relaxation paths for the correlated cluster model

In Barnes’ original picture, clusters are either ferromagnetic or antiferromagnetic.
If a pair is antiferromagnetically correlated, there is no net magnetization, so the only
effect on the Bloch-Hasegawa equations is a decrease in the concentration of free spins.
If the pair is ferromagnetically coupled, Barnes’ calculation shows that the Korringa rate
(l/T“) decreases by a factor of two, but the Overhauser rate (l/Tw) increases by a factor
of four, which favors bottlenecking.

For both bottlenecked and non-bottlenecked systems, the same high temperature

linear behavior is observed. As the temperature decreases, clusters become more

l 14
important. The thermal broadening and the residual width are found to be dependent on

the degree of short range correlation. As T, is approached, Barnes cites experimental
evidence of the importance of the DM anisotropy and assumes an EA phase transition
to obtain the divergent behavior. Two broadening mechanisms are present; the internal
fields due to the clusters, and a random-effective field due to the DM interaction.

The anisotropy can be accounted for in an effective moment to lattice relaxation time,
T,,. The broadening fields will be narrowed by cross relaxation between the clusters and
the free ions. In particular, the residual width will be proportional to l/T,L, where (using
Equation 5-34),

i ~ <(toa)2> 1:, 5-43
Til.

where r, is the mean relaxation time of the local moments and era is the anisotropy
frequency. This implies that the ESR linewidth is due to the contribution from the local
moments, and not the spins contained in correlated clusters. If an Edwards-Anderson

type phase transition is assumed, 1, ~ t", and

 

T - T -1
AH(T) ~ 7}— .. <(w.)”‘?[ IT ] 5_44
a. f

Barnes91 notes that, since there is little reason to expect measurements to yield mean field
values, the exponent should be used as a parameter. In particular, if the correlation

length goes like t", and the critical relaxation time goes like the correlation length to an

115

exponent, gr, we would predict a divergence which goes like t"‘. For antiferromagnets,
the relaxation time goes like 5"”, so the exponent governing the divergence of the
linewidth (it) would be equal to -3/2v.

Barnes offers a conceptual explanation for the frequency dependence of the ESR
linewidth in the bottlenecked system YzGd‘“, which is an antiferromagnet at higher
concentrations, and a spin glass at lower concentrations. The homogeneous relaxation
processes continue all the way down to T=0, so that broadening as the ordering
temperature is approached must be inhomogeneous in nature. The highly correlated
clusters generate an internal field distribution with a small average value, which results
in inhomogeneous broadening. If a system is strongly enough bottlenecked, the
narrowing effect due to cross relaxation will overcome the inhomogeneous broadening.
The strength of the bottleneck is proportional to the root mean square of the Knight shift
(which scales with frequency) and the Korringa rate, so that the bottleneck is more
effective at higher frequencies and explains the deviations from power law behavior as

a function of frequency.

Comparison of the Models

Although these two models predict the same behavior of the linewidth, their
foundations are very different, with the mean field theory attributing the ESR linewidth
divergence and lineshift to spin glass phenomena, and the droplet model attributing these
effects to paramagnetic (uncorrelated) spins. Both models agree that anisotropy is a

significant broadening mechanism in spin glasses, especially near T,.

116

The mean field approach does not present a functional form for the divergence,
so there is no obvious relation between it and other exponents. Other calculations have
been performed by Bruinsmam who determined it = vz by averaging the DM
interaction in a scaling context, and by Henley”7 who decoupled the four-spin correlation

functions and found it = uz - 28.

W

The theoretical treatment of the ESR of localized moments in metals indicates that
the linear behavior, AH(T) = A+BT is due to the temperature dependence of competing
relaxation rates. The correlation function formalism which predicts the linewidth and
lineshape in the case of strongly interacting spins has been outlined, with general
comments on how the cooperative behavior of the spins can narrow or broaden the
resonance line. Qualitative differences between ESR in low dimensional materials and
in three dimensions have been noted.

Experimentally, we have seen that the divergent behavior of the ESR linewidth,
along with the accompanying line shift, has been observed in one and two dimensional
antiferromagnets and ferromagnets. Anisotropy appears to be the dominant mechanism
for the behavior of the linewidth near T,, both in spin glasses and in other magnets. The
set of ferro- and antiferro- magnets which display this divergence are all anisotropic, with
the low dimensional materials being the limiting case of strong anisotropy. In lower
dimensional magnets, data taken with the field along the easy axis exhibits the standard

shift of the resonance field to lower fields; data taken with the fields along a hard axis

117

shifts to high fields. Measurement of the linewidth and shift at higher frequencies results
in a departure from the power law behavior for AH(T) at higher reduced temperatures
than measurements at lower frequencies. This frequency dependent has been seen in both
antiferromagnets and spin glasses.

Interpretation of ESR in spin glasses above T, has been primarily
phenomenological, in terms of Equation 5-2, due to the lack of specific theories (with
the exception of the AgMn analysis of Levy, Morgan-Pond and Raghavan which is valid
only for low frequencies) addressing this temperature regime. Two theories have been
presented. The first theory assumes that the divergent nature of the ESR linewidth is due
to critical slowing down near a phase transition. The second theory attributes the
linewidth to inhomogeneous broadening due to the formation of internal local fields of
correlated clusters, with cluster formation beginning at a few times T,. This second
theory is compatible with the existence of an intermediate phase between the
paramagnetic and the spin glass phase.

ESR investigation of thin film ferro- and antiferro- magnets have primarily
focused on behavior below the ordering temperature, so there has not been an extensive
investigation of ESR in multilayered magnets to which we could compare our spin glass

measurements .

MOTIVATION FOR THE EXPERIMENT DESCRIBED IN THIS THESIS

Now that the intricacies of spin glasses have been explained, the principles behind

the FER experiment and theory have been discussed, and the literature pertinent to ESR

118

in spin glasses has been reviewed, a statement of the motivation of the work presented
in this thesis is in order. As we have seen, spin glass properties are a function of
dimensionality, measuring time and magnetic field. The previous studies of multilayered
metallic spin glasses have primarily been of three types; ’DC’ susceptibility
measurements made at 100 G and measuring times on the order of 300 seconds, 2)
susceptibility measurements from 10“ seconds to 10‘ seconds at a fields of 10 G, and 3)
field dependent susceptibility measurements from 2 G to 15000 G at measuring times of
300 seconds. This group of experiments covers a small region of frequency-field space,
as shown in Figure 5-3. All of these experiments probe the behavior of one quantity -
the freezing temperature - as layer thickness is reduced. The observed effects have been
analyzed in terms of finite size scaling and dimensionality effects.

ESR experiments at 9 GHz have a measurement time of 1010 s, and require
magnetic fields from 1 to 6 kG. This represents a very different time regime than
probed by susceptibility measurements and investigates the behavior of a quantity (the
ESR linewidth and shift) different from the freezing temperature. We would like to
answer the following questions:

1) Does the expression (Equation 5-2) used to describe the ESR linewidth in bulk
samples apply for all layer thicknesses, or is an alternate expression needed to describe
the behavior of the very thin (2D) samples?

2) If the exponent x is a critical exponent, will it take on two different values in

the different dimensional regimes?

119

3) Are there substantial changes in the lineshape as a function of layer thickness
hich might be indicative of 2D behavior and therefore provide additional evidence of the

change in dimensionality in these multilayers?

 

 

 

log Measurement Time (s)
:1:

 

 

 

.4
.5 _
-3 —- ESR
40“—
l l//
0102/]

Magnetic Field (kG)

 

 

 

Figure 5-3: Measurement regimes of techniques used in the investigation of metallic
multilayers in the time-magnetic field plane

4) Does the lineshift behave in the same manner for all W30? If the shift and the

divergence of the linewidth arise from the same mechanism, we expect this to be true.

120

If they arise from a different mechanism, one of the mechanisms may be more or less
affected by the changing sample size.

5) Is the observed departure from power law behavior as T, is approached
dependent on W50? The mechanism for this departure in bulk materials is not clear.

6) Substantial evidence exists for the onset of correlations above T, in spin
glasses; these correlations have a broader spectrum of relaxation times than a
paramagnet, but nowhere near the extended time scales of the spin glass phase. Long
time probes like the DC susceptibility - and even the AC susceptibility - would be
insensitive to these phenomena. If there is indwd a ’Griffiths phase’ in spin glasses, the
time scale of the correlations would observable only to high frequency probes like uSR,
neutron scattering and ESR.

7) Most importantly, can the results of this study shed any light on the question
of whether the mean field or the correlated cluster model is the most accurate description
of ESR in spin glasses?

In summary, this study is designed to test the behavior of spin glasses in the
temperature regime above T,, in a field-frequency regime quite different from that of the
susceptibility experiments. We hope to extend our knowledge about ESR in spin glasses
to metallic multilayers, and hopefully to shed some light on the general behavior of spin

glasses on the approach to T,.

CHAPTER SIX

EXPERIMENTAL METHODS

INTRODQCTION

This section details the experimental techniques used in this study and is divided
into three parts. The first part describes the fabrication of the target and the sputtering
of the samples. This is followed by a discussion of characterization techniques used to
determine the quality of the samples, and is primarily a summary of work done by the
MSU spin glass group. The third section details the experimental techniques used in

sample preparation and data taking for the Electron Spin Resonance (ESR) experiment.

Sample Naming Convention

The samples sputtered are multilayers of the form W30 nm of spin glass and Wu.

nm of interlayer material, with N bilayers. The standard way of referring to these

samples is WSG/WIL/N, with all thicknesses given in nanometers. As discussed in

Chapter 4, W,L = 30 nm for copper and 7 nm for silicon.

121

122
SAMPLE FABRICATION

To have enough spins in a sample for a measurable signal, multilayers were
fabricated with Wso nm of the spin glass interlayered with a suitable amount of non—
magnetic material which prevents magnetic interactions between spin glass layers. The
original study“:72 investigated the effects of using both silicon and copper as the
interlayer materials. Samples with copper interlayers were chosen for the ESR
experiment because a) most of the measurements done on multilayered spin glasses at
the time focused on copper interlayers b) the silicon interlayered samples are not as close
to ideal multilayers as the copper interlayered samples and c) the copper interlayers

should not produce an independent ESR signal.

Target Production

The CuMn targets were alloyed in an RF induction furnace from 99.9999% pure
copper and 99.99% pure Mn purchased from Aesar, Inc. A cylindrical graphite boat of
radius 6.5 cm and depth 2 cm was painted with boron nitride to prevent contact between
the boat and the target. Two coats of boron nitride were applied, with each coat allowed
to dry overnight. Prior to alloying, the constituents were etched in nitric acid, rinsed in
deionized water and then in alcohol, and weighed. The Mn, which was etched
immediately before alloying to prevent oxidation, was placed at the bottom of the boat
and covered with the copper. The crucible and metals were immediately placed in a

quartz vacuum chamber encircled by the RF coils. The chamber is evacuated to a

123
pressure of approximately 10" torr, flushed three times, and then backfilled with a 90%

Argon/ 10% Hydrogen mixture. The Argon provides an inert atmosphere, while the
hydrogen helps remove oxygen which may outgas as the metals are heated.

The temperature of the boat was measured using an optical pyrometer. Because
of the difference in the melting points of the two metals (T,,,Cu = 1083 °C,
T,,,Mn = 1244 °C), the Mn is placed in the bottom of the boat, and covered with copper
so that the Mn is in contact with the heating graphite boat. The temperature is slowly
increased by increasing the current through the rf coils. As the temperature rises, the
copper melts and surrounds the Mn. As the temperature is increased, the Mn begins to
dissolve. At or near 1244 °C, the molten mixture begins to swirl in the boat. The alloy
is held at this temperature for approximately five minutes to encourage mixing. The
current is gradually decreased, and the sample is allowed to cool. A crust, probably
made up of contaminants and MnO, forms on the top of the liquid as the mixture cools.
The target is removed from the vacuum compartment after reaching room temperature
and turned on a lathe to a final radius of 5.7 cm and a thickness of approximately 1.25
cm.

The concentration of the target is determined by comparing susceptibility
measurements of the freezing temperature to known values of T, as a function of Mn
concentration. In general, a small (1 or 2 percent) excess of Mn is necessary in alloying
the target. We believe the loss of Mn in the alloying process is due to oxidation and
failure to alloy homogeneously, resulting in a larger concentration of Mn and MnO on

the surface which is then machined away.

124

Prior to being placed in the sputtering chamber, the target is washed with alcohol.
The target is presputtered for a few minutes prior to sample deposition to remove surface
impurities acquired during the machining process. The Cu target used for the interlayers

was purchased commercially from Varian, Inc.

Sputtering Apparatus

In sputtering, the surface atoms of a target (CuMn or Cu) are emitted as a result
of bombardation by energetic particles (Ar+ ions in this case). The sputtering chamber
is a stainless steel cylinder of radius 23 cm and height 48 cm. Four DC sputtering
sources are available, but we used only two targets at a time in the fabrication of these
samples. The sputtering sources are mounted so that target particles are sputtered
upward. A Substrate Positioning and Measurement Apparatus (SPAMA) is suspended
from the top of the tank at a height of about 10 cm from the targets. The motion of the
plate is controlled by a computer programmed stepper motor. The SPAMA holds up to
sixteen substrates, grouped in eight pairs of two. In earlier sample runs, two of each
sample configuration were simultaneously sputtered, with the midpoint between the two
samples centered over the sputtering source. Concerns about the angular distribution of
the sputtered ions changed this procedure to making samples one at a time, with each
sample centered over the source. Each removable part of the sputtering chamber is
etched in an acid solution to remove any metal left from the last sputtering run, then

rinsed with acetone and alcohol. Rubber or cloth gloves were worn at all times when

125
working inside the chamber, or with parts placed inside the chamber to reduce

contaminants which might degrade the vacuum.

Silicon (preferably 100, but also 110, or 111 orientation) was chosen as the
substrate material due to its smooth surface and ease of cleaving. After having been cut
into 1/2" x 1/2" squares, the substrates are cleaned in an ultrasonic cleaner first in
acetone and then in alcohol. In earlier runs, the substrates had been cleaned with
Alconox before the acetone and alcohol, but we were advised that this procedure
scratches the surface of the silicon, and this step was removed. The substrates are
mounted in the SPAMA and placed in the sputtering chamber. A capillary fed LN2
substrate cooler, which can cool the substrates to or below room temperature, was
installed to minimize diffusion during sputtering. Thermocouples attached to opposing
sides of the SPAMA are used to determine the temperature during the sputtering process.

Once the chamber is loaded and closed, it is pumped down by a roughing pump
followed by a cryopump. The chamber is baked for 24 hours to allow outgassing,
resulting in a base pressure of s 2 x 108 Torr. Ultra-pure Argon gas is passed through
a cold trap and a gas purifier before backfilling the chamber to a pressure of 2.5 x 103
Torr.

Argon atoms are ionized by accelerating electrons from a hot filament through the
gas flowing across the target. By holding the target at a negative potential, the positive
Ar ions are accelerated toward the target. The sputtering parameters are experimentally
determined so that the resulting CuMn layers are homogenous and preferential sputtering
of one component of the alloy over the other is avoided. Both Cu and CuMn are

sputtered at currents from 0.6 A to 0.7 A and voltages of about 400 V (which means an

126

incoming Ar ion energy of approximately 400 eV). Quartz crystal Film Thickness
Monitors (FTMs) were used to measure the deposition rate of both sources before
sputtering. Typical sputtering rates are 1.1-1.6 nm/s for both the Cu and the CuMn
targets.

In sputtering from alloys, one type of atom may be preferentially emitted from
the target. The transition temperatures of 500 nm films and target shavings showed no
more than i 1 K difference, which indicates that the degree of preferential sputtering
in CuMn is small or non-existent.

The sputtering process is inherently random, which we believe contributes to
achieving a uniform distribution of Mn atoms; however chemical clustering, especially
at higher concentrations, is a possibility. We have minimized both this problem, and the
problem of diffusion at interfaces by keeping the substrates cooled to room temperature
or less during the sputtering process. The effect of cooling on multilayer quality has
been demonstrated to be especially significant in CuMn/Si". The detailed analysis of the
freezing temperature in these multilayers has shown scaling behavior to hold for samples
with concentrations up to 14% , indicating that increased ferromagnetic interactions in

larger concentrations are not likely.

CHARACTERIZATION

An ideal multilayer structure has crystalline layers and perfect periodicity

(meaning flat, abrupt interfaces between layers with no intermixing). Although the

multilayers produced by sputtering are not ideal, the studies cited in Chapter Four have

127

shown that the quality of the samples is sufficient for the observation of numerous
interesting spin glass properties. Because knowledge of how closely these samples
approximate ideal multilayers is fundamental to further studies, many characterization
techniques were used to determine the integrity of the multilayers. The initial
investigations have provided a consistent picture of the quality of the sputtered
multilayers, so one or two of these techniques are now used as routine checks. In the
CuMn/ Cu samples, the primary techniques used were profilometer measurements, x-rays,
and susceptometry. Although investigation into the detailed structure of sputtered
multilayers continues, the sample production process should be regarded as consistent and
well-established. This section outlines the methods which have been used to establish the
quality of the samples and summarizes the conclusions of the spin glass group to date.

The step most conducive to intermixing of the layers is the actual sputtering
process. Ag-Cu multilayers, which were made because Ag and Cu are immiscible,
indicate that even with 0.8 nm/0.8 nm samples, the sputtering process must be good
enough to keep the two metals from spreading more than two atomic layers (.4—.5 nm).
The total spreading of the host materials is .81 nm for a given layer.

Diffusion of the Mn into the buffer layer is the most likely source of significant
deviations from ideal layering. Diffusion of Mn in Cu in defect free material at room
temperature and even at temperatures comparable to those reached during the sputtering
process is fairly negligible. The sputtered materials, however, are highly polycrystalline,
resulting in grain boundaries and other defects at a concentration of about 1%. If these

defects were all vacancies, then Mn diffusion into the bulk might be a problem.

128

Some susceptibility measurements indicate that subtle changes in the magnitude
and qualitative characteristics of x(T) occur over short (days or weeks) time frames after
sputtering, but no change in the value of T, is observed and these measurements are
highly preliminary. Measurements of samples on time scales of a few months to more
than two years after sputtering show no perceptible changes in either x-ray measurements
or the value of T,.

As a further test of the possibility of diffusion, samples were sputtered with the
substrates held at different temperatures ranging from ambient to below room

temperature. No significant changes in the values of T, were observed.
X-Ray Studies

X-ray diffraction uses Bragg reflection, in which periodic structures in the sample

scatter an incident x-ray beam of wavelength )s at an angle 0.

sin(0) = 1% 6-1

where 1 denotes the order of the reflection and a measures the periodicity of the sample.
Measurements are taken at large angles (LAXD) and at small angles (SAXD). In a
single crystal, measurement of the angle of a reflection peak allows calculation of the
spacing between the crystal planes. The individual lattice parameters of the layers are

periodic only within the layers, so the real periodicity is in the bilayer spacing, d

129

(d = W56 + W“). The lattice parameters for the spin glass and interlayers are aso and

ac“, with 11,6 and no, denoting the numbers of atoms in each bilayer, so that

d = "so ass + "Cu 004 6'2

In the limit of thick layers, several Bragg peaks corresponding to the bilayer
thickness will occur in the region where the Bragg reflections would have been seen for
the individual layer thicknesses, resulting in strong main peaks at these angles,
surrounded by satellite peaks. In the limit of thin layers, the two peaks corresponding
to the individual layer thiclmesses coalesce into one main peak at an average lattice
spacing.

Satellites are a result of periodic variations in electron density and lattice
parameters. By modeling these variations according to a square wave or sine wave
profile, a theoretical prediction of the position and intensities of the x-ray peaks can be
calculated, using the layer thicknesses as the parameters.

In CuMn/Cu, no SAXD satellites are found. LAXD satellites are seen, but are
weak for concentrations of Mn < 7 at.%. The lack of the SAXD satellites is not
surprising, because the electron densities are very similar in CuMn and Cu. In addition,
30 nm of copper are needed to decouple the spin glass layers. This means that the value
of d is at least 32 nm which, according to Equation 6-1, means that 20 = 0.25 degrees.
Experimentally, the direct x-ray beam obscures detection at this angle. The similarity
of the electron densities diminishes the intensity of the satellites, so that the first order

i=0 satellite intensity is 0.000006 of the main peak. In order to test the layering in the

130

CuMn/Cu system, samples were made with thinner values of both the copper and the
spin glass layers. Measurements performed on these samples showed weak SAXD peaks,
and we believe these measurements to be representative of the CuMn/Cu series.
Calculated values of the bilayer thickness in CuMn/ Cu from LAXD measurements are
less than 10% different from the value indicated by the deposition rates measured by the
film thickness monitors.

The crystallite dimensions can be estimated from the widths of the LAXD lines,
using

5 2 K). 6_3
0008(9)

 

where S is the crystallite size, K is a constant on the order of unity, A is the x-ray
wavelength, 0 is the Bragg angle, and B is the width of the peak in radians. In
CuMn/Cu, the LAXD reflections were from the < 111 > planes, indicating strong
crystallite orientation. Measured crystallite sizes for CuMn(7%)/Cu148 are 44 nm for

WSG = 2 nm, 43 nm for Wso = 3 nm, and 37 nm for WSG = 5 nm.

Microscopy

Kenning148 performed transmission electron and field emission scanning
transmission electron microscopy on a number of samples. The samples were coated in
epoxy and microtomed; if they were still too thick, they were ion milled. Direct TEM

imaging confirms the layered structures and polycrystalline nature of the material.

131
Energy Dispersive X-Ray analysis (EDX), which detects changes in chemical

composition, was used by scanning across the sample in the direction of layer growth.
Studies in CuMn(21%)/Cu again confirm the layered structure of the materials down to

the beam resolution of about 10 nm.

Resistivity

The resistivity of the samples was measured using a Van de Pauw149 technique.
The results were compared with the predictions of three models: 1) a perfectly layered
model, in which interfaces do not contribute to the resistivity, 2) a Fuchs model, in
which diffuse scattering occurs at the interfaces‘”, and 3) a uniform model in which
the Mn is assumed to have diffused uniformly throughout the sample. The diffuse
scattering model, in which an electron is scattered at the interface such that it loses
knowledge of its initial direction, is expected to be closest to the experimental situation
and this is confirmed by the resistivity measurements. The measured resistivities lie
between the values predicted by models 1 and 2, consistent with the assumption of
layered materials. Profilometer measurements of the total sample thickness were also
made to check for any gross inconsistencies.

To summarize the structural characterization: We believe our CuMn/ Cu samples
to be polycrystalline, with 111 texture along the axis perpendicular to the layers (i.e.
along the growth direction.) The layer spreading in these structures is probably not more

than 0.5 nm.

132
SQUID Measurements

Measurement of the susceptibility as function of temperature, x(T), is the primary
way of determining the freezing temperature. A typical x(T) was shown in Figure 1-1
for a sputtered CuMn film. The susceptibility measurements are made in a MPMS
(Quantum Designs) computer controlled SQUID, with a temperature range of 2 K to
400 K, and in magnetic fields up to 55000 G. Prior to measurement, the magnet is soft
quenched (brought to zero by alternating between consecutively decreasing positive and
negative magnetic fields) in order to eliminate any remanent fields which may affect the
zero field cooled susceptibility. The measurement protocol is as follows: after soft
quenching, the sample is brought to a temperature sufficiently above T,, and is lowered
to 5 K in zero field. A measuring field of 100 G is applied, and measurements are made
as the temperature is stepped back to about 100 K. The sample is then returned to 5 K
in the 100 G field. Field cooled measurements have been made in both directions - with
the temperature starting out low and increasing, and from higher temperatures down.
No changes in the value of T, are found, but a technical glitch in the SQUID requires us
to make measurements from low temperature to high. The freezing temperature is

extracted by visual inspection.

ELECTRON PIN E R TE HNI

Figure 6—1 shows a block diagram of the Varian 4500 ESR system. The

microwave source is a klystron driven bridge which operates at 9.3 GHz with a power

133

output of about 1 mW. The bridge is connected to a standard 3 cm microwave resonant
cavity which operates in the TE 110 mode. At the sample resonance, the resonant
frequency of the cavity will change, which changes the coupling of the ldystron to the
cavity. A sinewave modulation technique which compares the returning signal with a
reference signal is used to keep the klystron locked on the cavity resonance. This
procedure is called Automatic Frequency Control, or AFC.

The reflected ESR signal is detected using field modulation, in which a 10 kHz
ac magnetic field is superposed on the larger applied field via Helmholtz coils mounted
on either side of the resonant cavity. The resulting signal is proportional to the first
derivative of the power absorbed by the sample and is sent to the Y axis of the xy
recorder. A Fieldial field sweep allows us to sweep the applied magnetic field, H,, by
varying the amount of current sent to the magnet. The applied field is measured by a
Hall probe gaussmeter, and the output of the gaussmeter is fed to the X axis of the xy
recorder. The result is a plot of the first derivative of the power absorbed by the sample
as a function of magnetic field. This data is fed from the plotter through some
electronics and an A/D converter into a PC (not shown on the block diagram) for
analysis.

An Oxford ESR-9000 liquid Helium flow cryostat (not shown in Figure 6-1) is
used to control the temperature from 300 K to 4 K. A Au-Fe/constantan thermocouple
is mounted directly underneath the sample region to detect the temperature. The
temperature is regulated by controlling the power supplied to two resistive heaters in the
cryostat, which allows us to maintain the temperature at a stability of i 1 K for long

periods of time.

134

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ESR Sample Preparation

A stainless steel razor blade washed in acetone is used to carefully peel the
samples from the silicon substrates. Due to the overall thinness of the sample, this was
not straightforward. Wetting the samples with acetone periodically during the separation
helped in film removal. In most cases, the films could not be pulled off in one piece.

After separation, the films are mounted on flattened spectroscopic quality 4 mm
quartz rods using Apiezon ’N’ grease. Prior to mounting, the rods are cleaned in
Hexaflouric acid, acetone and hexanes, and checked in the ESR for contamination. The
total area of the mounted sample was approximately 4 mm by 1.1 cm. The sample rods
are placed in the cavity so that the plane of the layers in the film is parallel to the applied

magnetic field.

Skin Depth Effect

One problem associated with ESR experiments in these multilayers is a result of
the attenuation of microwaves in metals. The microwaves are exponentially attenuated
as the depth of penetration into the sample increases. In metals, the skin depth, 6 (in

units of meters), is given by

where 6 = 2 6‘4

 

136

where w is equal to 21w, (1: in Hz), or is in inverse Q-m, and I1 is in fl-m-s (note that 1
fl-m-s = l N/Az). At five skin depths, only 1 percent of the initial electric field
remains, and only 0.01 percent of the incident power.

The original investigations into the effect of the skin depth on ESR in metals were
pursued by Dyson‘”, who developed a theoretical expression for the lineshape, and by

Feher and Kip”2

, who experimentally confirmed the predictions. Dyson realized that
diffusion of conduction electrons in and out of the skin depth would alter the ESR
lineshape, as only the skin depth was (effectively) ’seen’ by the resonance experiment.
Dyson showed that the diffusion of electrons in metals throughout the skin depth changed
the shape and the intensity of the resonance line, resulting in a first derivative having a

positive lobe three times bigger than the corresponding negative lobe. Dyson derived the

following formula for the power absorbed by a metach sample of thickness 0

z _c_’ 2 i
PUD (41:) H1 R400

(F+%nvxa2TzGF2) 6'5

L2.

where

0
F u tan(u) u 26 (1 I)

I 2 COM

G = __ 2n + (“R—3“?) cotu + (wz-uz) cscz(u) 6-7

 

 

137

 

 

= l = 9. I2 6-8
w 2a(£+vn) a 6\ T2
1 a a a 6.9
5: sgn(x)[(l + 12)2 " 112 TI =sgn(x)[(1+ x2)2 +112
x = ——2 u: T’ (H, -H) 6-10

6 is the skin depth, T,, is the time it takes an electron to diffuse through a skin depth, and
T2 is the electron spin-spin relaxation time. The linewidth itself does not change, but the
shape and the intensity of the line do.

For layered samples, we can define an effective skin depth by summing the
attenuation in each layer. This does not take the complex effects of reflection and/or
refraction at the interfaces into account, but this detail is unimportant for the purposes
of this experiment. We do, however, have to consider the differences in the
conductivities of the CuMn and the Cu, as pure Cu, with its higher conductivity, will
more effectively attenuate the microwaves. The 30 nm thickness of copper required to
decouple the spin glass layers is the limiting factor on the skin depth. The resistivity
of sputtered Cu films is measured to be 2 j; 1 all-cm. The resistivity of Cu,,,,Mn,, is
about 8 all-cm for x = 0.04, and ranges to 41.43 all-cm for x=0.l4, where all

measurements cited were made at room temperature. The effective skin depths thus

138
range from 2100 nm for a multilayer with W30 = 1000 nm to 754 nm for multilayer with

WSG = lnm (for x = 11%).

When the time it takes an electron to traverse the sample is less that the time it
takes the electron to diffuse through the skin depth, the Dysonian lineshape reduces to
a Lorentzian. Dyson cites an upper limit to this approximation when the sample is less
than four times the skin depth, while Feher and Kip claim that the limit will
experimentally apply when the sample is ’less than’ the skin depth.

In our experience, we find that Lorentzian lines result if the sample thickness is
no more than 3/4 of the skin depth. This limit on the number of spins presents some
difficulty in acquiring data at high temperatures, as the signal to noise is poor in samples
with small WSG; however, we find this constraint preferable to the complications
inherent in analyzing the Dysonian lineshape. The average sample thickness ranges from
1000 nm (for spin glass films which have no copper layer) to 500 nm.

I We investigated a number of options which might reduce the constraints placed
on the experiment by the skin depth problem. From Equation 6-4, we see that we can
only change the conductivity of the sample or the frequency at which the measurements
are made. Increasing the Mn concentration increases the resistivity of the spin glass
layers, but the limiting factor is the requisite 30 nm of copper, so that no significant
advantage is gained by small increases in Mn concentration. Above 16% , clustering and
other factors begin to become important, so increasing the skin depth via an overall
increase in resistivity is not possible.

We sputtered buffer layers of 1000-2000 nm of copper onto the silicon substrates

prior to fabricating the samples in an attempt to increase the total sample thickness. The

139

very thick layer of copper would block out the microwaves entirely on one side of the
sample, so that we would still have a Lorentzian lineshape due to the microwaves
incident on the other side of the sample. Data taken on these samples appeared to
depend on the thickness of the buffer layer, and concerns that we might block out spin
glass behavior led us to abandon this idea.

Measurements made at the Wisconsin College of Medicine Electron Spin
Resonance Center (WCOMESRC) on Cuoanom/Cu samples at 1, 2, 4 and 9 GHz show
that samples which are Dysonian at 9 GHz are Lorentzian when measured at lower
frequencies. This confirms that the observed lineshapes are in fact due to the skin depth
effect and not a spin glass property. In standard resonant cavity ESR, the cavity size
increases as frequency decreases, which results in decreased sensitivity. A loop-gap
resonator system, like the ones employed by WCOMESRC, requires the same amount
of sample for 1 GHz as for 9 GHz measurements, with comparable sensitivity. Further
measurements at WCOMESRC were not pursued due to a lack of adequate temperature
control, but we have shown that the loop-gap system would be an ideal experimental

apparatus for measuring the ESR of multilayered spin glasses at low frequencies.

Background Signal

Samples are sputtered directly onto silicon squares, and then separated from the
substrates for measurement. The silicon wafers we buy are generally surplus wafers, the
(10th and concentration of which varies according to what the vendor has available.

Our study has shown that some of the (10th either diffuses into the sample, or adheres

140
to the back of the sample, resulting in a background to the ESR line. We believe that

this signal is due to magnetoresistive effects of the silicon dopants on the cavity
resonance. The magnitude of the background varies from sample to sample and, as
detailed records of which substrates were used for which run were not kept, it is difficult
to make any further correlations between substrate dopant and the extent of the
background. The background is significant only in samples with small W30, as the signal
from the spin glass in thicker layers is much larger than the background. In some cases,
background was seen on one ESR sample, but not on another ESR sample taken from the
same silicon square, indicating that this may be a result of how the sample is removed
from the substrate. The use of acetone in the removal was shown to have no effect on
the presence or lack of a background signal.

Figure 6-2 shows the background, as observed at T = 300 K in a Cu0,,9Mno.,,/Cu
3/30/20 sample. At this temperature, the spin glass signal is much weaker than the
background, so the signal seen here is primarily due to the background. The spin glass
signal can be seen as a very small feature centered at about 3300 G. Below 2000 G, the
background has a negative curvature and is nonlinear; over the rest of the field range,
the background is linear with a positive slope, allowing us to subtract it during the fitting

process.

Measurement Protocol

The data presented for a given sample was all taken during the course of a single

data run to avoid small changes due to sample position and orientation in the cavity,

141

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thermal cycling effects, and more significantly, hysteretic effects. Data is taken starting
at room temperature, with the field turned completely to zero while the sample is being
cooled to the next temperature. A residual field of about 20-30 G exists during the
cooling; however, the measurement fields are much larger than this field, and we expect
the residual field to have no effect on the data. The temperature increments are
dependent on the behavior of the line; at high temperature, data is taken every 10 K.
As the rounding of the AH(T) line begins, data is taken every 5 K. In the regime of the
divergent behavior, AH(T) has such a large slope that small temperature changes result
in very large changes in the linewidth, so that data must be taken every 2.5 K. Data
points are taken until the line becomes too broad to measure, or too weak to observe.
In cooling from T, to the next temperature to be measured, T2, the temperature
is not allowed to fall below T, to avoid hysteretic effects. Hysteretic behavior has been
routinely observed in these samples, but has not been investigated in any systematic
manner. As an example, consider a measurement at temperature T, (T, > T,) which
results in a linewidth AH,. The sample is cooled to temperature T,, where T, is also
above T,. After waiting some time at T2, the sample is returned to T, and the linewidth
is remeasured. The resulting value will be smaller than AH,. As time goes on, with the
system sitting at T,, repeated measurements of the linewidth will result in increasing

values of AH, until the original value is returned.

143
SUNIMARY

This Chapter has detailed the fabrication and characterization of the multilayers
used in the Electron Spin Resonance experiments. Based on the sum of the investigations
performed on these samples, we claim that the CuMn/Cu multilayers are polycrystalline
in nature, with the value of Wso modified by less than 0.5 nm due to diffusion of Mn
into the copper layers.

The ESR system has been described, and two problems — the skin depth effect
and the background - have been discussed. The skin depth problem limits the total
thickness of our samples to less than about 3/4 of the skin depth, so that the average
sample thickness is 600-1000 nm. A background, which is due to the magnetoresistive
behavior of impurities which are transferred from the substrate to the ESR sample, is
linear in the field range of 2000 G to 6000 G, enabling us to fit it with little difficulty.
The measurement protocol has been summarized, and the routine observation of

hysteretic behavior noted.

CHAPTER SEVEN

DATA AND ANALYSIS

"It appears that the quantitative analysis of such ESR measurements
(above T,) is a complicated problem "1

SUIVIIVIARY OF SAIVIPLES ANALYZED

Data was taken on three complete sputtering runs, with some additional
measurements made on other samples. The concentration regime over which data can
be taken across a wide range of temperatures and layer thicknesses is small, due to the
skin effect and background problems discussed in the last chapter. For lower
concentrations, the ESR line is undetectable at high temperatures, even in the thickest
samples. In samples with concentrations greater than 16%, the lines becomes extremely
broad, making reliable extraction of a linewidth and position difficult.

Tables 7-1 through 7—4 detail the configurations of the samples studied. All of
the samples studied were either single layer films, or had Cu interlayers of 30 nm, so the
samples will be referred to solely by the spin glass layer thickness, making note of any

exceptions. Tables 7-1 through 74 may be consulted for additional information.

144

145

Table 7—1: Configurations of Run 243 — Cuangna ,,

  

 

  

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146

Table 7-2: Configurations of Run 185 - Cuaanam

 

 

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185-2 50 30 6 36.0

 

185-4 30 30 15 35.0

 

185-3 7 30 16 25.0

 

185—10 5 30 16 23.5

 

185-6 3 30 17 17.5

 

 

 

 

 

 

 

 

147

Table 7-3: Configurations of Run 245 - Cua93Mna07

~ ‘tii‘nanztsiwtm. *

    

 

 

 

 

 

 

245-3 10 30 17 27.0 II
2454 9 30 17 26.0 H
245-6 7 30 13 23.5 "
245-7 6 30 18 22.0
ms 5 30 18 21.0

 

245-12 3 30 18 16.5

 

 

 

 

 

 

 

Table 7-4: Configurations of Additional Samples Run

148

 

'1 Other Samples Analyzed  

 

 

 

 

 

 

 

 

 

 

 

 

 

265-2 16 500 30 67.5 * ll
220-15 7 500 - 39.0
220—16 7 500 1 39.0

85-4 4 500 - 24.0
=___l_____ ______n

 

149

ANALYSIS OF THE FREEZING TEMPERATURE

Figures 7-1 through 7-3 show the freezing temperatures as a function of the log
of the spin glass layer thickness for runs 243, 185 and 245 respectively. This behavior

can be fit to the finite size scaling form

6 = 1}(°°) " Tf("ls(;) ~ W4
Tf(°°) so

 

where T,(oo) is the bulk freezing temperature. The value of A is obtained by the slope
of a linear regression of log(e) vs logONSG). The fits are limited to samples with
W86 2 30 nm, although data down to 9 nm had to be included in order to use the linear
regression program in fitting the 245 run. The correlation length exponent, u, is UK.

The fit values are summarized in Table 7-5.

150

Table 7-5: Parameters from Finite Size Scaling Fits of T,

 

 

 

 

 

 

 

 

 

 

 

Sputter Run , Smallest W30
' ‘ . Number . , .‘ '~‘
243 10 nm 0.98 i 0.10 1.02 :1; 0.10
185 7 nm 0.79 :t 0.08 1.27 :1; 0.12
245 9 nm 0.78 i 0.06 1.28 i 0.10
All 7% 10 nm 0.80 i 0.08 1.25 :1; 0.13
All samples 10 nm 0.78 j; 0.08 1.28 :1; 0.13

 

 

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155
Figure 7-4 shows a normalized plot of log(Tf(Wso)/Tf(oo)) vs logONSG) for all

samples. The solid line indicates the prediction of Equation 7 -1 for the value of
u = 1.25 i O. 13 found by fitting all concentrations. This value, along with the general
trends represented by this analysis, are in agreement with those determined by all of the
past work on CuMn/ Cu multilayers, indicating that these samples are of the same quality

as those described in Chapter Six.

LINEWIDTH ANALYSIS

As explained in Chapter 2 and Appendix A, the resonance line detected by the
ESR spectrometer in metals is proportional to a mixture of the real and imaginary parts

of the susceptibility. The Lorentzian dispersion and absorption signals are given by

 

 

 

I = x I, = 1 7‘2
X l + 1:2 X 1 + x2
where
H—Ho
x =

7-3

13m

2

The first derivative resonance lines captured by the computer are fit to
alx'+a2x", with the ratio of azlal held constant for each run. For the 185 data run,

012/011 = 9.0, while for all other runs, az/al = 7.0. The linewidth is relatively

156

insensitive to small changes in the value of azlal, while the resonance peak position may
change by as much as 20—30 G when azlal is changed by 1.

Figure 7-5 compares the lineshapes at 100 K for the smallest (1 nm) and the
largest (500 nm) W30 from the 243 (CuongnQu) run, along with the calculated fits. The
individual dots are the discrete values accumulated via the A/ D converter, and the thin
line is the fit. The linewidth of the 500 nm sample is 373 j; 10 G, and the resonance
position, H0, is 3294 i 10 G. The corresponding parameters for the 1 nm sample are
AH = 360 i 14 G, and H0 = 3315 i 12 G. In fitting the 1 nm sample, only the
portion of the data taken in fields greater than 2000 G was used in order to avoid effects
from the nonlinear part of the background. These lineshapes are representative of all
data taken.

Figure 7-6 illustrates the contributions from the real and imaginary parts of the
susceptibility and the background for a 500 nm CuaggMnm, multilayer. The solid line
is the total curve, the dotted line is the contribution from the azx” term, the dashed line
is the contribution from the alx' term, and the dot-dashed line is the linear background
contribution. Observed lineshapes are Lorentzian for all layer thicknesses. Small
changes in the character of the lineshape near Tf are seen; the shape tends to distort

from pure Lorentzian, and the peak height decreases.

157

 

Lineshape: for Cufln(11%) at '1' - 100 K
I I T l l

 

 

 

dP/dfl (Al-b. Units)

 

 

 

 

 

 

 

Figure 7-5: Representative lineshapes for a) a 1 nm Cua ”Mug, ,1 and b) a 500 nm
amneanon samPle

158

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159
Fitting Protocol

The standard expression used in the analysis of the ESR linewidth is

 

AH(T) = A + 37‘ + c[T'Tf‘WW)]" 74

T,( Wm)

Fitting to equation 7-4 can be somewhat complicated, depending on the
temperature range of the data to be fit. A standard nonlinear least squares computer fit
was used. We found that systematic errors and spurious divergences were avoided if the
value of x was fixed and the other three parameters allowed to vary. 1: was then
incremented, and the fit repeated, with the value of 1: resulting in a minimum reduced
chi-squared selected as the best fit. All values for T, are subject to an accuracy of
i 1 K. The fits are noticeably worsened if the value of T, is changed by more than a

few percent.
Departure from the Power Law Behavior Near T,

Departure from the power law behavior near T, has been observed in bulk
metallic spin glasses, with the departure occurring at higher reduced temperatures for
higher frequencies. This reduces the range of temperatures over which Equation 7-4 can
be fit. To determine where this departure begins, the fit was repeated, removing the

lowest temperature data point each iteration. At some temperature, continued removal

160
of points ceased to change the parameters or improve the reduced chi-squared value.

The temperature at which this insensitivity to continued removal of points occurs
(denoted T') is the minimum temperature used for the fit. The reduced temperature

corresponding to T’ is denoted t' (t' = (T'-T,)/T,).

Concentration Dependence

Figure 7-7 shows the concentration dependence of AH(T) for four bulk
(WSG = 500 or 1000 nm) Cu,,,Mn, samples (x = 0.04,0.07,0.11 and 0.16) and
illustrates the point made in the introduction regarding the concentration range over
which data can be taken for this experiment. For the 4% sample, data could not be
acquired above 200 K in a 500 nm film due to thermal effects. The data acquired on the
bulk samples is the strongest data as a result of the constraints on the total sample
thickness due to the skin depth effect, so this data indicates (as has been experimentally
confirmed) that our chances of acquiring meaningful data in samples with smaller Wso
are not good. The data for the 16% sample shows extremely broad lines, with AH(T)
having a minimum value of about 900 G. Fitting lines this broad is difficult, as
contributions from negative fields must be included in the fit to accurately represent the
lineshape. Also, the line becomes too broad to fit in the spectrometer range at
temperatures above 2 T,, which limits fitting in the critical area near T,. Table 7-6
shows the values obtained by fitting AH(T) for bulk samples (500 nm or 1000 nm) at

four different concentrations.

161

Table 7-6: Concentration Dependence of AH (T) in Bulk Spin Glasses

 

4 24.0 34 i 15 2.65 i 0.15 107 i 14 1.5 l

 

7 37.5 -132 j; 32 2.94 i 0.04 248 i 33 1.5

 

11 47.5 -245 i 13 3.36 i 0.01 328 j: 13 1.5

 

16 67.5 -l84 i 37 4.47 :1; 0.20 672 i 25 1.5

 

 

 

 

 

 

ll

 

All concentrations can be fit by Equation 7-4 with an exponent of x = 1.5 :1; 0.1.
We see that, with the exception of the 16% sample, the residual linewidth, A,
monotonically decreases with increasing concentration, while the thermal broadening
coefficient, B, monotonically increases with increasing concentration. The divergence
strength, C, also monotonically increases. The results for all parameters are in general
agreement with the findings of Mozurkewich, et. al.105 in AgMn. (See Table 5-1 for
Mozurkewich’s results). The residual width and the thermal broadening - with the
exception of data from the 16% sample - depend linearly on Mn concentration. The
failure of the 16% sample to follow this trend may be due to chemical clustering, or
inaccurate extraction of the linewidth in this run due to the very broad resonance lines.
The residual width decreases at a slope of 40 G/at. %, reaching a value of O at about 5

at. %. This can be compared to Mozurkewich, et. al.’s slope of 13 G/at. % and

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163

zero intercept at 2 at. % . The thermal broadening coefficient in our CuMn samples
increases at a rate of .1 G/K—at%, compared to 0.04 G/K at.%. in AgMn. The
divergence strength, C, is predicted106 to depend linearly on Mn concentration, but a
sublinear dependence has been observed by Mozurkewich,et. al. If the 16% data is
ignored, we also see a sublinear dependence in C. This may be attributable to the same
effects as are responsible for a sublinear dependence of T, on concentration, such as the
importance of preasymptotic RKKY corrections. Figures 7-8 through 7-10 plot the
magnitudes of the parameters as a function of concentration. The straight lines in
Figures 7—8 and 7—9 represent fits to the linear behavior.

The position and value of the minimum linewidth also change monotonically with
concentration, as shown in Table 7-7 and Figure 7—10. The temperature at which the
linewidth has its minimum value is defined to be Tm, and the minimum linewidth is

AHmin. Tmin can be fit to a power law in c; we find that Tmin ~ cm".

Table 7- 7: Concentration Dependence of the Positions and Values of the
Minimum Linewidth for Cuaana"

 

 

 

 

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The temperature at which AH(T) departs from power law behavior is tabulated

in Table 7-8. We see that lower concentrations obey a power law to lower reduced

temperatures than do higher concentrations.

Table 7-8: Dependence of t' on concentration

 

 

 

 

 

 

 

LAYER THI KNES DEP E

Figures 7-12 through 7—19 show AH(T) and H,,(T) for the 243 run, Figures 7—20
through 7-26 correspond to the same data for the 185 sample, and Figures 7-27 through
7-33 illustrate the behavior of the 245 sample. In all figures, the lower graph shows

AH(T) and the upper plot shows H,,(T). The error bars are less that the size of the

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Figure 7-32: AH(T) and H,,(T) for Sample 245-8 - ChamMnaa, - 5 nm

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191
symbols and have been omitted for clarity. In all cases, the same general behavior is

observed: the linewidth behaves linearly at high temperatures, reaches a broad minimum

and diverges as the freezing temperature is approached.

Qualitative N ates on the Raw Data

As can be seen from the raw data, the same general behavior is seen in samples
of all layer thicknesses. The data from sample 185—3 (7 nm) shows anomalous behavior
in both the linewidth and the resonance peak position; the minimum linewidth is much
larger than the minimum linewidths in the other samples from that run and the resonance
peak position as a function of temperature shows a hump not seen in other data. No
anomalies in the SQUID measurements were seen.

The lineshift as a function of temperature also exhibits the same qualitative
behavior for all layer thicknesses; a g-value 2: 2 is observed over the same region in
which the linewidth obeys a linear form. As the linewidth begins to broaden, the
position of the line shifts to lower fields. Over the region where the linewidth obeys a
power law, the fractional shift in the value of H0 is only about 5%; the majority of the

lineshift occurs below t’.

Reproducibility

Figure 7-34 shows two different data runs of sample 243-2, a 3 nm

CuMno.,,,,Mno.ll multilayer. The data runs were made about a week apart, and illustrate

192

the lack of significant run-to run variations. The data illustrated by the squares in
Figure 7-34 exhibited a strong baseline, while the data represented by the circles did not,
which confirms our ability to extract a reliable value for AH(T) in the presence of the
background discussed in Chapter 6.

AH(T) has been compared in samples with data runs taken as much as a year
apart, with the no significant changes in the behavior of the linewidth or shift as a
function of temperature. No differences in the values of AH(T) or HOCT) are found in
comparing data from ESR samples taken from different parts of the substrate. The
primary source of difficulty in making repeated data runs on a given sample is the
tendency of the mounted sample to attract dirt due to the presence of the Apiezon grease.
Great care was taken in the storage and handling of the samples to minimize
contamination. Contamination of the sample is observed as an additional resonance line

superposed on the spin glass resonance and is easily identifiable.

Effects Due to the Layered Structure

Figure 7-35 shows data from two samples which were specifically sputtered to test
the effect of the interlayers. The data shown in circles is data from a 500 nm
Cuo.93Mno.m film, and the data represented by squares is from a sample with a single
30 nm layer of copper in addition to the 500 nm of CuMn. Agreement between the two
sets of data is very good for all temperatures. A study over a larger number of
interfaces is difficult due to the trade-off between signal to noise and skin depth

constraints.

193

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194

Dependence of the Position of the Minimum Linewidth on WSC

The position of the minimum linewidth, denoted by Tm and the value of the
minimum linewidth, AH(Tm) for Run 243 (CuMgMnon) are shown in Table 7-9. The
position of the minimum, in terms of absolute temperature, gradually shifts as WSG
decreases. Viewed in terms of reduced temperature, the change is more dramatic,
increasing from 2.5 Tf in the bulk samples to 6.5 Tf in the 1 nm samples.

The minimum linewidth should be a comparison of the strength of the divergent
behavior compared to that of the linear behavior. The increased reduced temperature of
the minimum linewidth in thinner samples indicates that ordering phenomena become
more important compared to the linear behavior at higher reduced temperatures than in

samples with large W30.

Departure From Power Law Behavior

Tables 7-10 through 7-12 tabulate the temperatures at which AH(I‘) departs from
power law behavior as the spin glass layer thickness is reduced. We see from these
tables that departure from a power law occurs at higher reduced temperature for smaller
W80. Departures from the power law behavior may be due to 1) the effect of a large
field on the nature of the spin glass ordering, 2) regions of the material which, near T,,
are frozen into a spin glass state for times longer than the Larmor period or 3) failure

to remain inside the exchange narrowing limit - i.e. the frequency of the exchange is not

195

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Table 7-9: Characteristics of the Minimum Linewidth for Run 243 - Cuo, ”Mira ,,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

50 45.0 336 115 ‘ 2.6 ll

30 42.5 335 115 2.7

10 35.0 329 110 3.1

7 31.0 334 100 3.2

3 20.5 327 90 4.4

1 10.0 316 65 6.5
=L==————J___—_—__——L__——l___l

197

Table 7-10: Temperature at which AH (T) Departs From a Power Law for
Run 243 ‘ Cl‘agflnau/Cu

 

 

 

 

   

 

100 47.5 77.5 .63

 

50 45 .0 75.0 .67

 

30 42.5

 

10 35.0

 

7 31.0

 

3 20.5

 

1 10.5

 

 

 

 

 

 

198

Table 7-11: Temperatures at which AH (T) Deviates from a Power Law

for Run 185

’ cuo.93Mnao7/Cu

 

I   ‘

W* ‘stscuom _

 

37.5 60.0

 

38.8 65.0

.68

 

36.0 60.0

.67

 

35.0 68.0

.95

 

25 .0 57.5

1.30

 

23.5 ' 62.5

1.66

 

 

17.5 52.5

 

 

 

 

 

 

 

2.00

 

199

Table 7-12: Temperature at which AH (T) Diverges from a power law for
Run 245 " cuangrtam/Cu

 

 

  (nm)

   

11

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30

57.5

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.83

 

23.5

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1.34

 

21.0

60.0

1.73

 

20.0

62.5

2.00

 

 

 

16.6

 

 

50.0

2.00

 

 

200

much greater than the average width of the anisotropy mechanism. Mozurkewichw‘,
et. al. ’5 study of the effects of increasing the anisotropy by Sb doping showed that there
was no dependence of t’ on the Sb doping, as would be expected by the last of these
conditions.

The first case would imply that the formation of the spin glass state is being
affected by the application of a large field. The importance of field effects on spin glass
properties is characterized by the ratio guBH/k'l‘f, implying that field effects should be
less pronounced with smaller Tf’s. This would in turn imply that the value of t' should
decrease with decreasing concentration, as is confirmed by the data in Table 7-8. Since
samples with smaller WSO have smaller Tfs, t' should also decrease for decreasing spin
glass layer thickness - which is the exact opposite of what is seen.

If formation of clusters is responsible for the deviation from power law behavior,
the behavior of t' with Wso implies that cluster formation must begin at higher

temperatures relative to Tf in smaller WSG than in larger WSG.

lFits of AH(T) as a Function of WSG

Tables 7-13 through 7-15 show the values obtained by fitting to Equation 7-4 for
runs 243, 185 and 245. Figures 7-36 and 7-37 show representative fits of AH(T) to
Equation 7-4 for the smallest W80 (1 nm) and the largest Wso (500 nm) in the 243
CuMQMnQ1 1 run. The appearance of all fits is very similar, so only these two are shown
to avoid repetition. In both plots, data points which were not used in the fit (due to the

departure from power-law behavior) are shown by solid symbols. Figure 7-38 compares

201

the theoretical fits for a representative group of spin glass layer thicknesses in the 243
CuangnoJl. Figures 7-39 and 7-40 show the excess linewidth, AH“x = AH(T) -
(A+BT) for representative samples in the 243 and 245 runs. Error bars, which are

approximately 10—30 G, have been omitted to increase clarity.

General Behavior of the Fits

We can identify some broad trends as a function of layer thickness. The exponent
characterizing the divergence, It, increases with decreasing layer thickness for samples
with W30 2 5 nm. Below this thickness, the exponents decrease, but still have values
greater than the bulk value of x = 1.5. Both the residual width, A, and the thermal
broadening coefficient, B, tend to increase with decreasing layer thickness. From the
data in Table 7-6, the increase in the residual width with decreasing layer thickness might
be indicative of a reduced effective concentration. If spreading of the spin glass layer
into the interlayer were significant, the sample would have a larger W30, and a lower
concentration; however, the amount of spreading would have to be much larger than the
nominal .5-1 mm to explain the magnitudes of the changes seen. The graph of residual
width vs. concentration indicates that the CuomMnQll 3 nm sample from Run 243 would
have to have an effective Mn concentration of 7% to explain the observed value. The
concentration study also indicates that decreasing concentration should result in a smaller
thermal broadening, and the opposite effect is noted with decreasing W50. We conclude

that the systematic changes in A and B are not due to the spreading of the spin glass

202

Table 7-13: Fits to A+BT+CI* for Run 243 - Cuo,89Mno_,,/Cu

l

17A

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1113

 :1:=(66 

 

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-245 j; 13

3.36 :t 0.01

328 j; 13

1.5 :t 0.1 E

 

-245 j; 14

3.46 i 0.02

324 :1: 17

1.5 :t 0.1

 

-236 i 14

3.40 :t 0.02

366i16

1.5 fl: 0.1

 

 

-226 i 14

3.57 :l: 0.02

453 i 16

2.0 i 0.1

 

35.0

-201 i 14

3.54 :t 0.02

650 i 13

2.0 i 0.1

 

31.0

-183 i 14

3.70 :t 0.02

890:]:17

 

20.5

-145 :1: 15

3.73 i 0.03

1182 j; 21

 

 

 

10.5

 

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203

Table 7-14: Fits to A+BT+Ct" for Run 185 - Cu0_9,Mn0_07/Cu

 

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1000

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2.94 :1: .04

248 j; 33

 

100

-151 i 33

3.00 j: 0.10

280 j; 34

 

50

-153 i 33

2.92 i 0.03

294 i 35

 

30

35.0

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3.22 i 0.02

413 i 36

 

25.0

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3.45 :1: 0.14

566:1:45

2.1 :t 0.1

 

23.5

-115 j: 33

3.19 i 0.04

1504 i 56

2.4 i 0.2

 

 

 

17.5

 

—82 i 44

 

3.62 i 0.14

 

 

1340 j; 31

 

 

2.1 i 0.2

 

204

Table 7-15: Fits to A+BT+ Ct'“ for Run 245 - Cza,,Mna07/Cu

 

 

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280 :t 15

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3.38 i- 0.02

492 i 16

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23.5

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3.24 i 0.03

710 i- 14

2.0 i 0.1

 

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3.42 i 0.03

1888 i 46

2.8 i 0.1

 

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3.34 :t 0.03

2868 i- 65

2.8 i 0.1

 

 

 

16.6

 

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3.83 :t 0.03

 

1576 i- 20

 

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210

layers. The divergence strength, C, also displays a marked increase with decreasing
layer thickness.

The larger errors on the 185 run are due to three factors. This data was taken
prior to the renovation and fine—tuning of the ESR apparatus. These samples were also
fabricated before the substrate cooler was installed, and some of the samples were
sputtered as single samples with the substrate centered over the gun, and some were
sputtered two at a time, with the midpoint between the two substrates centered over the
gun. The susceptibility measurements were somewhat anomalous, with bulk samples
sputtered during the same run having Tfs which differed by as much as 3.5 K, (37 .5 K
to 41.0 K) which is outside the normal estimated errors of j: 1 K. The difference in Tfs
may be caused by changes in the sputtering rate over the course of a sputtering run,
differences due to the substrate positions with respect to the gun, or lack of substrate
cooling. Target shavings were measured to have a Tf of 38 K. The bulk freezing
temperature for this run was taken to be 39 K, which contributed to larger errors on the

fits, but allowed a consistent analysis of the run using a single bulk temperature.

INTERPRETATION OF THE MEANINQ OF T, IN mflATION 7-4

As is evident from the preceding chapters (and the opening quote of this chapter)
the interpretation of Electron Spin Resonance measurements above T,., even in bulk spin
glasses, is not straightforward. Neither of the two theories discussed (mean field and
cluster) has addressed ESR in spin glasses in lower dimensions, and there have been no

calculations regarding finite size effects on ESR measurements.

211

Use of Equation 74 in the analysis of multilayered spin glasses raises some
questions. The first problem is that our goal - observation of finite size and
dimensionality effects - is confused by the presence of T,(Wso) in Equation 7-4. We
must therefore determine which of the trends observed in fitting to Equation 7-4 are due
to the depression of T,, and which are indicative of effects on the ESR linewidth.
Knowledge of the behavior of T,(WSG) in different field, frequency and dimensionality

regimes will contribute to separating these effects.

Zero Temperature (2D) Scaling Forms

The first question is to determine if Equation 74, which has been shown to
describe the behavior of AH(T) in bulk spin glasses and is consistent with the predictions
of both the mean field and the cluster model, is suitable to describe samples with small
W36. The fits described by Tables 7-13 through 7-15 have shown that Equation 7—4 does
indeed describe the data for all layer thicknesses for T > T,. Systematic trends as a
function of WSO are seen in the parameters, but samples with Wso s 3 nm do not obey
these trends.

Analysis of ac susceptibility measurements of thin multilayers, and of 2D Ising
spin glasses, indicate that two dimensional behavior should be characterized by a zero
freezing temperature, so that our assumption of a finite Tf in Equation 7-4 results an
incorrect form for the divergence. The zero freezing temperature form corresponding

to Equation 7-4 is:

212

AH(T) = A + BT + c' T" 7-5

where C' is used to indicate that this coefficient should not be expected to equal the
values of C obtained from the fits to finite T,. Tables 7-16 through 7-18 show the results
of fitting to Equation 7-5.

Fitting the samples with WSO s 3 nm to Equation 7-5 does not substantially
change the values of A and B. The value of C' does not appear to have any obvious
relationship to bulk parameters, but does decrease with decreasing layer thickness. The
exponents obtained by this fit continue the trend of increasing exponents as the layer
thickness decreases.

Figures 7—41 through 7-44 summarize the dependence of the fitting parameters on
WSG for run 243. Figures 7-45 through 7-48 and 7-49 through 7-52 show the
corresponding information for runs 185 and 245 respectively. In the plots of A(WSG) and
«(WSGL squares indicate parameter values obtained by fitting to a zero temperature
transition and circles correspond to fits made with finite values of T,(WSG).

There are two issues to be considered in separating effects on T, from effects on
the ESR linewidth. The first issue is field and frequency effects on T,, and the second
involves the depression of T, with layer thickness. Thus far, all data has been analyzed
in terms of the values of T, measured at 100 G and a measuring time of 300 seconds.
The use of the bulk T, for all samples will not work; in very thin samples, the linewidth
is barely out of the broad minimum at this temperature. We consider these effects in the

following sections.

213

Table 7-16: Fits to A+BT+ C' T" for Run 243 - Cuaanau/Cu

 

 

 

 

 

No fit

 

No Fit

 

No Fit

 

No Fit

 

No Fit

 

 

31.0

-199 j; 14

3.70 :t 0.02

1.77(i.03) x 108

3.0 :1; 0.2 II

 

20.5

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l

3.73 i- 0.03

1.57(:.02) x 107

2.5 i 0.2

 

 

 

10.5

It

 

-76 i 14

 

 

 

4.18 i 0.03

__—__l|—_—___A.__—_

2.57(:t.06) x 10‘5

 

 

2.4 :t 0.2

 

 

214

Table 7—17: Fits to A+BT+ C’ T" for Run 185 - Cu0,9,Mn0_07/Cu

 

U Run185~Cu093Mnom

 

   

 

 

 

 

 

30 35 .0 No Fit

 

7 25.0 -27 _-_I-__ 13 3.45 i- 0.14 3.01(i.08) x 108 3.4 j: 0.2

 

5 23.5 -104 :I; 13 3.19 i 0.04 1.90(i.02) x 1010 3.7 i 0.2

 

3 17.5 -31 i 12 3.62 j: 0.14 5.93(i.08) x 108 3.7 i 0.2

 

 

 

 

 

 

 

215

Table 7-18: Fits to A+BT+ CT“ for Run 245 - Cuaanam/Cu

 

 

 

(G)

 

 

 

' (G) C ’

 

No Fit

 

No Fit

 

 

No Fit

 

23.5

No Fit

 

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6.35(;t.16) x 109

4.1 :1; 0.2

 

 

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3.34 :t 0.03

5.l9(i.12) x 109

4.0 :1: 0.2

 

 

 

16.6

 

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3.83 :t 0.03

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228
Field and Frequency Effects on TJWSG)

In general, the freezing temperature is a function of measuring field, frequency
and spin glass layer thickness, so that we must consider T,(WsmH,w). In ESR
measurements, the frequency determines the field used - a frequency of 1 GHz requires
a magnetic field of 300 G. For ESR at 9 GHz, a 3300 G field is required. We have
seen in Chapter 4 that field and frequency effects have opposite actions on T,; an
increased magnetic field results in a depressed value of T,, and an increased frequency
results in a larger value of T,. We have also seen that, as the layer thickness is reduced,
the form of the frequency dependence changes.

According to the prediction by Fischer”, and the experimental data by Kenning,

l. 83

et. a , the magnetic field dependence of '1‘f is different in low fields than in high fields,
with the crossover point experimentally found to be between 300 and 500 G in
Cuo_93Mn0m/Cu multilayers for WSG = 2 nm to 500 nm and fields from 2 G to 15000 G.
We can use this data to predict the freezing temperature over the range of magnetic fields
used in the ESR experiment. Concentration dependence can be taken into account by
scaling the freezing temperature like concentration to the 2/3 power. ‘53. Table 7-19
summarizes the values predicted for Cuo_89Mn0_u/Cu for the range of fields used in the
ESR experiment.

This table shows that the percent change of the value of T,(WsmH) from the value
measured at 100 G is much greater in the thinner layers than in the thicker layers. For

the 1 nm sample, the change from the ’DC’ measurement is a 40 to 68 percent

depression, while the corresponding change for the 500 nm samples is only 12 to 23

 

229
percent. In bulk AgMn, Wu, at. 01.107, claim that the freezing temperature is roughly

field independent for the magnetic fields used for ESR and assume a single value of T,
for all fields and frequencies in their scaling analysis, in contrast to the predictions of
Table 7-19.

The Uppsala group has measured the frequency dependence of the freezing
temperature over eight decades from 10“ to 10‘ seconds. Holtzberg, et. al."4 have
made a similar measurement on EuSrS from 17 Hz to 2100 Hz, and used a resonant
cavity technique to measure T, at 9 GHz. The EuSrS results indicate that a Fulcher law
with a characteristic time of 1013 s describes the data for all frequencies, including
9 GHz.

Assuming this extrapolation to 9 GHz is valid for CuMn, we can use the Uppsala
measurements to predict a value of T, at 9.3 GHz. Figure 7-53 shows the measured
values of T, from the Uppsala data, and the extrapolated T, at 9.3 GHz. Table 7—20
shows the values of T, at 9.3 GHz for Cumgan assuming both a finite T, and Tll = 0.
(Recall that T3 is the equilibrium freezing temperature.)

Figure 7-54 shows the ranges of predicted T,’s from the two field/frequency
regimes; the circles represent the Uppsala prediction [T,(WSG, 10 G, 10'10 s)], the
squares represent the magnetic field dependence for the fields used in the ESR
experiment [T,(WSG, 1000 G < H < 6000 G,3OO s)] and the diamonds represent the
values used in the fitting presented here, TKWSG, 100 G, 300 3).

Part of the problem in combining these two effects to obtain T, at conditions
appropriate to ESR is that the frequency dependent measurementswere made in the low

field regime, which should exhibit Ising-like behavior according to Fischer’s predictions,

230

Table 7-19: Predicted Dependence of T, on Magnetic Field for Cu0.89Mna ,I/Cu

 

 

 

    

...,

       

«00>

  

  

* (K)

 

47.5

41.7

39.0

36.7

 

100

47.5

40.4

37.8

35.4

 

50

45.0

39.8

37.2

34.9

 

30

42.5

35.3

32.8

30.6

 

10

35.0

25.8

23.5

21.5
1.

 

31.0

22.7

20.5

18.5

 

20.5

15.3

13.4

11.7

 

 

 

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232

Table 7-20: Ertrapolation of I} to 9.3 GHz for Cu,,.,,,,Mn,,L u/Cu

~ 1H2); «a»

       

_ “9}

 

 

 

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5 58.0 56.6

 

10 67.5 65.6
ll

 

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233

while the ESR data is taken well within the Heisenberg—like regime. If we were to use
the value of T, at 9.3 GHz as extrapolated from the Uppsala data in Fischer’s prediction
for the Heisenberg-like regime, we would substantially overestimate T,. Use of the
values of T, extrapolated from the Uppsala data with no field correction in Equation 7-4
is incorrect, as the linewidths continue to diverge for temperatures below these values.

Fitting to Equation 74 with the value of T, predicted to apply at 3300 G (roughly
the position of the resonance peak center) results in a systematic shift in the parameters
A, C and x. The thermal broadening coefficient remains roughly constant and the trends
with decreasing layer thicknesses are unchanged.

The analysis presented in this section leads us to conclude that, at present, we
have no better alternative to using the value of T, from the quasi-static SQUID
measurements at 100 G. All of the literature on ESR in bulk spin glasses uses quasi-
static measurements of the freezing temperature in analysis of AH(T), which will

facilitate comparison between this data and other reports.

Effect of the Depression of T, on Equation 7-4

The effect of dimensionality and finite size in depressing T,(WSG) in these
multilayers has been well documented in Chapter Four. The presence of T, in Equation
7-4 complicates interpretation of this expression. We must be very careful that the
effects of the depression of T, are separated from possible dimensionality and finite size

effects on AH(T). The similarity of the AH(T) curves for all layer thicknesses suggests

 

234

 

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that there might be a simple transformation - such as a temperature shift - which will
allow us to essentially remove the effects of the depression of T,.

In general, there exists some function which relates T,(Wso) to the bulk freezing
temperature, T,(oo). We can look at the specific case of finite size scaling as given by
Equation 7—1 , although the same analysis is applicable to any general relationship
between T,(WSO) and T,(00). We can solve Equation 7-1 for TKWSG) in terms of T,(oo)
and plug this result into Equation 7-4. This allows us to write a more general equation

which is expressed in terms of a shifted temperature, T'.

T’ = T + a w; T,,...) = T + 7,0») - T,(W,G> 7"

The shift in the temperature scale, T,, is just the difference between the freezing

temperature at W30 and the bulk freezing temperature. Equation 7-4 becomes

 

AH(T’) = A’ + B T’ +C’ T, - 7}(°°) 1 7-7
1,0»)

where the parameters in Equation 7-7 are related to those in Equation 7-4 by

AI =A - B a W33 Tf(00) =A + B(Tf(oo) _ Tf(Wso)) 7-8

cl=c(1_aWS‘G‘ "=c(W]-x 7-9

no»)

 

236

By using the relationships given by Equations 78 and 7-9, we can determine to
what extent the trends observed in fitting to the measured T,(WSG) (Equation 7-4) are due
to the depression of T,WSG. The values of A and C are predicted to change, but the
values of B and It should remain invariant under this transformation. Tables 7-21
through 7-23 compare the values of A and C under the transformations. Figures 7-55
through 7-60 graphically illustrate the effect of using T,(WSG) in Equation 7-4.

The changes due to this shift are very illustrative, especially in the case of Run
243. The residual width has the same value for all layer thicknesses when fit to Equation
7-7. The divergence strength, C, takes on roughly the same value for all layer
thicknesses 2 7 nm. Below this thickness, the analysis for Run 243 results in a lower
value for the divergence strength. The analysis for run 245 also indicates a departure
from systematic behavior at about Wso = 7 nm, but the values of C for samples less
thanthis thickness are larger than the bulk value. The data for Run 185 indicates a
failure between 30 and 7 nm; the quality of the data andthe lack of data between these
two layer thicknesses makes it difficult to localize where the failure begins. The values
of C are on average overestimated, with the exception of the 5 nm sample which has
already been identified as behaving anomalously.

Regardless of the model used to predict the depression of the freezing

temperature, the temperature shift will always be equal to the difference T,(WSG) and

237

Table 7-21: Parameters A and C after removal of T/WSG) dependence of
Equation 7-4 for Run 243 - Cuangna ,1/Cu

 

. , .

 

' = A

 

-245 :I: 13

-245 j; 13

328 j; 13

328 :t 13 1

 

 

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-245 i 14

324 i 17

324 _-t-_ 17

 

50

2.5

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337 j; 14

366i16

 

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-226 i 14

363 j: 14

453 i 16

 

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12.5

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353 :1: 13

650 :1: 13

 

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-183 i 14

349 i 13

890 j; 17

 

27.5

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-145 i 15

281 i 13

1182 j: 21 I

 

 

37.5

 

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166;]: 13

 

 

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238

Table 7-22: Parameters A and C afier removal of TKWSG) dependence of
Equation 7-4 for Run 185 - Cuo.9,Mn0_0,/Cu

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239

Table 7-23: Parameters A and C after removal of T/WSG) dependence of

 

W45 «2me

 

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246
T,( on ) . A’ , which is changed by an additive factor proportional to the magnitude of the

shift, will be the same regardless of the function describing the relationship between
T,(Wso) and T,(oo). The magnitude of C' will be affected by the form assumed to relate
T,(WSG) and Tf(oo), and the failure of samples with smaller Wso to take on the bulk value
of C’ may be due to the failure of finite size scaling in describing this relationship.

A further interesting observation is that the temperature at which deviations from
the power law behavior occurs (1") also becomes relatively constant in the shifted
analysis. If T' is shifted by T,, and the reduced temperature is calculated using T,(oo),
the resulting values of t' for all layer thicknesses range from .63 to .80 for Run 243
(Cuofigan), between 0.60 and 1.0 for Run 185 (CuomMnom) and between .65 and 1.1
for Run 245 (Cuo.93Mnom). The range of values represent scattered values for the shified
t', and do not represent systematic trends with decreasing W30.

The removal of the dependence of Equation 74 on Tf has shown that the
systematic changes in the values of the parameters A and C can be attributed to effects
on T, and not to effects on AH(T); however, the value of the exponent, x, and the

Thermal Broadening, B, are not due to the depression of T,.

COMPARISON WITH THEORY

The two theories developed to explain ESR in spin glasses represent very distinct
physical pictures, even though they predict the same functional form of AH(T). This
point is exceptionally important; in order to properly interpret these measurements, we

must understand what we have measured. The importance of this statement bears a

 

247

review of the differences between the pictures before considering the consistency of the
data with each.

We would like to look for data which would contradict one or the other of these
pictures. Because both pictures predict a power law behavior, but neither model predicts
the value of x, comparing the measured value of x with theory will not serve to
distinguish between the models. We must also consider if, near T,, the models are
consistent with either the general mean field/ hierarchical picture or the droplet picture

described in Chapter 3.

Comparison with the Mean Field Picture of ESR

In the mean field picture, the ESR resonance is due to critical slowing down as
the material approaches the spin glass phase. Physically, this implies that, at high
temperatures, the spins are all in the paramagnetic phase and contribute to the resonance.
As the temperature is lowered, the relaxation times of the spins begin to diverge, finally
going to infinity at T,, which results in an infinite relaxation time and an infinite width
to the resonance line. The ESR resonance thus measures the collective behavior of the
spins. Departure from the power law behavior near T, is attributed to the failure of the
exchange narrowing limit near T,.

The mean field ESR theory is consistent with the mean field picture of spin
glasses; this model assumes a phase transition at T,, and asserts that the transition exists
in large (1000 G to 6000 G) magnetic fields. This is inconsistent with the droplet theory,

in which a phase transition should not be observed in fields of this magnitude.

 

 

 

248
The mean field calculations of Levy, Morgan-Pond and Raghavan have been shown106

to describe low frequency ESR data with accuracy, but fail when applied to high
frequency (9 GHz) data. A direct comparison of their analytic expressions with this data
is therefore not possible.

One point which seems to have been neglected by proponents of the mean field
theory is the relationship of x to other exponents. If x is a critical exponent, relationships
between K and other critical exponents should exist. Wuw’ cites calculations which
indicate x: Zr, and also It: zu-ZB. Appendix B shows that measurements of zv vary,
even within measurements of metallic spin glasses. Values of zv obtained from scaling
measurements of the ESR linewidth in metallic spin glasses are between 1 and 3; those
obtained from AC susceptibility measurements are in the range 7 to 10. Mean field
theory predicts values in the range 2 to 4. Bulk samples are characterized by x = 1.5,
which corresponds to zu = 1.5 or zv = 3.5 from the predictions given above. These
values are consistent with other values obtained from ESR scaling, but are different from
other measurements of 21:.

If the measurement of the divergence of AH(T) corresponds to the measurement
of a phase transition, we would expect to see finite size effects. Finite size effects on
the linewidth should manifest themselves by departure from the bulk behavior at a higher
temperature, and a rounding of the divergence as the layer thickness decreases. We must
be careful to exclude the known effects of the depression of Tf(WSG). From the shifted
analysis described by Equation 7-7, we find that departure from a power law occurs at
approximately the same reduced temperature for all layer thicknesses. In addition, the

value of the exponent increases with decreasing layer thickness.

 

 

249

The effect of dimensionality on x is unclear. One difficulty with numerical
comparisons is that departure from the power law behavior occurs at higher reduced
temperatures for higher frequencies. The values of 1: obtained from our fits would be
an underestimate of the actual value if this effect were pronounced. In lieu of
measurements at other frequencies, our values can be taken as lower bounds on x. In
analogy with Equation 5-34 and 5-42, the linewidth due to an exchange narrowed
broadening should go like 1‘” in two dimensions. We have no a priori reason for
assuming that dynamic critical scaling applies in 2D; measurements in multilayers78 have

shown that the Fisher-Huse model is more appropriate.

Comparison with the Correlated Cluster Model

The correlated cluster model as originally presented by Barnes (See Chapter 5)
presents a very different picture from the mean field model. At high temperatures, the
spins are paramagnetic and all contribute to the resonance. As the temperature is
lowered, regions of the sample become correlated, so that when the relaxation time of
the correlated clusters becomes very long, these clusters will not contribute to the
resonance. This means that AH (T) is not a measure of spin glass properties, but is
actually a measure of the decreasing concentration of uncorrelated (paramagnetic) spins.

In order to apply the correlated cluster model to multilayered spin glasses, some
modifications must be made. Barnes’ original model was that the spins are paramagnetic
up until the time they become correlated. At that time, he assumes that their relaxation

times are longer enough so that they do not contribute to the resonance. Given the broad

250

spectrum of relaxation times above T,, this picture seems restrictive, as spins are either
paramagnetic and in the resonance, or are correlated and are not in the resonance. The
broad spectrum of relaxation times suggests that the resonance line will detect both
uncorrelated spins, and spins in clusters which relax with a time faster than the upper
limit of detection. Up to now, we have avoided using the word ’droplet’ to describe a
picture of this sort, but an extension of the Fisher-Huse theory to this regime seems
natural.

In the correlated cluster model, the linewidth is not due to the spin glass behavior,
and so there is no reason to expect dimensionality or finite size effects. Instead, the
effect of decreasing the sample size should manifest itself in how the correlated clusters
are formed and grow. As the sample size decreases, the ratio of surface to volume spins
increases. Surface spins will be missing nearest and next nearest neighbors, and this
could affect the formation and the shape of correlated clusters. Preasymptotic corrections
to the RKKY interaction are expected to be important when spins are separated by less
than 6 nm in CuMn, and should affect cluster growth and formation which result in a
concentration dependence. Chemical clustering in real materials would also contribute
to correlated cluster growth.

The fundamental role of anisotropy in ESR above Tf is further emphasized by the
fast that those ferro- and antiferro— magnets whose linewidths also diverge in this manner
are all anisotropic. Cubic (isotropic) antiferromagnets do not exhibit this behavior. This
indicates that anisotropy must play a fundamental role in the growth of correlated
clusters. This is especially applicable to highly anisotropic materials (such as ZnnMn;

see Chapter 5). In a one dimensional material, ordering occurs within the chain first,

 

 

251

so that we expect clusters to be oblate in nature and oriented along the chain direction.
These clusters would contribute to a strongly directional internal broadening field. This
indicates that anisotropy may be important to cluster formation and growth.

Relaxation between spins in clusters and the uncorrelated ions will exchange
narrow the inhomogeneous broadening due to the internal field distribution, so the
behavior of the divergence should be governed by how effectively the clusters and the
ions cross-relax relative to the strength of the internal field distribution. The thermal
broadening is a measure of the cross relaxation and is dependent upon the amount of
short range order. The value of B is inversely proportional to the relaxation time of the
electrons to the lattice. As the clusters correlate, their relaxation times to the conduction
electrons increase, which should lessen the bottleneck. The weaker bottleneck can not
effectively narrow the inhomogeneous broadening and the divergence should be stronger.
This is a possible explanation for the behavior of B and x as a function of decreasing
layer thickness.

We would expect that, at some layer thickness, the lack of neighbors would make
it harder to form clusters, so that the material would behave in the main as a paramagnet.
Awschalom” claims from susceptibility measurements of thin CdMnTe multilayers that,
when the behavior of a sample is determined by paramagnetic clusters, the dimensionality

of the sample is not important.

252
Summary of the Comparison with Theory

We believe our data to be better explained by the correlated cluster model than
by the mean field model. The major difficulty in using Equation 7-4 to analyze AH(T)
comes from the presence of a quantity (T,(WSG)) which itself is subject to finite size and
dimensionality effects. When the effects of the depression of TKWSG) are removed from
Equation 7-4, as reported in Tables 7-21 through 7-23, we see that the behavior of the
residual width and the divergence strength as a function of layer thickness becomes the
same as in the bulk. A conceptual explanation of the increase in B and x in the context
of the correlated cluster model has been forwarded.

The lack of correlation between the meaning of v and other critical exponents, as
well as the lack of finite size effects make this behavior difficult to explain within the
mean field model. The fact that the values of zv obtained from ESR scaling are quite
different from the values arrived at from other methods also calls into question the
relationship of the ESR linewidth divergence to the spin glass transition.

The question of dimensionality is open, due in part to the fact that our
measurements represent a lower bound to the values of 1: due to frequency effects. Other
measurements on multilayers indicate that we don’t expect conventional critical slowing
down to describe the data in 2D, so trying to compare the value of x with other measured
or calculated values is futile. The expected characterization of the different dimensions
by an exponent which takes on two distinct values is not seen. The value of the exponent
is unaffected by the temperature shift analysis, indicating that the increase with

decreasing layer thickness cannot be attributed to the depression of T,.

 

 

 

253

No mechanisms for the nucleation and growth of clusters have been suggested by
Barnes. The proximity of nearest neighbors - especially if the preasymptotic corrections
to the RKKY interaction are considered - should be fundamental to ordering. The
presence of this behavior in anisotropic ferro- and antiferro- magnets indicates that

anisotropy is also important.

INTERPRETATION OF AH!!! IN THE CONTEXT OF ACTIVATED CLUSTERS

The primary difference between spin glasses and regular, long-ranged magnets
is activated behavior. In a ferromagnet or antiferromagnet, we might expect that the
distribution of cluster relaxation times would be very small. In a spin glass, we know
that even above T, there is a broad distribution of relaxation times, so that some fraction
of the clusters will be partially frozen but will still contribute to the resonance.

Fisher and Huse predict (Equation 3-30) the relationship between the relaxation
time and the reduced temperature near the transition temperature. We can naively
appropriate this form, assuming that the linewidth diverges in the same manner as the
characteristic relaxation time. We would expect the divergent part of the behavior to

obey

[Alma~t-mexptww..>*"°m1

which is a modification of Equation 7-4 by an exponential factor. This equation holds

when the relaxation time is less than the measuring time, which corresponds to the case

 

254

in which the spins can contribute to the resonance. We can gather the exponents to result

in a general form.

AH(T) = A + BT +c t-"exp(Dt"')

This equation describes data with W80 2 7 nm consistently but results in lesser
quality fits for data 5 3 nm. The residual linewidth, A, and the thermal broadening,
B, are approximately the same as found by fits of the non-activated expression. The
values of C are approximately constant for all Wso 2 7 nm. The correction term,
characterized by the values of D and h, was not needed in fitting the samples with Wso
2 50 nm. (The best fit resulted when D = 0). The remaining samples are fit with an
exponent of approximately h z 1 and monotonically increasing values of D. The value
of x is 1.5 for all samples. These parameters are summarized in Table 7-24.

The form of Equation 7—11 appropriate to T, = 0 is

AH(T) = A” + BT + C”exp(D T“) 7'12

where one must be careful in comparing A" with the residual widths obtained in the

other fits; the high temperature behavior is described by A" + BT + C", so that the

255

 

 

 

 

 

 

 

 

 

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256
residual width, A, is actually A"+ C". Samples with W30 5 can be fit to the zero

freezing temperature form, as shown in Table 7-25.

Table 7-25: Fits to A"+BT+C” exp(D T”) for Run 243 - Cuaana ,,

 

 

 

(nm) (K)   (G)  (one 1 . ’ (G)

 

 

 

3 20.5 ~182 i 15 3.73 i 0.03 15 i 13

 

1 10.5 -l76 i 31 4.18 :1: 0.03 55 i 17

 

 

 

 

 

 

 

 

 

The resulting residual linewidths are -l67 G for the 3 nm sample and -121 G for the
1 nm sample. We note that the parameter h resulting from the zero temperature fit is
the same as the value for the thicker samples with non-zero T,.

An analysis of the divergence of the linewidth in terms of the relaxation of
activated droplets has been shown to consistently describe the data for all layer
thicknesses, but at the expense of two additional parameters for which we have no
physical feeling. The correction terms become significant for Wso s 30 nm, and are

characterized by an exponent 2: 1 for all layer thicknesses below this limit.

257
LE EHIFI‘ ANALYSIS

As has been mentioned, the behavior of the lineshift with temperature at 9 GHz
does not lend itself to simple quantitative analysis, which is why the majority of this
study has focused on the behavior of the linewidth. Over most of the temperature range,
the position of the resonance line has a constant value corresponding to g ~ 2. The shift
to lower fields is very small over the region in which the power law is obeyed. Equation
5-34 predicts that, in the low frequency limit, the divergence of the lineshift should go
like the reduced temperature to a power twice that of the power characterizing the

linewidth divergence. The values obtained by fitting the lineshift to

T-T -8
(Hom - H,(~)) ~ [71] 7-13
f

are given in Table 7-26 and compared with the values of x obtained from fitting the
linewidth. In making these fits, the same minimum value of t used in making the
linewidth fits was used in order to give a direct comparison. Values given in parenthesis
for the samples with smalle represent the values of I: obtained by fitting to a zero T,.
Fits of Equation 7-13 to a T,=0 form resulted in very large errors. The values of 2/2
are consistently larger than the values of x, but in the general expected range. Equation
7-13 should strictly hold only for the low frequency limit, so this analysis is mostly to
illustrate that there are no drastic changes in the behavior of the lineshift as a function

of W30.

Table 7-26: Fits to (H(T)-Ho(oo)) ~ t'3 for Run 243 - Cuo,39Mno.11

258

 

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259
SQMARY

The linewidth and resonance peak position as a function of temperature and spin
glass layer thickness have been measured and presented for the multilayered spin glasses
CumgMnQll and Cu093Mnom with spin glass layer thicknesses from 1 to 1000 nm. The
concentration dependence of bulk spin glasses with Mn concentrations of 4,7 , 11 and 16%
has also been studied. The data has been analyzed in terms of the standard equation
(7-4) used to analyze ESR in metallic ferromagnets, antiferromagnets and spin glasses.
The relevance of both mean field and correlated cluster interpretations of ESR has been
discussed, and the implications of each considered in light of the data presented here.
The observation is made that the systematic behavior observed in the residual width and
the divergence strength are the result of the depression of T,(WSG) with spin glass layer
thickness, as indicated by the shifted temperature analysis. The effects of assuming a
finite transition temperature in Equation 7-4 were considered, and samples with Wso
S 3 nm were found to obey a form of Equation 7—4 with T, = 0.

An alternate fitting form which contains a correction term to take activated
behavior into account was proposed, and shown to fit the data for all layer thicknesses
consistently, although at the expense of the introduction of two more parameters. The

conclusions of this study are presented in Chapter 8.

 

CHAPTER EIGHT

CONCLUSIONS AND SUGGESTIONS

This chapter summarizes the results of this thesis, and comments on some

experiments which have been suggested by this study.

SUMMARY AND CONCLUSIONS

Electron Spin Resonance (ESR) has been shown to be a useful probe of the
dynamics of spin glasses above T,. A straightforward interpretation of ESR results above
T, is difficult owing to the presence of multiple effects (i.e. bottleneck behavior,
frequency dependence, etc.), and even confusion as to the origin of the ESR signal in this
temperature regime.

The fabrication of multilayered metallic spin glasses with spin glass layer
thicknesses, W30, from 1 nm to 1000 nm has resulted in the observation and analysis of
both finite size effects and a crossover behavior from three dimensions to two as WSO is
decreased. These effects are observed as a depression of the freezing temperature,
TAWSG). Investigation of the frequency and field dependent behavior of multilayered
spin glasses has been limited to susceptibility measurements in the frequency regime from
10“ to 10‘ seconds at low fields, and from 2 G to 55000 G at 102 seconds. ESR
experiments, which probe a much different part of the field-frequency plane (as shown

in Figure 5-3), provide us with a means of testing the behavior of multilayered spin

260

261

glasses with a different type of measurement. The short time scale of ESR (1010
seconds) opens the door to observation of relaxation phenomena occurring on much
shorter time scales than the extended dynamics of the spin glass phase, but still short
enough times so that these phenomena cannot be seen by susceptibility measurements.

A large body of experimental evidence, as well as computer simulations, indicates
that correlations are evident in spin glasses well above T,. The two major spin glass
theories - the mean field and the droplet models - have by and large neglected this
temperature regime, preferring instead to label it as simply ’paramagnetic’. The
conclusions drawn from the behavior of the linewidth, AH(T) and the resonance peak
position, H,,(T), above T, depend to a large part on the origin of the resonance. The two
theories used in explaining the divergence of these quantities near T, can be classed as
’mean field’ and ’correlated cluster’ theories, with both theories predicting a power law
divergence, but for different physical reasons. The mean field theory attributes the ESR
signal to the divergence of a characteristic relaxation time of the sample as T, is
approached. The temperature dependence of the linewidth is due to an
exchange-narrowed anisotropy-broadened line which diverges with the same characteristic
exponent as the relaxation time (in 3D).

The correlated cluster model assumes that the dominant activity above T, is the
formation and growth of correlated clusters, composed of moments which interact more
strongly with each other than with other moments. We expect that there will be a
distribution of relaxation times as a result of a distribution of cluster sizes and correlation
strengths. When the relaxation rate of a cluster is sufficiently long, the clusters dominant

contribution to the ESR measurement is as a source of random internal magnetic fields.

 

 

262

The broadening of the linewidth as a function of temperature is due to two sources; the
random local fields due to the correlated clusters, and a random field due to the DM
anisotropy. The latter mechanism is introduced rather ad hoc as a concession to the body
of evidence that this anisotropy is indeed fundamental to broadening (of. Mozurkewich,
et. al. 105). The broadening due to both of these factors is narrowed by cross relaxation
between the clusters and uncorrelated spins.

If the mean field theory is correct, and the divergence of the ESR linewidth is
indicative of a phase transition, we expect to see finite size and dimensionality effects,
as seen in the behavior of T, in spin glass multilayers. If, as the cluster theory proposes,
the ESR linewidth actually measures some intermediate phase between the paramagnetic
and the spin glass phases, we expect the effects of reduced sample size will be evident
in how the correlated clusters nucleate and grow.

These measurements are the first observation of the Electron Spin Resonance
linewidth and lineshift in multilayered metallic spin glasses above T,. Qualitatively, the
same behavior is seen for all layer thicknesses; the linewidth decreases linearly at high
temperatures, experiences a broad minimum and diverges as a power law in reduced
temperature. The position of the minimum linewidth increases dramatically with
reference to the measured freezing temperature as WSO decreases. In bulk samples, the
minimum is at about 2.5 T,; in thinner samples, it moves to as high as 6.5 T,.

The position of the resonance peak corresponds to a constant value of g z 2 for
the same temperature range over which the linewidth obeys the linear form. As the
linewidth reaches its broad minimum, the position of the resonance peak begins to shift

to lower fields. This shift is much more pronounced in the region near T, where the

263
power law prediction for the linewidth fails. Analysis of the lineshift is limited to

qualitative study, due to the small temperature regime in which a value different than
g = 2 is seen, and in which the power law behavior holds. The expression used to
describe the behavior of the linewidth with temperature in the bulk is given by
Equation 7-4. At a reduced temperature t', the linewidth departs from this power law,
rising less rapidly than predicted.

The concentration dependence of the linewidth in bulk samples of Cu,,,Mn, (x =
4,7 , 11 and 16 at. %) has been studied. The linewidth of bulk samples diverges with a
value of x = 1.5 for all concentrations. The residual width, A, and the thermal
broadening, B, are found to depend linearly on concentration, with A decreasing with
increasing concentration, and B increasing with increasing concentration. The divergence
strength, C, is observed to have a sublinear (increasing) dependence on increasing
concentration.

We have shown that samples of all layer thicknesses (1 nm 5 W56 5 1000 nm
for concentrations of 7 and 11 at. %) can be fit with the linewidth expression used to fit
the bulk, although samples with W$0 less than about 3 nm are better fit to an expression
corresponding to Equation 74, but appropriate to T, = 0. This result is consistent with
susceptibility measurements on multilayers with small W30. The reduced temperature at
which departure from power law behavior is observed, t’, increases with decreasing WSG.
Systematic variation of all parameters is seen as a function of layer thickness. The
values of the residual linewidth, A, and the thermal broadening coefficient, B, increase
with decreasing layer thickness. The divergence strength, C, is also seen to increase

dramatically with decreasing layer thickness. Comparison of Wso - dependent data to

 

 

264

concentration dependent data allows us to exclude the possibility of layer spreading as
a source of these systematic changes. Considering the case in which samples with
Wso _<_ 3 nm are fit to the zero T, expression, the value of the exponent, it, increases
with decreasing layer thickness for all layer thicknesses studied. This general behavior
is obeyed for both concentrations studied.

In determining if there are finite size effects on AH(T), we must carefully
consider the parameterization of Equation 7-4 in terms of T,(WSG); the finite
size/dimensionality effects on T, may be misinterpreted as effects on AH(T). In general,
a relationship between T,(WSG) and T,(oo) (known to be finite size scaling over some
range of WSG) exists. We can use this relationship to remove T,(WSG) from Equation 7-4
so that the reduced temperature is expressed in terms of T,(oo) for all layer thicknesses
and the data for each W8,} is shifted by an amount equal to T,(oo) - TKWSG). This
essentially removes the known effects of the depression of T, from the linewidth
measurement. This analysis indicates that A and C are constant for all layer thicknesses
2 7 nm. The failure of the assumed form of finite size scaling in representing the
dependence of T,(WSG) on T,(oo) for samples with WSG < 7 nm is responsible for the
deviations in the thinner samples. The temperature shift analysis, however, does not
affect the value of the exponent, x, or the thermal broadening. The reduced temperature
at which departure from a power law occurs also takes on the same value for all WSG
upon shifting and using T,(oo) in calculation of t'. This analysis suggests that finite size
effects - which would be expected to appear as a rounding of the divergence, and a
departure from bulk behavior at some t' which should progressively increase with

decreasing WSG are not seen in the ESR linewidth. This could be due to the short length

265

scales which the ESR probes, or it may suggest that the measurement does not
correspond to a measurement of the spin glass phase. Our results indicate that no finite
size effects are seen in the linewidth; the appearance of these effects is due to the
presence of T,(WSG) in Equation 7-4.

We expect dimensionality effects to be characterized by changes in the exponent
x and possible by changes in the ESR lineshape and angular dependence of the linewidth.
The ESR lineshape remains Lorentzian for all layer thicknesses to 1 nm in the region
where the power law behavior is obeyed. The angular dependence of the lineshape has
not been systematically surveyed.

A value of x = 1.3 - 1.6 has been widely reported in bulk spin glasses,
ferromagnets and antiferromagnets. The only report of x in two dimensions (in an
antiferromagnetic salt) is x = 2. Frequency effects in the bulk‘“’ indicate that values of
x from high frequency measurements (such as the ones reported here) may underestimate
x. In the thinnest samples considered in this study, a value of well over 2 is found. A
gradual increase in x is seen as W30 is reduced, but the data is not sufficient to determine
if this change is indicative of a dimensionality crossover, or a result of the changes in
cluster growth and nucleation due to the larger surface spin to volume spin ratio.

An added difficulty with the mean field model is in a lack of identification of x
with other critical exponents. Calculations predict that x should be on the order of zv;
values of 11! obtained from dynamic scaling are on the order of 1-3, so values of x in this
region are not unreasonable. Most measurements of Z» in metallic spin glasses using

other probes find a value of Zn = 7 - 11, which is very different from the ESR

266

measurements. This may be a result of an incorrect interpretation of x as characterizing
a critical behavior.

We conclude that the cluster model is more likely to explain the behavior of the
ESR linewidth as a function of the spin glass layer thickness than the mean field model.
The evidence for correlations above T,, and the successful description of the stronger
correlations below T, via the Fisher-Huse droplet model suggests that a similar model
could describe the phase intermediate to the spin glass and true paramagnetic regimes,
and would provide a better theoretical understanding of the nature of the transition at T,.

As mentioned earlier, the divergent behavior of the linewidth is not unique to spin
glasses; however, we expect that the dynamics of the correlated cluster phase will differ
from those of long range ordered magnets. If the resonance is due to paramagnetic spins
and loosely correlated clusters, we might expect the dynamics of the clusters to obey an
activated expression. The behavior of the material can be consistently analyzed across
all ranges of W36, but lack of a theory gives us no feeling for the meanings of the
parameters and further interpretation of these results is difficult.

We believe that these ESR measurements are further evidence for the existence
of an intermediate phase, with a spectrum of relaxation times which are short enough that
they have gone undetected in the longer time susceptibility measurements. This raises
the question of the nature of this phase as a function of decreasing sample thickness. The
importance of the DM anisotropy to the broadening has been well documented. One
possible hypothesis for the increasing value of x with decreasing layer thickness is the
presence of regions of sample which have already condensed into the spin glass phase.

These regions would have macroscopic relaxation times and would be sources for the

 

 

267
DM anisotropy, which is known to be most important in the spin glass phase per se. We

might present a hypothetical phase diagram as shown in Figure 8-1.

For all sample thicknesses, a correlated cluster phase will begin at temperatures
in the area of the minimum of the AH(T) curve - about 2—3 T,. The onset of these
correlations is a property of a specific concentration of spin glass and should occur at
roughly the same temperature for all layer thicknesses down to some lower limit. At this
limit, the nearest and next-nearest neighbor distances between Mn ions will be large
enough to be prohibitive to the formation of correlations.

The ’correlated cluster’ phase is characterized by a distribution of relaxation times
which correspond to those of the weakly correlated clusters and the uncorrelated spins.
As the correlations within a cluster grow, the relaxation time of the cluster increases.
At some critical value of the correlation strength, the relaxation time of the cluster will
become macroscopic and undetectable by the ESR measurement. The cluster is then
essentially condensed into the spin glass phase. For a given concentration, this should
first happen (for all layer thicknesses) at T,(oo), with the caution that an absolute value
for the temperature must take into account the field/ frequency conditions of the ESR
measurement. The question mark in the diagram indicates that, at some layer thickness,
the number of neighbors will be so reduced that clusters may be harder to form, which
would prevent correlations at higher temperatures.

For samples with T,(WSG) < T,(oo) a phase analogous to the Griffiths phase
exists, in which macroscopic portions - but not all - of the sample will be frozen into the
spin glass state. This region of gradual freezing continues to TKWSG), at which point all

of the relaxation times are macroscopic. The only reason this Griffith’s-like phase could

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269

be observed in these materials is due to the depression of the spin glass transition
temperature via finite size and dimensionality effects. As W80 decreases, the fraction of
frozen regions which exist will increase relative to ’free spins’. The length scale over
which the critical temperature is significantly different from the bulk critical temperature
in ferro- and antiferro- magnets is very small and may entirely preclude observation of
this phase in those materials.

The presence of correlated regions suggests that irreversibility effects should be
seen above T,. These effects would be on time scales which are too short to be observed
in susceptibility measurements. Irreversible behavior in the ESR linewidth has been seen

in these samples, as described in Chapter 6, but not investigated systematically.

SU ESTIONS R EXP

The existence of an intermediate phase and the possibility of a Griffiths-like phase
suggests a number of experiments which could be important in further defining the nature
and dynamics of this phase. Because the nature of research entails both answering
questions and knowing which questions to answer, this final section proposes some tests

of the existence and the elucidation of the nature of the intermediate phase.

1. We have seen that field and frequency effects influence the value of T,(Wso).
The phase diagram shown in Figure 8-1 will be dependent of the values of T,(00) and
T,(WSG) at the field and frequency of the measurement. Extension of ac susceptibility

measurements to fields on the order of ESR fields would allow an extrapolation of

 

270

T,(WSG,H,w) which would give an absolute meaning to the parameter values which result

from fitting to Equation 74.

2. A frequency effect on the divergence has been shown to exist in bulk spin
glasses, so that lower frequencies follow the power law behavior to lower reduced
temperatures. Measurements of AH(T) at lower frequencies would result in better fits,
as a wider range of temperatures can be considered. This would more accurately
determine the value of 1:. Additionally, the behavior of the resonance peak position with
temperature is enhanced at lower frequencies, enabling a more quantitative analysis and

comparison with the AH(T) results.

3. As clusters form and ’drop out’ of the resonance, the intensity of the ESR line
should decrease. The total number of spins is proportional to the area under the ESR
resonance peak, and is further proportional to the susceptibility. Careful measurement
of the high temperature susceptibility will allow the extraction of the Curie constant and
the Weiss temperature for the multilayers. The ESR intensity should deviate from a
Curie—Weiss law as more and more spins drop out of the resonance and more slowly
relaxing clusters are formed. The reduction in the number of spins could be quantified
in this type of analysis, providing information about the growth of clusters, and the

temperature at which the correlations are first measurable.

4. The divergent linewidth and resonance peak position is seen in ferromagnets,

antiferromagnets and spin glasses. All spin glasses exhibit this behavior, but only some

271

ferro- and antiferro- magnets. The determining factor in the behavior of AH(T) seems
to be anisotropy. This intuitively follows, as anisotropies could influence the shape and
dynamics of the cluster, as well as the character of the resulting internal field.

Au and Pt are spin-orbit scatterers in CuMn. Addition of one of these to a series
of multilayer films would increase the anisotropy of the film without changing T,.
Mozurkewich, et. al. ‘°5 have studied the behavior of bulk AgMn doped with Sb (which
increases the anisotropy and relieves the bottleneck) and found that the exponent
governing the divergence increases with increasing anisotropy. Our cluster interpretation
would imply that the breaking of the bottleneck increases the effect of the broadening
relative to the cross-relaxation. The values of the thermal broadening and the exponent,
x, increase, in agreement with this analysis. This study could shed light on the
mechanism governing cluster formation and growth and the relationship between the

thermal broadening and the exponent x.

5) The importance of anisotropy also calls into question the effect of the growth
technique on the spin glass character of the material. Comparison with films grown in
some other way - such as MBE - would be interesting. Likewise, a similar analysis of

a short range spin glass would be of interest.

6) The dilute magnetic semiconductors (like Cd,,,Mn,Te, Cd,_,,Mn,,Se) are tunable
over a wide range of Mn concentration. At low Mn concentration, they are paramagnets,
intermediate Mn concentrations result in spin glass behavior, and larger Mn

concentrations are antiferromagnetic in nature. By studying a series of these materials

 

 

272

as they change magnetic character, the common (or different) elements of the behavior
as T, (or T,,) is approached can be identified. This would determine if the intermediate
state is specific to spin glasses, or if evidence exists in other materials as well. These
materials are semiconducting, so the skin depth problem would not limit sample size.
Multilayers are routinely fabricated by growth techniques which result in very low strain
samples, so data as a function of layer thickness can be taken and compared with

susceptibility results.

7) The intermediate phase should exhibit hysteretic effects. A study of these
effects would add to understanding of the dynamics of clusters, much as has been done
below T,. The advantage here is that the nature of the hysteretic effects are on relatively
short time scales (compared to true spin glass behavior), which are observable via ESR
measurements.

One study of Muse” showed distinct hysteretic behavior in the intensity of the
linewidth as a function of temperature, and only weak hysteresis in AH(T). MnSe
undergoes a structural phase transition as the temperature changes. On cooling, the
structure goes from a cubic phase, in which the Mn+ interact paramagnetically, to a NiAs
structure, in which the ions interact antiferromagnetically. The onset of
antiferromagnetism is indicated by a decrease in the intensity as the material changes
phase. On warming, the signal starts out with low intensity, due to the antiferromagnetic
correlations, and the intensity rises rapidly upon the change back to the cubic structure.

Some type of hysteretic effects should be seen in the intermediate phase.

 

 

273
8) The observation of the ’Griffith’s - like’ phase in these samples is possible

only because of the large WSO range over which T,(WSG) is different than T,(oo). It
would interesting to know if a similar range exists for random ferromagnets. More
theoretical work has been done on the possibility and characteristics of a Griffiths phase
in random ferromagnets, which might shed some additional light on the nature of this
phase. We might expect to see two phases; a true Griffiths phase, between Tc(p= l) and
T,(p), where p is the dilution from a pure ferromagnet, and a ’Griffiths like’ phase

between Tc(p, co) and Tc(p,W), where W is the width of the layered random ferromagnet.

9) A systematic search for changes in the lineshape and/or angular dependence
could determine if there is a difference in the approach to a three dimensional spin glass
vs. the approach to a two dimensional spin glass. A theoretical study of the effect of a
distribution of relaxation times on the lineshape could enhance our understanding of the

behavior of these materials in the immediate vicinity of T,.

 

 

APPENDIX A

SOLUTION TO THE BLOCH EQUATIONS

The Bloch equations are given by

 

 

dM M -M

d" = -y(fl?xfi¢z + o z A‘1
1

db! M

.710— : -Y(MXH¢)IJ - _XJ_ A'z

H,, can be considered to be the real part of a complex field, H’ = 2 H1 cos(wt), where
H1 is the properly rotating component of H,. The counter rotating part will have a
resonance at -wo; when the system is near the resonance for H,, the other resonance will
be far away, so we can neglect the counter-rotating field.

The x component of the magnetization is given by the real part of the complex

magnetization, M’x

M, = R(M,’) = xznlem‘ A-3

where the susceptibility, x, relates the complex magnetization to the complex magnetic

field. The susceptibility is now also complex, where

274

275

 

 

 

x = x’-1 x” A'4
so that
R(M;) = x’2chos(mt) + ;;”21111 sin(mt) A-5
The Bloch equations are (with the rotating field included)
dM . M
d: = 1(MzHlsrnwt — HOMQ — 7‘1 A-6
2
dM M
T; = “Hou, - Mzchosmt) - f A-7
2
dM . M —
d“ = y(M,H,cos(m:) - Mxfllsrn(wt)) + 0T1 '- A-8
We can define the operators N i;
Nt = e*""‘(Mx :1: (My) A'9
Adding and subtracting Equations A-6 and A-7, we find
t _ YHIMZ
- A-10

8
2

 

276
If we require a steady state solution, szldt = 0, and we find from Equation A-8 that

 

 

 

 

 

 

Mz—M0 = [ll-,1](N’ _ N') A-ll
T1 2:
Solving for M1, ‘5.
2
Mz = 0 l + 7'22“") A-12
1 + T2201»? + yzleTsz
:1
This is then plugged back into Equations A-9 and A-10 to give
TAmZHcoswt + 2Hsin cot
M, = 1101107012 ‘ 2 ‘ ( ) ; ( ) A-l3
2 1 + T220002 + 72111731;
1 T AmZH sin(wt) - 2H cos(mt)
M, = 5,030,012 2 ‘ 2 2 ‘2 A-14
1 + 132mm) + 7 11,131;
The real and imaginary parts of the susceptibilities are
- T
(“’0 w) 2 A-15

 

x’ = -21-x,(0.,T,)[

 

1+(m -m0)2 T,’ + «11,2111,

 

277

1
1+(to-wo)1 T,’ + yzleTsz

A-l6

 

1
x” = Emu!)

when H1 is small compared to H,,, which is usually satisfied, the last term is the
denominator can be ignored. For most nonmetallic samples, the real part of the
impedance will depend on x", the imaginary part of the susceptibility, so that the average

power absorbed is given by

P = 2mm: A-17

For metallic samples, the surface impedance, Z, is defined as

 

Z = 4n[fi'ExH.] A-18
cl H2

And the power absorbed from the rf field will be given by

4 2
P... = (7") IHI’MZ) A-19

APPENDIX B

CRITICAL EXPONENTS IN SPIN GLASSES

This appendix summarizes a representative group of measured and calculated
exponents, and defines the exponents. The dimensionality is denoted by d. Table B-1
defines some common exponents, and Table B-2 summarizes some important
relationships between the exponents. A list of calculated and measured static exponents
is given in Table B-3. The dynamic exponent is usually measured from dynamic scaling,
so a value of zu is found. Some authors distinguish between zu, which measures the
characteristic relaxation time and law, which is the exponent associated with critical
slowing down of the average relaxation time. A value of zym was cited only if the
author identified it as such, and was otherwise assumed to be zv. These results are

tabulated in Table B-4.

278

 

279

Table B—1 .' Definitions of Some Common Critical Exponents

 

A I Exponent andNam’e'..._, .. 7‘

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Order Parameter

   

 

 

 

 

 

 

 

 

 

 

 

B

'y Susceptibility X ~ t"

a Specific Heat C ~ t“

u Correlation Length E ~ t'

6 Equation of State H ~ M‘

77 Pair Correlation Function (I‘(r)) P ~ r“‘”2+"’
(In; Fisher-Huse Exponent (2D) 1 ~ exp(T‘”“")
2 Dynamic Exponent 1' ~ 5’

 

 

 

 

280

Table B-2: Summary of Relationships Between Critical Exponents

 

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