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FINITE SIZE EFFECTS
IN
COPPER MANGANESE SPIN GLASSES

Gregory George Kenning

A DISSERTATION

Submitted to
Michigan State University
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1988

 

 

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' ABSTRACT

FINITE SIZE EFFECTS
IN
COPPER MANGANESE SPIN GLASSES

By

Gregory George Kenning

Using UHV sputtering we have produced Cu1_,ilIn,,’Cu and Cladding/5i mul-
tilayered systems (MS) with 1:: .04, .07, and .14. Structural analysis of these
systems including SAXD, EDX, SAD, imaging, high angle x-ray, and parallel resis-
tivity confirm that these samples are layered and there is minimal chemical diffusion
between the layers in the CuMn/Cu MS, but some intermixing of the layers in the
CuMn/Si MS. We have shown that the 300 .4" Cu thickness used in the CuMn/Cu
Ms and the 70 A" of Si used in the CuMn/Si MS magnetically decouple the CuMn
layers. By systematically decreasing the CuMn thickness (LCuMn) we observe that
the CuMn layers retain their spin glass properties to LC“;" 3 20 A°. The spin glass
transition temperature T9 (as defined by the DC magnetic susceptibility) begins to
decrease from its bulk value T: (LC-“Mr. : 5000 A") at z 1000A° and approaches
zero at LCuMn z 10 A" in the CuMn/Cu MS and LC“!" 2: 36 A" in the CuMn 'Si
MS. These results have been interpreted in terms of finite size scaling analysis, the
droplet excitation model, and conduction electron mean free path effects. We be-
lieve that these results represent the first experimental observation that the Lower

Critical Dimension (LCD) of CuMn spin glass systems is between tw0 and three.

 

 

To my family, always.

111

 

ACKNOWLEDGEMENTS

I would like to thank Professor Jack Bass for his support and guidance during my
term at Michigan State University.

I would like to thank Professor Jerry Cowen for his guidance and for many helpful

discussions.

There were many other people who contributed to this thesis, either through help-
ful discussions or experimental assistance. Special thanks go to J. Slaughter for
teaching me about sputtering. I would also like to thank A. R. Day, W. Repko,
T. Kaplan, D. Leslie-Pelecki, L. Hoines, B. Leach, V. Schull, W. Abdul-Razzaq, D.

Gregory and C. Bruch for their contributions.

 

 

 

Table of Contents

Chapter ................................................................. Page
I. INTRODUCTION ............... ' ........................................... 1
1.1 Materials ................................................................ 4
1.2 Experimental Properties ................................................. 7
1.3 Concepts ................................................................ 9
1.3.1 Disorder ............................................................. 9
1.3.2 Frustration ......................................................... 11
1.3.3 Anisotropy ......................................................... 12
1.3.4 Lower Critical Dimension .. ......................................... 12

1.4 Thesis Outline .......................................................... 14

11. SAMPLE PREPARATION AND STRUCTURAL CHARACTERIZATION 15

11.1 Sample Production ..................................................... 16
11.1.1 Target Production ................................................. 16
11.1.2 Sputtering ......................................................... 17

11.1.2.1 The Sputtering Gas ............................................ 17
11.1.2.1.1 The Trap and Temperature Controller ...................... 20
11.1.2.2 Sputtering Process ............................................. 23

11.2 Structural Characterization ............................................ 26
11.2.1 Small Angle X—ray Diffraction (SAXD) ............................. 27
11.2.2 High Angle X-rays ................................................. 29
11.2.3 Cross Sections ..................................................... 33

11.2.3.1 Imaging ....................................................... 33
11.2.3.2 Energy-Dispersive X-rays (EDX) ............................... 41
11.2.3.3 Selected Area Diffraction (SAD) ............................... 41
11.2.4 Resistivity ......................................................... 44
11.3 Synopsis: Structural characteristics ..................................... 59

 

   

 

11.3.1 CuMn/Cu MS ..................................................... 59

11.3.2 CuMn/Si MS ...................................................... 59

111. THEORY ................................................................ 60
111.1 Disorder Theories ..................................................... 61
111.2 Site Disorder Models .................................................. 64
111.2.2 RKKY Interaction ................................................ 64
111.2.3 Quenched-Uniform Model ......................................... 66
111.3 Scaling Theory ........................................................ 67
111.3.1 Introduction ...................................................... 67
111.3.2 Finite Size Scaling ................................................ 69
111.3.3 Droplet Excitation Model ......................................... 70
111.3.3.1 Below Lower Critical Dimension: 2D .......................... 73
111.3.3.2 Cross-Over Between Three and Two Dimensions ............... 75

IV. DATA AND ANALYSIS .................................................. 77
IV.1 Magnetic Measurements ............................................... 78
IV.1.1 Squid Magnetometer .............................................. 78
IV.1.2 Magnetic Susceptibility Data ...................................... 79
IV.2 Finite Size Scaling Analysis ........................................... 85
1V.2.1 CuMn/Cu MS .................................................... 85
1V.2.2 CuMn/Si MS .................................................... 102
IV.3 Magnetization vs. Magnetic Field .................................... 107
IV.4 Universality .......................................................... 112
V. CONCLUSIONS ......................................................... 116
V.1 Structure of Multilayer Systems ....................................... 116
V.2 Suceptibility and Tg .................................................. 117
V.3 Comparison with Scaling Theory ...................................... 117
VA Conclusions ........................................................... 118
LIST OF REFERENCES ................................................... 119
APPENDIX 1: RKKY INTERACTION ...................................... 124

vi

 

LIST OF TABLES

TABLE ................................................................ PAGE
11-1 Comparison of methods used to establish target compositions .......... 19
11-2 SAXD analysis of Cu_9gl\/In,o4/Si MS ................................. 30
11-3 SAXD analysis of Cu_93ltIn,o7/Si MS ................................. 30
11-4 SAXD analysis of Cu.agMn.14/5i MS ................................. 31
11-5 SAXD analysis of Cu_79Mn_21/Si MS ................................. 31
11-6 High angle x-ray analysis of Cu,aeIlIn.14/5i AIS ....................... 32
11-7 High angle x-ray analysis of Cu,9oMn_04/Cu MS ...................... 34
11-8 High angle x-ray analysis of Cu_93Mn_o1/Cu AIS ...................... 34
11-9 High angle x-ray analysis of Cu,sollIn,14/C'u MS ...................... 35
11-10 Analysis of SAD line scans ........................................... 46
11-11 Resistivity of Cu_ggMn_o4/Cu MS ................................... 52
11-12 Resistivity of Cu,93Mn.o7/Cu AIS ................................... 52
11—13 Resistivity of Cu,35Mn.14/Cu MS ................................... 53
111-1 Critical exponents and their relationships ............................. 71

 

LIST OF FIGURES

 

FIGURE .............................................................. PAGE
1-1 Two different types of multilayer systems ................................ 3
1-2 Transition temperature Tg vs CuMn layer thickness LC" Mn .............. 5
1—3 Magnetization vs. Magnetic field for CuMn ............................. 10
11-1 Transition temperature vs. Mn concentration in bulk CuMn alloys ..... 18
11-2 Cold trap for purifying Ar gas ......................................... 21
11-3 Circuit diagram of the feedback temperature controller ................ 22
11-4 Measured and calculated sputtering rates vs. Ar energies .............. 24
11-5 Diagram of the sputtering process ..................................... 25
11-6 SAXD scans of Cu1-,A/In,/Si, A/IS .................................. 28
II-7 FE-STEM images of Cu.79AIn_gl/Cu (300 A°/300 A°) MS ............ 36
11-8 FE-STEM images of Cu_93Mn.o7/Sz' (200 A°/70 A°) MS .............. 38
11-9 FE-STEM images of Cu_g3Mn.,o7/Si (200 A°/70 A°) MS ............. 39
11-10 FE-STEM images of Cu,9gMn_o4/Si (200 A°/70 A°) MS ............. 40
11-11 EDX scan of a Cu,7oMn,21/Cu (300 A°/300 A°) MS ................. 42
11-12 EDX scan of a Cu,79AIn,21/Si (70 A°/70 A°) MS .................... 43
11-13 SAD pattern and line scan of CuMMnM/Si (200 A°/300 A°) MS . . . 45
11-14 SAD pattern and line scan of Cu_geMn,o4/Si (70 A°/70 A°) MS ...... 47
11-15 Sample geometry for two different resistivity measurements ........... 49
11-16 Graph of 1 vs resistivity ratio for Van de Pauw measurements ......... 50
11-17 Resistivity vs Lain“ for Cu.95AIn_o4/Si AIS ......................... 54
II-18 Resistivity vs 17:117. for CU.93Mn.o7/SI MS ......................... 55
11-19 Resistivity vs Z317; for Cu_33AIn.14/Si MS ......................... 56
11-20 Resistivity vs fl for Cu_7gAIn.21/Si MS ......................... 57
IV—1 Magnetic susceptibility vs temperature for the Cu_35Mn_14 / Cu M S . . .80

viii

IV-2

IV-3

IV-4

IV-5

IV-6 '
IV-7

IV-8

IV-9

IV-10
IV-11
IV-12
IV—13
IV-14
IV-15
IV-16
IV-17
IV-18
IV-19
IV-20
IV-21
IV-22
IV-23
IV-24
IV-25
IV-26

 

Magnetic susceptibility vs temperature for the Cu_aeMn,14/Si M S . .. 81

Magnetic susceptibility vs Tg/T: for the Cu_aoMn.M/Cu M S ......... 83
T9 vs Cu interlayer thickness for the Cu_95Mn,o4/Cu MS ............. 84
Tg vs Si interlayer thickness L51 ...................................... 86
T9 vs CuMn layer thickness LCuM" for the Cu,9¢AIn.o4/Cu MS ...... 87
T9 vs CuMn layer thickness Lcum,1 for the CuMA/Inyn/Cu M5 ...... 88
T9 vs CuMn layer thickness LCMMfl for the CuMMnM/Cu AIS ...... 89
Ta vs CuMn layer thickness LCuMn for the CulgoMnJM/Si MS ....... 90
T9 vs CuMn layer thickness LCuMn for the Cu_93Mn_o7/Si MS ...... 91
T9 vs CuMn layer thickness LCuMn for the CU.ggMn_14/Si MS ...... 92
T9 vs CuMn layer thickness LcuMn for the CunAInm/Si MS ...... 93
go} vs CuMn layer thickness LCuMn for the all of the CuMn/Cu MS . . 94
$5- vs CuMn layer thickness LCuMn for the all of the CuMn/Si MS .. . 95
Fit of e = T: 2“ vs Low" for cu,9.Mn,o.,/cu MS. .................. 97
Fit of e = FLT—fl vs Law" for Cu_93AIn_07/Cu MS. .................. 98
Fit of e = 3T?— vs Law. for Cu,35AIn,14/Cu MS. .................. 99
Fit of e = 3%: vs Lam. for all of the C'uAIn/Cu MS. ............ 100
Fit of e = 3.1;? vs LcuMfl for Cu.ggMn,o4/Si MS. ................. 103
Fit of e = 3%,5 vs Low" for 0119mm,07 /5i MS. ................. 104
Fit of e = 3-}?— vs Law. for Cu.35AIn,14/Si MS. ................. 105
Fit of e = 3;} vs LOW. for all of the CuMn/Si MS. ............. 106
Mean free path effects in Cu_geAIn,o4/Si MS. ....................... 108
Mean free path effects in Cu,93AIn,07/Si MS. ....................... 109
Mean free path effects in Cu_85AIn,14/Si MS. ....................... 110

Mean free path effects in CuMn/Si MS compared with CuMn/Cu MS 111

 

1V-27 Magnetization vs magnetic field in Cu_93/Mn.07/Si MS. ............. 113

IV-28 H“, VS LCuMn in Cu,93/Mn,o1/Si MS. ............................. 114

A—l 2D RKKY potential as a function of radial distance. .................. 128

 

 

CHAPTER I

INTRODUCTION

Magnetic phase transitions have been one of the most intensely studied areas in con-
densed matter physics this century."2 Experimentally, it has been found that most
ordered ferromagnetic and anti-ferromagnetic materials have discontinuities in such
important physical properties as their magnetic susceptibilities and specific heat at
a well defined magnetic transition temperature.3 In 1972 a type of disordered mag-
netic material termed a spin glass was found4 to have a cusp in its’ susceptibility
vs temperature curve, at a temperature defined as the spin glass transition temper-
ature T9. In general these materials are spatially disordered alloys or insulators,
composed of magnetic ions in a (generally) non-magnetic host. Several important
physical properties such as the specific heat and resistivity show no anomalies at
T9. There is therefore some controversy5 about the existence of a true phase tran-
sition and about the nature of the transition itself. Recent experiments on the
non-linear part of the magnetic susceptibility have shown power law divergences,
rather convincing evidence that the spin glass transition is a thermodynamic phase

transition in three dimensions.

6.7

Great theoretical interest in effects of dimensionality on phase transitionsS' , es-

pecially crossover effects in going from a higher dimension to a lower one have led

 

 

 

-2-

us to investigate how the spin freezing temperature T9, and other properties of a

spin glass (SG) change as the dimensionality is decreased from three to two. This

study was stimulated by:

1) The desire to explore effects of finite sample size on a magnetic
system with inherent spatial disorder.

2) The fact that a combination of experimental and theoretical evidence
in 3D and 2D suggests that the lower critical dimension (ie. the dimension
below which a phase transition no longer occurs at a finite temperature)
for a phase transition in both Ising and Heisenberg like SG’s with long
range RKKY interactions is likely to lie between 2 and 353.

Experimentally, SG’s thus provide an almost unique opportunity to study both
the static and dynamic behaviour across a lower critical dimension boundary, under
conditions involving intrinsic spatial disorder. We have chosen CuMn SG materials

to carry out this study for the following reasons:

1) These materials have spin glass properties over Mn concentrations of
40 ppm 3 c S 30percent,corresponding to a wide range of transition
temperatures T9, between z 10‘4K and z 120K 8.

2) CuMn is believed to be a Heisenberg SG system, with small spatial
anisotropy, and RKKY coupling between the magnetic Mn ions.

3) The metallic nature of the CuMn alloy allows for DC sputtering.

4) CuMn has been the most intensely studied spin glass material.

In order to achieve magnetization signals large enough to measure with a SQUID
susceptometer, we have had to produce Multilayer Systems (MS) in which many
layers of CuMn spin glass films are separated by a decoupling medium such as Si,
which is an insulator at low T, or a layer of Cu thick enough to decouple the layers
magnetically (Fig. 1-1). The advent of U.H.V. sputtering technology has made it
feasable to produce high enough quality MS to study the spin glass transition of

metallic spin glass films as the film thickness LCuMn is decreased so that the film

 

Figure I-1a: Multilayer System composed of alternating layers of crystalline ma-

terials.

   

 
 

. 90.0. of”.
0

.IO- .O-Oz.‘ O‘O‘O.“

Figure I-lb: Multilayer System composed of alternating layers of crystalline and

amorphous materials.

 

 

approaches a two dimensional state. We observed, for the first time, a dramatic

decrease in the spin glass transition temperature from its bulk value of T: to T: —» 0
at film thicknesses, L, of only a few atoms (Fig. 1-2), strongly suggesting that the
lower critical dimension of this type of spin glass is between two and three. We
can fit the shifted transition temperature, T: — TgL, as a function of LCu Mn with a
power law form suggestive of a thermodynamic phase transition and from this fit we
have obtained a value of the universal correlation exponent 11. It is the purpose of
this thesis to study the decrease in T9, determine the associated critical exponents,

and test for universality of these exponents as a function of Mn concentration.

The first step in understanding the sputtered CuMn/ Cu and CuMn/ Si MS is struc-
tural characterization of the layers and the layering. This includes determining
layer thicknesses, crystal structure in the layers, and the amount of interfacial mix-
ing between the CuMn layers and the decoupling layers (Cu and Si). Since we are
changing a fundamental parameter of the system (ie. the dimensionality), other
physical properties which help define the spin glass state (eg. hysteresis) are char-
acterized to see if there are any fundamental changes in the SG other then the

decrease in the transition temperature.

This chapter will be mainly concerned with outlining the experiments and concepts
necessary for understanding the current picture of the spin glass state. It starts
with a discussion of some of the spin glass materials, the defining experiments of
the state, and then discusses some of the major physical concepts thought to be
relevant to this thesis and to understanding the materials and experiments. These

concepts include disorder, frustration, anisotropy, and lower critical dimension.
1.1) Materials

Spin glass behaviour has now been observed8 in many different types of materials
with different types of interactions. Non-metallic crystalline materials such as
Eu35r1_,S show a cusp in the magnetic susceptibility, no long range order, time

dependent effects, and have other physical properties typical of spin glass materials.

approaches a two dimensional state. We observed, for the first time, a dramatic

decrease in the spin glass transition temperature from its bulk value of T: to T: —+ 0
at film thicknesses, L, of only a few atoms (Fig. 1-2), strongly suggesting that the
lower critical dimension of this type of spin glass is between two and three. We
can fit the shifted transition temperature, T: — T}, as a function of LcuMn with a
power law form suggestive of a. thermodynamic phase transition and from this fit we
have obtained a value of the universal correlation exponent V. It is the purpose of
this thesis to study the decrease in T9, determine the associated critical exponents,

and test for universality of these exponents as a function of Mn concentration.

The first step in understanding the sputtered CuMn/Cu and CuMn/Si MS is struc—
tural characterization of the layers and the layering. This includes determining
layer thicknesses, crystal structure in the layers, and the amount of interfacial mix-
ing between the CuMn layers and the decoupling layers (Cu and Si). Since we are
changing a fundamental parameter of the system (ie. the dimensionality), other
physical properties which help define the spin glass state (eg. hysteresis) are char-
acterized to see if there are any fundamental changes in the SG other then the

decrease in the transition temperature.

This chapter will be mainly concerned with outlining the experiments and concepts
necessary for understanding the current picture of the spin glass state. It starts
with a discussion of some of the spin glass materials, the defining experiments of
the state, and then discusses some of the major physical concepts thought to be
relevant to this thesis and to understanding the materials and experiments. These

concepts include disorder, frustration, anisotropy, and lower critical dimension.
1.1) Materials

Spin glass behaviour has now beer ‘ "cnt types of matcrials
‘ materials such as
range order, tinn-

pin glass materials.

 

 

 

 

 

 

 

-6-—

In this system magnetic Eu atoms (J = ;,g = 2) placed randomly on an fcc
lattice interact through direct ferromagnetic exchange with their nearest neighbors
and antiferromagnetic exchange with their next nearest neighbors. Amorphous
metallic and nonmetallic materials such as AIMGdu and MnO - A1203 - SiOg
(respectively) also show spin glass properties. Although the magnetic interactions
in these amorphous materials are not well understood, it is generally believed that a
competition between ferro and antiferromagnetic bonds, coupled with a disordered
array of magnetic ions, is essential in the formation of a spin glass state. This thesis
will discuss only CuMn which is a long range metallic spin glass. Such materials
are characterized by a concentration c of magnetic atoms (ie. Mn,Cr,Fe) randomly
dispersed on a lattice of a metallic host (ie. Cu,Ag,Au) and interacting through
long range, oscillatory RKKY exchange.

At very low concentrations, the magnetic impurities are far enough apart that they
are non-interacting. This is called the Kondo9 regime. For Mn in Cu this oc-
curs when cS 4 ppm. At concentrations larger than this, the impurities interact
through the long ranged oscillatory RKKY interaction which decays asymptotically
as 7%;10 from the impurity ions. When the average energy of interaction becomes
comparable to the ambient temperature, the local moments tend to ”freeze out”
in random directions. There is evidence11 that this ”freezing out” is a collective
process corresponding to the growth in correlations of the spins. As the concentra-
tion is increased further the statistical probability of the magnetic atoms becoming
nearest and next-nearest neighbors increases. The near neighbor position of Mn
atoms in FCC Cu strongly favors antiferromagnetic alignment while the next near-
est neighbor position favors ferromagnetic alignmentlz. X-ray studies”, in AgMn,
have shown that there is local chemical ordering which strongly prefers the next
nearest neighbor position. It has been proposed“ that ferromagnetic clustering
occurs and that these clusters interact in much the same way as the individual

magnetic ions in the dilute spin glass state. At a critical concentration cm.“ z 30

percent, a percolation network of nearest neighbors occurs and the material becomes

 

 

antiferromagnetic8 .

1.2) Experimental Properties

In 1972 Canella and Mydosh4 made a series of low field ac susceptibility measure-
ments on AuFe. They found a cusp in the susceptibility reminicent of a second
order magnetic phase transition. This behaviour was subsequently found in other
dilute magnetic alloys (CuMn,AuMn,etc) and also in some short range spin glasses
(ie. EuSrS).5 The non-linear terms in the ac susceptibility (ie. AI/H — x] =
Ax; +Bx5 + ) have been measured in AgMn by Levy and Ogielski“. They found
that x3,x5, x7 . .. diverge with an algebraic form [xzonumar ~ (T— Tc)_"7‘("‘1)5],
and measured values of‘y = 2.1 d: .1 and fl = 0.9i .2. This divergence in the suscep-
tibility, and hence in the free energy (F 2 —:—;§), is the most convincing evidence

to date that the spin glass transition is a real thermodynamic phase transition.

Low field ac susceptibility vs. temperature curves are similar for spin glasses and
anti-ferromagnets‘. Below the transition temperature, ferro and anti-ferromagnetic
materials show long range order as evidenced by Bragg peaks in neutron diffrac-
tion studies”. Until recently, no evidence of long range order below Tg had been
observed in spin glasses, suggesting that the spins freeze out in random directions.
Werner et. al.1“'17 have recently seen several satellite Bragg peaks in CuMn, sug-
gesting magnetic ordering on the length scale of 40A°. They have attributed this
ordering to a frustrated spin density wave in the Cu host. This work however is
still in its infancy and the interpretation is dependent on several significant extrap-
olations in the data. Nonetheless the observation of possible ordering in CuMn is

very interesting.

Since 1972, many measurements have been made of the specific heat of spin glass
materials to look for evidence of a phase transition. To date, no discontinuities
have been found in the zero field specific heat at the freezing temperature. The
specific heat exhibits a broad peak about 20 percent higher than Tg 5. It has been,

argueds, however that this does not necessarily rule out a phase transition. Several

 

possibilities exist: 1) The transition may show up as a non-linear effect in the specific

heat. 2) The width of the critical region may be very narrow and below present
experimental resolution. These arguments have been used to explain the lack of a

divergence in the specific heat at the Curie temperature of the dilute ferromagnet

CO;ZTE1-;(C5H5N05)(CIO4)2 18.

In 1976 Ford and Mydosh” measured the electrical resistivity of several long ranged
spin glass materials. The resistivity difference between the pure host and the alloy

AP(T) = Palloy(T) — Phost(T)

has a broad peak at a temperature approximately twice the transition tempera-
ture. Within experimental error no indication of any significant change in Ap(T)
or dAdTT was observed at the transition temperature. They interpreted this be-

haviour to mean that there is no long range cooperative magnetic ordering.

Time dependent effects have been observed using different techniques to measure the
relaxation times of individual spins and of the whole spin glass state. To measure
the former, three different techniques (neutron echozo'“, Mossbauer effect”, and
muon spin resonance (”SRY”) corresponding to different time scales (10'12 S t S
10'°,t _<_ 10‘7,t S 10"5 respectively) have been used. The results suggest that
the spin relaxation times increase rapidly near the critical temperature. These
data have been fit23 to a ln(t) dependence suggesting a gradually freezing glass-like
system. The data can also be fit24 to an algebraic form t‘f suggesting a phase
transition. Several different methods have been used to measure time dependence
of the spin glass state. Malozemoff and Imry25 made susceptibility measurements
on bulk CuMn and A153 Gd,“ on time scales from minutes to twenty four hours.
They found no shift in the spin glass transition temperature over these time scales.
The time scale of our susceptibility measurements vary from 102 — 103 seconds
per temperature point. The relaxation of the spin glass state below T9 has been
observed26 in the Thermoremnant Magnetization (TRM) (cooling in a field and then

removing the field) and the Isothermal Remnant Magnetization (IRM) (cooling in

 

 

-8—

possibilities exist: 1) The transition may show up as a non-linear effect in the specific
heat. 2) The width of the critical region may be very narrow and below present
experimental resolution. These arguments have been used to explain the lack of a

divergence in the specific heat at the Curie temperature of the dilute ferromagnet

Co,Zn1_,(CsH5NO¢)(CIO4)g 18.

In 1976 Ford and Mydosh19 measured the electrical resistivity of several long ranged
spin glass materials. The resistivity difference between the pure host and the alloy

Ap(T) = piracy/(T) - phost(T)

has a broad peak at a temperature approximately twice the transition tempera-
ture. Within experimental error no indication of any significant change in Ap(T)
or dAdTT was observed at the transition temperature. They interpreted this be-

haviour to mean that there is no long range cooperative magnetic ordering.

Time dependent effects have been observed using different techniques to measure the

relaxation times of individual spins and of the whole spin glass state. To measure

20,21 t22

the former, three different techniques (neutron echo , Mossbauer effec , and
muon spin resonance (pSR)23) corresponding to different time scales (10‘12 S t S
10‘°,t S 10‘7,t S 10"5 respectively) have been used. The results suggest that
the spin relaxation times increase rapidly near the critical temperature. These
data have been fit23 to a ln(t) dependence suggesting a gradually freezing glass-like
system. The data can also be fit24 to an algebraic form t'c suggesting a phase
transition. Several different methods have been used to measure time dependence
of the spin glass state. Malozemoff and Imry25 made susceptibility measurements
on bulk CuMn and Al,“ Gd,” on time scales from minutes to twenty four hours.
They found no shift in the spin glass transition temperature over these time scales.
The time scale of our susceptibility measurements vary from 102 — 103 seconds
per temperature point. The relaxation of the spin glass state below T9 has been

observed26 in the Thermoremnant Magnetization (TRM) (cooling in a field and then

removing the field) and the Isothermal Remnant Magnetization (IRM) (cooling in

 

 

_9_

zero field, applying a field for a short period of time, and then removing the field).
These measurements show a dependence on the temperature, field, and the sample
magnetic history suggesting that the zero field cooled (zfc) magnetization is not
a thermal equilibrium state. The field cooled (fc) magnetization shows only weak

time dependence leading the authors26 to suggest that it is an equilibrium state.

Below the transition temperature hysteretic effects are observed“ as a function of
magnetic field (Fig. 1-3, reproduced from ref. 27). The magnetization vs. magnetic
field curve is a shifted symmetric loop with hysteretic behaviour. This behaviour
is explained in terms of the Dzyaloshinskii Moriya (DM) and uniaxial anisotropies

discussed in the next section.

There has been one experiment, that we are aware of, to observe cross-over be-
haviour between three and two dimensions in a spin glass system. Awschalom28
has reported that the magnetic susceptibility cusp in CdMnTe rounds out and
eventually disappears as LadMnTe is decreased below 80 A°. He has interpreted
this as evidence for an LCD of three in this type of spin glass. It has been pointed
out29 however that even in the spin glass state there is an antiferromagnetic cor-
relation length of {Mufflmmagnmc z 100 A°, " is in the same direction that
the sample thickness was decreased. It is thus not clear that he has seen spin glass
finite size effects, as opposed to effects due to the reduction of this antiferromagnetic

correlation length.
1.3) Concepts
1.3. 1) Disorder

Quenched spatial disorder, a property of all spin glasses, means that the magnetic
atoms are fixed in space (substitutionally or interstitially) and are located randomly
throughout the material. Experimentally, randomizing the material constituents is
probably the biggest problem in the production of real spin glass systems. Ideally.

disorder is achieved by a series of annealing treatments to thoroughly randomize

 

 

 

 

 

 

 

 

Figure I-3: Magnetization vs. Magnetic Field for CuMn with 25 percent Mn
after field cooling (fc) and after after zero field cooling (zfc). (After Beck")

 

_11_

the alloy constituents and then quenching the sample to freeze in this disorder. For
a more complete discussion of how we obtain spatially disordered samples see 11.1.
Incorporating disorder into spin glass theories is the main theoretical challenge in

a comprehensive theory. This will be discussed in further detail in chapter 111.
1.3.2) Frustration

Spatially disordered magnetic atoms coupled through a long range oscillatory ex-
change interaction leads to frustration. D.Sherrington30 has described frustration
as ‘The global inability to simultaneously satisfy all local ordering requirements’.
On a global energy scale frustration allows the possibility of a highly degenerate
ground state. The evolution of the spin glass systems’ free energy as a function
of temperature is probably responsible for the observed time dependent effects5 as
the system seaches for its true equlibrium energy minima. On the macroscopic
scale, frustration combined with spatial disorder produces an absence of long range

magnetic order, and thus an apparent randomness of the spins.

The simplest example of a locally frustrated set of spins can be illustrated as follows:
1 f
2) )3

Consider a system of three spins with one degree of freedom (ie. up or down) all
interacting antiferromagnetically. If spin 1 is up then spins 2 and 3 want to point
down with respect to it. But spins 2 and 3 also want to point in opposite directions
hence frustration. This example was given for an Ising model (one spin degree of
freedom). In the case of a Heisenberg system (three spin degrees of freedom) it
is easier to satisfy local energy requirements since the spins have more degrees of

freedom. This lowers the tendency towardfrustration in the system.

4‘..- ”fin-“"1
‘

 

A m .fimm gm.--

 

-12-

1.3-3) Anisotropy

There are other interactions which increase the amount of frustration by limiting
the degrees of freedom of the individual spins, and which thus affect the formation
of the spin glass state. These are the anisotropic interactions. Without anisotropic
interactions CuMn is Heisenberg like which, early calculations suggested, had no
phase transition in three dimensions31 . With anisotropy, CuMn becomes more Ising
like, which has been shown to have a phase transition in three dimensions”. The
Dzyaloshinskii Moriya (DM) interaction is unidirectional in nature, coupling the
spin directionality to the host lattice atoms through spin orbit scattering of a third

atom32’33.

Below the transition temperature, in an applied magnetic field, the
DM interaction attempts to maintain the frozen in random spin alignment. This
alignment can only be overcome with a reversed magnetic field, large enough to flip
all of the by spins 180°. This ”flipping” or coercive field (Fig.1-3) is equal to the
displacement in a hysteresis loop34 and in experiments where impurities are added
to the SG material the coercive field is proportional to the impurity concentration
and the strength of the impurity spin orbit coupling. Another anisotropy found in
CuMn is uniaxial dipolar anisotropy, which is due to the direct magnetic coupling
of the moments. As mentioned above, early theoretical calculations”, have found
that without a ‘pseudo-dipolar’ anisotropy the three dimensional Heisenberg spin
glass does not ‘order’ (ie. have a phase transition); recent calculations suggest,
however, that anisotropy may not be fundamental to the ordering of the Heisenberg

spin glass".
1.3.4) Lower Critical Dimension (LCD)

The earliest calculations of systems undergoing phase transition have shown that
the system dimensionality is a fundamental parameter of the phase transition. In his
1925 thesis”, E. Ising showed that the one dimensional Ising model of a ferromagnet
has no phase transition at a finite temperature (it was later shown36 that there is

a phase transition at T20). Onsager" calculated, exactly, the solution to the Ising

 

_13_

model in two dimensions and found that it does support a finite temperature phase
transition. Therefore the Lower Critical Dimension (LCD) of this system is said
to be between one and two (ie. the dimension below which the system cannot
support a phase transition at a nonzero temperature). The theoretical evidence

which discusses the LCD of spin glass systems will be presented in Ch. III-1,2.
1.4) Thesis Outline
The rest of this thesis is organized as follows:

Chapter II is the experimental chapter. It is divided into three parts: The first
part includes target production, a description of the sputtering facility, a descrip-
tion of an argon purification system that was designed and built to clean the Ar
that is introduced into the sputtering chamber as part of the sputtering process,
and a description of the sputtering process itself. The second part of chapter 11
discusses structural characterization of the MS. This includes small and large angle
x-ray analysis, resistivity in both the 4-probe and Van de Pauw geometries, and
various characterization techniques using a Field Emission-Scanning Transmission
Electron Microscope (FE-STEM). The final section of this chapter is a synopsis of
the structural characteristics of the CuMn/ Cu and CuMn/ Si MS.

Chapter 111 is the theoretical chapter. Theoretical calculations on SG models in
three and two dimensions have been done within SG mean field theories; therefore
the first section of chapter 111 is a description of SC mean field theories. This
includes bond disorder models such as the Edwards-Anderson EA model and the
Sherrington-Kirkpatric model, and site disorder models. The second section of this
chapter outlines the uniform quenched model and the application of this model to
the layered geometry. We used this model to try to understand the effect on the
transition temperature of the diffusion of Si impurities into the CuMn layer. The
final section of chapter 111 includes an introduction to scaling theory and progresses
to the droplet-excitation model which has been used by Fisher and Huse to describe

our results..

 

-14..

In Chapter IV we analyze the magnetic susceptibilities of our samples. The first
section of this chapter describes the susceptibility data. The second section is an
analysis of the susceptibility data in terms of finite size scaling, mean free path

effects and the structural properties of our samples.

 

 

-15-

CHAPTER II

SAMPLE PREPARATION
AND
STRUCTURAL CHARACTERIZATION

The sample production goal of this thesis was to make as perfect as feasible Multi-
layered Systems MS using our sputtering system. The ideal sample would include
no chemical mixing at the interface, perfectly planar interfaces, and single crystal
layers, possibly with some lattice strain at the interface. The structural characteri-
zation experiments discussed in this chapter show that our samples have polycrys-
talline or amorphous layers, some chemical mixing at the interface, and irregular
interface boundaries. Although the ideal MS is difficult to make by sputtering,

an ideal sample is not essential to observe the important magnetic effects that are

discussed in Chapter IV.

The first section of this chapter describes the target production, sputtering facility,
sputtering, and sample production. The second section describes the experiments
done to characterize the samples and an analysis of these experiments. The final

part of this chapter is a synopsis of the inferred structural characteristics of the

CuMn/Cu and CuMn/Si MS.

 

 

_13_

11.1) Sample Production
11.1.1) Target Production

The CuMn sputtering targets were produced from 6 9" Cu and 4 9" Mn melted
into an alloy in a Lepel induction furnace. Selected amounts of each constituent
were etched in nitric acid and then weighed to give approximately the desired com-
position. The metals to be alloyed were placed into a cylindrical graphite crucible
of radius 6.5 cm. and depth 2 cm., which had first been coated with boron nitride
to ensure that the carbon from the graphite didn’t enter the target. The crucible
with constituents was placed in a vycor enclosure inside the RF coils and pumped

down to S 2x10‘° torr. The graphite served as the active heating element.

As the holder was heated, the Cu melted at 1083°C and the Mn floated to the
top. The Mn melted at 1244°C and the system was held at this temperature for
approximately five minutes to ensure good mixing. The target was then cooled
over several hours and turned on a lathe to a radius of 5.7 cm, the correct size
for the sputtering machine. Before use, the target was cleaned in alcohol and then
sputtered for several minutes to remove any iron that may have entered the surface
during the latheing process. After sputtering, it was found that the target was
composed of CuMn crystallites of diameter approximately 2 cm in the middle of
the target and approximately 0.5 cm near the edge of the target. Four different

concentrations of 0111-,an targets were made: x=.04, .07, .14, and .21.

Two methods were used to more closely establish the alloy compositions. Shav-
ings from the lathe were sent away for chemical analysis. Chemical analyses of the
same sample, by the same company (Galbraith Laboratories Inc.), yielded mark-
ably different results. Chemical analysis of a sample sent to a different company
(Schwarzkopf Microanalytieal Laboratory) gave a total percentage of constituents
significantly larger than 100 percent. We therefore, had limited faith in these re-
sults. Our main method of determining the Mn concentration in the targets was to

compare the bulk (shavings) spin glass transition temperature T g with the known

 
  
   
  

7 /'

,\

t' . / I II _1
j/I / a. '
1’ \‘

I ‘. I‘ . .25 V .M I u

- . if . i o '
E, .g , 7; I. ___._ . . , I ~ ' ' I

I ‘i *§ . f» '. ' y

I ._' ' . I - ' ‘ I.

-17..

T9 vs. concentration curve Fig. II-1. Results from this analysis are listed in Table

11-1 .

The Si target was commercially made from 6 9" Si which had a room temperature
resistivity much greater than 10 fl-cm. The semiconducting properties of this target

allowed for DC. sputtering at room temperature.
II.1.2.) Sputtering
II.1.2.1.) The Sputtering Gas

Sputtering is the emission“3 of surface atoms of a target by bombarding it with
radiation, generally particle radiation. In our case we use Ar+ ions to bombard the

target, removing several atoms of target material for every incoming Ar ion.

Sputtering is done in a cylindrical stainless steel tank of height 48 cm and radius
23 cm. Four L.M. Simard ‘Tri-Mag’ sputtering sources are mounted 90° from each
other, on a circle of radius 14 cm.. The sputtering sources sputter ions towards the
top of the tank onto substrates mounted on the Substrate Positioning and Moni-
toring Apparatus (SPAMA), 4” above the sputtering targets. The SPAMA holds
eight substrate holders containing two substrates each. Two samples of each MS
are made simultaneously; one for structural characterization, one for susceptibility
measurements. To make MS, the SPAMA is positioned alternately over two sputter-
ing sources by a stepping motor which is controlled by an IBM PC. The sputtering
tank is pumped down to S 2x10‘° torr first using a mechanical roughing pump
and then a CTI cryo-torr8 cryopump. The pressure in the tank is then raised to
2.5x10'3 torr by introducing Ar into the system through the sputtering guns. The
upper limit of the gaseous impurity percent per monolayer deposited by this system,

for a sputtering rate of 10 A°/s, has been estimated to be 1.7 percent”.

Prior to entering the guns, the ultra-pure Ar is further cleaned by passing through

a cold trap to be discussed next, and then through a Hydrox [MAT] gas purifier

 

 

 

-18—

Cutt-xil‘nm

 

H
N
O

H
O
O

80

60

40

+0

20

Transition temperature (K)
+

9

ViiUIIUUTrrrT'lIUIII'U'UITITU

 

 

llllljlllllljlglllllllll

10 15 20 25

 

O
0
0|

Figure II-l: Transition temperature vs. Mn concentration in bulk CuMn alloys.

Compiled from the Landolt-Bornstein tables“.

 

Target 1

Nominal constituent
composition

Chemical Analysis“

From Fig. 11-1

Target 2

Nominal constituent
composition

Chemical Analysis"

From Fig. 11-1

Target 3

Nominal constituent
concentration

From Fig. 11-1

Target 4

Nominal constituent
' composition

Chemical Analysis"

From Fig. 11-1

at: Galbraith
@ Schwarzkopf

-19-

Elements

Cu
Mn

Cu
Mn
C

Mn

Cu
Mn

1) Cu
Mn
2) Cu
Mn

3) Cu
Mn

Mn

Cu
Mn
Cu
Mn

Mn

 

Table II-l Comparison of methods used to establish target composition.

Percent

96
4

93.34
3.51
.078

4:1:1

90
10

90.61
6.03
.005

93.43
6.39

93.1
6.72

721:1

85
15

14i1

75
25

76.69
29.45
.02

21i2

 

 

 

 

mm”- -‘-‘"s="' ‘ '

 

 

 

-20-

which removes impurities such as 02 and N2 by reacting with a hot Ti filament.
The Ar gas then passes through a flow controller into the gun assembly. For a
more complete discussion of the sputtering tank, SPAMA, flow controllers and gun

assembly see .1. Slaughter, thesis”.
11.1.2.1.1.) The Trap and Temperature Controller

To remove impurities from the Ar gas entering the sputtering chamber, and increase
the lifetime of the Ti purifier, an Ar purification system was constructed. This
system has two parts: A trap which is kept a few degrees above the liquid N2
boiling temperature to freeze out H20 and other impurities, and a temperature
control system which controls the temperature at the bottom of the trap with an

accuracy of :lzl K.

A diagram of the Ar trap is shown in Fig. 11-2. The trap is made entirely of Cu to
ensure good thermal contact with the N2 bath. As the boiling temperature of Ar,
87.3 K“, is 10 K higher then the boiling temperature of N2, 77 K, it necessary to
always keep the trap at least 11 K above the N2 boiling point. Ar enters the trap
through a 1 / 2” stainless steel tube and flows down through to the bottom of the
trap. Simple calculations based on a flow rate of 120 cc/s of Ar at a pressure of
760 torr show that no significant deviation from the input pressure occurs in this
' tube, and that the Ar is cooled to within a few degrees of the trap temperature by
the time it leaves the input pipe. The trap is filled with Ag and Cu shavings to
provide a large surface area for impurity atoms to plate out on. The Ar then leaves

the trap by way of another 1 / 2” stainless steel tube located at the top of the trap.

A feedback temperature controller was constructed to ensure that the incoming Ar
never drops below its boiling temperature. A schematic of the controller is shown
in Fig. 11-3. The heater generates a maximum of 12 watts continous power. The
sensing thermocouple is located at the side of the trap. The distance between the
heater and the sensing themocouple gives the feedback loop an approximately tww

minute time constant. With three liters of N2 in the N2 reservoir, the system is

 

 

-21.-

Ar in 1 t Ar out
to tern erature -
p —— 1/2" stainless steel

 

 

 

 

 

controller . _ _ _
"ails
feedback will , Cu and Ag
thermocoup _l shavings
" Measuring
Heater thermocouple
PH '11-!
a
/ quurd N2
N2 dewa’ * ~ , solid
h- Cu rod

 

 

Figure 11-4: U010 trap 101' purifying Ar gas.

doses—So 833258» «32:.va 2: .3 Samoa. 33:0 5-: 0.5me

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

manoooEEE
0528 HI
I In I I. I.
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-23-

able to maintain a temperature of between 88-100 K, depending on the variable set

point, to within an error of :tl K, for about four hours.
11.1.2.2.) Sputtering Process

To create an Ar plasma, a current is passed through the Ar flowing over the target.
The target is then lowered to a negative potential, which causes the positive Ar ions
to accelerate and bombard it. To make our samples, the guns were programed to
produce incoming Ar ions with kinetic energies in the range 250-400 eV for sputter-
ing CuMn and Cu, and 300 eV for sputtering the Si target. This is enough energy
to knock several atoms out of the target per incoming ion, Fig. II-4. Keeping the
surface of the Si substrates cool was the main criteria which determined sputter-
ing gun parameters. We found that if the Ar energies or total flux of sputtered
atoms were too large, the samples were annealed (ie no layering was observed with
SAXD, see 11.2.1). Ideal sputtering parameters for making MS were found by trial
and error. The theoretical curve in Fig. II-4 is based on a collision cascade model
of sputtering“. The incoming Ar ion collides with a surface target atom, which
collides with other target atoms in a collision cascade. Some of this cascade mo-
mentum returns to the surface atoms through elastic collisions. If the surface atoms
recover kinetic energy sufficient to overcome their binding energy they eject from
the surface, Fig. II-5. Sputtering rates are therefore inversely proportional to the

surface binding energy and proportional to the incoming Ar energy.

Preferential sputtering of one type of target atom over another type of target atom,
in alloyed targets is possible if the binding energies of the different types of sputtered
atoms are different. We infer from the comparison of the spin glass transition
temperatures of target shavings and ‘bulk’ (5000 A°) sputtered films that little or

no preferential sputtering has occurred.

The magnetic atoms in a spin glass should be randomly located throughout the
material. We believe that we have achieved this in our samples through the ran-

domization of the sputtering process, and by keeping the substrates as c001 as

 

 

 

-24-

 

 

N‘%Cu

 

 

 

Figure II-4a: Measured and calculated sputtering rates of Cu as a function of

incoming Ar energies. S is the number of sputtered atoms per incoming Ar atom.

From Sigmund“.

 

2.0
1.5
ca 1.0

0.5

 

' o.o

 

l

 

L l l l l l

 

2

10 20 50 100 200 500
Ar ENERGY (keV)

Figure II-4b: Mesured and calculated sputtering rates of Si as a function of in-

coming Ar ion. S is the number of sputtered atoms per incoming Ar atom. From

Sigmund42 .

 

-25..

r ‘C‘T‘K\\\V

X

 

Figure II-5: Diagram of the sputtering process.

 

 

 

—26-

possible to minimize movement of the sputtered atoms on the substrate. In our
system the sputtered atoms are focussed through a chimney assembly onto room
temperature substrates located about 4” above the target. As the sputtered atoms
deposit their energies on the substrates, the substrate temperature increases. In
order to keep the substrates as cool as possible, only one or two samples were made

at a time and then the system was allowed to cool for several hours.

The sputtering rates were determined using Temescal FTM-3000 quartz crystal film
thickness monitors (FTM), which could be lowered into the substrate positions 4”
over the guns. The geometry of the sputtering system only allows measurement
of the sputtering rates prior to or directly after making a sample. Desired layer
thicknesses in the MS are programed into the software which controls the stepping
motor attached to the substrate plate. The stepping motor positions the substrate
over an individual gun for a prescribed amount of time corresponding to the required
thickness. Typical sputtering rates were 1 —3 A°/s for the Si target and 12— 16 A°/s
for the CuMn and Cu targets.

The substrates used in these studies were the [100] and [111] faces of Si, single
crystal sapphire, and cleaved NaCl. All substrates except the NaCl were cleaned in

the following manner. The substrates were:

1) Wiped with alcohol and visual observation was used as an aid to remove spots.
2) Cleaned for ten minutes in Alconox detergent in an ultrasonic cleaner.
3) Cleaned for ten minutes in distilled water in the ultrasonic cleaner.

4) Cleaned for ten minutes in alcohol in an ultrasonic cleaner.

11.2) Structural Characterization

We characterized the structural properties of our samples in several different ways.
Small Angle X-ray Diffraction (SAXD) was used to determine the average bi—layer
thickness. Cross-sections of the MS ~ 500A° thick were made for Field Emission-

Scanning Transmission Electon Microscope (FE-STEM) analysis‘3'“. Imaging of

 

 

   

 

 

-27-

these cross-sections and Energy Dispersive X-rays (EDX) helped determine the
integrity of the layering. Selected Area Diffraction (SAD) and high angle x-ray
studies were done to analyze the structure of the layers. The resistivities of our

samples were measured for two reasons:
1) To determine whether the thin films and metallic layers were continuous.

2) To determine the extent to which interfacial mixing deposited impurities in

the higher conductivity layers of the MS.

11.2.1.) Small Angle X-ray Diffraction (SAXD)

SAXD was used as a tool to confirm the layered structure of our samples and
to measure the bi-layer thickness. SAXD was done on a Rigaku [RIG] Geigerfiex
diffractometer and a Rigaku 1U-ZOOB series diffractometer. These machines have

an angular resolution of .1°, using Cu-Ka radiation 021.541 A°).

Fig. II-6. shows SAXD scans CuMn/ Si MS for four different Mn concentrations.
Usually between four and nine Bragg peaks were seen. The scans were typically
made from 2" to 8°. Angles below 2° are dominated by the main beam while the
diffraction intensity for angles greater than 8° are too small to interpret. It has been
pointed out“5 that the registry of layers defines the coherence length of the x-rays
in SADX on multilayer samples. Any deviations in the registry due to imperfect
interfaces, slight variations in the layer thickness, etc., can significantly alter the

intensity and width of the diffracted beam.

The bi-layer thickness, d, was determined from Bragg’s equation stin0 = 11).,
where d is the bi-layer thickness. A program was written which compared peak
angles through the equation 2d(sin0n_ — sine...) = (n, — ny)x\, where n, and "v
are the orders of the Bragg peaks. This analysis has the advantage of eliminating the
machine zero from the equation. This technique is only useful for determining bi-
layer thicknesses d <z 500A°, because the diffracted beam intensity drops rapidly
due to the smaller number of layers (which decreases the diffracted intensity), and

larger (1 (which decreases 0).

 

 

—28—

 

CuMn(4%)l SI
(100A! 30A)

 

 

 

 

CuMn(7%il SI
(140AI'IOA)

 
 
 

 

 

 

 

 

 

 

 

 

 

 

 

l>-- ACuMn(14%)ISI

(I) (70A/70A)
Z
UJ
I...
Z

CuMn(21%)l SI

(4OAI7OA)

2 THETA

Figure II-B: SAXD scans of Cu1-,Mn,/Si MS for :c = .04, .07, .14, and .21.

 

 

-29-

The electron densities of the CuMn / Si layers are sufficiently different for the x-ray
scattering to determine a bi-layer thickness. Unfortunately this is not true for the
CuMn/ Cu MS; we were unable to see any SAXD even for Mn concentrations as
large as 21 percent. SAXD scans on all CuMn/ Si samples (J <z 500 A°) used in
this study show good chemical layering. These range from samples with a (1 value as
large as 570.4", to a sample with Si layer thicknesses of L5.- = 30 A°. The absence
of SAXD intensities for L5.- < 30A° is interpreted as due to the lack of sufficient
coherence of layers due to interfacial mixing of the constituents. Tables II-2,3,4,5
compare values of bi-layer thicknesses determined from the FTM and SAXD for
the different Mn concentrations. It can be seen that these values are within a few

percent of each other.
11.2.2) High Angle X-rays

High Angle x-ray analysis was done on Cu,aoMn,14/Cu MS and Cu,33NIn,14/Si
MS of varying CuMn layer thicknesses. Several high angle peaks are observed in
each scan. Most of these are due to the Si substrate. After the Si substrate peaks
are subtracted from the scan we observe only the < 111 > peak of Cu or CuMn,
indicating stong preferential orientation of crystallites in the plane perpendicular to
the layers. As the CuMn layer size is decreased in the CuMn/ Si MS, the intensity
of the < 111 > peak decreases and the width of the peak increases. We interpret
these effects as due to a reduction in the crystallite sizes. Table II-6 displays the
intensity and widths of the peaks as a function of CuMn layer thickness, and esti-
mates the crystallite sizes. The lack of superlattice lines, usually observed in MS
with alternating layers of crystalline material, suggests that the Si interlayers are

amorphous.

The situation in CuMn/ Cu is more difficult to analyze quantitatively as there are
now two peaks very close together, one from the CuMn and one from the Cu. In
the samples with large LcuMn (LCuMn > 300 A°), and large Mn concentration

(ie. 14 percent), the peaks are far enough apart to obtain a reasonably accurate de-

 

 

 

 

‘ x
_ /

y \_ r
z ‘ a. a
. - I ‘ / '._\
.l . 1"--
‘W I' . ‘ - . ‘ _
~ . I I . I' ‘ ‘
‘.T h. i f ‘ . ‘ .= 1' 9' ; I?) .. “ .'.
El! . - - ' _ . r (‘. ~‘ I
u‘ ‘ ’ ’ .. . "' . '-. ‘ h 1‘s.“ \.
, ' a ' - " — g.... o.. atlsvvct- « ~~ -' ‘ s" '

   

-30-

Table II-2: SAXD analysis of Cu,"M 11,04 / Si MS. Layers = total no. of bilayers
in MS. Peaks = no. of observed Bragg peaks. dam“ = nominal bilayer thickness.

d.-,.,. = bilayer thickness determined from x-ray analysis.

 

 

 

011.950/[1204/52
SampleNo. Layers Peaks dam“ A° dkmy, A°
913-58. 100 5 120 135:1:7
86-6a 71 9 140 1452i:7
86-6b 71 7 140 138:1:7
86-2a 50 5 170 177:1:9
86-8a 33 7 220 230:1:12
86-4a 25 5 270 2702i:14
86-7a 10 5 570 614i31
96—5a 125 3 110 128i7
96-6a 50 4 130 140i?
96-7a 50 0 120 —
96-8a 50 0 110 —

 

 

 

 

 

 

 

Table II-3: SAXD analysis of Cu,93Mn,o1/Si MS. Layers = total no. of bilayers
in MS. Peaks = no. of observed Bragg peaks. dam“ = nominal bilayer thickness.

d,-,..,, = bilayer thickness determined from x-ray analysis.

 

 

 

Cu.93Mn.o7 /Si
SampleNo. Layers Peaks dam“ A° dbmy, A"
30-h 60 6 100 101i5
32-b 84 6 120 119i6
56-h 60 4 120 122i6
28-h 67 4 130 134i7
55-h 43 6 140 139i7
33-b 44 9 210 210i11
24-h 31 5 170 169i8
23-h 25 10 270 277:}:14
55—h 15 3 270 256i13

 

 

 

 

 

 

 

 

-31-

 

 

Table II-4: SAXD analysis of Cu,5¢NIn,14/Si MS. Layers = total no. of bilayers

in MS. Peaks = no. of observed Bragg peaks. dun“ = nominal bilayer thickness.

d..,,,, = bilayer thickness determined from x-ray analysis.

 

 

 

 

 

 

 

 

Cu.86M”l.14/5i
S ampIeN o. Layers Peaks d "m“ A0 d343,” A0
120-1a 60 5 120 122i6
120-3b 43 5 140 152i8
120-5b 30 6 170 150i8
120-5b* 30 6 170 170i9
120-5a 30 5 170 167i8

 

* (rotated 90°)

Table II-5: SAXD analysis of Cu,79Mn,21/Si MS. Layers = total no. of bilayers
in MS. Peaks = no. of observed Bragg peaks. dun,“ = nominal bilayer thickness.

d,_my, = bilayer thickness determined from x-ray analysis.

 

 

 

011.79 Mam/Si
SampIeNo. Layers Peaks dam“ A" dz_my, A"
106-6b 75 4 110 110i5
106-2b 60 5 120 135:}:7
106-4b 50 3 130 136i7
101-2b 43 4 140 132i?
101-5b 30 4 170 154i8
101-7b 15 4 270 264:1:13

 

 

 

 

 

 

 

 

 

-32..

 

Table II-6: High Angle X-ray analysis of CuMM n_14 / Si MS. Bragg peak inten-

sity, width and calculated crystallite size are displayed.

 

 

 

 

 

 

 

Sample No. LCuMN A" Intensity Width Size A°
120-13. 50 153 2.37 40
120-1b 50 175 2.28 42
120-3b 70 684 1.69 56
120-5a 100 937 1.48 64
120—5b 100 1280 1.45 65
120-4a 500 2034 0.81 150
120-4b 500 2213 0.78 150
120-23. 1000 1531 0.64 190
120-2b 1000 1258 0.70 170
120-6b 5000 3084 0.56 210

 

 

 

-33-

termination of crystallite sizes. Samples with small LCuMn (LCuMn _<_ 30 A°) have
a relatively sharp Cu peak and correspondingly large crystallite sizes (> 300 A°),
suggesting that the CuMn layer is forced into large Cu crystallites with the Cu
lattice parameter. In between these two extreme regions the data are more difficult
to analyze but estimates of crystallite sizes have been obtained by a simple peak
extrapolation method. Tables 7,8,9 display peak intensities, estimated peak widths,
and estimated crystallite sizes for the CuMn/Cu MS.

II.2.3.) Cross Sections

In order to analyze the structural and chemical compositions of the MS in the FE-
STEM, a technique was developed for preparing thin film cross sections. Initially
we attempted to prepare the cross sections using a technique developed by Sheng
and Marcus“. This technique was very time consuming and the samples usually
broke before the sample and substrate could be thinned enough to be transparent

in the electron microscope (S 500 A°).

A much simpler procedure using a Reichert-Jung Ultracut E microtome was devel-
oped by J. Heckman at the MSU Centre for Electron Optics Studies. The layered
sample was coated with epoxy while it was still on the substrate. The epoxy and
sample were then removed from the substrate and placed in an epoxy mold. The
mold was allowed to harden and then shaped to fit into the microtome. The micro-
tome sliced the sample on a diamond knife edge to thicknesses S 1000 A°. These
slices were then floated on water and picked up on Ni microscope grids. The grids
were then placed into the microscope. If the thin films were opaque to the electron
beam the grid was removed from the microscope and the sample further thinned in

a VCR Group Inc., Model 306, Ion Reactive Gas Milling System.
II.2.3.1.) Imaging

Selected MS cross sections were imaged with a VG HB501 (FE-STEM) and a JEOL
J EM-100CX 11 Transmission Electron Microscope (TEM). Imaging was used both in

 

 

-34..

Table II-7: High Angle X—ray analysis of GugoM 11,04 / Cu MS. Bragg peak inten-

sity, width and calculated crystallite size are displayed.

 

 

 

 

 

 

Sample N o. LCuM N A° Intensity Width Size A"
107-2b 20 (194629 0.24 400
97-6b 70 14877 0.28 330
102-2b 200 13006 0.33 *
107-3b 300 6696 0.41 *
102-5b 300 7003 0.42 *
107-5b 500 4044 0.48 *
107-43. 1000 3834 0.46 *

 

 

 

* (Peak composed of two unseparated peaks)

Table II—8: High Angle X-ray analysis of Cu,93Mn,o7/Cu MS. Bragg peak inten-

sity, width and calculated crystallite size are displayed.

 

 

 

 

 

 

Sample No. LCuMn A" Intensity Width Size A0
121-5b 20 99619 0.21 440
121-6b 30 51696 0.22 430
121-1b 50 47498 0.27 370
121-4b 100 14121 0.37 *
121-2b 500 4156 0.54

 

 

* (Peak composed of two unseparated peaks)

 

-35-

 

 

Table II-9: High Angle X-ray analysis of Cu,saMn,14/Cu MS. Bragg peak inten-

sity, width and calculated crystallite size are displayed.

 

 

 

 

 

 

Sample N o. LcuMn (A°) Intensity Width Size A°
113-31) 30 86992 0.30 320
113-6b 50 8244 0.56 *
113-4a 70 15294 0.60* *
113-1a 100 4260 0.61 *

113-5a (Cu) 300 2640 0.6 200 .
(CuMn)) 200 1440 0.5 200
113-21) (Cu) 300 2574 0.4 200
(CuMn) 500 4021 0.4 200
113-8a 10000 4542 0.65 180

 

 

* (Peak composed of two unseparated peaks)

 

 

   

 

_ 2000 A
Figure II-7a: Dark field image of a CngMngl/Cu (300A°/300A°) MS taken
on the FE-STEM. Magnification = 100k.

 

Figure II-7b: Bright field image of a CujgllInJI/C'u (300A°/300A°) MS taken
on the FE-STEM. Magnification = 100k.

 

 

 

view the integrity of layering and to check the individual layers. Figs. II-7 displays
both the bright field and dark field images of a Cu_79M'n_21/Cu (300 A°/300 A°)

MS, at a magnification of 100k. This sample was microtomed thin enough so that
ion milling was not necessary. The bright field image is produced by the directly
transmitted electron beam. It is therefore sensitive to thickness fluctuations in the
cross section. The dark spots in the bright field image are believed to be crystallites
that have been preferentially cut along grain boundaries, by the microtome. We
infer from this image that the sample is composed of crystallites S 500A°. The
dark field image is produced by a scattered beam. It is therefore sensitive to dif-
ferences in the electron density of the cross section. The most direct observation of
sample layering comes from dark field images. Fig. II-7a shows that there is lay-
ering in the electronic densities consistent with layering of two different materials.
The thicknesses of the individual layers, as estimated from the magnification, are
consistent with thicknesses programmed into the sputtering control system during
sample preparation. Beam broadening in the sample limits the use of this technique

for quantitative layer thickness determination.

Figs. II-8 and 9 are bright field images of a Cu.93Mn_o7/Si (200 A°/70 A°) MS,
taken at different magnifications on the TEM. The TEM has better resolution then
the STEM for this kind of imaging. The darker layer is electronically more dense
then the lighter layer. We therefore conclude that the darker layer is CuMn. The
CuMn layers appear to be composed of crystallites with diameters approximately
equal to the layer thickness. A great deal of structure is observed at the interface
where the CuMn crystallites have apparently grown into the Si layer. This probably
occured during sputtering. We do not observe any crystallites in the Si layer, from

which we conclude that the Si layer is amorphous.

Fig. II-10 displays bright field images of a. Cu,93AIn.o4/Si (40 A°/40 A°) MS. The
tranverse thickness variations of this cross section are probably due to ion milling.
Both the CuMn layer (dark) and the Si layer (lighter) appear to be continous, but

there is some evidence of deformation of the layers by the microtome (Fig.II—8).

 

 

 

4 A
l 000 l

Figure II-Ba: Bright field image of a Cu,93}VIn_o7/Si (200A°/70A°) MS taken on
the TEM. Magnification = 72k.

    

l 1500 A I
Figure II-8b: Bright field image of a Cu_93Mn,o7/Si (200A°/70A°) MS taken
on the TEM. Magnification = 190k.

:1

 

    

 

1500 A
l—__l

Figure II-Oa: Bright field image of a Cu,93Mn,o7/Si (ZOO/1° / 7011") MS taken on
the TEM. Magnification = 190k

I 500 A I

Figure II-Ob: Same as above with but with magnification 700k.

 

-40-

 

1500 A

|__——l

Figure II-10a: Bright field image of a Cu_95Mn_o4/Si (40A°/40A°) MS taken
on the TEM. Magnification = 190k.

 

1000 A
|______l

Figure II-lOb: Bright field image of a CuMMnM/Si (40A°/40A°) MS taken
on the TEM. Magnification = 270k.

l
s
l 5‘
\ ‘\‘ /

   

I . ‘ 'r
‘ I
Q .‘ ‘V
R: 'g a“ . l'l '1
II I h A 0
I -- I'II' I . .. ‘ I . 'l
_. i‘ . _ . .4 . 7‘ ‘ .1 -- ' ‘_'. -_ "

 

 

-41_

SAXD analysis of this sample confirms a bilayer thickness, d, of approximately
80 A°. The TEM images suggest that the CuMn layer is significantly thicker than
the Si layer. We believe, however, that this thickness difference is an artifact of

greater beam broadening in the electronically denser CuMn layer.
II.2.3.2.) Energy-Dispersive X-Rays (EDX)

Energy-Dispersive X-ray analysis was used to check the spatial variation of the
chemical composition of specially chosen CuMM nm/ Cu and Cu,nMn,21/Si lay-
ered systems, The similarity of Cu and Mn electron densities, combined with the
fact that the Cu layers were 300 A° thick (so that d is always > 300 A°), did not
allow SAXD studies of the CuMn/ Cu MS.

High energy electrons (100kev) irradiating a small volume of the sample will knock
core electrons out of their shells. As the atoms relax to their ground states, via
the conduction electrons filling the empty core levels,they emit x-rays characteristic

of the particular atom. An EDX detector (lithium-drifted Si) collects the emitted

x-rays and converts them into current pulses proportional to the x-ray energy.

For our EDX analysis, the electron beam is scanned over the entire width of the
cross section perpendicular to the layers. The intensities of the core energies of each
element analyzed are plotted as a function of scanning distance, starting at the edge
of the cross section. Sample line plots of a GugmM 71,21 / Cu. (300A°/300A°) MS and
a Cu,79Mn,31/Si (70 A°/70A°) MS are shown in Fig. II-11 and II-12. It can be
seen that the variation in the chemical composition of the elements is consistent
with chemical layering. Beam spreading and poor resolution of low element con-
centrations (< 21 percent Mn) limited the application of EDX for microanalysis of

our layered structures.

II.2.3.3.) Selected Area Difi'raction (SAD)

Selected Area Diffraction (SAD) was performed on both CuMn/ Si and CuMn/ Cu

-42~

 

 

40

20

 

Mn

 

 

200

Total Counts

100

 

Cu

lL1800 A

l
l

 

Figure II-ll: EDX scan of a CuggMnm/Cu (300A°/300A°) MS taken on the

FE-STEM.

Position

 

 

 

 

 

_43-

 

 

 

 

 

 

 

 

 

Mn
50‘
O
100
(D
‘25.
0 Si
0 0
7.3
,2
500
Cu
200 A
0

 

 

Figure II-12: EDX scan of a Cu,79Mn,,21/Si (70A°/70A°) MS taken on the FE-

STEM.

 

 

 

 

 

 

_44_

MS. This technique involves sending a parallel beam of electrons through a small
area of the cross section z 1pm“. The diffraction pattern which is produced allows

analysis of the crystal structure orthogonal to the beam.

Fig II-13 shows the SAD pattern and corresponding line scan for a Cumll/I n,“ / Cu,
200 A°/ 300 A" MS. There is a systematic multiplicative factor between the mea-
sured diffraction ring diameter D05. and the calculated diameter D“; due to an
incorrect setting on the camera focal length. The line scan has been analyzed in
table II-10a. The observed diffraction peaks correspond to a fee lattice with the lat-
tice spacing of Cu, 3.61 A°. The SAD pattern is grainy indicating that the sample is
composed of crystallites. The non-uniform nature of the diffraction rings indicates

that there are preferred crystallite directions in the plane of the layers.

Fig. II-14 shows the SAD pattern and a corresponding line scan of a Cu,” M n,” / Si
(70 A° / 70 A°) MS. The data indicates that the dominant structure is fee with the
lattice spacing of Cu. Preferred crystallite directions in the plane of the layers are
also observed in these samples. The broad smoother rings observed in this pattern
indicates that the crystallite sizes are significantly smaller than in the CuMn/ Cu
MS. No evidence of crystalline Si was found.

II.2.4.) Resistivity

The resistivities of the CuMn/ Si MS were measured to determine layer continuity
and to estimate the extent to which Si impurities penetrated the CuMn layers.
The resistivities of the CuMn/ Cu MS were measured to determine if Mn diffused
into the Cu interlayer. The resistivities of our samples were measured with two
techniques; standard 4-probe measurements on samples specially prepared with a

7

4-probe geometry, and Van de Pauw4 measurements on sample films of arbitrary

2D geometry.

The Van de Pauw technique is a 4-probe method of measuring the resistivities of

homogenous thin films of arbitrary shape. It was deduced in 1958 by L.J.Van de

 

 

_45-

 

Figure II-13a: SAD pattern of a Cu_55Mn,14/Si (200A°/300A°) MS taken on
the FE-STEM.

 

Figure II-l3b: Line scan of above SAD pattern.

-46—

 

Table II-lOa: Analysis of SAD line scan for (711ng n.14/ Cu. (200A°/300A°)

 

 

 

 

 

 

 

 

 

MS.

CU.3sllln.14/Si 200 440/300 A0
Line ID 0 Deal (cm) Dob, (cm) factor
(111) 0.510 1.775 2.35 1.324
(200) 0.590 2.050 2.69 1.312
(220) 0.83" 2.900 3.83 1.320
(311) 0.970 3.401 4.53 1.332

(222) 102" 3.552 absent
(400) 1.170 4.102 5.43 1.324
(331) 1.280 4.470 6.10 1.365

(420) 1.31" 4.587 absent
(422) 1.440 5.026 6.65 1.323

Table II-lOb: Analysis of SAD line scan for Cu_geMn,o4/Si (70A°/70.4°) MS.

 

 

 

 

 

 

 

 

CU.96Mn.o4/Si 70 140/70 A0

Line ID 0 Deal (cm) Dob, (cm) factor
(111) 0.510 1.775 2.42 1.363
(200) 0.59" 2.050 absent
(220) 0.830 2.900 3.92 1.352
(311) 0.970 3.401 4.58 1.347
(222) 1.020 3.552 absent
(400) 1.17" 4.102 absent
(331) 1.28" 4.470 6.07 1.358
(420) 1.31" 4.587 absent
(422) L44" 5.026 6.75 1.343

 

 

 

 

-47—

 

Figure II-l4a: SAD pattern of a C'u_g¢Mn,o4/Si (70A°/70A°) MS taken on the
FE-STEM.

 

Figure II-14b: Line scan of above SAD pattern.

m
."e' ‘7'" ‘ '-
_ live. . .T ‘3th "—

 

 

-48..

Pauw from conformal mapping arguments. This technique requires the following
conditions:
1) The current and potential contacts are at the circumference of the sample.
2) The contacts are sufficiently small.
3) The sample is homogenous in thickness.

4) The surface is singly connected.

The sample was cut to a size and geometry that would fit onto a holder that could be
inserted directly into a helium dewar. The holder is a simple rod shaped device that
connects four leads from one end (the sample end) to the other end (the measuring
end). The sample can then be lowered into a N; or He dewar and measurements
made at 77 and 4.2 K, respectively. Four small scratches were made on the sample
circumference with a diamond scribe. This ensured contact to all the layers. The
four leads were attached at the scratch points with SCZO silver micropaint Fig. II-
15a. Two resistances were obtained as follows. First current was put through leads
1 and 4 and the voltage drop was measured across leads 2 and 3, giving R1433.
Current was then put through leads 1 and 2 and the voltage drop between 3 and 4

measured giving 1212,34. The resistivity was determined from

 

1rd (R1433 + R12,34)f(312,34)

P 2 111(2) 2 R1433

where d is the total sample thickness and f is obtained from Fig. II-16. Errors
are estimated as follows; geometrical error in the determination of d, 3 percent;

resistance measurement error, 1 percent; and error in determining f, 3 percent.

The 4-probe technique is a standard method for measuring resistivities. We used
this method on several samples to check the Van de Pauw method. Our samples
were prepared by sputtering through a mask of the 4-probe geometry. This mask
had the dimensions shown in Fig. 15-b. Current was passed from lead 1 to lead 4
and the voltage drop across leads 2 and 3 was measured. The resistivity was then

Rwd V

determined from the equation p = I where R = T and I is the current, V is

 

n

-v—.av Aa—fi . m“

   

 

_49_

   
  

 

 

Diamond Si'Ve'
Scratches Epoxy
4 3

Figure II-15a: Sample geometry for Van de Pauw resistivity measurements.

 

 

 

 

.17cm
/ 2
/ Potential
Current Leads
Leads
.4 5cm
/
‘ 3
4',

Figure II-l5b: Sample geometry for 4—probe resistivity measurements.

.SBzem up 55

88m .maqoaouameofi 335 01 =e> Ham 0:8 33m???” m> a mo £9.30 57: whammh

0N4;

 

V0.3

-50-

or m macaw No. m N.

No
to
so ..
ad
3

 

 

-51_

the measured potential, (1 is the sample thickness, w the sample width, and l the
sample length. Errors were determined as follows: error in d, 3 percent; error in w,

10 percent; error in l, 10 percent; error in V, .1 percent; error in I, .1 percent.

Resistivity measurements were made on four different types of samples: CuMn/ Cu
MS on Si substrates, CuMn thin films on Si and sapphire substrates, CuMn/Si MS,
varying LCuMn, on Si and sapphire substrates and CuMn/ Si MS, varying L5,- on

Si substrates.

Tables II-11, 12 and 13 compare the total resistivities (pTot) of the CuMn/Cu MS
with: 1) pm“; a model for the CuMn/Cu MS assuming only specular reflection
from the interface between the CuMn and Cu layers. 2) paw; a model for the
CuMn/Cu MS assuming only diffuse scattering at the interfaces. 3) pungform; a
model for the CuMn/ Cu MS, assuming that all of the Mn is distributed uniformly
throughout the sample.

With only a few exceptions all of the values of PTot are within the limits set by
pane and pain. We infer from this result that there is little diffusion of Mn out of
the CuMn layer. The samples with large LCuMn appear to be closer to the values
of of pd”, while the smaller LouMn are nearer in value to pwec. This shift in pTo‘
coupled with the high angle x-ray data and imaging data suggest a model for the
growth of the layers during sputtering.

The measured crystallite size is strongly dependent on LCuMn in CuMn/Si MS.
LCuMn is determined by the length of time the substrate is held over the sputtering
gun. Energy deposited at the surface of the MS, from the sputtered ions, causes
the temperature of the layer to increase. The larger the layer the longer the layer
atoms have to anneal and the larger the crystallites. The tendency towards diffuse
scattering in the layer suggests that the growth of large crystallites causes a rough

interface. This is confirmed by TEM images (see Fig. II-9).

Graphs of the resistivity vs. inverse layer thickness for the CuMn/ Si MS are shown

 

 

 

 

Sample LouM. 4° pass; p?“ p?” p4"
102-4 500 8.2i.4 4.36 6.57 9.90
102-5 300 3.6:i:.2 3.52 5.53 8.32
102-2 200 3.0:t.1 3.06 4.91 7.06

97-7 70 3.0i.1 2.39 3.97 4.39
102-1 30 2.2:t.1 2.17 3.64 3.14
102-5 20 1.6i.1 2.10 3.55 2.7

 

 

 

 

 

 

Table II-ll: Table of the total resistivity pTog for the Cu,95Mn.o4/Cu MS, com-

pared with a specular scattering model, a diffuse scattering model, and a uniform

model for the MS. All resistivities are in 110 — cm

 

 

 

 

 

 

Sample Low. 4° p325?” W“ p?” pat"
121-3 5000 19.0i.8
121-2 500 7.0i.3 4.54 7.00 15.83
121-4 100 4.3i.2 2.57 4.27 7.53
121-1 50 4.3i.2 2.29 3.84 5.16
121-6 30 3.4:t.2 2.18 3.66 4.01
121-5 20 3.2i.1 2.11 3.57 3.38

 

 

 

 

Table II-12: Table of the total resistivity pTot for the Cu,ggllIn,o7/Cu MS, com-

pared with a specular scattering model, a diffuse scattering model, and a uniform

model for the MS. All resistivities are in [1.0 — cm

 

 

 

 

 

 

 

-53..

 

 

meas

 

 

Sample LCuMn A° pro. p3?“ 1033’ f pit"
113-8 10000 41.12.
1132 500 6.31.3 4.93 7.96 29.65
113-4 70 3.41.1 2.44 4.10 8.37
113-3 30 3.01.1 2.19 3.70 6.02
113-7 20 2.11.1 2.13 3.59 2.77

 

 

 

 

 

 

 

 

Table II-l3: Table of the total resistivity p10; for the Cu,5¢Mn,14/Cu MS, com-
pared with a specular scattering model, a diffuse scattering model, and a uniform

model for the MS. All resistivities are in [Lil — cm

   

 

_54_

 

125 CuMn(4%)/Si 1.

100 — 5

75*-

 

I
t I.

p (MO-cm)
0|
0
' l
|

p

[ULlllilllllllLlllllllllLlll I

o
0 0.6 1 1.5 2 2.5 O 40 80
1/L x100 (11“) 1' (K)

Figure II-17: Resistivity vs. 1 for CuMM 11.04/ 51' MS. The open circles

LCuMn

 

 

 

 

 

correspond to MS measured with the Van de Pauw technique. The solid circles
correspond to MS measured with the 4-probe technique. The open squares corre-
spond to thin films measured with the Van de Pauw technique. The solid squares
correspond to thin films measured with the 4-probe technique. The dashed line is
the fit to the Fuchs’ model. Inset on the right; p at 4.2 K and 77 K.

 

_55-

 

125 .—CuMn(7%)/Si
. 100—
' l) °"‘~-e
: b-‘--.
5 75L- 4)
I ‘ 32::
3 : :1: ‘l’ 3:::
°‘ 50 — {
25+— _. -o
ti c,__..__--e>
unlnnlnnlnnlanlnn I l I I

 

 

 

 

 

0O 0.5 1 1.5 2 2.5 0 40 80
1/L x100 (A4) T(K)

Figure II-IS: Resistivity vs. 1 for Cu,93Mn.o7/Si MS. The open circles

LCuMu

 

correspond to MS measured with the Van de Pauw technique. The solid circles
correspond to MS measured with the 4-probe technique. Inset on the right; p at
4.2 K and 77 K.

 

-56-

 

150 _ CuMn(14%)/Si
. + p~ ~ ~
1' ‘0
A 1001— + D__ _£
5 r-
‘1’
" - - - -o
3 _ t t
a. ' ..
5O '1 '0— - " '°
3 33333
p
llllllllL

 

 

 

 

O
0 0511.5 2 2.5 O 40 80

 

-l
1 /L x100 (A ) 1'(K)
Figure II-19: Resistivity vs. Lciun for Cu,35Mn,14/Si MS. The open circles

correspond to MS measured with the Van de Pauw technique. Inset on the right;

pat 4.2 K and 77 K.

 

._57_

 

150 L-

 

100‘
A ......... o
a " f \ \
O \
I - ‘9
3 - 1 .
a. '
50 — '9

- CuMn(217.)/Si

 

 

 

o JlllllllllllLlLllllllllllllll I I I I
o 0.5 1 1.5 2 2.5 0 40 so

1/L x100 (4“) T (K)

Figure II-20: Resistivity vs. 1 for CuMM n.04/ Si MS. The different sym-

LCIMn

 

 

 

bols correspond to samples made during different sputtering runs.Inset on the

right; p at 4.2 K and 77 K.

. _ - MEWZ'IP'L." “‘1' .

 

 

-58-

in Figs. II- 17, 18, 19 and 20. The comparison of the resistivities of CuMn thin
films with the resistivities of CuMn / Si MS in Fig II-17, show that there is definitely
some penetration of Si into the CuMn layer. Using the value for [’5qu measured
on samples 5000 A° thick, the thin film data were fit to a. simple Fuchs’ model“8
with completely diffuse surface scattering, in the limit LCu Mn >> I (where I is the
estimated bulk conduction electron mean free path). As can be seen in Fig.II-17
the data fit the Fuch’s curve fairly well. In contrast, similar thin films of CuCr and
AgMn measured by Vraken et 31.49 show large increases in the resistivity as the
sample size is decreased. The differences in the resistivity between our samples and
their’s may be alloy dependent but it is more likely that it is preparation dependent.
Their films were prepared by quench-condensing small pieces of a master alloy onto
a liquid nitrogen cooled glass substrate in a residual pressure of S 10‘5 torr. They
have attributed the increase in resistivity with decreasing layer thickness to disorder
inhanced electron-electron scattering. Considering the data just presented, they

may be seeing the effects of island formation in their samples.

The resistivity measurements presented above have been made at 4.2 K. We have
also made resistivity measurements at 77K. The resistivity of the Cu_93Mn_o7/Si
MS show a systematic change in 41% from positive (metallic) in our thicker samples
to negative (nonmetallic) in the thinner samples. This result led us to investigate
possible localization effects in the magneto-resistance of these dirty samples. The
characteristic signature of localization50 is a small negative magneto-resistance for
magnetic fields perpendicular to the thin layers and much larger (at least an order
of magnitude) magneto-resistance for a field parallel to the layer plane. No evidence

of localization was observed in these experiments.

 

-59-

11.3) Synopsis: Structural Characteristics

11.3.1) CuMn/Cu MS

1) EDX and dark field images show chemical layering in the sample
Cu,7gMn,21/Cu (200A°/300A°) consistent with layer
thicknesses programmed during sputtering.

2) Dark field images show that interface topology is fairly rough.

3) N o SAXD peaks are seen for any concentration.

4) Resistivities are consistent with CuMn/ Cu layering.

5) High angle x-rays, SAD and the dark field images show evidence
that the layers are polycrystalline with crystallite sizes ranging
from z 30 .4" in the 40 A°/ 70 A° MS to z 350 A°

in 5000 A° sample.
11.3.2 CuMn/Si MS

1) SAXD confirms compositional modulation in samples (d S 50011") to
within a few percent of the thicknesses selected during the
sputtering process.

2) Dark field images and EDX on samples with thick layers show compositional
modulation of layers with layer thicknesses consistent with SAXD.

3) Dark field images show that the topology of the interfaces is not
planar.

4) High angle x-rays and line scans of SAD show fcc Cu lines with preferential
directions in the layer, from which we infer that the layers are
polycrystalline with crystallite sizes ranging from
m 40 11° in the 50 A°/70 A" MS to z 200 A"

in 5000 A" sample.

5) Lack of observed crystallites in the TEM images indicate that the
Si is amorphous.

6) Resistivities are finite down to at least LcuMn :2 4OA° implying
that the layers are continous.

7) Comparison of CuMn/ Si and thin film CuMn resistivities imply
diffusion of Si into the CuMn layers.

 

 

—60—

CHAPTER III

THEORY

The experimental motivation to study the three to two dimensional crossover in
spin glasses was provided by theoretical studies of three and two dimensional spin
glasses within SG mean field theories. SG mean field theories fall into two major
catagories: 1) Bond disorder models such as the Edwards Anderson51 (EA) model
and Sherrington Kirkpatrickfi2 (SK) models incorporate magnetic disorder by ran-
domizing the bonds between magnetic ions on a fully occupied lattice. These models
and their implications on dimensional crossover are discussed in the first section of
this chapter. 2) Site disorder models incoroporate magnetic disorder by random-
izing the location of magnetic ions on a lattice. These models have been used to
analyze Heisenberg spin models with RKKY interactions in three dimensions. We
have used a static site disorder model -the uniform quenched model— to help un-
derstand the effects on T9 of the diffusion of Si impurities into the CuMn layers in
the CuMn/ Si MS. Site disorder models including the uniform quenched model are
discussed in the second section of this chapter. For a more complete discussion of

spin glass mean field theories, see ref. 5.

The third section of this chapter starts with an introduction to phase transition

scaling, and is based on lectures by M. Fisher”, presented at the Advanced Course

 

-61-

on Critical Phenomena. Through a modification of the early work of Landau, some
of the present day scaling relationships are derived. There is then a discussion of
finite size scaling and its relationships, derived from a scaling ansatz. The final
part of this section is a fairly detailed discussion of the application of scaling in
the droplet excitation model (Fisher and Huse“) as applied to the three to two

dimensional cross-over behaviour of spin glasses.
III.1) Disorder Theories

In general, a mean field theory in a statistical mechanical treatment of many body
problems attempts to average over all possible statistical probabilities. The major
question as this applies to spin glasses is; how does one average over all possible
states of a disordered system? The first problem is the technical difficulty of carrying
out the proper type of disorder average and the second difficulty is that many
equivalent ‘ordered’ (frozen) states exist. The details of these states depend on the
exact nature of the physical interactions in the sample and are therefore sample
dependent. The free energy equivalence of these states is an accidental consequence
of the system randomness. Binder and Young5 argue that all macroscopic properties

are the same for these degenerate states.

In general the free energy of a statistical mechcanical system is calculated by aver-

aging over all possible states in phase space.
F = —kBTln[Z]¢.,

This averaging can be done only when the fluctuation time scales of the system
are very small compared to experimental time scales, 77¢“c << Tap. Under these
circumstances all possible states in phase space have an equal probability of be-
ing sampled and the average is an equilibrium thermal average. This is called an

annealed average.

In the case where 1'", << Tfluc the experiment cannot average all possible states

so averaging over the partition function does not make sense. Instead the averaging

 

—62—

is done over an extensive variable such as the free energy. This type of averaging
will'encompass all possible random configurations with the same free energy. This

is called a quenched average, F = —kBT[ln Z]a.,.

One possible solution employed by EA, to calculate the average over (17.2 , comes in

the form of a mathematical identity known as the n=0 replica trick.
[In 2]“ = lim m
n=0 n

If x is some random variable (ie concentration) describing the system then we can

write
Z"(1:) = H242) = emp[— Z[H(z,$f‘)/kBT]
a=l 0:]

where the Hamiltonian is of the Heisenberg form;
-1 I —
H=7§U:Jng;-Sj—h§i:si Ill 1

In a variation of Marshall’s55 mean field distribution EA proposed a model consist-
ing of random magnetic bonds interacting between the nearest neighbors of a fully
occupied lattice. The exchange interaction J5,- between nearest neighbor spins i and

j is chosen according to a. fixed gaussian distribution.
PM.) = [2«(AJ.-.)’J-%ezp[(—J.-.- — jsj)/2AJ?,—

Sherrington and Kirkpatrick (SK)52 proposed extending the sum in the Hamilto-

nian, over all sites i and j.

In addition EA51 (1976) suggested that while there does not appear to be any spatial
correlations, between the moments, in a spin glass, there may be correlations in time.
They proposed an autocorrelation function in time to describe the ordering of the
spin glass state

4 = tlirgo((51(0)5i(t)>)
Where the inner bracket denotes a thermal average and the outer bracket a spin

average. This ‘order parameter’ measures the correlation of spin i at time t = 0

F

 

-63_

with the same spin at time t. As t -» 00 all possible time scales of the system are
averaged and q converges to a constant value. If q is non-zero below a certain T9

then correlations exist and a phase transition has occured.

Both of these models (EA and SK) have been studied extensively in Ising like
systems. These models predict a second order phase transition and show a cusp
in the zero field susceptibility and also in the specific heat. The EA correlation
parameter q is found to decay to zero for T > T9 (indicating paramagnetism) and
decays to a finite value for T < T9 (indicating temporal correlations). While these
models are clearly different from real spin glass materials they may be seen as a
reasonable starting point in a proper treatment of the inherent disorder. Ogielski
and Morgenstern56 (1986) did Monte Carlo simulations of a three dimensional short
range Ising model with lattice sizes 163 , 323 , and 643 . They have found the existence
of a spin glass state at a finite temperature, through the convergence of the EA order

parameter. Fitting the average correlation function
as) = V-1 Z < 3.5... >2
Z

to a standard three parameter form

C [8314 :39]

1.8

C(r) =

 

7

they find the growth of the correlation function 5 described by the algebraic form
5 ~ CIT — Tgl",

with u = 1.2 :l: .1. Other groups have found using different methods, and smaller
lattices, values for the Ising model of u = 1.8 :l: .557, u = 3.3 d: .6”, u = 1.3 :l: .359.
For a more complete discussion of short range Ising spin glass models in 2, 3,
dimensions, see ref 5. The lower critical dimension of the short range Ising spin

glass model is not yet known but it has been suggested that it lies in the range

dc ___ 2 __, 460,151.62.

 

 

 

 

 

-64_

Recently Reger and Young7 extended the EA model to three dimensions with RKKY
interactions by making the exchange interaction in Eq. III-1 spatially dependent,
(ie. 15,- ~ W). Using lattice sizes of L3 = 43, 63, 83, 113, 163, they found
that they could not rule out a non zero transition temperature and suggest that the
LCD of a long range Heisenberg spin glass is z' 3. They also infer from their results
that this RKKY Heisenberg model is in a different universality class then the short
range model discussed by Ogielski and Morgenstern. Finite size scaling analysis7 of

these small systems gives a lower bound of V > 2.3.
111.2) Site Disorder Models

Chakrabarti and Dasgupta63 applied the Hamiltonian

 

=oJ Z[cos( ZkZRij)]S;- SJ

j<i

to an LxLxL fcc lattice, where the Heisenberg spins (5;) are at randomly chosen
sites on the lattice. Using a Monte Carlo technique, on lattice sizes with 20, 44,
81, 161, and 312 spins, they conclude that the RKKY model has a critical point
at T9 = O, and that the correlation length exponent is v = .87 :i: .08. Monte
Carlo simulations of RKKY CuMn spin glasses in three dimensions show no spin
glass ordering, at finite temperatures, if only pure isotropic exchange is employed.
Walstead and Walker31 have found a non-zero EA order parameter using a dipolar
anisotropy. They find, fitting to experimental transition temperatures, that the

strength of this anisotropy is much greater than experimental values.

Bray, Moore and Young“ studied vector spin glasses with RKKY and anisotropic
interactions, within a site disorder framework. They have concluded that the
‘isotropic vector spin glass in three dimensions is at its lower critical dimension

and lies in a different universality class from the case of short range interactions.’

Larsen” has developed several site disorder models of varying complexity includ-

ing the uniform quenched model. Although this is a static model, it should give

 

 

_65-

reasonable estimates to the spin energetics and hence the transition temperature.

This model relies on an RKKY interaction between lattice sites.
III.2.2) RKKY Interaction

Magnetic atoms in a. dilute metallic alloy couple through the RKKY interaction.
This interaction was first used by Ruderman and Kittello to explain the coupling
between nuclear moments through the hyperfine interaction. Kasuya“ later ex-
plained how localized d electrons in transition metals scattered conduction electrons.
Yosida" combined these to derive the Ruderman Kittel Kasuya Yosida (RKKY)
interaction. The RKKY interaction couples separated magnetic atoms together
through the conduction band electrons.

Ii ' IJlAkrkr Izm

J(Rij) = 4 ~(21r)3R?j . h’

[ZkFjoc08(2kpR,-j) - sin(2kFR,~J-)] Ill-'2

It has an oscillatory nature, and decays as fly away from the impurity atom. For

a complete derivation of the RKKY interaction in three and two dimensions see
Appendix 1. Recently Levy and Zhang68 went beyond the free electron RKKY
derivation by incorporating the (1 state resonance of Mn in Cu, in a Friedel Anderson
model. They found that the interaction is oscillatory and decays as F1; in the

asymptotic region but only decays as E1,- in the preasymptotic region, R < 20A".

The effects of non-magnetic impurities on the strength of the RKKY interaction is

now a matter of some discussion. DeGennes6

9 incorporated mean free path (MFP)
effects into the RKKY interaction by giving the free electrons, in the second or-
der perturbation expansion, a finite lifetime. This has the effect of exponentially

damping the RKKY interaction.

R
1:535?“ = JRKKYe—T Ill-3

where l is the mean free path of the conduction electrons.

Larsen“ has used this damped RKKY interaction to explain the effects of added

non-magnetic impurities studied experimentally by Vier and Schultz”. Recently

 

-66-

Jagannathan, Abrahams and Stephen"1 have claimed that the DeGennes method is
an incoherent phase average and therefore smears out any phase information. They
claim that if the phase information is averaged ‘properly’ (ie by averaging the
square of the oscillating potential) the disorder has little effect on the strength of
the interaction. This suggests that the DeGennes model sets an upper bound on

impurity effects.
III.2.3) Quenched-Uniform Model

Larsen65 has developed several models to obtain the local energy scale of individual
spins in the spin glass. This local energy scale is of the form of a root mean squared

sum over the RKKY interaction between spin sites.
i
4.- =[Z[J(R.-.-)1’]
1'?“
This can be modified to approximate a uniform environment by replacing the sum

with an integral. Including the RKKY interaction A.- has the form:

3.12 °° 6 i
71,-: [ agc/ 122%) dB] Ill-4

To

 

where c is the concentration and a is the lattice parameter.

Larsen points out that ‘the essence of quenching is that a given spin never experi-
ences a closer encounter then a definite nearest neighbor.’ Quenching can therefore
be incorporated into the model by letting the lower limit of integration r0 be con-

centration dependent and equal to nearest neighbor impurity distance.

Applying the quenched-uniform model to a layered geometry requires some modifi-
cation to Eq. III-4. The layered geometry has cylindrical symmetry which can be
used to simplify the derivation. This geometry can be employed with several small
approximations. The volume that a single magnetic impurity occupies, defined by
ro in spherical symmetry, must remain the same as in the spherical system. It

would be a difficult task to construct a sphere in the cylindrical geometry, so an

 

 

~67—

approximation is made to a cylinder of the same volume. The best approximation
can be made by realizing that a sphere minimizes the surface of a given volume. A
minimum area cylinder is therefore employed in the calculation. It is easily shown
that the radius r3" (c )— - height hcy(c)— - ——}— where a is the lattice constant of

(81m)
Cu (3.61 A°) and c is the impurity concentration.

The second problem that arises is that unlike the infinite spherical model all im-
purity sites in the layer are not equivalent. The atoms near the edge of the layer
do not have the same number of near neighbor impurity atoms as those near the
ceBntre of the layer. An average must therefore be taken over all possible lattice

sites. Equation III-4 therefore becomes:

N a
SEEN/‘00 f4. Hh J2(r, z) )drdz lll’5
N- r c a
3:0 0( )
L

00 3- :51:
+ f [/ J2(r,z)dz + / 12(r,z)dz] dr]
0 art-ho Gi-‘ho first

legs,

 

T+jT-a1

+ [.4221 Pm, M f... 120,412,] w “H

layer.

where p is the density of lattice sites, L the layer thickness of the alloy, T the
interlayer thickness, a.- the distance fron the center of the layer to the middle of the
cylinder and J (r, z) is the RKKY interaction given by Eq. III-2 or with exponential
damping Eq. III-3.

Above To, the temperature that corresponds to the energy Ag, thermal fluctua-
tions dominate and the spin looks paramagnetic. Below To the RKKY interaction
dominates and the spin is ‘frozen’. The transition temperature is therefore defined
T9 = To = AoA; where A0 is a proportionality constant found by equating A to a
bulk experimental value of T9.

It must be pointed out that this model assumes perfect boundaries and does not

account for distortions of the Fermi energy due to the finite film size.

 

 

—68-

111.3) Scaling Theory
III.3.1) Introduction

Scaling theory is a phenomenological theorygwhich mathematically models diver-
gences in real systems. Scaling theory was first applied by Landau” to describe
discontinuities near the critical temperature in systems undergoing phase transi-
tions. The essence of a thermodynamic phase transition is nonanalytic behaviour
in the free energy F. This nonanalytic behaviour manifests itself in physical quan-
tities such as the specific heat and magnetic susceptibility. The divergences have
been found, theoretically and experimentally, to be symetric about the critical point
and completely reversible in the absence of a symmetry breaking field. The leading
order of this nonanalytic behaviour manifests itself in the algebraic form (T — Tc)”
where m is called a universal critical exponent. It was found experimentally that
many different systems have the same critical exponents and are said to belong
to the same universality class. An example of universality is the solubility phase
transition of binary liquid mixtures. These systems are characterized at low tem-
peratures by insoluble phases and at temperatures higher then the consolute point
(ie. transition temperature) by a homogeneous liquid. The molar fractional differ—
ence between the two phases is found to scale as (16" — :67) ~ IT — Tclf’. The same
exponent fl = .33 has been experimentally determined for the coexistence of C'C'l.
and C7F14, Na, Li, Ca dissolved in N H; 73, and many liquid vapor coexistence
systems. The similar form of the divergence suggests that the underlying physics
for all of these different transitions is similar and therefore they all belong to the

same universality class.

The Landau theory” assumes that near the critical point the system orders and
that there is an identifiable order parameter which describes the growth in order
near the critical point. The order parameter in ferromagnets is the magnetization.
The ferromagnetic system will be used to elucidate the scaling concepts. Landau’s

prescription is as follows: 1) Expand the free energy in powers of the order param-

 

 

 

-69—

eter.

F(T, M) = F0(T) + F;»,(T)M2 + F4M“ + . ..

From thermodynamic arguments we know that g—f-l— = H the magnetic field. There-
fore x" = 2F2(T) + 0(117I2) 2) Assume F2(T) can be expanded in powers of

t = (T — Tc) where Tc is the critical temperature. Therefore one obtains
X.;= 2F2,0 + 2F2’1t-i- 0(t2)

or the magnetic susceptibility scales as x ~ (T — T)" = (T — Tc)‘l The equation of
state can also be found from (€51— which gives H = M(cT + uMz) where c = 2F“
and u :2 4F“). At the critical temperature H = uM‘ = 11M3 or 6 = 3.

The critical exponents obtained by Landau’s theory are wrong when compared to
real or simulated systems. The reasoning is that the expansions made by Landau
assume that F is analytic, which is clearly not the case at the transition tempera-
ture. Widom” proposed that nonanalytic behaviour could be accounted for in the
phenomenological approach by simply grafting in new critical exponents. For ex-
ample the new equation of state would be H = M (ct +uM 3) where [3 is found from
experiment or simulation. The scaling postulate states that the order parameter
scaled by some appropriate temperature is a function of a scaled field.

- M _ RE Ill-6
zc. W—BW(lt|A)

where W is a universal function.

Starting from Eq. III-6 we obtain

8M

:— =tfl-ABDW'0
x 8H ”:0 ll ()

The susceptibility scales as
x ~ 0

therefore

7=A—fl

 

 

-70-

This is the first of the so-called scaling relationships. A list of critical exponents
and their relationships to physical quantities is given in Table III-1.

III.3.2) Finite Size Scaling

In thermodynamics, phase transitions and their associated singularities occur only
in the limit that both the volume V and the number of particles N approach infin-
ity while the density p = «(5;- remains constant. In real experimental systems this
thermodynamic limit is never reached. Quantities like the specific heat or magnetic
susceptibility may appear to be divergent, but they are not. If one could follow
the curves with perfect resolution it would be found that the the ‘divergence’ only
grows to a finite height. One therefore asks ‘What determines the height of the

curve?’

Phase transition are characterized by an ordering in the system. The system orders
by growth in correlations among the constituents. As the critical temperature is
approached the spatial extent (correlation length) of these correlations approaches
infinity. In real samples the growth in the correlations is limited by the finite size

of the sample, limiting the divergent behaviour of the phase transition.

Finite size scaling is the theory that addresses the concepts of phase transitions
and their singularities in thermodynamic systems of finite size. The development of
Renormalization Group Theory (RGT) has lead to a very physical way of looking
at this ‘phenomenological’ theory. From RGT the concept of universality follows
naturally from intrinsic spatial symmetries and the form of the Hamiltonian. For a

more complete discussion of RGT see ref. 53.

The discussion that follows is based on the phenomenological approach starting
from the scaling ansatz’s. This ansatz states that ‘In the vicinity of the bulk
critical temperature the behaviour of a system with at least one large but finite
dimension is determined by the scaled variable y = 3L7; where 5 (t) is the correlation

length and L is the characteristic length scale of the system’. For our samples L is

 

 

 

 

-71-

Name

Correlation Exponent
Magnetization Exponent
Magnetic Susceptibility Exponent
Specific Heat Exponent

Shift Exponent

Rounding Exponent

Tc is the bulk critical temperature.
M is the magnetization.

H is the magnetic field.

C is the specific heat.

T‘(L) is the rounding temperature.

Table 111-1 Critical exponents and their relationships.

Exponent

t

EVQQ

 

Relationship
£~ (T - Tc)”
M ~ H6

x ~ (T — T.)—1
0 ~ T — Tc)‘°‘

 

 

-72..

the thickness of the individual layers of spin glass. In general the argument to be

presented is based on the review article by Barber.75

Let PL(T) be some thermodynamic function that diverges at the critical point (ie
magnetization in ferromagnets, non-linear susceptibility in spin glasses). In the

thermodynamic limit this function scales as
Poo(T) ~ Coot‘”

, .
where t = [%], and p is an appropriate critical exponent. In the layered geom-

etry a phase transition occurs but at shifted temperature TCL. Therefore
PL(T) ~ 011-",

where t ~ [T - TCL] and p' is an appropriate two dimensional exponent. In general
p sé ,5 and one should expect to see crossover behaviour at some finite L. The finite

size scaling ansatz asserts that the behaviour of P is determined by y.

PL(T) = LwQQ/l

where Q is a universal function and the L out front scales the magnitude of the
physical property with the system size. A more general way to write this is PL(T) ~
L‘“Q,(L9t). By requiring that

Q(:c) ~ Coon-P L —> 00
it is easily determined that
w = p9 = a, or 0 = —
u u
The correlation length 6 is postulated to grow as

£~ (T — Tc)”.

Fisher and Ferdinand" have asserted that the criteria for determining finite size

effects in the critical region is found by matching the correlation length to the

 

 

 

 

-73-

thickness. Therefore the critical exponent A which determines the shift in the critical
temperature away from the bulk critical temperature T: as a function of layer

thickness should be equal to the inverse of the correlation exponent (ie.) = '17).

In the layered geometry we have been working with the quantity t. We can redefine

the universal function Q(t) to be a function of t such that
PL(T) = 1950141390

by defining Qp(x) = (2(2) — 23¢) where :éc = Law or

C

[TCL _ T: éc . —A
T" ~ 17; = ”CL lll'7

C

This is the scaling relationship which we have used to determine the critical expo-

nent A from the shift in the transition temperature.
III.3.3) DropletExcitation Model

Recently Fisher and Huse“ used their droplet excitation model to understand our
layered spin glass systems. In this model one looks at an infinite lattice (in a
spin glass ground state at T20) with spins placed randomly on the lattice. If one
takes a volume in this system defined by the length scale L and flips all of the
spins in that volume then that ‘droplet’ aquires a free energy which according to
their fundamental ansatz scales as F ~ YL" where Y is the surface tension of
the droplet. The theory then analyzes the relaxation of these excited droplets (of
various L) back to the ground state within a scaling framework. The density of

states of these excitations scales as D ~ 13‘”.
III.3.3.1) Below Lower Critical Dimension: 2D

Below the LCD, 02 is assumed to be negative so that many large scale excitations
exist at arbitrarily low energy, destroying the spin glass ordering. At temperature T

all excitations of the scale YL"2 and larger occur. The correlation length is therefore

 

 

 

 

-74-

defined as the length scale below which the excitations do not destroy the ordering.
As T -1 0 the correlation length diverges as

5.. Q)” ~ L , m-s

where V2 2 “91—”.

Fisher and Huse assume that the energy barriers that must be overcome by a droplet

of size L for it to relax back to it’s unflipped state is
BL ~ L‘l’a a
where L S 5 and the newly defined critical exponent 0 S 1,!) S d -- 1.

The relaxation time of these droplets is given by the Boltzman function

T BL
L ~ “4;;5] - m—s

Since the droplets of size 5 are the largest in the system the relaxation time of the

full system will be
1112 1

MT z T ~ W.
Experimentally one makes measurements on a time scale tap. For experimental
time scales 1' << tap, the experiment measures the equilibrium properties of the
system. On time scales 1' >> tap the system falls out of equilibrium. This be-

haviour becomes important in the measurement of the susceptibility.

For time scales on the order of ln(tezp) << ln(r) the experiment senses droplets of
the size L ~ [Tln(t"p)] 31?. The contribution of these droplets to the susceptibilty
is

x0...) ~ L9 ~ [Tzn(t,,,,)) If:

which is a positively increasing function of time. At time 1' >> t", the system

relaxes as if it is at equilibrium and shows Curie-like behaviour X ~ %. x will then

show a maximum when 1' ~ t“? (ie. when the system falls out of equilibrium).

 

 

 

 

 

 

-75--

The time scale of the measurement is thus very important. If the dynamic critical
exponent z; is large so that 1' diverges rapidly then there will be a peak in the
susceptibility. From Eq. III-9 and the consideration that the peak occurs at t ~ 7'
there should exist a peak for d < dc at
130...)- (—’——) _ , Ill-10
ln(tup)

which approaches T9 = 0 as t“up :> 00.

Recent experiments by Dekker et al." on a two dimensional Ising spin glass sys-
tem, RbgCunaComgF4, have been analyzed within the spin droplet excitation
model. As predicted they find a peak in the susceptibility at a temperature which
is strongly time dependent. Analyzing their ac susceptibility in a phenomenological
Cole-Cole approach, they are able to fit the relaxation time vs. temperature in
Eq. III-9 over sixteen decades of time and have found a value of $21!: = 2.2 :i: .2.

III.3.3.2) Cross-over Between Three and Two dimensions
There are two regions of interest: 1) T << T9 2) T ~ T = T:

To start the discussion of T << T9 consider a film of thickness w. For length scales
of excitations L < w the system will behave three dimensionally so that F ~ w".
Setting Y(0) ~ T9 = 1 and rescaling the temperature (free energy) to length scale w
one obtains TR(w) ~ Tut—9'. For L > w the system will behave two dimensionally
and the excitation size can be broken up into regions of size w. The temperature
can then be rescaled as a 2D system as

TR(L) ~ T(£)o’w'9° .

w
The 2D system becomes disordered (from III-8) at length scales

0
£~ T-ali'wl‘t'a'g' : T—Ugwl-i-V393 .

Similarly the barrier energy can be scaled as

36 "‘ (57)” * Ill-11

 

 

 

-76-

yielding relaxation times of order

[”7- ~ T-(1+VI¢1)w¢s+V:¢ios .

The system will appear to freeze at a temperature Tf(t¢,,) where ln(tnp) ~ (111'.

For Tf << Tf

 

Tf(te=p) ~ w¢a+¢iwoi Hui"? Ill-12
T9 ln(tezp) ’

This equation describes how the transition temperature behaves in the quasi two

dimensional region (ie. very thin films).

We will now examine the thick film region where T ~ T9 For length scales L << w
the system looks three dimensional and the free energy scales as F2 ~ YaLa' where
Y3 ~ |e|""" and e = (T7221). For length scales L >> w the system looks two
dimensional and the free energy scales as FE ~ YgLe’ The two dimensional surface

tension can be found by matching the free energies at the crossover point L a: w.
131119 ~ Y3w°~
Using Eq. III-8 this yields a 2D correlation length

£~ w(H Va9sl|€lvaasva _

From this and Eq. III-11 it is easily seen that the activation barriers for lengths
L z E are

Bf ~ [lelmeV’s-H’z'l'fls

At long measuring times tnp >> 1' ~ w" the shift freezing temperature will be
given by

we 2 we: [ln(tfl) , Ill-13

T9 w“

from Eq. III-10. To first order this equation is the same as Eq. III—7.

] l(¢a+91¢293)Vs]‘ 1

 

 

 

_77-

CHAPTER IV

DATA AND ANALYSIS

The preceeding chapters have summed up the experimental characterization of the
layered samples and the theories which exist on three to two dimensional crossover
in metallic spin glasses. The first section of this chapter starts with a description of
the SQUID susceptometer which was used to measure; 1) the magnetic susceptibil-
ity as a function of temperature for all of our samples, and 2) the magnetization as
a function of magnetic field data for the ngaM 11.07 / Si MS. There is then a discus-
sion of the magnetic susceptibility vs temperature data on the 011.1%an /5i and
Cu1_zMn,/Cu MS where x = .04, .07, .14 and .21. The transition temperature
T9 as a function of spin glass layer thickness is plotted and its main features are

discussed.

The second section of this chapter is a theoretical analysis of the magnetic suscepti-
bility data, beginning with a finite size scaling interpretation of the CuMn/Cu and
CuMn/Si MS for all concentrations. Mean free path effects on T9 in CuMn/Si MS
are looked at in the Quenched Uniform Model.

The third section of this chapter is a discussion of the magnetization vs. magnetic

field data.

   

 

-78_

The chapter is concluded with a general discussion of the relation between the
susceptibility results and the concepts of lower critical dimensionality, universality,

and dimensional crossover.

IV.l) Magnetic Measurements
IV.1.1) SQUID Magnetometer

Magnetic susceptibility vs temperature and magnetization vs magnetic field mea-
surements were made on a commercial S.H.E. corporation V.T.S. 800 series Super-
conducting Quantum Interference Device (SQUID) susceptometer. This machine is
capable of measuring sample magnetic moments over a wide range of temperatures

and magnetic fields with a resolution of 5 1x10‘7 emu.

The 800 series magnetometer is a form of magnetization measuring device known
as a fiuxmetric magnetometer. In these devices the sample is pulled at a constant
rate through two oppositely wound coils. The change in magnetic flux induces a
current in the coil circuit related to the sample moment. The advent of SQUID

technology allows for very sensitive detection of the coil current.

We have measured magnetic susceptibility vs temperature curves for all of our lay-
ered samples. The sample plus substrate had to be cut into .3 cm x 1cm strips in
order to fit in the measuring coils. Initially the magnetic field was set to zero and
the samples were loaded into the space between the measuring coils, which was at
a temperature of 5 K. The magnetic field was then set at 100 gauss or 200 gauss.
Magnetic susceptibility measurements were then made at various temperatures be-
tween 5 K and 100 K. These measurements are called the zero field cooled (zfc)
data. The temperature was then reduced to 5 K, with susceptibility measurements
made at various temperatures along the way. These measurements are called the
field cooled (fc) data. It was shown by Nogata et al.78 that the spin glass transition
temperature could be determined from the peak in the zfc ‘D.C.’ susceptibility vs.

temperature curve. D.C. susceptibility usually refers to susceptibility measurements

   

-79-

made on time scales greater then ls. In most types of magnetic systems, measur-
ing time scales of > 1.9 correspond to equilibrium measurements. As Edwards and
Anderson”1 have pointed out (Ch.III.3.3), time scales of this magnitude do not nec-
essarily correspond to an equilibrium state in a spin glass. The rounded spin glass
transition peak limits the determination of the spin glass transition temperature to

11 K.

Magnetization as a function of magnetic field was measured on several of our
Cu_g3Mn_o7/Si MS, to look for changes in the anisotropy field as a function of
layer thickness, These measurements were made at a fixed temperature in fields up

to 6 kG.
IV.1.2) Magnetic Susceptibility Data

Figs. IV-l and IV-2 show sample plots of the zfc and fc data for several of the
01%le 11,14 / Cu and 011,351)! 71,14/ Si MS, respectively. These curves are represen-
tative of the susceptibility data of all of our CuMn/ Cu and CuMn/ Si MS. It can
be seen that all of the zfc curves have peaks reminiscent of typical spin glasses.
The peak position is defined as the spin glass transition temperature T,. Starting
from T = 0 K the zfc curves increase with increasing temperature up to T9 , with
occasional evidence of anomalous increases in the susceptibility. We interpret these
anomalous increases as time dependent effects (caused by different measuring times
per temperature point) as the susceptibility tries to approaches the (nominally)
equilibrium fc susceptibility (in the regime where the relaxation time approximates
our measuring time) as time —-» 00. Above T9, the zfc susceptibility exhibits Curie-

like behaviour (ie. x ~ 11;).

Above T9, the fc susceptibility is also Curie like but slightly larger than the zfc
susceptibility at the same temperature. By comparing the zfc and fc curves of a
standard Pt sample we have found that this difference between the curves is due
to a systematic machine error. In bulk spin glass materials the fc susceptibility

below T9 is nominally constant‘. It has been suggested that the fc state is the true

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equilibrium state of the spin glass". In our samples we generally observe that the fc

susceptibility decreases slightly below T, and then increases as T approaches zero.

The sharpness of the susceptibility peak appears to increase as the LCuMn is de-
creased. This however is an artifact of the decrease in T9 with decreasing LcuMn
(since x = 0 at T = 0 for the zero field cooled curve). A more appropriate mea-
surement of the peak width would involve normalizing the width by dividing each
temperature by the transition temperature. When the normalized zfc susceptibility
curves are plotted (Fig. IV-3) we cannot resolve any difference in the widths of the

susceptibility peaks, to within experimental resolution.

The absolute values of the fc susceptibility (T >> T:) of all the CuMn/ Cu MS
appear to follow similar Curie behaviour, corresponding to approximately the same
number of paramagnetic moments. The differences in the susceptibility may be due
to differences in susceptibility of the individual Si substrates backing the samples.
These substrates are diamagnetic and have an approximately constant susceptibility

of —11:10"‘ emu/ gm, over the range of temperature considered in this study.

In contrast the absolute values of the susceptibility of the CuMn/ Si MS are quite
different from each other and do not appear to depend on the CuMn layer thickness.
At this point it is unclear as to what is causing these differences. It is clear, however,
that the absolute values of the susceptibility are the same order of magnitude as
the CuMn/ Cu MS, suggesting that the formation of silicides (MnSi; = lip-g,

Mn; p = 5-5I‘i30) is not extensive.

To determine what interlayer thicknesses of Cu and Si were sufficiently large to
magnetically decouple the spin glass layers from each other, samples with varying
interlayer thicknesses were made. Fig. IV-4 shows a plot of T9 vs. interlayer
thickness for a fixed alloy thickness of 100 A° in the Cu.951lIn,04/ Cu MS. It is
observed that no significant deviation from the transition temperature T: occurs
down to L0" = 100A°. 300A° of Cu was used to magnetically decouple the spin
glass layers. Fig. IV-5 shows the variation of T9 with Si thickness in the CuMn/ Si

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MS. SAXD confirms good layering down to L5,- : 30A" but no layering is observed
for L5,- = 20A". The transition temperature T: remains constant down to 30 A°
(Fig IV-5) but begins to shift upward towards the bulk transition temperature for
L5; = 20A° and 10A°.

If one examines the susceptibility curves in Figs. IV-1,2 one sees that all samples
show a downward shift in T, as a function of decreasing layer thickness. Plots of Tg
vs. LCuMn show that this fundamental behaviour is representative of all the data
for the three concentrations of Mn in CuMn/ Cu MS, Figs.IV-6,7,8 and for all four
Mn concentrations in the CuMn/Si MS, Figs.IV-9,10,11,12.

As mentioned previously the bulk value T: of target shavings compares well with
the value of T9 for 5000A°. This thickness is therefore defined as the bulk thickness.
Figs IV-6,7,8,9,10,11, and 12 show that there are deviations from the bulk transition
temperature for thicknesses as large as 500 — 1000A°. The transition temperature
decreases in a continous fashion as a function of decreasing layer thickness going to
T9 = 0 at approximately 12A° in the CuMn/Cu MS and 40A° in the CuMn/Si MS.
To compare curves of varying concentrations, the normalized transition temperature
% is plotted as a function of layer thickness for the CuMn/ Cu MS in Fig. IV-13,
and the CuMn / Si MS in Fig. IV-14. To within experimental resolution the samples
with similar interlayers show the same behaviour. The differences between the

CuMn/Cu MS and the CuMn/Si MS will be discussed in the next section.
IV .2) Finite Size Scaling Analysis
1V.2.1) CuMn/Cu MS

We believe that the CuMn/ Cu MS most closely approximate the ideal layered spin
glass system. Fig. IV—4 shows 300A° of Cu effectively separates the spin glass layers
magnetically from each other. The systems are not contaminated by impurities (ie.
Si) and the Fermi surfaces should be similar to three dimensional Cu with small

deviations due to the Mn impurities and grain boundaries due to the polycrystalline

 

 

 

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nature of the MS. The structural characterization of these samples, presented in
Chapter II, suggests that the layers are polycrystalline and that the diffusion of Mn
out of the CuMn layer is small. For the following analysis we will assume that this
diffusion is negligible. The data for the three Mn concentrations in CuMn/ Cu MS
have been fit (Figs.IV-15,16,17) to the form

Tb—TL
_ 9 9 —A

All three of the critical exponents A are within error of each other. The non-
universal parameters A have values of 5.1, 2.9, and 4.7 (for the 4, 7 and 14 per-
cent Mn samples, respectively). The CuMn thicknesses where LCuMfl -9 0 are

12.9 11", 8.6 A°, and 9.3 A° for the three concentrations (respectively).

Although the only spin glass used in this study was CuMn, in the range of concen-
trations 4 - 14 percent the material changes a great deal. Statistically, the average
number of impurities in the nearest neighbor rm, and next nearest neighbor nnn
sites, to a selected impurity increases by 350 percent. If the analysis of J aganathan
et a1."1 is correct, the RKKY interaction should be similar in all of these materials
with some deviation due to impurity concentration effects on the Fermi surface.
The increase in rm and nnn increases the bulk transition temperature T: by about
300 percent. The amount of frustration felt by each spin increases as the number
of rm and nnn neighbors increases. The increase in the nnn will also increase the

amount of short range ferromagnetic ordering.

The aforementioned differences in materials of different concentrations coupled with
the similarity of the exponents A leads us to conclude that the exponent A is a
universal critical exponent as postulated.” The best value of A, obtained by fitting
all of the data to a single exponent is A = .7 i .05 (Fig.IV-18). We therefore expect
to see similar behaviour in all metallic spin glasses such as AgMn, AuFe etc., which
should all belong to the same universality class. The value of V = ~1- = 1.72 :l: .15

A
that we have obtained is compared with the value of V = 1.3 :l: .2 obtained by Levy

 

 

 

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et al.” from non-linear susceptibility experiments, and the value of u = 1.2 :t .1
obtained by Ogielski and Morgenstern“3 on simulations of Ising spin glasses. This
similarity to the Ising model is intriguing, but it is not clear that these systems are
in the same universality class. Finite size effects should become evident when the
correlation length equals the limiting sample size. In our samples this occurs in the
region LCuM,1 = 500 —> 1000 A°. This agrees well with the value offi z 2000 A°

which Levy and Ogielski11 have infered from their experiments on AgMn.

Finite size scaling analysis of phase transition systems where the dimensionality is
reduced predicts rounding of divergent behaviour, due to restriction of the correla-
tion length in the directions of the reduced dimensions. As mentioned previously
we see no evidence of rounding in the widths of the normalized D.C. susceptibility.
These peaks however are not divergent even in zero field. A more appropriate ex-

periment to measure rounding behaviour would be be the non-linear susceptibility.

The lack of noticeable rounding in the susceptibility is disconcerting, especially since
the exponent which determines rounding behaviour, 9, is postulated to be equal to
the shift exponent A, and we have applied the scaling analysis over a large variation
in LCuMn- One possible explanation to the lack of observed rounding is given by
the analysis of Fisher and Huse.54 They suggest that the scaling analysis we have

used (5 ~ Léim) is relevant only for our thickest samples.

For an infinite measuring time in a 2D film, the susceptibility should show Curie-like
behaviour all the way down to T20. In the thin film region (L50 —’ 0), the peak
in the susceptibility is a manifestation of the system falling out of equlibrium with
the measuring time scale. In this limit the apparent freezing temperature has been

derived as a function of measuring time and film thickness (Ch.III.3.3),

 

Tf(tm) [L¢a+¢w=9s]n+u‘m
Tg ln(tm)

We observe behaviour in Fig. IV-l reminiscent of the CuMn/ Cu MS following

similar Curie-like curves and falling away from the Curie curve as a function of

 

 

—102—

LouM". Assuming that LCuMfl S 200 A" is the thin film region and knowing that

our measuring times are approximately constant (z 400.5), we obtain a value of
$3 + dbl/293

1 + ”2%
Fitting our data to the scaling relation 6 ~ L‘A in the large LCM.“I region (LCuMn >

= .38i .03.

200 A°) we obtain a value of A = 1.1 :l: .3. Unfortunately we have not been able to
measure TgL accurately enough to observe the logarithmic corrections predicted for

the crossover between the thin and thick film regions.
IV.2.2) CuMn/Si MS

Figs. IV-19, 20, and 21 shows fits to A of the Cu1_zMn,/S’i MS for 232.04, .07 and
.14. All values of A obtained in these fits are slightly larger then those determined in
the CuMn/ Cu MS and within experimental error of each other. The fit to all of the
CuMn/ Si data is shown in Fig. IV-22. The CuMn thicknesses where LCuMn —» 0
are 36.6 A°, 37.8 11°, and 33.3 A" for x=.04, .07 and .14 (respectively).

We have excluded the Cu_7gll/In_31/ 51' MS from this scaling analysis for the fol-
lowing reasons. 1) After sputtering, the target showed signs of inhomogeneities.
2) The resistivities were extremely dependent on the sputtering run in which they
were made. 3) Although the T: show behaviour qualitatively similar to the other
CuMn/ Si MS, there appears to be a sputtering run dependence.

It has already been established (II.2.4) that the Si in the CuMn/ Si MS is enter-
ing the CuMn layer. These impurities increase the resistivity of the alloy layer,
thereby decreasing the average elastic electron mean free path. To get an estimate
of the effects of the increased resistivity on the transition temperature T: we have

developed the Uniform Quenched Model (UQM) in a layered geometry (III.3.3.).

It has been shown” that the introduction of non-magnetic impurities into an RKKY
mediated spin glass reduces the transition temperature from that of the pure alloy.
Estimating this reduction can be done by introducing an exponential damping of

the form

 

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where l is the mean free path of the conduction electrons, and a is a damping
constant which goes to the de Gennes limit (Eq. 111.3) for a = 1. At this point in

time no theoretical estimate of (1 exists.

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When this boundary effect is incorporated into the UQM for a non damped RKKY
interaction (0: = 0) it is seen to have negligible effect down to approximately 30 A°,
Fig. IV-23, 24, and 25. The transition temperature begins to decrease at this point

and must approach zero as LCuMn => 0.

Alternatively, setting a = 1, the de Gennes limit, we find that mean free path
effects ( due to the increasing resistivity as LCuMn decreases in the CuMn/ Si MS)
significantly reduce the transition temperature from the bulk value. The results of
this damping are shown in Figs. IV-23, 24, and 25. These data shows that mean
free path effects become significant below about 500 A" but do not account for the

entire decrease in T, as a function of LcuMn.

If we take the difference between the CuMn / Si data and the estimated reduction in
Tg, due to mean free path effects the resulting data are found to fit to the CuMn/ Cu
data fairly closely (Fig. IV-26). The 4 and 14 percent CuMn/ Si MS (with estimated
mean free path effects subtracted out) are larger then the CuMn/ Cu data in the
samples with thinner LcuMn, consistent with the assertion by J agannathan et al."1 ,
that the de Gennes limit” is an over estimation of the effects of a reduced mean

free path on the RKKY interaction.
IV.3) Magnetization vs. Magnetic Field

The magnetization vs. magnetic field was measured by W. Abdul-Razzaq“ for

 

 

 

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-112—

various LCuMn in our Cu,93Mn,o7/Si MS. The zfc magnetization of these sam-
ples were measured in fields up to 6kG. The magnetization was then measured in
fields between 6kG and ~6kG, and then in fields between -6kG and 0G. Although
the maximum fields were not quite large enough to saturate the anisotropic fields
(Abdul-Razzaq has estimated the saturation field of these samples to be z8kG)

useful information can still be obtained from them.

After ‘saturation’ the magnetic field is reversed. At a negative field the magne-
tization displays a rapid reversal. This occurs at a field termed H ,5. Although
true saturation has not occurred, H .15 should display behaviour qualitatively sim-
ilar to the coercive field needed to flip all of the spins 180°. It is noticed (IV-27)
that H“. is finite for all samples measured. We infer from this that the layered
samples are qualitatively similar to bulk CuMn (ie. have DM anisotropic interac-
tions). The coercive (IV-28) field remains approximately constant (~ -2kG) down
to LCuMn = 300 A" and then decreases rapidly down to a value of ~ -.3kG at
Loamn = 100 A°. Abdul-Razzaq has interpreted the reduction in the the coercive

field to be due to a reduction in the DM interaction.
IVA) Universality

From the relationship A = i we have obtained a value of u = 1.72 :l: .15 From
the exponents fl and 7 found directly from non-linear susceptibility experiments,
Levy and Ogielskill deduced from the hyper scaling relationship, du = Zfl + 7, a
value of u = 1.3 :l: .1, We expect the material they used (AgMn) to be in the same
universality class as CuMn due to the similarities in the materials and their magnetic
interactions. These values of :1 compares well with values reported for Ising Model
simulations (see III.1). Until recently, theoretical Heisenberg like systems in three
dimensions have yielded only transition temperatures equal to zero, suggesting a
lower critical dimension greater then three. In contrast the recent work by Reger
and Young7 includes the possibility of a non-zero transition temperature and sets a

lower bound of u > 2.3. These simulations were done on small lattices Law," 5 163,

 

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10
r . ,9
1' / ’ H‘h" 16.19
3r
+ l I 1 j
30 100 1000 5000

CuMn thickness(A)

Figure IV-28: H“. vs LCuMn in Cu,g3/Mn,o7/Si MS.

 

 

 

-115-

which may be too small for an accurate determination of u. Walstead and Walker
have found a non-zero transition temperature for Heisenberg systems with a dipolar
anisotropy. This suggests the possibility that the anisotropy in the real materials
causes them to belong to the same universality class as Ising spin glass systems.
Levy and Ogielskill have also measured the suSceptibility exponent, “y = 2.1:1: .1, the
order parameter exponent, fl = .9:l:.2, and the dynamic exponent 21/ = 6:l:.6. These
compare less well then V with the values ‘7 = 2.9”, fl = .559, and 21/ = 6 :1: 1“,
obtained from Monte Carlo simulations on Ising spin glasses. The comparison
of CuMn with Ising spin glass models is therefore unclear and in need of more
experimentation and larger scale simulations. A comparison of the other critical
exponents (see Table III-1) should resolve the question; Which universality class

does the CuMn spin glass transition belong?

 

 

-116-

CHAPTER V

CONCLUSIONS

V.1) Structure of Multilayer Systems

Using UHV sputtering we have produced Cu1_,Mn,/Cu MS with :c = .04, .07

and .14, and Cu1_,Mn,/Si MS with a: = .04, .07, .14 and .21. Imaging and EDX
analysis of selected samples indicates chemical layering in both types of samples
consistent with thicknesses programmed into the sputtering control system during
sample preparation. Quantitative SAXD analysis of the CuMn/ Si MS indicates
bi-layer thicknesses within a few percent of programed bi-layer thicknesses. SAD,
TEM imaging and high angle x-ray studies indicate that the CuMn layers in the
CuMn/ Si MS are composed of crystallites approximately half the diameter of the
layer thickness in the thinner samples (LCuMfl S 200 A°) and saturating at a
diameter of a: 300 A" in the thickest samples. In contrast, we infer from the lack of
superlattice lines in the high angle x-rays, and the lack of observed crystallites in
the TEM images that the Si layers are amorphous. The resistivities of the CuMn / Si
MS indicate that some Si is entering into the CuMn layers, probably at the grain
boundaries. Resistivities of the CuMn/ Cu MS support a layered geometry with

minimal diffusion of Mn into the Cu interlayer.

 

 

/.

'fi’f vim.

"'-~ .

fl . "v

' ' / _.
1‘ I" i \

 

-117—

v.2) Susceptibility and T,,

We have measured the zero field cooled (zfc) and field cooled (fc) susceptibility
curves of all of our samples from 5 K to about 100 K. All zfc and fc susceptibilities
show behaviour typical of spin glasses. The susceptibility measurements coupled
with the low T (T << T9) hysteresis curves indicate that the CuMn layers in our
samples remain spin glass-like to LCuMn = 20 A°. Following convention we have
defined the peak in the zfc susceptibility to be the spin glass transition temperature
T9. We observe, for all concentrations and for both the Cu and Si interlayers,
that this transition temperature decreases systematically with decreasing LcuMn.
This reduction in Ta begins at layer thicknesses as large as 1000 11° and apparently
goes to zero at LCuMfl z 12 11° in the CuMn/Cu MS and LCuMn z 36 11° in the
CuMn/ Si MS. Above Tg absolute values of the fc susceptibility in the CuMn/ Cu
MS appear to follow similar Curie-like behaviour. Scaling the measured transition
temperature Tgl’ by the appropriate bulk transition temperature T: we observe that
all of the different concentration CuMn / Cu samples follow similar behaviour. This

is also true, separately, for the CuMn/ Si MS.
V.3) Comparison with Scaling Theory

For each interlayer, we can fit the reduced temperature

5 = (31%;) ~ ALoian
over the whole length scale studied. We find values of A consistent with each other,
for all values of x, for a common interlayer. By fitting all of the data for a given
interlayer, we obtain values of A = ..58 :l: .05 for the CuMn/Cu MS and A =
.8 j: .05 for the CuMn/ Si MS. The value of the correlation exponent u = 1.72 3: .15
obtained through the scaling relation A = % is compared with the value of u =

1.3 :t .2 obtained by Levy and Ogielski from non-linear susceptibility measurements
on AgMn. '

Within experimental resolution, no change is noticed in the widths of the normalized

 

 

-118-

zfc peaks, as predicted by finite size scaling theory. Fisher and Huse suggest that
finite size scaling should only be valid in the thickest CuMn samples. Applying
finite size scaling analysis to CuMn/ Cu MS for LCuMn > 200 A" we obtain a value v
of A = 1.1 :l: .3 For thinner samples Fisher and Huse predict that the apparent
freezing temperature should scale like

Tf(tm) ~ LVN-tibiae. 1"+"u—1,'¢—,
T, ln(tup)

Fitting our data to this form for CuMn/ Cu MS with LCuMn S 200 A° (assuming

t... constant) we obtain a value of

$3 + 111211293

2 .38 i .03 .
1 + 112$:

 

v.4) Conclusions As we thin down CuMn layers in MS samples, we find that the
spin glass transition temperature T, decreases and apparently approaches zero as
the system approaches two dimensions. We believe that this is the first experimental
evidence that the Lower Critical Dimension of this universality class of transition
is between two and three. The data is generally consistent with finite size scaling

theory and we have obtained a value of the critical exponent A.

 

 

 

-119-

REFERENCES

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Ed. Domb and Green,
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6. Kac, M.: Statistical Physics Phase Transitions and Superfluidity, vol.1
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16. Werner, S.A., J.J. Rhyne, J.A. Cotes: Soild State Comm.50, 5, 457, (1985).
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—120-

19. Ford, P.J. and J.A. Mydosh: Phys. Rev. B 14, 5, 2057 (1976).
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24. Heffner, R.H., M. Leon and DE. MacLaughlin: Hyperfine Interactions, 18,
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25. Malozemoff, A.P. and Y. Imrg: Phys. Rev. B 24, 489, (1981).
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29. S. Jackson: private communication

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34. Prejean, J.J., M.J. Joliclerc and P. Monod: J. Phys. (Paris) 41, 427 (1980)
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36. McCoy, B.M., and T. T. Wu: The Two Dimensional Ising Model, Harvard
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-121—

37. Onsager, L.: Phys. Rev. 65, 3 and 4, 117 (1944)

38. Hofer, W.0.: Erosion and Growth of Solids Stimulated by Atom and
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40. Fischer, K.H.: Landolt Bornstein Tables 15, 289, (1982) (and references

therein).
41. Handbook of Chemistry and Physics 56, CRC Press (1976).
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43. Watt, I: The Principles and Practice of Electron Microscopy, Cambridge
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44. Thomas, G., and M. Goringe: Transmission Electron Microscopy of Ma-

terials, John Wiley and Sons, Inc., (1979)

45. Slaughter, J .: private communication.

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47. van de Pauw, L. J.: Philips Res. Rep. 13, 1, 1 (1958)
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50. Van Haesendonck, C., L. Van den dries and Y. Brunsereade: Phys. Rev. B 25,
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-122—

Fisher, D.S., and D.A. Huse: Phys. Rev. B 36, 16, 8937 (1987).

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. Ogielski, O.T., and I. Morgenstein: Phys. Rev. Lett. 54, 9, 928 (1985).

McMillan, W.L.: Phys. Rev. B 31, 340 (1985).

Bray, A., and M.A. Moore: Phys. Rev. B 31, 631 (1985).

Bhatt, R.N., and A.P. Young: Phys. Rev. Lett. 54, 9, 924 (1985).
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Yosida, K.: Phys. Rev. 106, 5, 893 (1957).

Zhang, Q., and RM. Levy: Phys. Rev. B 34, 3, 1884 (1986).
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—123—

Ed. E. V. G. Cohen, North Holland (1975)

75. Barber, M.N.: in Phase Transitions and Critical Phenomena, edited by
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80. Moorjani, K. and J. Coe: Magnetic Glasses, 299, E. Sevier, (1984).

81. Abdul-Razzaq, W: To be published.

 

—124—

APPENDIX A.l
3D RKKY INTERACTION

The RKKY interaction is an indirect interaction between two magnetic ions with moments
I, and 13. These ions are placed in a metal sufficiently far apart so that their local electronic
wavefunctions do not overlap. The ions are coupled via the conduction band electrons
which have a spin monent S. The interaction is therefore calculated as a second order
perturbation calculation with the sum over conduction band states. App. is the overlap
matrix element between conduction band states k and lie. The second order perturbation

expansion is:

.. __ , . "' 11L °° dk’ A“. Awash-wins
J(Ru)- (5 Il)(s Ij)‘/0 (2703i (2“); E(k')—E(k)

In a free electron model the fermi surface is spherical. The above integrations can therefore

 

P

be done in a spherical geometry.

 

' 2 hp 2!
_Il Ii lAk" l A / jo- e'l‘R°°'0sin0d9 dd) kzdk

(21r)°
2w e—ih' Reosfl' I
sing I r12 I
f... f. /. End) -E(k) “”4”“ d"

The integration over 4; is trivial.

2* 1r 1r
Q = j / eil‘Rcmasin0dcfi = 211' / eichosOsinOdO
o o o

 

The integration over the 9 of Eq. A-l can be done by substitution. Letting u = ichosO,
du = —ikRsin0d0 and doing the integration we obtain

_i27l' fiL/h' e—ikszdk— A” 6“:sz die]

Take the first integral f0" e“"‘dek and let k1 = —k

h:
:> f 8.11de
o

 

 

—125—

0
=> —/ eil‘dek
-1,,

_ ' 5r ,
= 2m / ad'dek
-1,,

 

R

Therefore Eq. A-1 becomes

. . — Ii . Ij lAkh'lz lb! eikflu ff: foo e-il: Rij ' I
J(R._7 _ (210412 h chk h E(k') _ E(k)k dk

 

 

Using the argument of RKlo this becomes

1,1, |A,,,,.|2 f“! ,R, U” e-ih'Ru , 'l
JR,-- =— e' ”India India
( ” (mm;- -.. -.. Bad) — Em

 

 

In a free electron model E(k) = lat—1. Therefore we obtain

 

 

Il' Ij IAwPZm eihR. e-ik' 11.5
J(R;j) = — (21,412,” 1.... . chk ”Ht “k'dk'

The integral in brackets can be evaluted by using the identity

 

1 __1_ 1 _ 1
k'3—k3—2k lc'-k Ic'+lc

————'=lcdk' k'dk’——

I I
cow—1c: 2k _... Ic'-lc 2k _... hunch“

 

 

0° e—ak RU ___/091 e—s’h’Rq 1 0° e—ik'RU

Evaluate the first integral by letting z = k'Rij
co e—ib’Ru 1 w e-i‘
k'dk' = - d
lav—k RAM—m”

Jordan’s Lemma states that the line integral around an analytic region in the complex

 

 

plane is zero (ie. f = 0 ) Therefore

e—iz e—iz 0° e—iz
d = d P
/ z—kRzz fgaz—kR22+ [002+kR2dz

 

 

 

 

 

-126—

where P is the principal part of the integral.

°° e"z .
P/ zdz = —1rie"z

= m'e‘WkR)
00 z — 1:12

Hi

 

 

°° e“ .
P/ zdz : 1rze"z

 

é —7rie_i"R(kR)

 

 

.00 z + IcR _kR
Therefore
00 e—ik'Ru —7|' i _..
f... n+1. ’“"“°' = 7“?” _. “9
and

. hr . hr , hp '
if. [/ e2tthdk _ f ’6ko = 171/ e2sthdk
2 —hp hr 2 -’¢r

 

 

 

 

_ 3 (2m) _ (22'12)2 _h, — 2 4 SR h
_ 1r ' ' 7rsin(2kpR)
__ 4R [2kpcos(2kpR R ]
Therefore the RKKY interaction in 3D is:
— .. _ It‘ll lAlele'l2 m .. 1rsin(2kFR.-,-)
_ J(R:J) — _ (21"),jo 4,12 kacos(2kpR,J) — R".

 

—127—

APPENDIX A.2
2D RKKY INTERACTION

The following is a derivation for the RKKY interaction in a metal with a two dimension
conduction band. The sum in the second order perturbation expansion is now over a tw.

dimensional fermi surface.

hp dk co dkl Akh' Ak,kei(k‘k')'nfl
J(Rij) = ‘(5 'II)(S “1.1)/o (EU—2 lu- (2a)“ E(k') — E(k)

 

 

 

Ii ' I) lAhk' l2 f," [2* 'lcR 0 2 foo f2” e—‘h'flcow' I :2 I
= — e' c” d0 k dlc d0 lc dk
(27')4 o o I... 0 Elk') - E(k)

The integral over 0 is a Bessel function:

1 21'
10(2) 2 EA e-chosOdo

Therefore

 

. l 2 5! °° 1c)
—1‘ 1’ IAuI 2’" Jo(kR)lc’dk ——J°(

__ 2 :
(2102),: o k, kn k dk

HRH) =

Using the asymptotic expansion for J,(Ic'R) for kR >> 1

to) = (Em — 2)

.. __ 1;.13 IAWI22m\/—2-/h' 2 °°cos(k'—§ k’ ,2 ,
J(R.,)— (21r)3h’ 7r 0 J,(IcR)lc die I" x/R—Ic’)—k k dlc

 

 

This expression was analyzed numerically. The results are plotted in Fig. A-1. The RKKY

interaction in 2D is found to decay as J (R) ~ 513-.

 

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