3....w.
3.5...
Ann'i bf. w 1
at) (an...
53.5....
9:...
tn... 9

r... l... x I".
I :vsvut.

 

 

 

 

 

 

 

 

.na..v.u..,...\
: . :.. .3.

 

THEE!

MICHIGAN STATE UNNER [TY UBRAHIES

I 1W! U/I/I/II/NIIWI/M li’i/llfl/I/l/IIHW

i r
3 1293 01029 6550

i!

 

 

 

 

 

 

This is to certify that the

dissertation entitled
Electrical and Magneto-resistivity
Measurements on Amorphous CuTi Alloys
at Low Temperatures

presented by

Renyong Fan

has been accepted towards fulﬁllment
of the requirements for

Ph.D Physics

degree in

 

 

Major professor

 

 

Date 0"": z /¢?2

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

 

 

 

 

LIBRARY
Mlchlgan State
University

 

 

 

PLACE ll RETURN BOX to roman this checkout from you rocord.
TO AVOID FINES rotum on or baton dot. duo.

DATE DUE DATE DUE ‘ DATE DUE

   
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MSU Is An Atﬂnndtvo Actionleol Opportunity Institution
a -- WE‘

ELECTRICAL AND MAGNETO-
RESISTIVITY MEASUREMENTS ON
AMORPHOUS CuTi ALLOYS AT LOW

TEMPERATURES

By

Renyong Fan

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1992

ABSTRACT

ELECTRICAL AND MAGNETO-RESISTIVITY MEASUREMENTS
ON AMORPHOUS CuTi ALLOYS AT LOW TEMPERATURES
By

Renyong Fan

The anomalous transport properties of highly disordered metallic glasses, which
require corrections to the classical Boltzmann theory, are due to quantum interference
effects of the scattered electron waves. These corrections provide new contributions
to the resistivity: “weak localization” and “electron-electron interaction”. To study
these quantum interference effects, we have made the highest-precision measurements,
so far, of the resistances of the amorphous CusoTiso and CumTim ribbons at much
lower temperatures than before (15mK< T <6K) and in small magnetic ﬁelds ( 0T<
B <0.2T). To measure the resistance and temperature accurately, we developed a
novel method: measuring the resistance perpendicular to the ribbons with potassium
as the non—superconducting glue between the CuTi ribbons and two Cu electrodes in
order to make excellent electrical and thermal contact. With this method, we were
able to measure the resistances with a relative precision of Ap/ p = 10-7 — 10"8 and

temperatures reliably down to l5mK with an error of less than lmK.

The zero ﬁeld resistances and magnetoresistances were analyzed using weak local-
ization theories that include the Zeeman splitting and electron-electron interaction
theories. Possible background contributions from the K layers, the Cu electrodes,
and their boundaries are quantified in the analysis. In zero ﬁeld, these background
contributions were negligible for T < 3K. At zero magnetic ﬁeld and T < 0.15K, we
found that electron—electron interaction dominates the resistance, while weak local-
ization makes a nontrivial contribution to the resistance for T > 0.15K. In contrast,

at the lowest temperatures, the magnetoresistances were dominated by weak localiza-

tion with Zeeman splitting and Maki-Thompson superconducting ﬂuctuations. For
higher magnetic ﬁelds and lowest temperatures (B / T > 1 T / K), we ﬁnd discrepancies

between our data and the theoretical calculations.

We found that most of the parameters of the theoretical ﬁts to the data were
similar for both Cu5oTi50 and CueoTiw alloys. The two important exceptions were
the inelastic and spin-orbit lifetimes: their zero-ﬁeld values were about an order
of magnitude smaller than those from the magnetoresistances. Also the inelastic

lifetimes tend to saturate for T < 0.1K in non-zero magnetic ﬁelds.

Finally, we were also able to estimate the expected superconducting transition
temperatures of both Cu5oTi50 and CusoTim alloys: less than 15mK and 5mK, re-

spectively.

Our novel technique can, in principle, be used to make high precision resistance

measurements down to 15mK on any ribbon or ﬁlm-like high resistivity metal.

DEDICATION

This thesis is dedicated to my mother, mother in law, especially to my wife, Zhenli,

and my son, Dan, whose assistance and encouragement were greatly appreciated.

iv

ACKNOWLEDGEMENTS

My sincere thanks go to Dr. William P. Pratt, Jr. for his supervision and invalu-
able aid and suggestions throughout the writing of the thesis. I am very grateful for

his help during many late-nights measurements.

I would also like to express my thanks to Dr. Dennis Greig for his cooperation
and advice. I wish to thank Dr. Peter A. Schroeder for his help in taking data.
Special thanks to Dr. Reza Loloee for his help in the ion milling and sputtering, and
to Thomas Palazzolo, Jim Muns, and Tom Hudson for their help in the construction
of the apparatus. I wish to thank Dr. Zhao Liang and Yang Qing for their friendship

and encouragement.

I acknowledge the ﬁnancial support of the National Science Foundation.

Contents

1 Introduction 1
2 Quantum Interference Theories 5
2.1 Classical Transport Theories ....................... 5
2.1.1 Deﬁnition of Resistivity ..................... 5

2.1.2 Electrical Resistivity ....................... 6

2.1.3 Magnetoresistivity ........................ 8

2.2 Quantum Interference Effects ...................... 8
2.2.1 Weak Localization ........................ 9

2.2.2 Spin-Orbit Interaction ...................... 12

2.2.3 Electron-electron Interactions .................. 14

2.3 Zero Field Electrical Resistivity ..................... 17
2.3.1 Elastic Scattering and Dephasing ................ 17

2.3.2 Inelastic and Spin Scattering ................... 18

2.3.3 Weak Localization with Spin Orbit Scattering ......... 19

2.3.4 Electron-electron Interactions .................. 20

2.3.5 Superconducting Fluctuations .................. 21

vi

2.4 Magnetoresistivity ............................ 24
2.4.1 Weak localization with magnetic ﬁelds ............. 24
2.4.2 Electron-electron Interaction ................... 25
2.4.3 Superconducting Fluctuations .................. 27

3 Experimental Techniques 29

3.1 Main Apparatus .............................. 29
3.1.1 The Dilution Refrigerator .................... 29
3.1.2 High Precision Resistance Bridge ................ 30
3.1.3 Glove Box and Vacuum System ................. 30

3.2 Sample Preparation ............................ 30
3.2.1 Preparation of CuTi Alloy Ribbons ............... 30
3.2.2 Sample Sandwich ......................... 31
3.2.3 Electrical and Thermal Contact and Cu Sputtering ...... 35
3.2.4 Magnet and Current Supply ................... 36
3.2.5 Making and Mounting Samples ................. 39

3.3 Thermometry ............................... 42
3.3.1 Sample Thermometers ...................... 44
3.3.2 Reference—Resistor Thermometers ................ 45

3.4 Measurements ............................... 47
3.4.1 Resistance Measurement (B = O) ................ 47
3.4.2 Magnetoresistance Measurement ................. 51

3.5 Residual and Boundary Resistances ................... 53

vii

3.6 Heat Generated in our Samples ..................... 53

4 Experimental Results 55
4.1 Sample Information ............................ 55
4.2 Temperature Dependences of the Zero Field Resistivities ....... 60

4.2.1 Background Resistances ..................... 60
4.2.2 Fitting Formula and Parameters ................. 69
4.2.3 p(T) for Cu5oTi50 and CumTiw Alloys ............. 70
4.3 Magnetoresistivities ............................ 83
4.3.1 Background Magnetoresistivities and Fitting Formulas . . . . 83

4.3.2 Analyses of the Magnetoresistances due to Various Scattering

Mechanisms ............................ 87

4.3.3 p(B) for CusoTi5o and CumTiw Alloys ............. 91

4.4 Comparisons of the Fitting Parameters ................. 102

5 Summaries and Conclusions 107
A Calibration of xold 109

B Temperature Dependence of of the Reference Resistance 110

viii

List Of Tables

2.1

4.1

4.2

4.3

4.4

4.5

4.6

4.7

A.1

B.1

B(T) vs T ................................. 28
CuTi Sample Preparation Conditions .................. 57
Cu/ K/ Cu Sandwich Preparation Conditions .............. 58
The Fixed Parameters .......................... 70
The Fitting Parameters for KCusoTisoCu—ll .............. 72

The Fitting Parameters for the KCusoTiwCu and KCusoTiquu Sand-
wiches ................................... 75

The Fitting Parameters from the Zero Field Resistances and Magne-

toresistances ................................ 102
The Fitting Parameters for Comparisons ................ 105
Xold VS T .................................. 109
R"! VS C11 ................................ 111

ix

List Of Figures

2.1

2.2

2.3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.1

4.2

4.3

Low temperature resistivity for a typical simple metal .........
Electron scattering ............................

Interactions of two diffusion-channel electrons through the hologram .

X-ray diffraction of amorphous CuTi ribbons ..............
Sample sandwich .............................
View of superconducting magnet with the sample sandwich. .....
Magnetic ﬁelds produced by the magnet with I =2 4A .........
Press used for aligning holes and pressing the Cu electrodes .....
Mounting of sample ............................
Rn, vs. C11 ................................
Low temperature circuit .........................

Circuit for MR measurement .......................

The superconducting proximity effects in the Cu electrodes. .....

Normalized resistances of the Cu/K/Cu sandwiches and the pure K for

T<6K ..................................

Temperature dependent resistances of the Cu/K/Cu sandwiches. . . .

X

11

15

32

33

38

43

46

48

52

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

The relative resistances vs x/T for KCusoTi5oCu sandwiches with dif-

ferent CuTi ribbon thicknesses ......................

The relative resistances vs \/T for KCusoTiwCu sandwiches with dif-

ferent CuTi ribbon thickness .......................

The resistances vs \/T for the KCusoTiwCu-ll and KCu-7 sandwiches.

The relative resistances vs \/T for the KCusoTisoCu sandwiches
The relative resistances vs x/T for the KCusoTiwCu sandwiches
The resistances vs \/T for the KCuGoTi4oCu-5 and KCu-7 sandwiches

The contributions to the relative resistances due to weak localization

and electron-electron interaction .....................

The temperature dependences of the resistances at different magnetic

ﬁelds for sample KCusoTisoCu-16 .....................
The magnetoresistances vs 82 for the Cu/ K / Cu sandwiches. .....

The magnetoresistances vs B2 at different temperatures for the KCu-9

sandwich. .................................

The theoretical relative magnetoresistivities due to different scattering

mechanisms .................................
The scattering ﬁelds vs T and B for various scattering mechanisms. .

The relative magnetoresistance data. and ﬁts for the KCusoTisoCu-l6

sandwich. .................................

The modiﬁed magnetoresistance data and their ﬁts for the KCusoTisoCu-

16 sandwich .................................

The relative magnetoresistance data and ﬁts of the KCueoTiwCuJ

sandwich. .................................

66

67

71

74

77

78

81

82

84

85

88

89

92

93

4.19 The modiﬁed magnetoresistance data and their ﬁts for the KCusoTiwCu-
7 sandwich. ................................ 99

4.20 The background magnetoresistances due to K, Cu and the boundaries

in the KCuTiCu sandwiches ....................... 101

4.21 1/1', vs T for CusoTiso and CusoTim alloys ............... 103

xii

Chapter 1

Introduction

Disordered metals and alloys have the following unusual transport properties:
(1) The residual resistivity p is much higher than in the crystalline phase.

(2) The temperature coefﬁcients of the resistivity are very small and usually pos-

itive for p < l50uflcm and negative for p > 150p9cm.

(3) Magnetoresistance may be positive or negative, depending on chemical com-

position of a material.

The classical Boltzmann theory has been very successful in explaining the trans-
port properties of the crystalline metals. However, this theOry failed when it was
applied to disordered conductors. Obviously, the structural difference between the
two systems is a major reason. Why does the structure of the system strongly affect
its transport properties? In 1958, Anderson [And 58] pointed out that, in a strongly
disordered system, an electron wave function is localized and therefore the system
becomes an insulator. For a weakly disordered system, Abrahams et al. [Abr 79] ﬁrst
introduced the concept of quantum interference effects that become important if the
electron mean free path is of the order of inter-atomic distances. This interference
effect in which electron-electron interactions are ignored is called weak localization
(WL). In addition, quantum interference causes an enhanced electron-electron inter-

action (EEI) that also makes an anomalous contribution to the resistivity. There have

1

2

been a number of review articles [Alt 85, Fuk 85, Ber 84] which deal with WL and EEI
using the diagrammatic Green’s function method. Bergmann [Ber 82, Ber 83, Ber 87]
has presented excellent reviews with very clear physical pictures of these quantum in-

terference effects.

Since quantum interference changes the scattering probability of electrons, which
is proportional to the resistivity, it is natural to calculate and measure the resistance

of the system in order to study these quantum interference effects.

Two dimensional disordered systems have been under extensive study since the
quantum corrections to the resistivities are relatively large and thus easier to measure.
In contrast, much less work has been done in three dimensional systems since the
quantum corrections produce a temperature dependence of only Ap/ p ~ 10’5 for
a temperature change of 0.1K at T S 1K, which can be very difﬁcult to measure

accurately.

Among the three dimensional amorphous systems that have been studied, CuTi
alloy is one of the most extensively studied systems because of its stable structure and
properties. Most experiments were done at low temperatures where weak localization

and electron-electron interaction contributions are most signiﬁcant.

Three major groups have studied the CuTi alloys extensively. The English group
(Greig, Howson, Hickey et a1.) [Hic 86, Hic 87, How 86, How 88] studied a series of
CuxTi1_x ribbons at temperatures between about 0.1K and 300K and in magnetic
ﬁelds up to 30 Tesla. The relative precision of their resistance measurement was about
10‘5. Schulte et a1. (German group) [Sch 86, Sch 90] took their data at T = 1.5—60K
and for B < 40T with about the same relative precision. Lindqvist and Rapp[Lin 88]
is the only group (Swedish) that attempted to make measurements down to 20mK
with similar precision and B < 5T. However their thermometry was not good for

T < 0.3K because of poor heat conduction between their samples and refrigerator.

3

Although these three groups did obtain very useful data, further work is needed for

the following reasons:

As this thesis will show, much lower temperature and magnetic ﬁelds are required
in order to clearly disentangle the WL and EEI contributions to the resistivity. Rel-
ative precision of about 10’5 is not high enough for useful measurements at very
low temperatures and with small magnetic ﬁelds. As clearly observed by Lindqvist
and Rapp, use of an insulator as thermal contact agent between the samples and the

refrigerator leads to unreliable thermometry below about 0.3K.

To overcome these problems, we have developed a novel method for measuring

the resistance of the amorphous CusoTi5o and CumTim ribbons:

(1) By using the high precision current comparator with a superconducting quan-
tum interference device (SQUID) as a null detector, we could measure the resistance

to a relative precision of about 10’? — 10’8 for B < 0.2T.

(2) We measure the resistances of CuTi alloys by applying a current perpendicular
to the ribbons. We use potassium as a non-superconducting glue between the ribbons
and the refrigerator, so that excellent thermal contact is obtained at all temperatures.

The error in the temperature is much less than lmK at T m 15mK.

(3) With the above improvements, we have obtained very precise measurements
of the resistances of the CuTi alloys at very low temperatures (15mK< T <6K) and
small magnetic ﬁelds (0T< B <0.2T).

In this thesis, we are going to present our zero ﬁeld resistance and magnetoresis-
tance measurements, compare our results with those of other workers, and analyze
the data using appropriate theoretical models. Since our new measurements show
that both CusoTi5o and CumTiw are superconductors with transition temperatures
below 15mK and 5mK, respectively, we will also brieﬂy discuss the paraconductivity

due to superconducting ﬂuctuations when T is close to, but above, the transition

temperature (Tc).

Chapter 2

Quantum Interference Theories

In this chapter, we will describe the quantum interference effect (QIE) of electrons in
disordered three dimensional systems. A brief description of the classical Boltzmann

transport theory will also be given.

2.1 Classical Transport Theories

2.1.1 Definition of Resistivity

When an electrical ﬁeld is applied to a metal, a current proportional to the electrical

ﬁeld will ﬂow.
J = 0E . (2.1)

In general, the conductivity 0 is a tensor, but for a metal with cubic symmetry, it
reduces to a diagonal tensor in which all the elements along the diagonal are equal.
Because of this reduction, 0 can be represented as a scalar. The electrical resistivity

p, deﬁned as the inverse of the conductivity, is given by

= E . (2.2)

If we also apply a magnetic ﬁeld to a metal, the resistivity, given by the above formula,

one

p:

is called magnetoresistivity.

2.1.2 Electrical Resistivity

In Boltzmann transport theory, an electron is described as a quasiparticle with wave
vector k and energy E = h2k2/2m‘. Electrons are scattered by impurities and lattice
defects. Between collisions, the electrons are assumed to move in a classical trajectory.
For kpl >> 1, where I is the electron mean free path, the interferences between electron
waves can be neglected. In a simple metal with only 3 or p electrons, the conductivity

or resistivity due to electron scattering is given by

 

 

a = "e T , (2.3)
m.
1 m‘
p — 3 — nez'r ’ (2.4)

where m‘ is the electron effective mass, n is the electron number density, 6 is the
electron charge, and T is the relaxation time, which is inversely proportional to the
scattering probability of the electrons. If there are more than one scattering mech-
anisms such as electron-impurity, electron-phonon, and electron-electron scattering,
then according to Matthiessen’s rule [Bla 68], the total resistivity can be written as

a sum of the different scattering terms, that is

P = pa + pee + pep a (2-5)

where p0 is the temperature independent resistivity due to electron impurity or defect
scattering. pee and pep are due to electron electron and electron phonon scattering, re-
spectively. At low temperature, for a simple metal, pcc is proportional to T2 [Zim 60],
and pep has a stronger temperature dependence. There are two kinds of electron-
phonon scattering, which are called normal and Umklapp processes. Theoretically,
the resistivities are proportional to T5 for the normal process and exp(-O / T ) for the
Umklapp process [Zim 60]. The resistivity at low temperature for a typical simple

metal is shown in Figure 2.1.

 

 

 

 

 

 

Figure 2.1: Low temperature resistivity for a typical simple metal

2.1.3 Magnetoresistivity

When we apply a magnetic ﬁeld, because of the Lorentz forces on electrons, the
electrons will deﬂect and travel a longer path before arriving at a destination, so one
would naively expect the resistivity to always rise. But the theory is not that simple.
Details have been given in many texts [Zim 60, Bla 68]. For a simple metal with a
spherical Fermi surface and with the magnetic ﬁeld parallel to the current, the Lorentz
force, F=quB, is zero. So there is no magnetoresistivity. For the magnetic ﬁeld
perpendicular to the current, electrons will be diverted from their paths along the
specimen and accumulate along the specimen sides (Hall Effect). Thus the charges
created on the specimen sides will exert forces on the moving electrons and cancel
the Lorentz forces. Therefore the electrons will continue to travel down the specimen
along the original paths, and the resistivity is not affected. However for a metal
with two or more electron energy bands, since the electrons in these two bands have
different effective masses, their drift speeds in an electrical ﬁeld will be different. Thus
the Lorentz forces on the two carrier types differ and the electrical forces due to the
Hall electrical ﬁeld will not cancel the Lorentz forces exactly. Therefore a transverse
magnetoresistivity can exist [Bla 68]. Ziman showed that this magnetoresistivity is
proportional to B2 for a small magnetic ﬁeld. But for the parallel magnetic ﬁeld,
experiments show that the (longitudinal) magnetoresistances are of the same order
as the transverse magnetoresistances in many metals. Theorists have found that this
longitudinal magnetoresistivity is due to non-spherical Fermi surfaces. For K and Cu,

a 82 law is approximately obeyed at low magnetic ﬁelds [Pip 89].

2.2 Quantum Interference Effects

For the amorphous systems such as liquid metals, since kpl >> 1 (e.g. kpl ~ 100 for

liquid potassium), the Boltzmann assumption is still valid. With some modiﬁcations,

9

their transport properties have been explained by the Ziman liquid model [How 88]..
However for systems such as metallic glasses, the conventional theory is not adequate

due to the shortness of the electron mean free path.
The metallic glasses show some unusual properties:

(1) Resistivity is very high at all temperatures, typically on the order of 100uﬂcm
to 300pﬂcm.

(2) The temperature dependent resistivity is very small, typically on the order
of a few percent in a temperature range from 4.2K to 300K. When ,0 < l50chm,
the temperature coefﬁcients are usually positive. For p > 150uﬂcm, they are often

negative at low temperatures.

(3) The positive and negative magnetoresistances are observed depending on the

materials chosen. They are usually proportional to \/B at high magnetic ﬁelds.

The conventional Boltzmann theory failed to explain these properties. The basic
reason is that the Boltzmann assumption is invalid. Electrons after scattering do not
move in classical trajectories, but travel like waves. So quantum interference between

the scattered electrons must be considered.

2.2.1 Weak Localization

In this section, we consider only non interacting electrons. In a highly disordered
metal, an electron is scattered at almost every lattice site. Its mean free path is of the
order of an atomic distance which is not very large in comparison to its wave length.
Thus the quantum interference of scattering amplitudes for different scattering sites

become important.

Now let us consider an electron in a conductor, which can move along paths 1,
2, 3, and 4 from A to B as shown in the top part of Figure 2.2 [Alt 85]. The total

transferring probability w for the electron moving from A to B is the square of the

10

sum of all the amplitudes along all possible paths:

w: .214.“

where the ﬁrst summation is the classical probability, and the second one is due

= 2 p1,]? + 2 AA; , (2-6)

#1

 

to interference between the different paths. For paths 1, 2, and 3, because their
path lengths differ signiﬁcantly, the phases of the electron wave functions also differ
substantially along these paths. Therefore in summing over all the paths, the mean
value of interference terms vanishes because of its oscillating nature. These paths
are called incoherent paths. However there is a special kind of path, namely self-
intersecting path as shown path 4 in Figure 2.2. It is phase coherent. Because an
electron moves diffusively in the disordered metal, there is ﬁnite probability for the
electron to be scattered back to its original position such as in path 4. In order to
examine this speciﬁc kind of path, path 4 is enlarged and detailed in the bottom
of Figure 2.2 [Ber 83]. Assuming an electron with the wave vector k has a wave

‘(k‘r‘W‘), then an electron at position 0 with wave vector k0 is scattered

function 6
into two partial waves, which travel in clockwise (solid lines) and counterclockwise
(dashed lines) directions on the loop. Bergmann claims that there is ﬁnite probability

these two waves are scattered back to the position 0 with the wave vector —ko.

To examine the interference of the two partial waves after going around the loop,
we must ﬁnd the phases of the two back-scattered partial waves. Since the scattering
is elastic, for the wave traveling in clockwise direction with initial wave vector k0,
the phase change due to the direction change of the wave vector k at the ith center
is (k,+1 — k.) - r,-. The phase change due to the propagation between center i and

center (2' + 1) is k,+1 - (n+1 -— r,). The total phase change after sequence scattering

0—t1—>---—+8—+0is

A43 = (kl—k0)'r0+kl'(l‘1—r0)+(k2—k1)~1‘1+k2-(r2—l‘1)+°"+

(k8 _ k7) ° r7 + k8 ' (1‘0 _ 1‘7) 'f‘ (—k() — k8) ' ['0 = —2k() ' To , (2.7)

11

 

Figure 2.2: Electron scattering

12

where the ﬁnal wave vector of the ﬁrst partial wave is -ko. Similarly the other partial
wave propagating in the opposite direction also from kg to —ko suffers the same phase
change. These two partial waves have the scattering amplitude, where |A1| = IAgl
because of symmetry. Thus the total probability of ﬁnding the electron at position
0 is IA; + Ag]2 = 4|AI2, which is twice as much as the incoherent (classical) values.
So the electron tends to be weakly localized at its origin, and this phenomenon is
called weak localization. Since the phase difference between the two partial waves
of an electron can be changed by phonons, magnetic ﬁelds, and spin impurities,
etc, anything dephasing the partial waves will reduce the probability of the electron
returning to its origin. Because the scattering probability is directly related to the
resistivity (p ~ 1/ T =scattering probability), the quantum interference effects can be
detected by measuring resistance vs temperature, magnetic ﬁeld, and spin impurities

etc. All these will be discussed in Section 2.3 and 2.4.

2.2.2 Spin-Orbit Interaction

In Section 2.2.1, the spins of two partial electron waves were not considered. But
if there is an interaction with the electron spins such as the spin-orbit interaction
between the electron spin and its orbit, the weak localization correction to resistivity
could be changed dramatically or even reversed [Hik 80, Mac 81]. Bergmann [Ber 82]
gave a vivid picture of how the spin-orbit interaction changes the sign of the quantum
interference effect. Assume that an electron with spin up is scattered into two partial
waves, then in the absence of spin-orbit coupling, these two scattered waves are in
phase and interfere constructively and the probability of the electron returning to its
original position is twice as much as the classical value. When there is a spin-orbit
coupling, which depends on the angle between the electron spin and its momentum
or velocity, the spins of the two partial electron waves will be rotated. Since two

partial waves with parallel spins have opposite velocity, the spin-orbit interactions

13

on the two waves have different signs. Thus their spins are rotated in the opposite
direction. If we assign a matrix R for the spin rotation, when one partial wave reaches
its original position after going round a closed loop, its spin state Ix) will be rotated
to Rlx) while the other partial wave with the same initial spin state will reach a
spin state R’llx) after going round the same loop. The interference of the two ﬁnal
states at their origin position gives a factor (leflx) before the interference terms
in Equation 2.6. When the spin-orbit coupling is weak, this factor is close to one.
For the case of dominating spin-orbit coupling, the electron spin completely loses its
original orientation and its ﬁnal orientation is statistical. From quantum mechanics,

the 1/ 2 spin rotation matrix R, in three dimensions expressed in Euler angles, is

__ cos(0/2)e[‘(¢+¢)/2] isin(0/2)e[‘(¢’¢)/2] (2 8)
isin(9/2)el-i(¢—w)/2l cos(0/2)e["(¢+¢’)/’] '
Taking the square, we get
R2 _ cosf(0/2)e[‘(¢+"’)l — sin2(0/2) ésin(0)(e"¢ + 6"”) (2 9)
— ésin(0)(e'f + e““’) cosz(0/2)c[’i(¢+¢)l — sin2(9/2) '

If for the initial spin state I x)=[a,b] and la]2 + lb]2 = 1, then the expectation value of
R2 is

0 . -.-
(Xlelx) = cos2(-) Ia|2e[‘(¢+w]+|b|26[ own
2

— sin2(-g) + (i/2) sin(0)[ab'(ef¢ + emf”) + c.c.] . (2.10)

If there is no spin-orbit coupling, there are no spin ﬂips. So all the angles are zeros,
and the factor (xlﬂzlx) is 1. Thus total probability of returning remains twice as
much as the classical values. If the spin-orbit coupling is strong, ﬁnal spin states
are statistical. The statistical average of the factor (le2lx) is —1/2. Thus the
interference reduces the total probability of returning to the half of its classical value.
An important effect on the quantum interference effect due to the strong spin-orbit

coupling is that anything which dephases the scattering will increase the probability

14

of an electron returning to its origin, and therefore increase the resistivity. Some of
these dephasing mechanisms are electron-phonon interactions, spin scattering, and

magnetic ﬁelds, which will be described in Sections 2.3 and 2.4.

2.2.3 Electron-electron Interactions

The weak localization effects that we have so far considered involve only single elec-
trons and could take place even if electrons do not interact each other. For the case
of electron-electron interactions, the interference between two electrons, which may

have different energies, should be also considered.

How do electrons interact with'each other in the presence of the weak localiza-
tion? The answer is very complicated. Much work has been done in terms of the
diagrammatic Green function method [Fuk 85]. A comprehensive review paper was
published by Altshuler and Aronov [Alt 85]. In general, electrons interact in two
diﬂerent ways: called the diffusion channel and Cooper channel. In the diffusion
channel, the two interacting electrons have parallel momenta, while in the Cooper
channel, their momenta are antiparallel. If the interaction between two electrons in
the Cooper channel is attractive due to exchange of virtual phonons, the system will
cross over to the superconducting state. Even in the normal metallic state (T > Tc)
or in the case of net electron repulsion, this electron-electron interaction still results

in a nontrivial contribution to the resistivity.

In order to obtain a physical picture of the Coulomb interaction, Bergmann
[Ber 87] has given a highly simpliﬁed interpretation. Beginning with the diffusion
channel, Figure 2.3 shows how electrons interact in the presence of quantum interfer-
ence effects. An electron wave which propagates from C is scattered by scattering cen-
ters (impurities, defects, or phonon, etc) and travels a loop: C—3 1” -—> 2” —-* -- - —+C.
At C the electron interferes with itself and forms a charge pattern. This charge pat-

tern depends on the total phase shift that the electron suffered in going around the

 

Figure 2.3: Interactions of two diffusion-channel electrons through the hologram at C

16

closed path, and it is called ‘an electron hologram’ by Bergmann. A second electron,
starting from O, can go round the same path in same direction. During its trip, the
second electron suffers the same phase change as the ﬁrst electron from the scattering
by impurities, phonon etc, but it also interacts with the charge hologram formed by
the ﬁrst electron. Bergmann claims that there is a ﬁnite probability that the phase
change of the second electron on the closed loop can be precisely reversed by the
interaction with the hologram so as to restore its original phase. So the phase differ-
ence of the two partial waves of the second electron at O is zero in comparison to the
same electron without the electron-hologram interaction (which has nonzero phase
difference due to inelastic scattering, etc.). Therefore waves of the second electron
will interfere constructively at its original position O and enhance its backseattering

probability.

For the Cooper channel, in which the two electrons travel in the opposite directions

along the closed loop (Antiparallel momenta), the same conclusion is still valid.

Only electrons with energies within kBT of the Fermi energy are involved in an
interaction. Because of this possible energy difference between the two electrons, the
phase change of the second electron due to various scattering on the closed path will
not be completely compensated by the hologram formed by the ﬁrst electron on the
same path. So as temperature goes higher, the dephasing of the second electron will
grow. Therefore, the probability of ﬁnding the second electron at its original position
(or resistivity) will decrease with increasing temperature. Since the electron energy
difference is the order of kBT, the thermal coherence time and related diffusion length

are

h 15
~ — ~ — 2.11
T“ AE kBT ( )

(2.12)

17

and

Lth = \/DTth , (213)

where D is the diffusion constant. Only electrons that return to their starting point

within a coherence time can contribute to constructive interactions.

2.3 Zero Field Electrical Resistivity

The concepts given in Section 2.2 are correct for both two and three dimensional
disordered systems. Since our sample dimension is above 20pm, which is much larger
than thermal diffusion length Lu.(< 0.02pm for T >10mK), we can treat our sam-
ple as a three dimensional system for electron-electron interaction. We will show
in section 4.4 that our samples are also three dimensional for weak localization. In
this section, we will discuss the temperature dependence of the resistivity for three
dimensional disordered conductors due to various dephasing mechanisms at zero mag-

netic ﬁeld. The paraconductivities due the superconducting ﬂuctuations will also be

described.

2.3.1 Elastic Scattering and Dephasing

In disordered systems, electrons are scattered by the disordered lattice sites. At very
low temperature and zero magnetic ﬁeld, the scattering by these lattice sites gives
the constructive or destructive interferences depending on the strength of spin-orbit
coupling. Assuming no spin-orbit coupling, anything which dephases the scattering
will reduces the probability of an electron returning to its original position. Therefore
the resistivity will be lower. In order to estimate the effects of the dephasing on the
resistivity, we assign two time constants To and rd. To is the minimum time for a
scattering process and “rd is a maximum time for coherence. Usually To is called the

elastic scattering time and “rd is called the dephasing time. For an electron moving

18

diffusively in a disordered metal (assuming that the electron at position 1' = 0 is
scattered into two partial waves at t = 0), the probability of ﬁnding the electron at a
distance between r and r + dr at a time between t and t + dt is given by the classical

three dimensional diffusion Equation [Dug 87]:

p(r, t)drdt = exp(—r2/4Dt)drdt , (2.14)

1
(41:19:)3/2

where D is the diffusion constant. The probability of the electron returning to a small
ﬁxed volume around the origin (r = 0) at a time between t and t + dt is proportional

to

l
p(t‘,t)dt ~ mt“ . (2.15)

Since the two partial waves interfere constructively in the time interval [10,14], the
excess probability of the electron returning to the origin due to the interference (that
is, the correction to the classical resistivity) can be obtained by integrating Equa-

tion 2.15 from To to 3rd,

 

Ap n 1 1 1
—~ ——~ —— . .1
p /. (003/3 w: w: (2 6)

We see, ﬁrst of all, that when dephasing is so strong that rd and elastic scatter-
ing T0 are equal, no correction to the classical resistivity is required. If there is no
dephasing(‘rd‘1 = 0), the correction to the classical resistivity is positive and propor—
tional to 1/ \/T—o Between the two extremes, the correction decreases with decreasing
rd. Since 74 is usually temperature dependent, we can observe and study the quantum

interference effect by measuring the temperature dependence of the resistivity.

2.3.2 Inelastic and Spin Scattering

The electron phonon interaction can cause inelastic scattering. At a ﬁnite tempera-

ture, two partial waves of a scattered electron interact with phonons, so their energies

19

are changed. This energy difference (AE) introduces a phase difference of an order
AE t / h for the two partial waves scattered back to the origin. When this phase change

is the order of 1r, the dephasing time, which is called the inelastic scattering time, is

7rh

~ —A—E . (2.17)

Ti

Since the electron-phonon interaction energy (AE) increases with temperature, T,- usu-
ally decreases with temperature. A simple power law 75 = noT” has been suggested
by some authors [Alt 85, Ber 84], and experiments have provided some evidence for

such a power law [How 86, Sch 86].

The spin scattering is due to the magnetic impurities in the host metals. The
spins of the partial waves of a scattered electron interact with those of the impurity
atoms and change the relative phases of the two spin states. The dephasing time,
when phase difference is 1r, is called spin scattering time (7,), which usually decreases

with increasing the concentration of magnetic impurities.

2.3.3 Weak Localization with Spin Orbit Scattering

In Section 2.2.2, we learned that the spin-orbit coupling could reverse the sign of weak
localization correction to the resistivity. If we consider all the scattering mechanisms
such as elastic, inelastic, spin, and spin-orbit scattering, then we have for the dephas-
ing probability 1 / T ~ 1/T,o + 1/T,- + 1/T,. The detailed calculations were performed
using the Green function method by many theorists [Alt 85, Fuk 85, Ber 84]. The

correction to the resistivity is

 

 

Ap 62 1 l 1 1 1

— = —— —— —- 3 —— —— — , 2.18

P2 21r2th ( 473' + 47.: \/4Ti + Ts + Tao) ( )
where 7',- is inelastic relaxation time. T, and 7,0 are spin and spin-orbit scattering

lifetimes respectively. We see that when there is no spin-orbit coupling (7,31 ~ 0),

the resistivity decreases with decreasing T,‘ and 'r, (or increasing temperature and spin

20
impurity concentration). If the spin-orbit coupling is very strong (T;1 >> “nil, Tfl),

the resistivity increases with temperature and spin impurity concentration.

2.3.4 Electron-electron Interactions

Since the electron-electron interaction involves two electrons near the Fermi surface,
the energy difference of the electrons is the order of kBT. So the dephasing time
Tee ~ h/kBT. From Equation 2.16, its correction to the resistivity is

1 k T
2 _ ~ _ -B_.
A” \/—D¢¢. hD

In fact, the electron-electron interactions are very complicated and different in the

 

(2.19)

diffusion and Cooper channels. Equation 2.19 only gives the general feature of the
electron-electron interactions. The quantitative calculations have been done using

the Green’s function methods.

In the diffusion channel, electrons have parallel momenta, but they may have
parallel or antiparallel spins i.e the two electrons may be in either in singlet (spin
=0) or triplet (spin=l) states. Including this spin splitting, Altshuler et 01 [Alt 85]

calculated the Coulomb correction to the resistivity in the diffusion channel

Ap 1.382 4 3 ~ ] ICBT
_ = __ _ _ -5; —— , 2.20
,o2 4x/21r2h 3 2 ) VhD ( )

where E, is, in Thomas-Fermi approximation, given by

 

‘ _ Q l 3/2 2
F, _ 3F (1+2F) 1 41"], (2.21)
F : ln(l+x), X=(2kF)2 ,

X ks

where kp is the Fermi wave vector and k, the Thomas-Fermi screening length. Here
0 S F S 1 and 0 S F}, S 0.93. If the electron-phonon coupling is considered, the

effective mass of an electron is then enhanced [How 88] by

m' = (l + A..,,)me , (2.22)

21

where Ac, is called the electron-phonon mass enhancement constant. Howson claims
that B, should be replaced by (F, — A3,.) when the electron-phonon coupling is con-

sidered.

In the Cooper channel, the two electrons have antiparallel momenta. If the elec-
trons attract each other via virtual phonon exchange, they will form Cooper pairs
and the system will cross to a superconducting state. This interaction will enhance
the conductivity. For |ln(T/ T 6)] >> 1, the correction to the resistivity in the COOper
channel [Alt 85] is

 

(2.23)

p2 = -4\/2«2n1n(T/T.) hD

where Tc is the superconducting transition temperature.

_A_p 1.3.22 2 (nary/2

In the case of net electron repulsion, the system will not go to a superconducting
state, but the Cooper channel interaction still results in a nontrivial contribution to
the resistivity. For the non-superconducting case, TC is replaced by Tp, the Fermi

temperature, in Equation 2.23.

2.3.5 Superconducting Fluctuations

If the disordered metals or alloys are superconductors, the superconducting ﬂuctua-

tions will change temperature dependence of the resistivity especially near Tc[Sko 75].

For the temperature above Tc, the superconducting ﬂuctuations will contribute
excess conductivity, especially at T ~ Tc. In general there are two kinds of contribu-
tions from the superconducting ﬂuctuations: Aslamazov-Larkin (direct) and Maki-

Thompson (indirect).

Aslamazov-Larkin contribution was ﬁrstly derived by Aslamazov and Larkin in
1968 [A5168]. This excess conductivity is due to the direct acceleration of the
ﬂuctuation-induced superconducting pairs in an electric ﬁeld. Since the supercon-

ducting pairs decay very rapidly at T > Tc, this contribution is important only at T

22

close to Tc. In terms of the Ginzburg—Landau theory [Sko 75], this contribution to
the conductivity in three dimension for T very close to T c is

e2 (T — T.)-‘/'~’

A” = 32thL(0) Tc

(T)

 

(2.24)

AL

where {GL(T) is the coherence length of Cooper pairs at temperature T. In the dis-

ordered (“dirty”) superconductors [Sch 90], it reduces to

_ -1/2
can") = 0.85[:.(0)11‘/’ (Li—T) . (2.25)

At T = OK, we have

661(0) = 0-35(€o(0)’)1/2

0.181? ht);
= , I
0 85\/ 2 kBTc

 

 

 

hD
kBTc a

0.783

 

(2.26)

where 50(0) is the coherence length in the clean crystalline case, I is the mean free path,
and vp is the Fermi velocity. We see from Equation 2.24 and 2.26 that A0“ 0: Tc.

Thus for our samples with T6 < 15mK, AO’A will be intrinsically small.

L

In addition, for c = (T — TC)/Tc > 0.1, experiments [Toy 84, Job 78] show that
the Aslamazov-Larkin ﬂuctuation contribution to the conductivity decays much more
rapidly with increasing 6 than the Equation 2.24 indicates. Johnson et a1. [Job 78]
interpreted this behavior as being due to a breakdown of the slow-variation approx-
imation in the GinzburgLandau theory, and they introduced a short wavelength
cutoff QC ~ (”1(0) in the ﬂuctuation spectrum. With this assumption, the effective

Aslamazov-Larkin contribution 0’ for c > 0.1 is given by

0’ 2 2x, 5:1:C

GAL—3W (1+:1:;f)2 (l-l-xg)

 

 

+ 3tan'1(xc) , (2.27)

where a“ is given by Equation 2.24 and are ~ 1/\/Z for dirty superconductors. For
example, with T/Tc = 2, Equation 2.27 gives that 070“ 2 0.08. Thus if T is kept

sufﬁciently larger than Tc, the Aslamazov-Larkin term can be neglected.

23

The Maki-Thompson (MT) contribution is an indirect effect. The direct accelera-
tion of the ﬂuctuation-induced superconducting pairs leads to the Aslamazov-Larkin
contribution. These superconducting ﬂuctuations then decay into pairs of quasipar-
ticles which remain in a state of small total momentum even after scattering from an
impurity potential and continue to be accelerated as much as they were while they
were a superconducting ﬂuctuation. The quasiparticle life time, limited by the scat-
tering from impurities and decaying back into a superconducting ﬂuctuation, can be
much longer than that of the superconducting pairs. So the Maki-Thompson contri-
bution can persist to temperatures much higher than Tc. Maki [Mak 68a, Mak 68b]
ﬁrst calculated this contribution, but there was a divergence in two or lower dimen-
sions. To remove the divergence, Thompson [Tho 70] introduced a pair breaking

parameter a. For T ~ Tc, the Maki-Thompson contribution is

40162 T—Tc ””2
Aamm ’ 32h50b(0)( T. ) ’ (2°28)

 

where a is related to the inelastic scattering time [Pat 74, San 85] by

h 60
a — m — 2k3—TBJT) , (2.29)

and B,- = h/4eD'r,-, is called dephasing or scattering ﬁeld.

Some experiments [Joh 78, Toy 84] indicates that the Maki-Thompson contribu-
tion for T > Tc is absent due to a strong pair breaking effect. In our case, the pair
breaking parameter 0 ~ 0.03T’"l with p > 1.6. Thus for T > T. ~ IOmK, the

Maki-Thompson contribution is very small.

In summary, at zero magnetic ﬁeld and for ln(T/Tc) >> 1, where Aslamazov-
Larkin and Maki-Thompson ﬂuctuation contributions are negligible, the corrections to
resistivity due to the weak localization and the electron-electron interaction, assuming

that they are additive, are

ﬂ _ 1.3.22 4 3 - 2 kBT ”2
hD

 

— - — Ae - a
p2 (“/5th 3 + 2( p F )+ ln(T/Tc)

 

24
+——ez—— (/i+l 3\/-1—+i+-1— (230)
2N2h\/ID- 472' Ta 473' To Tao . '

2.4 Magnetoresistivity

2.4.1 Weak localization with magnetic ﬁelds

As we know from the previous discussions, anything which dephases the closed loop
scattering will change interference pattern of the two back-scattered waves. If we
apply a magnetic ﬁeld, the phase change on a wave function going around a closed
loop is

2685
h 9

 

26
A3 = 277 f A-dl = (2.31)

where h / 2c is one ﬂux quanta and S is the area of the closed loop. Since electrons
move diffusively in a disordered metal, we have S ~ Dt. Because the two partial waves
of an scattered electron travel in opposite direction, the phase changes for the two
partial waves are :tZeBDt/h respectively. Thus the phase difference between these
two waves is 4eBDt/h. When the phase difference is ~ 7r, the magnetic dephasing
time is given by

T h
B 4608 '

 

(2.32)

Including also the spin-orbit coupling, the weak localization correction to the magne-

toresistivity is
A B 1
_P(2 ) ~ ;, /— ~ :fé . (2.33)
P 7'3

—’ sign corresponds to weak or no spin—orbit coupling and ‘+’ sign to the

L

where

strong spin-orbit coupling (see Equation 2.18).

Another dephasing mechanism is called the Zeeman splitting effect. In the pres-

ence of the spin-orbit interaction, the spins of the two back-scattered partial waves

25

of an electron are no longer parallel. An external magnetic ﬁeld introduces an energy
difference between the two waves with different spin orientation, which then produces
a phase difference. If there is no spin—orbit coupling, there is no Zeeman splitting.
The general formula, including weak localization, spin-orbit coupling, and Zeeman

splitting, was given by Fukuyama [Fuk 81]:

(3).. = giﬁl- 451+) ff (:1) f:(-§E:)l}
+2;:,(/46_f:[.+1

where B is the magnetic ﬁeld, t = 7,0/(4T,), '7 = (gpgr,:B/2h)2, ti = t + 0.5(1 :l:
\/1- 7), B: = h/(4eDr,),B¢ = B,- + 4B,O,Bi = 4B,..ti, and

MW. )2 (n+g)W—(n+-;-+:>*”1-

n=0

 

 

(2.)34

 

           

The ﬁrst term (inside the curly brackets) is the weak localization contribution in the
presence of spin-orbit coupling and Zeeman effect. The second term (inside the square
bracket) is due to the Zeeman splitting in the presence of the spin-orbit coupling.
Please note that the above expression is independent of the magnetic ﬁeld orientation
with respect to the measuring current in three dimensions [Alt 85, Kaw 80a, Kaw 80b,

Kaw 81].

2.4.2 Electron-electron Interaction

In the presence of a magnetic ﬁeld, electrons interact with each other also through
both the diffusion and the Cooper channels as in zero ﬁeld, but the interactions act
differently in non-zero ﬁeld. First, the energies of the two interacting electrons are
split due to the magnetic ﬁeld—Zeeman effect. Second, the magnetic ﬁeld directly
introduces a dephasing for the two electron orbits. Note that “ two electron interac-
tion” means that the second electron interacts with the hologram formed by the ﬁrst

electron on the same loop, as shown in Figure 2.3.

26

For the diffusion channel, the momenta of the two interacting electrons are parallel.
In a magnetic ﬁeld, if their spins are also parallel, the magnetic ﬂux through the closed
loop that electrons traveled are same for both electrons. So there is no phase difference
between the two electrons. The magnetoresistivity is not affected by this part of the
interaction. If the spins of the two electrons are antiparallel, the energy difference of
the two electrons is about (1238). Thus the hologram of the spin up electron does
not perfectly compensate the phase shift of the spin down electron and vice versa.
This effect is also called Zeeman splitting effect. The correction to the resistivity due

to the Zeeman splitting is given by Lee et aI [Lee 82]

Ap _ ~ 62 (CRT (9/183)
(ﬁlm ’ E’wr. 207.93 kBT ’ (2'36)

where F}, is given in Equation 2.21, g is the Lande g-factor for electrons, p3 is the

 

 

Bohr magneton, and
g3(:r) =/0°°dw [(1:22 (8:: 1)] (x/w-l-x-l-y/[w—xI—Zx/S) (2.37)

In the Cooper channel, the momenta of the two interacting electrons are antipar-

 

allel; i.e., they travel around the same loop in opposite directions. The magnetic
ﬁeld introduces the phase difference of the order 4eBDt/h between the two electron
orbits, which dominates the dephasing. If the electron interaction is attractive via
virtual phonons, the system crosses over to the superconducting state. The magnetic
dephasing and the attractive interaction enhance the conductivity for T > Tc, and

the correction due to the Cooper channel [Alt 81] is

 

 

 

Ap e2 eB (2068
._ = _ I)
where
. ’ 1/2 00 t1/2 $1
$3“): (22:) /(i sinh_2—t—( )(1 — sinh(:ct)) dt ’ (2'39)

(1 is a dimensionless constant, which equals 1 for weak spin-orbit coupling and 0.25

for the strong coupling, 1/g(B,T) = —ln(T"'/Tc) with T’ =max(T,4DeB/k3) for

27

superconducting metals, and l/g(B,T) = l/Aep + ln(7Tp/1rT‘) for normal metals
[Bax 89]. In this last expression, Tp is the Fermi temperature, 7 is the Euler’s constant

(0.577), and Ac, is the electron phonon coupling constant.

The analytical approximations with 0.1% error for f3, o3, and 93 have been given

by Ousset and Baxter et a1. [Ous 85, Bax 89].

2.4.3 Superconducting Fluctuations

In the presence of a magnetic ﬁeld, the superconducting transition temperature is de-
pressed. Like impuritias and defects, the external ﬁeld is also a source for the decay of
the superconducting ﬂuctuations and the quasi-Cooper pairs. Therefore with increas-
ing magnetic ﬁelds, the paraconductivity due to the superconducting ﬂuctuations will
be reduced. As in zero magnetic ﬁeld, there are two kinds of superconducting ﬂuc-
tuations: Aslamazov-Larkin and Maki-Thompson ﬂuctuations. These ﬂuctuations

usually depend on the relative direction between current and magnetic ﬁeld.

Aslamazov-Larkin effect. This contribution is only important for T close to Tc.
Many authors [Usa 69, Ber 69] have calculated this contribution for thin ﬁlms and

bulk material. For magnetic ﬁeld parallel to the current, Usadel [Usa 69] derived

 

 

62 ”0 1/2 00 CB
(A0“)AL = ( ) . (2.40)
16h 8hk3T ":0 [ln(%) + 22g¥(2n +1)]3/2

The expression for the ﬁeld perpendicular to the current can be found in the paper
by Usadel [Usa 69]. Bergmann also calculated the same contribution, and he claimed
that both transverse and longitudinal magnetoresistivity for three dimensional super-

conductors are same to the ﬁrst order in small magnetic ﬁelds.

Maki~Thompson effect. As in the zero ﬁeld. this effect can persist to temperatures
much higher than Tc. For two dimensional systems, this contribution was ﬁrst calcu-

lated by Larkin [Lar 80]. He showed that the Maki-Thompson contribution has the

28

same magnetic ﬁeld dependence as weak localization, but with the opposite sign and
a coefﬁcient B(T) which diverges at Tc. Although a lot of theoretical work has been
done for the two-dimensional magnetoresistance, little work has been done for the
three dimensional case. In the limit of small magnetic ﬁeld, the relation that Larkin
obtained may be extended to three dimension [Alt 85, Bax 89]. From Bergmann’s
work on the Aslamazov-Larkin effect [Ber 69], we assume that the same relation is

applicable to the longitudinal magnetoresistance, which is

A4 _ .2 e E
(7)MT—5(T)2W2h hf3(B,-) , (2-41)

where B.- = h/4eD‘r,. f3 is the same function given in Equation 2.35. B(T) has been

 

derived by Santos et a1 [San 85]:

 

= T)”T EH- 1))mr(|m| —Zr"(2n+ 1) , (2.42)

m=—oo 71:0

where
_ —1 Ll _ .1. __ I 1 [ml 1 -1

F(|ml)—]- -g “((7 + 2) 44(2) 4 (2+ 2 45143731, . (2.43)

In the case of small applied ﬁelds and inelastic ﬁelds

kBT T

- — . 4
B,B<< <4Deln(T—C-) , (24)

5(T) reduces the function given by Larkin, which is tabulated in Table 2.1 and will
be used in the analysis for our magnetoresistances. From the table, we see that 6(T)

decays very rapidly for T 2 Tc and varies as ln’2(T/Tc) for T >> Tc.

Table 2.1: 5(T) vs T

 

 

T/TC 4.5x10"5 22000 148.4 28.0 7.38 3.49 2.72 1.65 1.22 1.11
1n"(§:) -0.1 0.1 0.2 0.3 0.5 0.8 1 2 5 10
6(T) 0.017 0.015 0.06 0.13 0.33 0.73 1.05 3.0 9.8 22

 

 

 

 

 

Chapter 3

Experimental Techniques

In this chapter, we will describe the major apparatus and experimental methods used
in this study. Details will be given about the sample sandwich, magnet, sample
preparation, thermometry, and measurement procedure. In addition, electrical and
thermal contacts, residual and boundary resistance, and heat ﬂow in the system will

be discussed.

3.1 Main Apparatus

3.1.1 The Dilution Refrigerator

Since we are interested in the resistivity of amorphous CuTi alloys below 6K, a locally-
built dilution refrigerator [Ime 74, Ime 75, Hei 83] was employed. It takes the refrig-
erator 12 hours to cool from room temperature to liquid nitrogen temperature (77K),
and 2 hours from 77K to 4.2K with about 18 liters of liquid helium. The lowest tem-
perature (about 13mK) can be reached after about 5 hours of continuous 3He/4He
mixture circulation. A detailed description of a dilution refrigerator is given by O.

Lounasmaa [Lou 74].

29

30

3.1.2 High Precision Resistance Bridge

For amorphous CuTi alloys at temperature below 6K, the relative change of the resis-
tivity, Ap/p, is about 10‘5 at 1K with AT=0.1K. Therefore a high precision current
comparator system with SQUID (Superconducting Quantum Interference Device) null
detector [Edm 80] was used. As you will see in Section 3.4.1, in the most interest-
ing temperature range (T<6K), this system can resolve the quantity (Ap/ p) to a

precision of better than 0.1ppm.

3.1.3 Glove Box and Vacuum System

As you will see in Section 3.2.2, we used potassium as a non superconducting glue for
making good electrical and thermal contact between CuTi ribbons and Cu electrodes.
Since potassium reacts strongly with oxygen and water, a commercial argon-ﬁlled
glove box, made by Vacuum Atmospheres Company, was used for making samples.
The inert atmosphere inside the glove box was maintained by a locally built argon
puriﬁcation system. Although we have not directly measured the contamination due
to water vapor and oxygen, we know that the water and oxygen concentration is less

than 0.4ppm since potassium stays shiny for a few hours inside the glove box [Yin 87].

3.2 Sample Preparation

In this section, we will describe the techniques for sample preparation and for precision
perpendicular resistance measurement on amorphous-metal ribbons with very reliable

thermometry down to 15mK.

3.2.1 Preparation of CuTi Alloy Ribbons

CuTi alloys ribbons were prepared at University of Leeds, UK by the melt spinning

method. High pure Copper(99.999%) and Titanium(99.98%), mixed in desired con-

31

centration, were melted and poured to a fast spinning copper wheel which was in an
argon atmosphere. The CuTi liquid mixture, cooled at a rate of 106K/s, formed a
amorphous alloy on the surface of the wheel. The amorphous structure of the alloy is
conﬁrmed by x-ray diffraction shown in Figure 3.1 (ACu = 1.5421). The alloys made
by this method are ribbon-like with a width of 3—5mm and thickness around 40pm.
Their resistivities are in the range of 180 to 220pﬂcm, depending on the relative

concentration of Cu and Ti.

3.2.2 Sample Sandwich

Our measurements take place at temperatures between 15mK and 6K. In order to cool
the sample to very low temperature (15mK), the heat generated in the system should
be as low as possible. The current through the sample from the high precision current
comparator is about 20mA. In order to use the high precision current comparator and
not to produce noticeable heat (<4nW) in the sample or its reference resistor, the
resistances of both sample and reference must be less than lOpﬂ. Since the resistivity
of the CuTi alloys is about 200p9cm, the resistance of a CuTi ribbon is about 0.29 if
measured parallel to the plane of the ribbons. The heat produced by this resistance
with 20mA current is about 10“ Watt and is high enough to warm the sample
to well above 0.5K. To reduce the heat load, either current or resistances must be
lowered. Lowering current will reduce the precision of our measurements. Therefore
the resistance of the sample should be lowered dramatically. We have developed a
novel method to reduce the resistance to about 10110 by measuring the resistance
perpendicular to CuTi ribbons. A special sample sandwich was designed and built
to achieve our goals. The sandwich is made of two cylindrically-shaped Oxygen Free
High Conductivity (OF HC) Cu electrodes placed on the opposite sides of a CuTi

ribbon. Figure 3.2 gives the reader a detailed view of the sample sandwich.

In order to passivate the magnetic impurities and thereby the Kondo effect, the

32

 

Counts

 

 

 

' -’ _._ , _- P _' ,
u

26 (degrees)

Figure 3.1: X-ray diffraction of amorphous CuTi ribbons

 

33

AgWires \ I

Top Cap ]] l3

 

 

 

CuElectrode

Voltage Lead

Insulating
Tape

Voltage Lead
Cu Electrode

 

 

Bottom Cap

Figure 3.2: Sample sandwich. Solid black regions on voltage probes and bottom cap
indicate presence of Pb(Sn) superconducting solder.

34

OFHC copper was annealed in oxygen atmosphere with a pressure 4.8x10"torr at.
950°C for four days. The RRR (Residual Resistivity Ratio = R(300K)/R(4.2K)) of
the annealed Cu was about 400 in comparison to 100 for unannealed Cu. The ratio of
diameter to length of each copper electrode is less than 1 / 3 in order to reduce current
convergence or divergence. The diameter of each voltage lead is much bigger than
its thickness, so that currents ﬂowing into voltage leads are negligible. The distance
between a voltage lead and the nearby surface of the Cu electrode is about 0.13mm.
The contribution from this layer of copper is less than 0.01% of the total resistance
of the Cu/ K/ CuTi/ K/ Cu sandwich between the voltage probes. To eliminate the
superconducting proximity effect in the Cu, the voltage leads were nickel plated before
the superconducting Pb(Sn) solder was put on as discussed later in Section 4.1. The
two outside ends of the Cu electrodes are attached to two Cu caps using potassium
as a glue. The top cap, which also served as a current lead, was spot welded to a
silver wire. The silver wire is used for making a connection to a silver mount at the
bottom of the mixing chamber, so that excellent thermal conduction between the
sandwich and the refrigerator is guaranteed. If required, the bottom cap, can be used
to mount a G heater so that the thermoelectric G ratio (see Section 3.4.1) can be
measured. The nylon insulating tape, glued on both sides of a CuTi ribbon, is used
for preventing electrical shorting between the voltage leads. Potassium is used as non

superconducting glue to make good electrical and thermal contacts at the interfaces.

The resistances of CuTi ribbons have been measured by many researchers. All of
them used the parallel method by applying current parallel to the plane of a CuTi
ribbon. The precision of Ap/ p in their measurements was about 10'5. One of the
disadvantages of their measurements is that they used varnish as an agent for non-
electrical heat conduction. In the varnish, the heat is carried by phonons. At very low
temperatures, the thermal conduction by phonons is poor, and their sample will not

cool well. In comparison to the parallel method, our method has two unique features:

35
1) A high precision, Ap/ p = 10’7 — 10‘8, has been achieved by using our high

precision resistance bridge.

2) The direct contacts between CuTi and the Cu electrodes using potassium as a
glue allow the thermometers, the CuTi sample, and the refrigerator to have excellent
thermal contact each other. At very low temperature, where thermal conduction by
phonons is poor, most of the heat is carried by electrons in a metal. Thus using a
non-superconducting metallic glue assures excellent thermal contacts at temperatures
below 0.25K. In principle, our method is applicable to any high resistivity metal or

alloy in a ribbon or ﬁlm form.

3.2.3 Electrical and Thermal Contact and Cu Sputtering

The contacts between CuTi ribbons and Cu electrodes are crucial to our measure-
ments. Very good electrical and thermal contacts have been achieved with the fol-

lowing two methods.

(1) Direct Cleaning: Both the Cu electrodes and CuTi ribbon were cleaned in air
with acetone and alcohol. Then they were put into the Ar ﬁlled glove box. After
staying in the glove box for about 2 hours, the surface oxide layers of CuTi ribbon
and Cu electrodes were removed with knives and ﬁles, and potassium was rubbed on

the cleaned CuTi surface immediately.

(2) Cu Sputtering: A better way to clean the surface of the CuTi ribbons is to use
an ion milling machine in a high vacuum. This was done in the follow way: The CuTi
ribbons were ﬁrst cleaned by ultrasonic waves in acetone, then they were mounted
on a plate inside the vacuum chamber of the sputtering machine. The chamber was
pumped to 10‘8 torr and the ribbons were actively cooled to —20°C. The surface oxide
layers of the CuTi ribbons were removed by bombarding the surface with an Ar ion

beam for less than 60 seconds. The ion beam current was 2—6 mA and acceleration

36

voltage was 50V. Then, the high purity copper was sputtered onto the cleaned surface
for about 4 minutes. The thickness of the Cu overlayer was about 3000A. The above
procedure was repeated for the other side of the CuTi ribbons. Finally, following
steps in (1), good contacts between the Cu coated CuTi ribbons and potassium could

be made.

One disadvantage of this method is the heating of the ribbons during the ion
milling and Cu sputtering. By mounting a thermocouple directly on a CuTi ribbon,
we observed that the temperature increase mainly due to the ion milling was S 100°C.
By doing x-ray diffraction on these ribbons (with Cu etched away), we saw the same
x-ray diffraction pattern as that of a bare CuTi ribbon (see Figure 3.1). Thus this

heating does not anneal and recrystallize the amorphous CuTi ribbon.

3.2.4 Magnet and Current Supply

Magnetoresistance measurement at low temperatures and low ﬁelds is one of the most
important measurements to make. In order to get stable magnetic ﬁelds below 6K,

we designed and built a small superconducting magnet.

The magnet shown in Figure 3.3 is made of copper. It is a hollow cylinder, 3cm
long by 1.0cm in diameter. About 1200 turns of single-ﬁlament N bTi superconducting
wire were wound on the Cu cylinder in eight layers. Between the superconducting-
wire layers, Apiezon-N—grease coated mylar was wrapped to prevent possible electrical
shorting among the wires and to conduct away heat produced in the wires. The
outermost layer was tied with threads and painted with a liquid insulating tape,
made by GC Electronics. The sample sandwich can be inserted from one end of the
magnet. Good thermal contact was achieved by pressing together the magnet and the
upper Cu cap of the sandwich with three screws. The position of the CuTi ribbon
between the Cu electrodes in the sample sandwich is very near the middle of the

magnet. The magnetic ﬁelds produced by this magnet were measured by a low-ﬁeld

37

 

 

 

  

 

 

 

 

 

O
O
O

 

   

O
O
u..u.¢.‘4g-.o-~o-‘aooo.ooom-...“

AconA—LLA

 

 

 

Figure 3.3: View of superconducting magnet with the sample sandwich.

38

 

 

 

 

2000 I I WI ] l I I I ] I F l I ] l I 7 F ]
’ I
.— +++++ _‘
_ OM -]
_ + q
1750 — 000+ °°¢° ‘—]
' °°+ 04 1
.. O o ..,;
A 00* 000 T]
U) 0 *o ‘1
U) -— o __)
:3 1500 .. 09 0° 41
CG " o O "‘3
O
m 1250 — ° . o i
- o 0 Calculation o ']
- ° + Measurement 0 -
.. 0 O —:
1000 :- 0 o ..
"' O O _.
- 0 j
750 l L I l l l l l l l l l l l l l 1 l l
—0.02 -0.01 i 0.00 0.01 0.02

X (m)

Figure 3.4: Magnetic fields produced by the magnet with I = 4A. X: 0 represents
the center of symmetric axis of the Cu cylinder in Figure 3.3.

39

magnetometer. By applying lOmA of current, we measured the magnetic ﬁelds by.
inserting the probe of the magnetometer into the magnet. The scaled values of the
measured magnetic ﬁeld on the symmetric axis of the magnet cylinder are shown in
Figure 3.4. The theoretical calculation is also shown in Figure 3.4, Good agreement
is obtained, especially near the center of the magnet. Magnetic ﬁeld (B) produced at

the position of the CuTi ribbon is about 1900 Gauss with 4A of current.

3.2.5 Making and Mounting Samples

Sample preparation and mounting in our our experiment are pretty complicated and
time consuming. It takes about 12 hours to make and mount a sample by hand. The

procedure for making a sample is as follows:

1) The Cu electrodes, Cu caps, CuTi ribbon, press, and all other tools were cleaned
in air with acetone and 200 proof alcohol. They were placed in a metal pan carpeted
with a clean sheet of aluminum foil. Circular holes, with a diameter of 2.7mm, were
cut in the centers of two pieces of square shaped insulating tape using a special drill.
If necessary, the diameters of the holes could be changed by using different drills.
With a specially designed press shown in Figure 3.5, the two pieces of the tape were
aligned so that the two holes in their centers were exactly on opposite sides of the
CuTi ribbon. Then they were stuck on each side of the CuTi ribbon, which also had
a square shape. The size (3.5mm) of CuTi ribbon square was chosen to be smaller
than that of insulating tape (6mm), so that the potassium on the both sides of the

CuTi ribbon could not be electrically shorting

2) All the cleaned items and taped CuTi ribbon were now ready to be put into
the Ar ﬁlled glove box. In order to outgas all these items, they were placed in the air
lock of the glove box, then ﬂushed with Ar, and pumped on for 2 hours. Before the
sandwich was made, we waited another two hours to allow the puriﬁcation system to

remove any contamination brought with these items.

40

Brass
Screws

     

Brass Rod

CuTi Ribbon
With Tape

Figure 3.5: Press used for aligning holes and pressing the Cu electrodes

41

3) Surface oxide layers on the CuTi ribbon were removed with ﬁles for the naked
CuTi ribbon and with knives for the Cu coated CuTi ribbon. The high purity potas-
sium (RRR=1500), extruded from a stainless steel die, was immediately put on the
cleaned surface. The same thing was done for the other side of the ribbon and for
the Cu electrodes. Then the potassium coated Cu electrodes and CuTi ribbon were
pressed together using the press. Next, the outside end of one electrode and the hole
in the top cap were cleaned with knives, coated with potassium, and pressed together
by hand. Because of the potassium inside the hole in the top cap, the electrode and
the top cap were in excellent thermal and electrical contact. A similar procedure was
used to mount the lower Cu cap onto the other Cu electrode. The total thickness
of the potassium layers between the Cu electrodes and the CuTi ribbon, which con-
tribute to the total resistance of the sandwich, is estimated to be less than 0.3mm.
The sandwich was then dipped into melted parafﬁn wax and pulled out. After repeat-
ing this a few times, a layer of the parafﬁn wax with thickness of l-2mm was formed
around the sandwich. The parafﬁn layer will keep the potassium layers from oxidizing
for a period of few hours while in the ambient atmosphere. The parafﬁn wax is a 1:4
mass-ratio mixture of standard parafﬁn wax and a special parafﬁn called n-Eicosane,
made by Humphrey Chemical Company. This mixture has a melting point of 40°C,
which is lower than that of potassium (65°C), so that potassium will not melt during
the dipping. After the parafﬁn layer cooled down to room temperature (about 40
minutes), the sample sandwich was ready to be taken out from the glove box and

tested.

4) The sample sandwich was tested with a stainless steel dip stick. A hollow
nylon cylinder, attached to the bottom of the dip stick, was used for mounting the
sandwich. The sample sandwich was placed into the hollow nylon cylinder, so that it
was protected from accidently being hit. The cylinder with the sandwich was slowly

dunked into liquid helium. The resistance of the sandwich at 4.2K was measured

42

using a current source and a nanovoltmeter. If the effective resistivity was about
the same size as that of CuTi, the sample was declared good and was ready to be
mounted on the dilution refrigerator. If the resistance was too small or too large
in comparison to that of the CuTi, it indicated that either the Cu electrodes were
electrically shorted by potassium or the contacts between CuTi, potassium, and Cu

electrodes were bad. In either case, the sandwich was rebuilt.

The tested sample sandwich was warmed up slowly from 4.2K in a pure helium gas
environment to prevent cracking of the parafﬁn. The magnet was attached to sand-
wich using three screws. Then the sample sandwich was mounted on the refrigerator

as shown in Figure 3.6.

Since our SQUID null detector is very sensitive to voltage noise produced by a
magnetic ﬂux change in the circuit, relative vibrations of the sample, magnet and a
superconducting shield on the inner vacuum can had to be minimized. In order to
reduce these vibrations, the silver wire (spot welded on the top cap of the sample
sandwich) was rigidly connected to the bottom mount of the dilution refrigerator. At
the bottom of the magnet, the three stainless steel wires were tied to a stainless steel
cylinder which was ﬁrmly connected to the refrigerator. Wires for the current and
voltage leads were tied and varnished down to minimize vibrations. The inner vacuum
can with p metal and superconducting shield was mounted to minimize the external
magnetic and electrical interference. This inner vacuum was rigidly attached to the
still of the dilution refrigerator, which in turn was rigidly attached to the mixing

chamber.

3.3 Thermometry

Since accurate temperature measurements are very critical in our experiment, several

thermometers were used in the measurements. The temperatures of sample and

43

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

—- Screws
Stainless Steel
Cylinder —'
Ag Mount for A8 Mount for
Sample Reference
Bran Screw
A8 Win Voltage and
Current Leads
Sample and
/i __Sta1n1eu Steel
_] J Wires

 

 

Figure 3.6: Mounting of sample

44

reference were measured by two germanium resistance thermometers, one carbon.
resistor, and two Cerrous Magnesium N itrate(CMN) susceptibility thermometers (see
Figure 3.8 and 3.6. The calibration of these thermometers was done by C. W. Lee and
V. Heinen et a1. [Hei 83]. The two germanium resistors were mounted in the holes of
the bottom piece of the sample mount underneath the mixing chamber using Apiezon
N grease for thermal contact. The susceptometers of the CMN thermometers were
mounted against and thermally insulated from the plastic wall of the mixing chamber.
Silver wires of l-mm-diameter provide thermal contact between the bottom piece of

the sample mount and both the CMN sensors and their susceptibility coils [Yin 87].

3.3.1 Sample Thermometers

Two germanium resistors called R5, R7, and a CMN thermometer called x0“ were used
to measure the sample temperature. For temperature above approximately 1.5K, R6
was used, which was a Lakeshore Cryotronics germanium resistor. For temperature
between approximately 50mK and 1.5K, R7, a Cryocal CR50 germanium resistor
was used. The calibrations for R6 and R7 were done by Heinen [Hei 83]. Below
approximately 50 mK, X01.) was used, which was a CMN thermometer [Hei 83, Yin 87].

For details, please see Heinen’s thesis.

Since the susceptibility vs temperature for CMN obeys Curie’s law, it follows a
straight line on a plot x vs l/T. Since X013, R7, and R6 are mounted in the same
silver mount, they have the same temperature. Therefore X01.) was compared with R7
and R6 at the temperature above 50mK and calibrated as follows: Xold and R7, or
x01.) and R6 were recorded at the same temperature. The temperature was calculated
from R, or R6. Then we plotted X014 vs 1 / T and ﬁtted Curie’s law to the data using
a least-square ﬁtting program. The temperature below 50mK can be obtained by

extrapolation of the ﬁt of Curie’s law. For more details, please see Appendix A.

45

3.3.2 Reference-Resistor Thermometers

As we discuss later, our high-precision resistance circuit requires a very stable refer-
ence resistor. A carbon resistor called R11 and a CMN thermometer called xnew, were
used for measuring reference-resistor temperature. Since the reference resistor has a
resistance that is slightly temperature dependent, we needed to measure its temper-
ature. At temperature above approximately 50mK, R11 was used, but its reciprocal,
the conductance called C11, was measured since we used an existing automatic con-

ductance bridge for R11.

For temperature below 50mK, an was used since the CMN has a much better
temperature response in this range. The calibration was done in exactly same way
as for x0“. A SQUID was employed to measure the susceptibilities of both x0“ and
Xnew Simultaneously. Details about the measurements can be found in V. Heinen and

S. Yin theses [Hei 83, Yin 87].

The slight temperature dependence of the reference resistance can be calibrated
as follows: Another resistor was installed as the new reference and was held at a
constant temperature slightly above 4.2K. A germanium thermometer (R5) was used
to measure its temperature. Our old reference served as the “sample” and was mea-
sured as a function of C11 for T S4.2K. The relation between R", and C11 is shown

in Figure 3.7. The data were ﬁtted using the empirical equation

A1
(A2 - Cu)”

 

R=&+ on

From the ﬁt (solid curve in Figure 3.7), R”, was corrected to an precision of AR/R S

0.6ppm. More details about the data and ﬁtting parameters can be found in Appendix
B.

7.2655

7.2650

7.2645

ref (“0)

01 7.2640

7.2635

7.2630

46

 

IIIIIIIII[IIII|IIFI'IIFI

 

 

h"lllllllllllllllllllllllllllll“

lllllllllllllllllllLllllLl

 

 

1.0 1.5 2.0 2.5 3.0 3.5

C11 (ma—1)

Figure 3.7: R", vs. C11

4.0

47

3.4 Measurements

This section is intended to explain how the zero-ﬁeld electrical resistance and mag-

netoresistances of the samples were measured.

3.4.1 Resistance Measurement (B = 0)

First, we measured the resistivities of CuTi ribbons at room and liquid He tempera-
tures. The measurement was done by passing a current I parallel to the ribbons and
measuring the voltage V across two points separated by a. distance a. The length a,
width b, and thickness t of the ribbon were measured with a micrometer (precision

0.0001 inch), giving for the resistivity of the CuTi ribbon

Rbt
p = — (3-2)

a

Once the sample was mounted on the refrigerator and cooled to 4.2K or lower,
the circuit shown in Figure 3.8 was used. This low temperature potentiometer circuit
consists of the SQUID null detector and the two resistors, R3 and Rn}. R, is the
sample resistor to be measured and R", is a dilute Cu(Ag) alloy resistor mentioned
in Section 3.3.2. The wires connecting these resistors are made of Niomax ON, a
multi-ﬁlament Nb-Ti superconducting wire with Cu-Ni cladding made by Imperial
Metal Industries. To minimize the noise introduced by stray magnetic ﬁelds, most of
the wires except those near the sample were shielded in superconducting lead-tubing.
The wires leading to the sample were all carefully tied or varnished down to reduce

magnetically induced currents and heating due to vibrations.

In this experiment, we did not measure a sample resistance directly, but instead
we measured its ratio with respect to a reference resistance in order to use our high
precision current comparator. The reference resistor is made from a dilute Cu(Ag)

alloy and has a 4.2K resistance of 7.26110. The resistance ratio C was measured by

48

Mixing Chamber

 

Figure 3.8: Low temperature circuit

49

using the current comparator together with the SQUID as the null detector mentioned
in Section 3.1.2. The current comparator generates two currents Im and I, with the
ratio C = I./I,,, being stable to better than 0.1ppm precision. When comparing
resistances, one current 1", passes through the master side resistor R... (either R, or
12",) and the other current 1. passes through the slave side resistor R, Since Im is
greater than 1., Rm must be less than R.. The master side current can be ramped
slowly to a. predetermined value. The slave side current is C 1,, where C is the switch
setting of current comparator. C is adjusted until the SQUID output signal indicates

a null condition at its input.

Any thermal electro-motive forces (EMFs) generated in the circuit can be elimi-
nated by the following procedure. Let V, be a stray voltage in the circuit due to a

thermal EMF. Then for the current going one direction, the SQUID measures
V(+) = LR, — ImR... + V. . (3.3)
When one reverses the current, one has
V(—) = —I,R, -+- ImRm + V, . (3.4)

When C is adjusted until V(+) = V(—), one gets

 

I,R, - ImRm + V, = I,R, — I,,.Rm + V, . (3.5)
Finally one obtains
I, Rm _
E — Rs — C , (3.6)

which is independent of V,. Once we obtain C, the resistance of the sample can be

calculated from the above equation, where C = Rr/Rn; or its reciprocal.

The temperature dependent resistance was measured in the following ways: For
temperatures above 1.5K, the sample temperature was regulated at a desired value by

regulating on R6 through the mixing chamber heater. When C11, which indicated the

50

reference temperature, reached a stable value, it was recorded; and C was obtained
by balancing the SQUID. For temperature below 1.5K, we cooled the sample directly
to the lowest possible temperature (zl5mK). Then we took the measurements while
warming up. For temperature below 50mK, xnew (reference temperature) was regu-
lated through the mixing chamber heater, and x0), was used to measure the sample
temperature. When x,“ was stable enough, xnw, X0“, and C were recorded. For
temperatures above 50mK, C 11 was used for the regulation, and R7 was used for mea-
suring the sample temperature. Following the same procedure, R7, C11, and C were
obtained. Once we know C 11, the resistance of the reference resistor can be corrected

using the ﬁt given in Figure 3.7.

Since the reference resistance is temperature dependent, this problem can be
solved by measuring the differential of the sample resistance with respect to its tem-
perature while keeping the reference temperature constant. A thermal link (R1,)
between the refrigerator and the silver mount was used to produced the temperature
difference easily (see Figure 3.8). The procedure is as follows: We keep the reference
temperature (Trey) constant by regulating it through C" for TZ50mK and xnw for
TS 50mK. Then we apply a steady current to the lower left heater (L1) in the sample
side and measure C at that temperature. Next we increase the power to the L1 heater
so the sample temperature rises to T + AT and C is adjusted to C + AC to balance
the SQUID. Here C is deﬁned as

Rx

C =
Rref

 

(3.7)

Because the reference temperature is kept at a constant temperature T”), we have

 

AR,
AC: . 3.8
R“; ( )
Hence
AC _ ARJr (3.9)

 

CAT — RIAT '

51
If AT is small enough, this leads to

AC _._,.

m = pdT . (3.10)

The thermoelectric ratio can be measured using the above circuit if we put a G

heater onto the bottom of the sample sandwich (see Figure 3.2). For details, please

see Yin’s thesis [Yin 87].
3.4.2 Magnetoresistance Measurement

The magnetoresistance (MR) measurements were performed in the same way as for

the zero—ﬁeld resistance. Figure 3.9 shows the circuit for applying magnetic ﬁelds.

Since the sample was mounted at the bottom of the refrigerator, two single-
ﬁlament NbTi superconducting wires connected the magnet via polyethylene tubing
to a superconducting persistent switch (SPS) located at the top of the outer vacuum
can. In order to prevent heat ﬂow from the SPS near 4K to the 1K pot and from 1K
pot to the sample, the Cu cladding on the superconducting wires connected between
the SPS and the sample was etched away except in one place near the 1K pot where
the electrical connection was made for applying current to the magnet from a external
current source. The SPS is made of a single ﬁlament superconducting wire wound on
a heater (called SPS heater) which has weak thermal connection to the 4K bath via
a piece of brass. To make a superconducting connection, the wires from the SPS and
the wires from the magnet were mechanically pressed onto two superconducting Nb

strips with six screws.

At a ﬁxed temperature, the procedure for a magnetoresistance measurement is
as follows: First we apply 1.5 Volts to the SPS heater, so that the superconducting
wires wound on the heater become normal. Then we slowly apply a current to the
superconducting magnet. Once the desired current is reached, we turn off the power

to SPS heater, and the SPS becomes “closed”. Next we slowly reduce the source

52

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Outer Vac.
Can Heater
Nb Sta-1P8 I TS ' ] ] Screws
[ To
1K Pot * ' External
r ‘l I Current
Magnet / Source
Wires _ l
Ag Sample L 1
Mounts ' L
Magnet Stainless
and Sample Steel Can

 

 

 

 

 

 

 

 

Figure 3.9: Circuit for MR measurement

53

current for the magnet to zero and turn off the power supply. The magnet and SPS
are all in a superconducting state so a very stable persistent current ﬂows through
magnet coil. Then temperatures and resistance ratio C can be obtained using the

procedures described in Section 3.4.1.

We made magnetoresistance measurements only at the following sample temper-

atures: 4.2K, 1K, 0.5K, 0.25K, 0.1K, 52mK, and 20mK.

The heat generated by eddy currents during energizing and de-energizing of the
magnet is signiﬁcant only at temperature below 50mK. In order not to heat the sample
too much (AT < lmK), the time rate of change of the current was set to less than

lOmA/s. A more detailed discussion of eddy currents will be given in Section 3.6.

3.5 Residual and Boundary Resistances

Both the Cu electrodes, the K glue, and various boundaries can contribute a back-
ground resistance to the total resistance of our CuTi sandwich that might mask the
actual resistance of the CuTi sample. In Section 4.2.1, we will show that these back-
ground resistances are not a problem and in most circumstances can be ignored or at

least corrected for.

3.6 Heat Generated in our Samples

A major heating problem during the measurements is caused by eddy currents during
energizing and de-energizing the magnet. When the current in the magnet changes,
currents are induced in Cu cylinder of the magnet and in the top cap of the sandwich,
which then produce Joule heating. At T < 50mK, this heating might signiﬁcantly
warm up the sample, and it might take a long time to cool back down. To limit

this cooling time to less than one minute, we restricted the temperature increase to

54
AT < lmK at T = 15mK. Let Q be the heat ﬂow through the thermal link RL, then'

- AT

where W is the thermal resistance of RL. According the Wiedemann-Franz law, we
have
HI.

= L—f , (3.12)

where L is Lorenz constant (2.45 x 10‘8). Since R; = 50pﬂ, Q is 8nW. For the
hollow Cu cylinder, the relationship between the change rate of B and the heat ﬂow

Q is given by

Q - —’ ZPQ W (313)
dt — 1rI(b" —a‘) ’ i
where I is the length of the Cu cylinder, 0,6 are the inner and outer radii of the

cylinder, and p is the resistivity of Cu for T < 4.2K. We obtain dB /dt S 5 (Gauss/s).

Since B is proportional to the magnet current I, we have dI/dt S 10 (mA/s).

Chapter 4

Experimental Results

In this chapter, we will present and discuss the measured electrical and magneto-
resistances of the amorphous CusoTiSO and CumTiw ribbons at temperatures between
15mK and 6.0K and in magnetic ﬁelds up to 0.2 Tesla. Possible systematic errors
in both zero ﬁeld and B 74 0 resistance measurements will be discussed and taken
account of. The theoretical models given in Chapter 2 will be ﬁtted to the data. The
superconducting proximity effects at low temperatures and electron phonon Umklapp

contributions to the resistivity at the temperature above 3K will also be discussed.

4.1 Sample Information

Since our samples are sandwich-like as shown in Figure 3.2, the total measured resis-
tance of a sandwich includes the contributions from the K layers, the Cu electrodes,
the CuTi ribbons, and their boundaries. The contributions from the K layers, Cu
electrodes, and the boundaries are called background contributions. Even if they
are very small in comparison to those of the CuTi ribbons in the sense of absolute
resistance, they may affect the temperature or magnetic ﬁeld dependences of the re-
sistances dramatically because the resistances of the CuTi ribbons only change by
less than about 0.2% between 4K and 15mK and by 0.02% between OT and 0.15T.

In order to measure the resistances of CuTi ribbons accurately, we have to quantify

55

56

these background contributions and correct for them if necessary. Therefore, we have
measured a series of CusoTiso and CueoTiw ribbons with different sample preparation
conditions. A few sandwiches without CuTi ribbons were also measured. Table 4.1
lists the preparation conditions for the sandwiches with CuTi ribbons. The samples,
which were originally about 30-40pm thick, were thinned to various thicknesses with
sandpaper. The thicknesses were measured with a micrometer with an uncertainty of
about 1pm. The contact areas are the areas of the holes in the two pieces of insulating
tape, which were used for the contacts between K and CuTi ribbons as described in
Section 3.2.2. The actual areas may differ by a few percent, but this does not matter
because only relative changes of resistances are important. The Ion Milling and Cu
Sputtering columns in Table 4.1 indicate how the surface of each CuTi sample was
prepared, as described in Section 3.2.3. Some of the samples used bare CuTi ribbons

and they are listed as “No” under these columns.

Since two Cu electrodes, which are used as both current and voltage leads in our
sandwich, were glued to both sides of the CuTi ribbons with K, the Cu electrodes
and K layers do contribute to the total resistances. In our measurements, several Cu
electrodes have been used. In order to study their contributions to both the zero ﬁeld
resistances and magnetoresistances, we made some Cu / K / Cu sandwiches by gluing

only the Cu electrodes together with K.

Table 4.2 shows the preparation conditions for these sandwiches. The contact
areas have the same meaning as in the Table 4.1. Samples KCu-3, 4, and 5 are
made of unannealed OFHC copper, which has the RRR about 100 and exhibits a
Kondo effect [Ken 64] below about 1K, as shown in Figure 4.2 and discussed later.
Thus there are some magnetic impurities in the OFHC copper. To reduce the Kondo
contributions, the OFHC copper was oxygen annealed at 950°C and 4.8 x 10“ torr
for four days before the Cu electrodes were machined. The RRR for the annealed

Cu was 375. The Cu electrodes used in samples KCu-7, 9, and 10 were made from

Table 4.1: CuTi Sample Preparation Conditions

57

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Samples Thickness Contact Area Ion Milling Cu Sputtering
(pm) (cm?) Vb,Va,t ‘ Vt,D,t b
KCusoTisoCu-l 43 0.080 No No
KCusoTisoCu-2 43 0.080 No No
KCll5oTl5oCll-7 36 0.080 NO NO
KCU50T150Cll-8 36 0.080 NO NO
KC050T150C1P9 36 0.080 NO NO
KCusoTisoCu-m 25 0.080 No No
KCD50T150C11- ll 36 0.080 N0 N0
KCu50T15oCu-12 38 0.058 Vb=600V Va=60V Vt=400V t=4005
t=10min Ib=6.0mA D=5000A
KCusoTisoCu-13 18 0.061 Vb=65OV Va=50V Vt=400V 4:240s
i=304 Ib=10.0mA D=3000A
KCusoTisoCu-ls 30 0.057 Vb=600V Va=50V Vt=400V 4:2404
t=805 Ib=4.0mA D=3000A
KCu5oT15oCu-16 30 0.057 Vb=600V Va=50V Vt=400V t=2408
t=805 Ib=4.0mA D=3000A
KCusoTiwCu-l 42 0.058 No No
KCusoTimCu-Z 36 0.059 Vb=600V Va=50V Vt=400V t=1805
t=4min Ib=6.0mA D=2500A
KCusoTi4oCu-3 16 0.059 Vb=600V Va=50V Vt=400V 4:1805
t=4min Ib=6.0mA D=2500A
KCueoTimCu-tt 20 0.059 Vb=700V Va=57V Vt=00V t=2606
t=4min Ib=5.0mA D=3500A
KCugoTimCu-S 27 0.059 Vb=700v Va=57V Vt=400V t=2608
t=4min Ib=5.0mA D=3500A
KCusoTi4oCu-6 18 0.062 Vb=650V Va=50V Vt=400V t=240s
t=308 Ib=10.0mA 0:3000A
KCueoTi4oCu-7 31 0.058 Vb=640V Va=50V Vt=400V t=2208
t=80s Ib=4.1mA D=3000A

 

‘) Vb: ion beam voltage, Ib: ion beam current, and t: time of the ion milling.
b) Vt: target voltage, t: time of Cu sputtering, and D: thickness of Cu overlayer.

 

58

Table 4.2: Cu/ K /Cu Sandwich Preparation Conditions

 

 

 

 

 

 

 

Samples Contact 02 RRR Ni used in
Area(cm2) Annealing Plating Sandwiches
KGB-3 0.080 NO 106 N0 KcusoTlsoCU-I,2,7
KCu-4 0.080 No 106 Yes KCu5oTi5oCu-8
KCu-5 0.080 No 106 Yes KCusoTisoCu-QJO
KCu-7 0.080 Yes 375 Yes KCusoTisoCu-ll,l2,l3
KCusoTiwCu—l,2,3,4,5,6
KCu-9 0.080 Yes 375 Yes KCusoTisoCu-l4,l5,16
KCUGOTI4oCU-7,8
KCu-10 ‘ 0.080 Yes 375 Yes

 

 

 

 

 

 

 

 

a) Same Cu electrodes as KCu-9, but with much thicker layer (1 — 2mm) of K for checking its
contribution to magnetoresistances.

the annealed Cu. No Kondo effects were observed in these sandwiches, as shown in

Figure 4.2.

In order to obtain high sensitivity in our measurements, the resistance of the volt-
age leads should be minimized. So superconducting Pb(Sn) solder was put around
the leads before the superconducting wires were soldered on (see Figure 3.2). Because
of the superconducting proximity effect [Mot 89], part of these leads may become su-
perconducting at very low temperature. Figure 4.1 shows the measured resistances of
sample KCu-3 between 40mK and 6K with 10mA and 50mA currents. The very rapid
drop of the resistances below 0.4K indicates that the KCu sandwich was becoming
superconducting, which is due to the solder. The rather strong current dependences
of the resistances in the same temperature range also indicate the presence of this
proximity effect. To prevent this effect, the edges of the voltage leads of the Cu
electrodes were plated with ferromagnetic nickel before solder was put on the leads.
Note that the nickel plated KCu-4 sample in Figure 4.1 shows no proximity effect.

The same is true for all the nickel plated sandwiches shown in Figure 4.2.

59

 

1.10IIIIIIIII IIIIIIIIIIIII

IIII

 

llll

 

 

c
1.05 -- 00 -—
: ¢ ’1 j
_. e gt 1

00 e 0
01.00—xmxsemoxx x x xxx 1‘
_ + _
1— O -I
c _
0.95 —- + OKCu-3 (I=10mA) —:
- +KCu—3 (I=50mA) -
- XKCu-4 (I=10mA 8c 50mA) -
0.90 —- + o —4
.. + ..
.. + o -
0.85 .- 1 Cl 1 l l l l L L l l l l l I l l l l l l l M-
0.0 0.5 1.0 1.5 2.0 2.

Figure 4.1: The superconducting proximity effects in the Cu electrodes.

60

4.2 Temperature Dependences of the Zero Field
Resistivities

In this section, we will present and discuss the temperature dependences of the resis-
tivities of CuTi ribbons at zero magnetic ﬁeld. The temperature dependences of the

resistivities at ﬁnite magnetic ﬁeld will also be brieﬂy described.

4.2.1 Background Resistances

As mentioned before, the background contributions to the resistance of a CuTi sand-
wich basically have two parts: Resistances of a Cu / K / Cu sandwich and of the bound-
aries between Cu and CuTi or K and CuTi. Although in this section we will analyse
these background resistance in detail, we state at the outset that their magnitudes and

temperature dependences are negligible in comparison to those of our CuTi ribbons

for T < 3K in zero ﬁeld.

As discussed in Chapter 2, the resistance of a crystalline metal is due to scatter-
ing of electrons from impurities, defects, other electrons, and phonons. Usually elec-
tron impurity scattering produces a temperature independent contribution. Electron-
electron scattering gives a T2 dependence, but it is usually very small in magnitude.
Above about 1K, usually the most temperature dependent part of the resistivity
comes from two electron-phonon scattering processes: normal and Umklapp. For

pure K at temperature below 5K, its resistivity has the form [Hae 83].
px =p0+ATeXp(_eK/T) s (41)

where the last term on the right hand side is due to the electron phonon Umklapp
process, OK is an effective Debye-like temperature (220.3K), and p, is the residual
resistivity due to the electron impurity or defect scattering. For T > 1K, the electron-

electron interaction T2 term is negligible in comparison to the Umklapp term.

61

For pure Cu, normal electron phonon scattering is signiﬁcant. Theoretically it is
proportional to T5 for T < Ocu = 320K [Zim 60, Ros 65], but experiments [Ste 81]

show a T4 dependence for 2K< T < 7K. That is, the resistivity of pure Cu obeys

pct. = pa + CT4 ° (42)

The ﬁrst part of the background contributions to the resistances of the CuTi sand-
wiches is determined from the Cu/ K / Cu sandwich data shown in Figure 4.2. The
data are taken between 15mK and 6K, and their ordinates are shifted by a constant
value for clarity. The dotted line represents the resistivity of a pure K sample which
has the same p, as in our sandwiches. The resistances of the Cu/ K / Cu sandwiches
show similar behaviour to that of the pure K. For T < 3K, these resistances are nearly
constant, as expected. The slight increases at very low temperature for KCu-4 and
KCu-5 are due to the previously-mentioned Kondo effect. The Kondo contribution
gives a resistance proportional to — ln(T) at low temperature. However, the sand-
wiches KCu-7 and KCu-10 show no resistance increase even at the lowest temperature
since the Cu electrodes used in these two sandwiches were made of oxygen-annealed
OFHC Cu. For T > 3K, the increases in resistance with increasing temperature are
due to electron phonon interactions in the K and Cu. Since these sandwiches were Ni
plated, there is no superconducting proximity effects even at 15mK. Finally, we will
show in Figure 4.6 that the temperature dependence of the Cu / K/ Cu sandwich resis-
tance is negligible in comparison to that of the CuTi samples for T < 3K. Therefore

our Cu-electrode system is reliable at least down to 15mK.

The residual resistance of a Cu / K / Cu sandwich is the order of 20 — 30nf2, which is
negligible compared to the typical 10119 resistance of the CuTi sample. The resistance
of a Cu/K/Cu sandwich has three sources: the Cu electrodes, the K layers, and their
boundaries. To estimate the three contributions for the sample KCu-7, let us ﬁrst

calculate the resistance due to the Cu electrodes. Since the total thickness of the

 

 

 

 

 

1.10 T I I I I I I I I I I I T I I I I I F I I T I I PT I I
L. ..
_ o KCu-4 (R,=o.oa p0) q. I
- D KCu-5 (Ra-0.03 an) ..
~ + KCu-‘7 (Ra-0.02 an) ..
.' q.
1.05 — e KCu—10 (no-0.11 an) _
o '- ' Pure K (Ra-1.00 an) a. ..
0: - _
\ ...' q. o d
D: '4. o
e e an or: 5 e ~ar ”1' o d
O
1.00 ooooooo o o 0 ° Do++ _.
D a a +
' D 0 D D c: c) D + _
+ +
.. + + _
4» +++++++ + 4+ + + +
o 95 l l l l I l l l l I l I l l l l l l l l L l l I l l L I;
' 0 1 2 3 4 5 6

T (K)

Figure 4.2: Normalized resistances of the Cu/ K/ Cu sandwiches and the pure K for
T < 6K. The data have been displaced along ordinate for clarity.

63

Cu electrodes that contributes to the resistance is about 0.03cm, the area is 008ch
and the resistivity of the Cu electrode at 4.2K is about 5nﬂcm, the total resistances
due to the Cu electrodes is the order of 2nQ, which is very small compared to the
total resistance of the sandwich (20nf2). The contribution from the K layer can be
calculated in the same way. The thickness of the K layers varies from sample to
sample and is not exactly known. It is estimated the thickness is of the order of
0.03cm. Since the resistivity of our K at 4.2K is the order of 4nﬂcm and its area is
same as that of Cu electrodes, the resistance from the K layers is about 1.5nﬂ. Thus
the total boundary resistance between Cu and K is estimated to be <17nﬂ depending

the quality of the contacts between Cu and K on the boundaries.

The most interesting resistances in our Cu / K / Cu measurements are the temper-
ature dependent resistances, which also have three sources: Cu, K, and their bound-
aries. To estimate these, we redraw the Figure 4.2 in the manner shown in Figure 4.3.
From Equation 4.1, we know that if K dominates the temperature dependent part
of the sandwich resistances, we should see linear dependences for ln(Ap/ T) vs 1 / T
in Figure 4.3. Indeed we see that all the data follow straight lines of negative slope
that are parallel to that of the pure K sample (dotted line). We also ﬁtted the data
to Equation 4.1 and 4.2 using a least-squares ﬁt, and we we got much better ﬁt with
Equation 4.1. Therefore we know that the Cu does not contribute signiﬁcantly to
the temperature dependent resistance. However, at this point we do not know if
the temperature dependence is dominated by the d = 0.03cm-thick K or a “K-like”
K/ Cu boundary resistance. To resolve this uncertainty, we scaled the pure-K data
(dotted line) in Figure 4.3 to conform to the case of a K layer with d = 0.03cm and
A = 0.08cm2 (dashed line). Since the dashed line agrees reasonably well with the
KCu-5 and KCu-7 data, we conclude for these samples that the K-layer dominates.
Note, however, that the KCu-4 sandwich exhibits a temperature dependence that is

about 2.5 times larger than for the K layer. Thus this sample appears to have a signif-

10"1

H
‘1
N

(R—Rog/T (MD/K)

H
on
1h

10'5

Figure 4.3:

64

 

 

 

 

E' I F I I I I T T 1 l I 1 1 r 3
E 0 KCu—4 (Ro=o,03 H0) 3
' U KCu-5 (Ro=0.03 p0) -
' + KCu-7 (R,=0.02 an) "
.— 5 KCu-10 (R,=o.11 m)__
E -. ' Pure K (Ro=1.00 11.0) E
: - Pure K (d=O.3mm) ‘
L 3 -
$ —t
a:

:- i‘ __=
E 'i‘ E
I 5, j
T *I' ..
o ..
:— \+~+ 00 -§
: 55. ,. 3
- D\ 0 *1. _,
P \ .-

J l I ' 1 1 \L I_O.L I l l 1 1 1 114

0.2 4 0.5

1/1?(1/1<)0'

Temperature dependent resistances of the Cu/ K / Cu sandwiches.

65

icant K / Cu boundary contribution, perhaps due to a severely reduced effective area
of contact between the K and Cu. Hence, there remains some uncertainty about the
relative importance of the “bulk” and “ boundary” contribution for K. Fortunately,
it is really of no consequence because, as we show later in Figure 4.6 and 4.9, the
temperature dependence of the Cu/K/Cu-sandwich total resistance is negligible in

comparison to that of our CuTi ribbons for T S 3K.

However, at this point, we do not know if the temperature dependence of the re-
sistance between K and CuTi or between Cu and CuTi is also negligible. To estimate
this resistance, we measured the resistances of the CuTi sandwiches with different
CuTi ribbon thicknesses. Because the sandwiches with different CuTi ribbon thick-
nesses have the same number and kinds of boundaries, the resistances due to these
boundaries should be very similar. If these boundary resistances have large enough
temperature dependences, the temperature dependence of total relative resistance of

the sandwich will depend upon the thickness of the CuTi ribbon.

Figure 4.4 and 4.5 show that temperature dependences of relative resistances for

the KCusoTisoCu and KCuGoTiwCu sandwiches with various CuTi ribbon thicknesses.

For the KCusoTisoCu sandwiches shown in the Figure 4.4, with T < 3K, the
behaviors of the data for the sandwiches KCusoTisoCu-12 and 13, which have CuTi
ribbon thicknesses of about 38pm and 18pm, respectively, are almost the same. With
the same CuTi ribbon thickness, the Cu coated sandwich KCusoTisoCu-l2 shows
the same relative resistance behavior as the non Cu coated sample KCu5oTi50Cu-11.
Although the thicknesses of K layers in these sandwiches may differ, from Figure 4.2
we know that the K layers contribute only very small constant resistances for T < 3K
and thus will not affect the behaviors of the data for the CuTi sandwiches. Since we
plot AR/RO vs \/T, the absolute resistances due to the boundaries must also be very
small in comparison to that of CuTi ribbons, otherwise the shapes of the curves in

the Figure 4.5 will be changed.

66

 

 

KCUsoTisocu -
1 001 - # B. d=36pm, no Cu
' °#11, d=36um, no Cu '—
0#12, d=38pm, Cu .
A +#13, d=18um, Cu 1 _
be: I
30. Q) / ‘
F". 4. 000 20 I ..
02'. 1.000— + o °o, / _
\ _ +4259) 0. / A
m ++ O \O‘ /
+ 0 "
.. + 0 o _
+++ O O o 0
1- o -
++ oo M000
r ++ o
+

p

0.999 —— *m
1%

 

 

lllllilllLJllllllllllILl

0.0 0.5 1.0 1.5 2.0 2.5

 

Figure 4.4: The relative resistances vs \/T for KCusoTisoCu sandwiches with different
CuTi ribbon thicknesses. For the clarity of presentation, the ordinates have been
displaced.

67

 

 

 

 

 

1.0015
’ KCusoTiwCu —
1.0010 :- . ”#4» d=20ﬂm .L l
- +#5. d=27pm _
A IX x446. d=1B/.1,m :
M _ 1:1 °#7. d=31ttm -
LO 1.0005 _
.—i Z :
v
D: : 1
\10000
D: ' _— _
- WM;
— M
0.9995 — = a
09990 1 1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1 . 1 I 1 1 1 1
0.0 0.5 1.0 1.5 2.0 2.5

Figure 4.5: The relative resistances vs \/T for KCusoTiwCu sandwiches with different
CuTi ribbon thickness. The ordinates have been displaced for clarity

68

However at the temperature near 20mK, the behavior of the data are a little
different for sandwiches with different CuTi ribbon thicknesses. We believe this is
related to the onset of superconductivity of the CusoTiso ribbons. It is found that the
sample with thinner CuTi ribbon shows a lower Tc than that of the thicker sample.

The origin of this TC dependence upon thickness is not understood.

Therefore we can conclude that the boundary resistances do not give any appre-
ciable temperature dependent contributions at T < 3K for the samples with different
preparation conditions, and that they are very small in comparison to that of CuTi

ribbons.

For T > 3K, the temperature dependence of the resistances is mostly due to
the electron phonon interaction in K. Its magnitude will depend on the thickness
of K, as expected, but it also seems to depend on the quality of the contacts at
K/Cu or K/CuTi interfaces. For samples KCusoTisoCu-ll, 12, and 13, the data
show very similar temperature dependences over the entire temperature region, for
which we say that the K-contact is “good”. However for sample KCusoTiso — 8
(dashed curve in Figure 4.4), we say that the high temperature dependence of the
resistance for T > 2.9K is related to having “poor” contacts between K and CuTi
since this kind of problem rarely happens for the Cu-coated CuTi samples. As you
will see later, whenever we have this poor contact problem at zero magnetic ﬁeld, the

magnetoresistivity also shows strong background contributions.

For the KCueoTiwCu sandwiches, which are all Cu coated, the relative resistances
are shown in Figure 4.5. We see that the temperature dependence of the resistance
for the sandwiches with different CuTi ribbon thicknesses are almost the same over
the entire temperature range. Since no onset superconductivity is seen at the lowest
temperatures, we conclude that CumTim alloy has a much lower Tc. Small positive
temperature dependences at higher temperatures are due to the electron phonon

interactions in K. Poor contacts rarely happen for these sandwiches.

 

69

4.2.2 Fitting Formula and Parameters

At zero magnetic ﬁeld, the corrections to the resistivities of the amorphous CuTi
alloys due to the weak localization and electron-electron interaction were given in
Equation (2.30). Let us assume that our samples contain no spin impurities, then

1 /1', ~ 0. If we also assume 1.- = 1,-OT ", then the corrections to the resistivity for

ln(T/Tc) >> 1 are
A 1.3 2 4 3 ~ 2 k T "2
__)O = " 6 " + “(Aer ’ F0) + B
p2 4\/§1r?h 3 2 ln(T/Tc) IiD

,_;3_ (T; 3/1‘1,_1_ (.3)
2172II\/5 4710 4710 Too , .

where the electron phonon coupling constant A,,, is related to superconducting transi—

 

 

tion temperature and Debye temperature by MacMillan’s expression [Mac 68, Sam 82]

1 _ 1.04 + ,1: hue/1.4573.)
‘P "' (1 - 0.62p')ln(O/1.45Tc)—1.04 ’

 

(4.4)

where p' is the effective Coulomb coupling constant, which is usually set to p“ = 0.13;
and O is the Debye temperature, which is about 300K for both CusoTiso and CumTim
alloys [Miz 83]. Since A,,, is insensitive to Tc, we estimate that A,,, = 0.26 — 0.29 for
Cu5oTi5o with TC = 4 — 15mK, and Ac, — 0.25 for CusoTiw with T, = 0.5 — 4mK. The

diffusion constant D can be calculated from the Einstein relation

1
D - m , ‘45)

where p is the resistivity of the CuTi alloy and N,(Ep) is the phonon—enhanced
density of the states. From speciﬁc heat measurement [M00 84, How 86], the values
of the enhanced densities of states (N,(Ep)) are 1.8 and 1.6 states/(eV atom) for
CusoTim and CumTiw alloys, respectively. The residual resistivities for the CusoTi50
and CueoTiw alloys were measured at 4.2K by the typical four probe method with

current parallel to the ribbon, and they are about 214p!) and 193710, respectively.

70

So the diffusion constants D are found to be 0.24cm2/s and 0.28cm2/s, respectively.
Therefore, in all of the following analysis, Aw, p, and D are ﬁxed at the values shown

in the Table 4.3. The ﬁtting parameters are Fa, Tc, no, 7,0, and p.

Table 4.3: The Fixed Parameters

 

 

sample p(4.2K) D A6,, GD
(#9) (ch/s) (K)
CusoTiw 193 0.28 0.25 326

 

 

 

 

 

 

 

At the temperature close to Tc, we have to consider the paraconductivity due to
Aslamazov-Larkin and Maki-Thompson superconducting ﬂuctuations as discussed in
Section 2.3.5. Due to the previously-mentioned short wavelength cutoff and strong
pair breaking effect, both the Aslamazov-Larkin and the Maki-Thompson contri-
butions to the resistivity are less than about lppm in comparison to the residual
resistivity for T/Tc > 5. To avoid these superconducting contributions, we have con-
servatively chosen to ﬁt our data by Equation 4.3 for T / Tc > 5. Please also note that

the Equation 4.3 is only strictly accurate for ln(T/Tc) >> 1.

4.2.3 p(T) for CU50Ti50 and Cll6oTi40 Alloys

In this study, the resistances for different CuTi sandwiches were measured at the
temperature between 15mK and 6K. Figure 4.6 shows the resistance of the sandwich
KCu5oTi50Cu-ll. The lowest temperature for this sample is about 15mK. This is the
ﬁrst time that any measurements have been made on for CusoTiso alloys to these low

temperatures.

First, our data show that the resistance decreases with increasing temperature,
which is generally true for the amorphous metals at low temperature due to weak

localization and electron-electron interaction. The interaction term, which is propor-

 

11.02

’- KCUsoTisocu- 1 1 " 1
11.01 — — ‘- 11
A P _
c: _ . r
3. 0

V - u-l
D: 11.00 '— —

- 0 4

0
I- o q

'- c-

10.99 —- KCU-7 -4
Sggfggwwm 71
0.0 0.5 12.0 1.5 2.0 2.5
1
T / (Kl/2)

 

 

 

Figure 4.6: The resistances vs x/T for the KCusoTisoCu-ll and KCu-7 sandwiches.

72

tional to -\/T, dominates the temperature dependence of the resistance at the lowest

temperatures.

Secondly, for T~22mK, the resistance of the sandwich starts falling, which indi-
cates that the sample is becoming superconducting. The drop in the resistivity at
17mK is only about 2%, so the transition temperature must be lower than 17mK.
The downturn, which starts at about 22mK, is due primarily to the superconduct-
ing ﬂuctuations although the attractive electron-electron interaction in the Cooper

channel also contributes a signiﬁcant negative curvature as T -> Tc.

Thirdly, the data have a positive curvature for T > 0.2K. Since the electron-
electron interaction contribution is proportional to -\/T and the weak localization
term goes as +T" with n > 0.8 for T < 10K [Lin 88], the positive curvature of the
data vs \/T indicates that weak localization makes a non trivial contribution. For
T > 3K, the background contribution due to the electron phonon interaction in K
also contribute to the positive curvature. For a comparison, the bottom data show
the typical resistance of a Cu / K / Cu sandwich. Its magnitude is very small compared

to that of the CuTi sample and its temperature dependence is negligible for T < 3K.

Table 4.4: The Fitting Parameters for KCusoTisoCu-ll

 

 

P Tia Tao T Fa X2

(psKP) (ps) mm
1.4 4.3 :1: 0.3 0.01 :t 0.0 7.9 :t 0.4 —0.64 :1: 0.03 72
1.6 10.6 :t 6.3 0.06 :1: 0.08 7.0 :1: 0.9 —0.27 :1: 0.19 66
1.8 21.1 :t 1.0 0.18 :1: 0.03 5.9 :1: 0.4 —0.08 :1: 0.01 74
1.9 28.1 :1; 1.4 0.27 i 0.04 5.3 :t 0.4 —0.02 :1: 0.01 85
2.0 36.5 :t 1.8 0.39 :t 0.06 4.6 :t 0.5 0.03 :t 0.01 100
2.1 46.5 :1: 2.5 0.54 :1: 0.09 3.9 i 0.6 0.08 :1: 0.01 122
2.2 58.2 :1: 3.3 0.72 :t 0.13 3.2 d: 0.7 0.11 i 0.01 150
2.3 71.7 :1: 4.2 0.92 i 0.17 2.3 :1: 0.9 0.14 :1: 0.0 185
2.5 100 :1: 3.9 1.40 :t 0.20 0.5 :t 2.5 0.16 :1: 0.06 284

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

73

The solid curve in the ﬁgure is the best ﬁt of the Equation 4.3 to the data. The

ﬁt was done only between 52mK and 2.7K in order to avoid the electron-phonon
contributions at the high temperatures and superconducting ﬂuctuations at the low
temperatures. In the ﬁt, A3,,D,p are ﬁxed as in Table 4.3, and p is set at some
discrete values such as 1.4, 1.6, 1.8 - - - etc. The ﬁts are done for each p, and the best
ﬁtting parameters are obtained as shown in Table 4.4. However the ﬁts for p < 1.9
are not good because F, < 0 (the physical meaningful values: 0 < F, < 0.93).
If we chose the ﬁts with xz/xfm-n < 2 as acceptable ﬁts, the best parameters are:
1.9 < p < 2.3, 28psK” < no <72psK’, 0.3ps< 1,0 <0.9ps, 2.3mK< Tc <5.3mK,
and 0 < F, < 0.14. These parameters systematically change as p varies from 1.9 to
2.3. The best-ﬁt curve is shown in Figure 4.6, and it ﬁts the data very well and is
unaffected for 1.9 < p < 2.3. Note that the X2 of these ﬁts is based on our estimates

of the random uncertainties for our resistance measurements.

To check in part the reproducibility of our samples, we studied a number of
CusoTiso and CusoTim ribbons, which were made under conditions and with thick-
nesses as listed in Table 4.1. The temperature dependences of relative resistances for
the KCusoTisoCu sandwiches are shown in Figure 4.7. The ordinates of each data set
have been displaced by a constant for clarity. For the earlier samples KCusoTisoCu-7,
8 and 9, the data were taken between 40mK and 5K. For samples KCusoTisoCu-ll

to 16, the measurements were carried out from 15mK to 6K.

At temperatures between .1K and 3K, the data for all the sandwiches agree with
each other pretty well. At temperatures below about 30mK, the differences in re-
sistivities among the sandwiches are probably due to variations of TC. It is found
that the Tc is lower for the sandwich KCu50T150Cu-13, whose CuTi ribbon thickness
is just about half of the other samples. Note that the resistance of KCusoTisoCu-l6
increases somewhat faster than the other samples with decreasing temperature and

has a lower Tc. We believe the data for this sample were not taken at zero magnetic

 

 

 

 

 

1— _,
1.002 "— KCUsoTisocu _l
L . 0 :
A - d
324 ~ 0 ..
LO. 1.001 -— o A
H _ _
v 1. + o
E 0 ° 7
m - -
h- + q
1 000 —- .x * + 1 _
‘ o KCum'I‘iwCu-B . s 2 xx; t
- +KCu50'I‘iwCu-11 “ , “an x
- x KCusoTiwCu-IZ ° 0 0°
.. " KCumTiwCu-IS
.. o KCuw'I‘iwCu-lﬁ 000%”
0999 1X1 1 1 l 1 J 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1n 1
0.0 0.5 1.0 1.5 2.0 2.5

Figure 4.7: The relative resistances vs \/T for the KCusoTisoCu sandwiches. The
ordinates have been displaced for clarity.

 

75
ﬁeld. Further details will be presented at the end of this section. The differences in
behaviors of the resistances for the T > 3K are due to the electron-phonon interaction
in K. For the poor-contact sample KCu5oTi5oCu-8, the resistance increases rapidly

with increasing temperature.

The solid curves are the best ﬁts using the same method as in KCusoTisoCu-ll

for 52mK< T < 2.7K. The best-ﬁt parameters are listed in Table 4.5.

Table 4.5: The Fitting Parameters for the KCusoTisoCu and KCueoTiwCu Sand-

wiches

 

 

 

Sample p T10 Tao Tc Fa
(1)st) (PS) (mK)

 

KCuwTisoCu-S 2 1.6 — 2.5 40 — 172 0 — 3.1 6.8 — 5.7 0.14 — 0.35
KCusoTisoCu-ll 1.9 - 2.3 28 — 72 0.3 - 0.9 5.3 — 2.3 0.0 — 0.14
KCusoTisoCu-12 b 2.0 —- 3.2 60 - 441 0.28 — 3.8 8.7 — 4.1 0.12 — 0.32
KCusoTisoCu-l3 1.6 — 2.5 24 — 187 0.03 — 1.4 4.2 - 0.3 0.0 - 0.28
KCU50TI50CU-l6 2.3 50 0.57 0.3 0.0
KCueoTiwCu-Z b 2.2 — 3.1 61 — 160 0.08 - 1.2 3.3 — 2.3 0.27 — 0.39
KCuwTiwCu-4 1.8 — 2.4 29 — 77 0.0 — 0.34 2.0 - 0.3 0.15 — 0.31
KCueoTiwCu-S 2.0 — 2.5 41 — 89 0.06 - 0.47 2.1 — 0.9 0.26 - 0.37
KCueoTimCu-G 1.7 — 2.2 21 — 51 0 - 0.20 1.8 - 0.5 0.110.28
KCumTimCu-7 1.7 — 2.6 22 — 105 0 — 0.63 2.8 .— 0.2 0.14 — 0.39
Cu65Ti35 2.0 — 2.5 0.08 — 0.25 0.0 0.0 — 0.32

from Lindqvist

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a) Temperature ﬁtting range: 50mK< T <1.5K.
b) x2 is large and insensitive to p.

From the Table 4.5, we see that the parameters for all the KCusoTisoCu sandwiches
are in the similar ranges even if the samples were made under different conditions. The
best ﬁts are obtained for p between 1.7 and 2.5. 7,0 changes from about 30 to 200psK”,
depending on the choice of p. Similarly, the spin orbit scattering time T30 has values
between 0 and 3ps, which is rather small. The screening parameter If}, is in the range
of 0 to 0.3, which is consistent with Lindqvist [Lin 88]. Tc decreases substantially with

increasing p. As Equation 4.3 shows, the resistance is weakly dependent upon Tc via

m“
‘ .
.

ﬂ

76

the ln(T/Tc) term. Also note that Equation 4.3 is only accurate for ln(T/Tc) >> 1.
Thus it is difﬁcult to obtain an accurate value of Tc from the data with T/TC > 5.
The sample KCu5oTi50Cu—16 behaves a little differently from the other samples and
was more difﬁcult to ﬁt. For p < 2.2, we could get a very good ﬁt, but only with
I", < 0, which is unphysical. For p 2 2.3, the x2 of ﬁt is larger than other samples and
diverges rapidly as p increases. It is possibly that Equation 4.3 is invalid at nonzero
magnetic ﬁeld. It is remarkable that “no and 7,0 are so similar for all the samples at

the same value of p.

For T < 50mK, we must consider the paraconductivity due to the Aslamazov-
Larkin and Maki-Thompson superconducting ﬂuctuation discussed in Section 2.3.5.
Since the theoretical formula for the Maki-Thompson contribution to the conductivity
is not exactly known in this temperature range, we will omit this analysis for these

temperatures.

Figure 4.8 shows the relative resistances vs temperature for the KCueoTiwCu
sandwiches in zero ﬁeld. The data were taken between about 16mK and 6K. Although
all the samples are Cu coated, the poor contact occurred occasionally, as shown by
the KCusoTi4oCu-8 sandwich, whose residual resistance is ﬁve times bigger than other
samples. In comparison to Figure 4.7, these data behave similarly to the KCusoTisoCu

sandwiches, with the following exceptions:

(1) There is no downturn in the resistivities even at the lowest temperature, but
very slight negative curvatures in the resistances are observed at temperatures below
about 25mK. This feature indicates that the CuGOTiw alloy is a superconductor with

a much lower TC than for the Cu50T150 alloy.

(2) The resistivities for the different KCusoTiwCu samples are much more similar
over the entire temperature range in comparison to those of KCusoTisoCu sandwiches,

particularly at very low temperature. Note, however, that sample KCusoTi4oCu-8

 

 
 
  
   

 

 

 

1.0015
Kcu60T14ocu '7
— o KCuwTiwCu—2(d=36pm) -
1.0010 '— '3 KCuwTiwCu—4(d=20pm) - l
- +KCuooTiwCu-5(d-27um) -
- X KCuwTiwCu-6(d=18um) -
g - o KCuwTiwCu-?(d=31p.m) -
' 1.0005
4—1
v
D:
\
03
1.0000
0.9995
—Lllllllllllllllllllllll
0.0 0.5 1.0 1.5 2.0 2.5

Figure 4.8: The relative resistances vs \/T for the KCuGoTiwCu sandwiches. The
ordinates have been displaced for for clarity.

 

     
 
  

 

 

 

9.3050 n _ . _;
: iii; 0 Kcu30T14ocu- 5 .1
b “.1“ +KCu-7 1 '
9.3025 __ 0Cu80T14o Lmdqwst et a1 _: g 1
I— mxlm 0“ ﬁ : 1|
A 9.3000 L ‘ _: l
C : :
1 h 4
v _ 4
Cr. 9.2975 -— M’
- .0000 5
- o :
9.2950 —- o 0—
0.0205
0.0195F—'+1 7‘ L+§M4+ 111 +1 + 33,411“ :1
0.0 0.5 1.0 1.5 2.0 25

T1/2 (Kl/2)

Figure 4.9: The resistances vs x/T for the KCusoTi4oCu-5 and KCu-7 sandwiches.
The ordinates have been displaced for clarity and easy comparison.

 

79

does behave somewhat differently due to its “poor” electrical contacts with K.

(3) For 20mK< T < 4K, the total change of the resistance for KCusoTiwCu
samples is only about half of that for KCusoTisoCu samples. This is due to the

differences in D, p, T;, and Tc.

In Figure 4.9, the data at the top and the bottom of the ﬁgure are for the
KCusoTi4oCu-5 sample and for the KCu-7 sandwich, respectively. The open circles
represent the data of a CumTiw sample from Lindqvist et aI [Lin 88], which were the
only data for CumTiw alloys measured to these low temperatures. These data have
been suitably renormalized for a comparison. Between 0.25K and 1.7K, their data
agree beautifully with ours. At T > 4K, our resistances are bigger than theirs, which
is mostly due to the previously-mentioned electron-phonon Umklapp contribution in
the K of our samples, as shown in the bottom of the ﬁgure. At T < 0.25K, the
resistance of their sample becomes a constant while the resistance of our sandwich
continues to increase. We have concluded that their sample probably did not cool
much below 0.25K, due to the poor thermal conducting properties of the varnish that
they used as a thermal conduct agent between their sample and their cold ﬁnger with
its attached thermometers. Thus Figure 4.9 clearly shows the superiority of our new

method.

The solid curves in Figure 4.8 are the best ﬁts by Equation 4.3 over a temperature
range 15mK< T <2.7K, using the same method as for the KCusoTiwCu samples. The
best-ﬁtting parameters are also listed in Table 4.5. From the table, we see that the
parameters for the diﬂerent CueoTi4o sandwiches have similar values. The values
of 1,0,T,O,Tc, and F, systematically change with p. The most favorable p is about
1.8 - 2.5, which is about the same as for the KCusoTisoCu sandwiches. The ﬁtting
values for 7,0,T,O,Tc, and 13}, are 30 - 100ps, 0.0 — 0.5ps, 3 - 0.2mK, and 0 — 0.37
respectively. It is clear that T,,,,-,,/TC > 5 is satisﬁed in our ﬁts (15mK< T < 2.7K).

For the same p, the values of the 7,0, 7,0, and F0 for different CusoTiw sandwiches are

80

very close. As observed for the KCu5oTi50Cu sandwiches, Tc is lower for the sandwich
with the thinner CuTi ribbon. Since Lindqvist et al. did not list the parameters for
their CumTiw sample, we list in Table 4.5 the parameters of their Cu65Ti35 sample
for a rough comparison. We see that their parameters are consistent with ours, as

discussed later in Section 4.4

We know that weak localization can make a signiﬁcant contribution to the resis-
tance even at very low temperatures, but how large is this contribution? Figure 4.10
compares the data with the computed total resistivity and the Electron-Electron
Interaction (EEI) contributions. The dashed and dotted lines represent the EEI
contributions for the Cu5oTi50 and CerTlgo samples, respectively, which are calcu-
lated from Equation 4.3 using the best ﬁt parameters in Table 4.5. The upper and
lower dashed and dotted lines correspond to the extreme values of p: 2.5 and 1.8,
respectively. The differences between the data and the lines are the weak localization
contributions to the resistivities. From this ﬁgure, we see that the weak localization
makes a non trivial contribution (APWL/Apee > 10%) to the resistivity even at the
temperatures down to 0.15K. The total resistivity computations (solid curves) are

unaffected by this range of p on the scale of this graph.

As we mentioned earlier, Figure 4.7 shows that resistivity of the KCusoTi5o—16
sandwich increases faster than any other sample with decreasing temperature below
about 0.3K, which indicates that the T; for this sample is smaller than for the other
samples. The ﬁts in the Table 4.5 also showed a much lower T6 for this sample. This
difference is at least partially due to the non-zero ﬁeld trapped inside the supercon-
ducting magnet. For this sample, we ﬁrst measured the magnetoresistances at 4.2K,
reduced the magnet current to zero, and then cooled the sample down to the lowest
temperatures. Next we measured temperature dependence of the resistance, as shown
in Figure 4.7. Once the superconducting solenoid is energized, there can be a nonzero

magnetic ﬁeld trapped inside the magnet even the current though the magnet is zero.

“‘4'"

(I

81

 

r
AR/R

L 2x10“

 
 

 

V I V V

lllllLLlllll

 

lllll

 

lllleLlllll

 

 
  

AR/R

   

\
1— \ \
. \ \
" O Kcu5oTlmcu-1 1 \ \
'— 0 KCuaoTiwCu-5 \

b

\

\

 

0.10 0.15 0.20 0.25 0.30 0.35 0.40—
: -. 101/2 (KI/2) -1

lllLLLllllLlllllllllllll

 

 

 

0.0 0.5 1.0

Figure 4.10: The contributions to the relative resistances due to weak localization

1.5
T1/2 (Kl/2)

2.0 2.5

and electron-electron interaction. The ordinates have been displaced for clarity.

.“' 'Nl v

on
{G

 

 

 

 

14.02 I I I I T I I I I I I I I I I l I I I l I T I
KCUsoTisocu- 16 "

I B Top to Bottom Points
14.01 —— 0.1472 '1' ——
r i 3 0.1103 'r a
r 5 0.0004 r —
A 7 3 0.0500 7 —
g r 5 0.0230 r 4
v 14.00 —- 3 0.0027 7' —-
0.1 ' 5 0.0000 *1 -
13.99 — __
’ I l 1 . . . 1‘. . . .‘

13.98 1 1 L L 1 1 1 i. l 1 1 1 1
00 05 10 15 20 25

Figure 4.11: The temperature dependences of the resistances at different magnetic

fields for sample KCusoTisoCu-16.

83

Based on the typical behaviors of the superconducting solenoids, we estimate that
the residual ﬁeld is less than about 10 Gauss. This residual ﬁeld seems to suppress
Tc appreciably, which will enhance the resistivity especially at temperatures close to
Tc. In a future experiment, it would be useful to cool the sample ﬁrst in zero ﬁeld

and then energize the magnet.

To observe the overall magnetoresistance, the temperature dependences of the re-
sistances at different magnetic ﬁelds are plotted in Figure 4.11. At low temperatures,
the field dependences of the resistance are signiﬁcant, and these ﬁeld dependences

decrease with increasing temperature.

4.3 Magnetoresistivities

Magnetoresistance measurement is another important tool for studying the amor-

phous alloys. In this section, we will present and discuss our magnetoresistance data.

4.3.1 Background Magnetoresistivities and Fitting Formu-
las

Like the zero ﬁeld resistances, the magnetoresistances for our sandwiches also have
two parts: resistances of the CuTi ribbons and the so-called background resistances,
which are due to the Cu electrodes, K layers. and the boundaries among the K layers,

Cu electrodes, and CuTi ribbons.

Experiments and theories have shown that at small magnetic ﬁelds, the magnetore-
sistivities of pure Cu and K are proportional to 82 [Zim 60, Pip 89]. In order to ﬁnd
the magnitude and ﬁeld dependences of the magnetoresistances due to Cu, K and their

boundaries, we measured the magnetoresistances of the Cu/ K / Cu sandwiches listed
in Table 4.2. Figure 4.12 shows the magnetoresistances AR(B, T) = R(B, T) - R(T)
for the Cu/K/Cu sandwiches at T =0.5K, where R(T) is for B = 0.

AR(O.5K,B)(,uQ)

84

 

 

 

0.020[VIIITIIIIIIITIIIIIIIIIIIIIOT
‘ OKCu-IO (Ro=0.11 p0) "‘
' 'FKCu-Q (R030.013 [1.0) o "'
0.015 — —
.— 0 .1
[— -1
0.010 —- -—
- o -
.. o -
0.005 —— —
D o I!
.. o ..
.. 0 -1
0
O'COOW‘JI111il1111li‘111

 

0.000 0.005 0.010 0.015 0.020 0.025 0.030
2
B (T2)

Figure 4.12: The magnetoresistances vs 82 for the Cu/ K/ Cu sandwiches.

85

 

 

 

 

0.0133 '- I I I I I I I I I I I I I I T I I1
: KCu-9 .:
0.0132 :— at:
_ ° 4.2K x _
5 + 0.5K e j
. 31 __ X 70mK I! _
A O 01 - 0 26mK o 4
C ~ G x s
:1 - . ~
v - u —1
0.0130 ‘— ° —:
m : . . 7 .
.— 0 X -1
" o + .-
0.0129 — o ° ‘ —
1.00 + x 3
O
0*): x _
0.0128 " ° —
0..
o "l ‘
I l l I l l l l 9 l l 1‘
0.0127 ' 1 1 '
.00 0.01 0.02 0.03

132 (T2)

Figure 4.13: The magnetoresistances vs 82 at different temperatures for the KCu-9
sandwich.

86

From the ﬁgure, we see that the magnetoresistances of the two sandwiches increase
with magnetic ﬁeld and obey approximately a square law, but the magnitudes are
very different for the two samples. KCu-9 has a very thin layer of K, whose zero
ﬁeld resistance is about 0.02110. Its resistance increases very slowly with magnetic
ﬁeld. At 0.15 Tesla, the resistance has risen only by about 3%. For sandwich KCu-10,
which has a much thicker layer of K and a resistance of about 0.1110, its resistance
increases much more rapidly with B than that of KCu-9. Thus in order to reduce
this part of the background magnetoresistances, the thickness of the K layer in the

sandwich must be minimized.

To examine the temperature dependences of the background magnetoresistances,
we show in Figure 4.13 the magnetoresistances of the KCu-9 sandwich vs 82 at dif-
ferent temperatures. We see that all the data have almost the same slopes. That is,
at most ﬁelds, the magnetoresistances at different temperatures only differ by con-
stants. This means that the field dependent parts of the resistances of the Cu/ K / Cu
sandwiches are temperature independent. However for the data at 4.2K, the resis-
tance drops suddenly for B > 0.13T. We believe that this is related to the presence
of superconducting PbSn solder on the voltage leads of the Cu electrodes. At 4.2K,
the critical ﬁeld of the superconducting solder will be lower because T is closer to its
Tc of about 7K. Thus it is not surprising that this resistance drop occurs at lower
temperatures with B > 0.17T since the critical ﬁeld will be higher at these lower
temperatures. Similar drops in resistance are seen for the CuTi sandwiches. So in

our analyses and ﬁts, we keep B (0.15 Tesla.

The other background contribution to the magnetoresistance is due to the bound-
aries between the sputtered-Cu coatings and the CuTi ribbon, which we have not
directly measured. Since we know, from Figure 4.13 and the above arguments, that
the magnetoresistance (AR(T,B) = R(T,B) - R(T,B = 0)) of a Cu/K/Cu sand-

wich is temperature independent, we are going to assume in our analysis that the

87

Cu/CuTi boundary magnetoresistance has the same property. We will then see if

this assumption is supported by the magnetoresistance data for CuTi sandwiches.

4.3.2 Analyses of the Magnetoresistances due to Various
Scattering Mechanisms

From Chapter 2, we know that the corrections to the magnetoresistivities of an amor-
phous metal due to weak localization, electron-electron interaction, and supercon-
ducting ﬂuctuations are given in Equations (2.34), (2.36), (2.38), (2.40), and (2.41).
These formulas are fairly complicated, and it is very difﬁcult to analyze the data
by directly applying these formulas. In order to simplify the ﬁtting, we will present
an approximate analysis concerning the magnitude of the magnetoresistance and its

temperature and ﬁeld dependences for each scattering mechanism.

Figure 4.14 shows the magnetoresistivities due to weak localization (ﬁrst term
in Equation 2.34), Zeeman spin splitting (second term in Equation 2.34), electron
electron interaction (EEI), and superconducting ﬂuctuations at T = 52mK and 0.5K
for a set of typical parameters: Tc = 5mK, D = 0.24cm2/s, and a = 0.25, 15,, = 0.3,
7,0 = 700psKP, 7,0 = 15ps, p=1.7, g = 2, and 6 is from Table 2.1.

At both temperatures, we see that the weak localization and Zeeman spin splitting
make the major contribution to the resistivity at low and high ﬁelds, respectively. The

Maki-Thompson contribution is important only for T/Tc < 20 (B > 0.1).

The contribution from the diffusion channel (DC) in the electron-electron interac-
tion depends on B0 and becomes important only for T < 0.1K and B > 0.05T. The
Cooper channel (CC) gives a very small negative contribution at all temperatures,

which can be neglected.

Another quantity which indicates the strength of the scattering at some particular

temperature is the scattering ﬁeld, which is deﬁned by BM, = h/(4eDTm, ). The scat-

88

 

 

 

E T=52mK 3»
0.000100 :- xWeak Localization 00°31
: oZeeman Splitting ( o o o 2
L oMaki-'1'hornpson Effect MT) 00 _.
0.000075t ,CC in EEI 1
:‘DCinEEI 0° 2
0.000050:- M...
: 00° :1
0.000025“— n"“:3335"”3””"”:::=u
Ex:°:8. 000'..... 3
Q _xo°o ......O _‘
\0-000000_ ‘....’2+++¢+¢++++¢++++++¢0+s—J.
Q. : :
< - .4
0.000100:— T=0.5K .3
0.000075 :— 006.3»
: 0° :
P o -I
0.000050 L— °°o° .1
1- 0° .1
b o d
0000025:— xxxga.g§:xxxxxxxxxxxxx;z
: "i‘xx°°° ooooooo
0.000000905.080888332252222222000000
1.
llJlllllllLllllLlLLlllllllllg

 

 

0.000 0.025 0.050 0.075 0.100 0.125 0.150
B (T)

Figure 4.14: The theoretical relative magnetoresistivities due to different scattering
mechanisms.

89

 

 

 

 

 

 

B (T)
0.0010002 0.005 0.010 0.020 0.050 0.100
I IllllIIl ITIIIIIII

A
E I- d
(I)
(D 0
El 10
U)
Q
.4
1:151
E:

10"2
(D
Z
H
0:
El
E3
< 10"4
U
U) /

1 1 L1/111' 1 1 llLllll 1 1 1
0.01 0.05 0.10 . 0.50 1.00 5.00
T (K)

Figure 4.15: The scattering ﬁelds vs T and B for various scattering mechanisms.

 

90

tering ﬁelds for weak localization with the Zeeman splitting and the Maki-Thompson

effect are B,-, B”, and B-, where we have
— 2 81+ 7800 9 (4'6)

and 7 = 7.32 x 10-3T3032. For the Cooper and diffusion channels of the electron-

electron interaction, the scattering ﬁelds are

T
300 — 1.3555 , (4.7)

800 0.745T . (4.8)

The ratios of these characteristic ﬁelds to the actual applied magnetic ﬁeld B
determine the magnitudes of functions f3,¢3, and g3 in the weak localization, the
Maki-Thompson ﬂuctuation, and the electron—electron interaction contributions. For
B < 3...", the magnetoresistivity due to each scattering process is proportional to
B2 and for B > Bum, it is proportional to x/B. However the scattering ﬁelds for each
scattering mechanism are different, and they can be temperature and ﬁeld dependent

as shown in Figure 4.15.

In this plot, the parameters are same as for Figure 4.14. The ordinate is the
scattering ﬁeld; the bottom abscissa is the temperature; and the top abscissa is the
magnetic ﬁeld, which applies only to 7B”. The region between the two horizontal
dotted lines represents the range of the actual magnetic ﬁelds that we used. We see
that the scattering ﬁelds B” and Boo are generally much larger than B, so their
contributions can be evaluated in the small ﬁeld limit or be neglected. For T > 0.1K,
diffusion-channel interaction contribution can be also approximated as a small ﬁeld
limit. Weak localization contribution is related to B,- and B- = B,- + 78,0. The
dashed line shows the ﬁeld dependence of 73,0, and it is always much smaller than
the applied magnetic ﬁeld. So at T < 0.1K, where B > B,, B_, the weak localization
contribution may be approximated as a large ﬁeld limit, but for T > 0111’ it is difﬁcult

to make any approximations.

91

At magnetic ﬁelds and temperatures where we can not make any approximations,
we will calculate the magnetoresistivities using the approximate analytical expressions

for f3, (113, and 93 given by Ousset and Baxter [Ous 85, Bax 89]

4.3.3 p(B) for Cu50Ti50 and CusoTi40 Alloys

Figure 4.16 shows the relative magnetoresistances of the KCu5oTisoCu-16 at seven
different temperatures. Data symbols are our results, and the solid curves are the best
ﬁts, which will be discussed later. The data sets from the top to bottom correspond
to the temperatures shown in the ﬁgure. Except for 4.2K, the ratio of adjacent

temperatures is about a factor of two.

First, from the ﬁgure, we see that the magnetoresistivities increase with increasing
magnetic ﬁeld at all the temperatures, with the exception of the 4.2K data for B >
0.08T which was described in Section 4.3.1. The 20mK data increase faster at low
ﬁeld than at any other temperature, which is mainly due to the Maki-Thompson

superconducting ﬂuctuations.

Secondly, for B < 0.03T, the curvature of the magnetoresistivities changes from
negative to positive at 0.25K. We believe that this sign change is mostly likely due to
the inelastic scattering. Since B, ~ T”, for T S 0.1K we have B/B, >> 1, for which
the resistivity is proportional to \/B. For T > 0.5K, we have B/B, < 0.6, and the

resistivity is proportional to B3/2 (see also Figure 4.15).

Thirdly, for B >0.05T, the resistances increase with magnetic ﬁeld more or less
linearly, which indicates that the contributions from weak localization, Zeeman split-
ting, electron-electron interaction, and the background are all competing with each

other.

Since we do not know the magnitude and ﬁeld dependence of the background resis-

tivity, we can not ﬁt our data directly to the equations given in Section 2.4. However,

 

 

 

 

bIIlIllITIIIIITIITIIIIIITIIIII‘
_ KCu5OT150Cu-15 1
0.0003 -- x 0.020 x _
.. 0 0.052 K
D 0.10 K 04
O ’ + 0.25 x x
m — x 0.50 x .
\ __ 4- 1.00 K x
A x 4.20 K .
II; 0.0002 — x x .
e - x ._ 1
E? - x .
<1 - .. -
0.0001 *— , '- __
:3” x 8 ‘
00000 9"!“ 1 1 1 1 1 l 1 1 1 1 l Lg1_1 l 1 1 1 13
0.000 0.025 0.050 0.075 0.100 0.125 0.150
B(T)

Figure 4.16: The relative magnetoresistance data and ﬁts for the KCusoTisoCu-IG
sandwich.

93

 

 

 

 

0.00015 I I I I l I I I I I I I I I I I I I I l I 1 I I l r I 7 r
L x 0.020 K 1
.- 0 0.052 K -1
n 0.10 K x x X x x

O * 23:53 E x "

D: ’ .1. 1.00 x O 0*

\ 0.00010 +— * 4-194 K o —

A O

A - .

En. 1' I . 7

O. . ' .

F-l . " 4

v ' . a ‘ +

m 0.00005 — ' a -"

<1 . - .

I , .' -

A 2."

m 1 ;.' .

£4 2:" , 4

5; 0.00000

<1 ~ «

v _ .1
__ X 8 x Kcu50T150CU" 16 q
r 5‘ 1

_0.00005 1 1 1 1 l 1 1 1 1 L1 1 1 1 l 1 1 1 1 1 1 1 l 1 1 1 1

0.000 0.025 0.050 0.075 0.100 0.125 0.150
B(T)

Figure 4.17: The modiﬁed magnetoresistance data and their ﬁts for the KCusoTisoCu-
16 sandwich.

94

in Section 4.2.1 and 4.3.1 we assumed that the background magnetoresistances are
temperature independent. Therefore, if we take the differences of our data points
between T and 1K at the same magnetic ﬁeld (AR(T, B) — AR(1K,B)), the resulting
data (called modiﬁed data) should be free of the background contributions. Thus the
modiﬁed data can be ﬁtted by the differences of the equations given in Section 2.4
between T and 1K. Figure 4.17 shows the modiﬁed data vs B, where R, is the residual

resistance.

According to Figure 4.14 and 4.15, the Cooper channel contribution is very small.
In the ﬁts we set a = 0.25, and g(B,T) = —1/ln(T'/Tc), T‘zmax(T,4DeB/k3)
[Bax 89]. Thus there is no free parameter for the Copper channel, whose contri—
bution to the resistance can be calculated for each temperature and ﬁeld. For the
diffusion channel, ¢3(B/BDC) can be calculated using the approximate analytical

expression [0115 85]. The only variable parameter is F}.

The other contributions to the magnetoresistivity are due to weak localization with
Zeeman splitting and Maki-Thompson ﬂuctuations. In the Maki-Thompson contribu-
tion, 5(B, T) is the function in Equation 2.42, and it depends on temperature as well
as magnetic ﬁeld. However in the low ﬁeld limit, when B, B,- < (k3T/4De) ln(T/Tc),
we have ﬂ(B,T) = ,6(T) [Ric 88], which is tabulated in Table 2.1. To simplify the
computation of ﬂ(B,T), we have used the approximation that ﬁ(B,T) = 5(T). In
our measurements, this low ﬁeld limit is satisﬁed except for the lowest temperature
and the highest ﬁeld. For the weak localization, since B,- is not exactly known and
f3 converges very slow even for B > B,, it is inaccurate to make any simpliﬁcations.
Thus B,- (or 7,) is chosen as a ﬁtting parameter. The approximate expression for f3
is used for the calculation. So in our analysis, the ﬁtting parameters are 7,, 7,0, Tc,
and Ba, and the ﬁxed parameters are D, Aep, and p (as listed in Table 4.3), the Lande

g-factor g = 2 for electrons, and a = 0.25 for the strong spin-orbit scattering.

Since we ﬁt the modiﬁed data by the differences of the equations between T and

95

1K, we are required to know the ﬁtting parameters at 1K. Since Tc ,7”, and B, are
temperature independent, the only parameter we are required to know is 1,, which is
equal to “no at 1K. So we preset the no to different values until the best ﬁt is obtained
for all the data at different temperatures. Then parameters 1,,T,O,Tc , and B, are
obtained, and the best ﬁts are shown in Figure 4.17. The ﬁts (solid curves) are very
good except for the modiﬁed data at 20mK and B > 0.04T (B / T >2T/ K), which are
almost constant for B > 0.05T. We tried and failed to get a good ﬁt to this set of the
modiﬁed data over the whole ﬁeld range. The theoretical formulas in the Section 2.4
give values that are too large for us to obtain a good ﬁt to the data. We now suggest
some reasons for this disagreement, although at this time we can not make a deﬁnitive

choice among them.

(1) Dilute spin impurities. Baxter et al [Bax 89] have indicated that magnetic
impurities will introduce a magnetoresistance which is proportional to -B2 at low
ﬁelds and saturates at high ﬁelds. Since Ap/p for CuTi alloy is typically of order of
~ 10" between 0 and 0.15 Tesla, a tiny amount of magnetic impurities may reduce

the magnetoresistance signiﬁcantly.

(2) Since B is not much less than (k3T/4De)ln(T/Tc)for B > 0.05T and T <
40mK, [3(B, T) will decrease with increasing magnetic ﬁeld [San 85], and it is smaller
than MT), which we used in our ﬁts at all ﬁelds and temperatures. Therefore the

actual magnetoresistances at higher ﬁelds will be lower than our ﬁts indicate.

(3) Magnetic ﬁelds lower the Tc. With such small Tc, a small magnetic ﬁeld
will suppress Tc substantially [Sch 90]. This effect will make the Maki-Thompson

contribution to the magnetoresistance decrease with increasing magnetic ﬁeld.

However, for weak spin-orbit coupling systems (e.g. CaAl), theoretical models
agree with experimental data even for B / T = 23T/ K. [Lin 90, Sam 89]. It is unclear

why there are discrepancies between theoretical predictions and experimental data

96

for strong spin-orbit coupling systems.

The the best ﬁtting parameters for this sample are 7,0 = 680 — 750psK" and 1,0 =
13 - 25ps, which are much bigger than the the values from the zero ﬁeld resistances.
At low temperature, 7,- shows tendency to saturate, which will be discussed later. The
bigger 7,0 is mainly due the the data at 4.2K. For T > 1K, the contributions from
the weak localization, electron-electron interaction, and Maki-Thompson ﬂuctuations
are very small. The big difference in the magnetoresistivity between 1K and 4.2K
is mainly due to Zeeman splitting, which makes the magnetoresistance increase with
increasing 7,, for 0.1ps< 1,0 <50ps [Lin 88]. p and B, are similar to values obtained
at zero magnetic ﬁeld but with a smaller range. Tc drops by half, which may be

suppressed due to the magnetic ﬁeld.

For the KCusoTi4oCu-7 sandwich, the raw data are shown in Figure 4.18. As with
KCusoTi5oCu-16 sandwich, the magnetoresistance increases with the magnetic ﬁeld

over the entire ﬁeld range. There are some differences between the two samples:

(1) For T S 0.1K and B < 0.03T, the magnetoresistivity increases much slower
with B than that of the KCu5oTi5oCu-16 sandwich. Since the CumTiw alloy has a
much lower Tc than CusoTiso, the Maki-Thompson contribution to the magnetore-
sistance, which is proportional to B(T) ~ In‘2(T/Tc), is much smaller than for the

CusoTiso sample with T S 0.1K.

(2) The magnetoresistances for this sandwich become temperature independent
for T $0.1K while no obvious low temperature saturation is observed for sample
KCusoTisoCu-16. From (1), we know that the Maki-Thompson contribution is very
small for sample KCusoTi4oCu-7. Thus from Figure 4.14 we know that weak local-
ization with Zeeman splitting will contribute a major part to the magnetoresistivity.
From Figure 4.15 with T S 0.1K, we see that B > B,, B-, and the weak localization

contribution becomes insensitive to temperature change. This insensitivity to the

 

97

 

     

 

 

 

 

0.00025 — l I l I I
: KCUsoTl4ocu-7
0.00020 — X
- o
"' D
O - +
Q: - x
\ 0.00015 — '1'
A __ a
0:1 _
E... -
ES "'
< 0.00010 7
7 4
0.00005 L-'
5 5
' 4
0.00000 Nu"1111l1411l1111l1111l1111
0.000 0.025 0.050 0.075 0.100 0.125 0.150

13(1)

Figgre 4.18: The relative magnetoresistance data and ﬁts of the KCueoTimCu-7 sand-
w1c .

98

temperature accounts partially for the weak temperature dependence of the magne‘
toresistivity. Another possible mechanism is due to the dephasing by electron spin
scattering by impurities as described by Baxter et aI [Bax 89]. This scattering will
make B.- saturate below some temperature T,, which depends on the concentration
of the spin impurities. Thus the saturation of B,- causes the function f3(B/B,-) in the
weak localization and Maki-Thompson contribution terms (i.e. magnetoresistance) to
become temperature independent for T < T,. A rough calculation shows that amor-
phous alloys with a 10"7 relative concentration of magnetic impurities could have a
spin scattering lifetime 1', comparable to T,- at 0.1K. Thus the effective inelastic scat-
tering time, l/T: = Tp/T,o + 1/1',, may be saturated at T < 0.1K, as shown later in

Figure 4.21.

As with the KCusoTisoCu-IG sandwich, background magnetoresistances also ex-
ist for the KCuGoTi4oCu-7 sandwich. Using the same procedure as with sample
KCusoTisoCu-lﬁ, the differences of the magnetoresistances between T and 1K were
taken and are shown in Figure 4.19. Based on the same arguments for sandwich
KCusoTiwCu-lﬁ, the ﬁxed parameters are a = 0.25, D = 0.28cm2/s, and p = 193110.
The set of ﬁtting parameters is same as before. The best ﬁts are obtained as shown

in Figure 4.19.

We see that the ﬁts are pretty good for T > 0.1K, but once again we are unable
to get good ﬁts for T < 0.1K and B > 0.25T. Since TC is very small, the effect of
13(B, T) on the magnetoresistances is much smaller. Thus the most probable reason

for these poor ﬁts is due to impurity-spin scattering, as discussed earlier.

From the best ﬁts, the ﬁtting parameters we obtained are 7,0 = 870 -— 950psK”,
7,0 = 7 — 15ps, p = 1.6 —1.8, and F}, = 0 — 0.3, and T, < 0.1mK. As with the
CusoTiso alloy, 7,, and 7,0 are much greater than their zero ﬁeld values, and Tc is
extremely small. For T <0.1K, 7', saturates, and this will be discussed in more detail

in Section 4.4. We notice that, except for Tc, these parameters are very close to the

99

 

   

 

 

 

4.0'10—5 — I I I I I I I IT? I I I 9 I I9 I I I'd I I [a
. a a
’ I x X X t
a _ Z x + + I r
\ ‘- ﬂ,“ .-
2 2010—5 .— 1,» -
III. - -
O. - ~
4—1 -3." ,. _
v f" q
0:
I 1 _
A
E. _ KCusoTi4oCu—7
E - x 0.020 x -
_. . -5 0.0 0.052 x . _4
3 2'0 10 o 0.10 K ..
” + 0.25 x ,_
- x 0.50 x -
,_ s 1.00 x x _
x 4.199 K _
_40.10—5 l l l l l I l l l l L 1 Lil L1 1 l l l l l l l l l

 

 

0.000 0.025 0.050 0.075 0.100 0.125 0.150
B(T)

Figure 4.19: The modiﬁed magnetoresistance data and their ﬁts for the KCueoTiwCu-
7 sandwich.

Whrq' “ ‘-".

100

parameters for the KCusoTi5oCu-16 sample.

Using the best ﬁtting parameters, the background magnetoresistances, which were
removed in the modiﬁed data, can be estimated by taking the differences between
the data in Figure 4.16 or 4.18 and values from the theoretical expressions at the
same temperatures. The background contributions for T = 0.5K, which include the
various boundary resistances, are shown in Figure 4.20 as a function of B. For the
other temperatures, the background magnetoresistances are found to be almost the
same as for T = 0.5K, which means that the background magnetoresistances are

temperature independent, as we assumed.

For a comparison, the magnetoresistances of the KCu-9 and KCu-10 sandwiches
are also plotted in the same ﬁgure. Indeed the background contributions follow
rather well a B2 dependences. The magnitude of the background resistances for
KCusoTisoCu-l6 is different from that for KCueoTi4oCu-7, which is understandable
because the K layers in these two sandwiches may have different thicknesses, and
the boundary resistances maybe also be different. But they are both rather small

compared to KCu-10 sample, which is the worst of the Cu/K/Cu sandwiches.

By simply adding these 0.5K background magnetoresistances to the best-ﬁtting
theoretical expressions for all temperatures, we obtained the predicted ﬁts for the
raw data, which are the solid curves in Figure 4.16 and 4.18. We see that the ﬁts
are remarkably good except at lower temperatures and higher ﬁelds, where the ﬁts
to the modiﬁed data were also not so good. Note that these good ﬁts for T #
0.5K in Figure 4.16 and 4.18 again conﬁrm our assumption that the background

magnetoresistances are temperature independent.

i‘o’ﬁ.» '. f'. o -

win-u: ‘

101

 

 

 

 

0.005 [- I I I I I I I I I I I I ﬁ ﬁ I I Id
I o oKCu—io (nor-0.11 110) :
UFrom KCu Ti Cu-7 __
A 0004 __— +From KCuzrigm-is _
C3 _ 'I'KCu-Q (120:0.013 1.11) d
:L - -
v .. .4
a 0.003 —- o —
.. [- -1
ﬁ : .3
° .. O D _.
8 0.002 -— o —
DZ 1- D -
Q h o -
L. 0 t-
- :1 + ..
0.001 :7 1: + "j
+
:0 n '3 D I .1. a:
o D + i.

7 9* f 111 'I‘ ‘

0 000 j 1 l 1 1 1 1 l 1 1 1 1 l 1 1

0.00 0.01 0.02 0.03

132 (T2)

Figure 4.20: The background magnetoresistances due to K, Cu and the boundaries
in the KCuTiCu sandwiches

 

102

4.4 Comparisons of the Fitting Parameters

For a comparison, the best ﬁtting parameters obtained in the Section 4.2 and 4.3 are

presented in Table 4.6. From the table, we see the following:

(1) Except for Tc, the parameters for CusoTi5o and CusoTiw alloys are rather
close either at zero or ﬁnite ﬁelds. The differences in T6 are due to the different Ti

concentrations in these alloys.

(2) For the same alloy, p and F, from both zero and ﬁnite magnetic ﬁeld are
about same, but no and 7,0 from the magnetoresistance measurements are about an
order of magnitude larger than their zero ﬁeld values. This is, perhaps, not very sur-
prising because, at low temperatures, the zero ﬁeld resistance and magnetoresistance
are, respectively, dominated by the different scattering mechanisms—electron-electron
interaction and weak localization. The big differences in T; may be due to the sup-
pression of the Tc at ﬁnite ﬁeld. We are unable to ﬁt magnetoresistance data for

T < 0.25K if we use the values of Tc from the zero ﬁeld ﬁts.

Table 4.6: The Fitting Parameters from the Zero Field Resistances and Magnetore-
sistances

— 4-

 

 

'

r_S-aﬁmples B T p 73-, 1,, Tc F,
(T) (K) (Pst) (PS) (mK)
CusoTiso 0.0 0052—27 1.7—2.5 30—200 0.06—1.4 9—0.3 0—0.3
0—0.15 0.02—4.0 1.6—1.8 680—750 13-20 5 0—0.2
CusoTim 0 0.015—2.7 1.8—2.5 30—100 0.0—0.5 3—0.2 0—0.37
0—0.15 0.02—4.2 1.6—1.8 870—950 7—15 0.1 0.0—0.3

 

 

 

 

 

 

 

 

 

 

 

 

The inelastic scattering time T, is one of the parameters which was not completely
explained in the Table 4.6. Usually T,- is temperature dependent, and its reciprocal is
proportional to TV. From the best ﬁts to the magnetoresistances at each temperature

in Figure 4.17 and 4.19, the values of T,- are obtained and shown in Figure 4.21. The

103

 

 

 

 

 

1.0—1 : I T I I Imr I ﬂ I IIIIII /I" I I
: /." 3
- /..' '1
_ /.' a
- 0 KCuuTi”Cu-16 /,-' -
10-2 E" + KCugoTi“Cu-7 ('4' :3
A I ' I
I - 2
U) _ -
D. - 3

V

.2 10"3 E'- _-1
1‘ : E
\ t :
v—I _ ..
10-4 :— ._:.
C I
. l.' -

10—5 1 11'.1i111'1l 1 1 1 l1111l 1 1 1

0.01 0.05 0.10 0.50 1.00 5.00

T (K)

Figure 4.21: 1/1', vs T for CusoTiso and CueoTim alloys

104

data sets are from the magnetoresistance ﬁts, and the solid curves are the best ﬁts to

the inelastic scattering times by

P
l=A+T+, (4.9)

T: 7:0
which yield the parameters p and “no in the Table 4.6.

The dotted and dashed curves, respectively, are the corresponding 1',- values for
CusoTiso and CusoTiw alloys from zero ﬁeld resistance ﬁts with p =2.1, and 7,0 =
65psK" and 50psK’, which are mean values from Table 4.5. From the Figure, we see

the following features:

(1) For T > 0.1K, r, at zero ﬁeld is much smaller than at ﬁnite ﬁelds, and they

cross over at about 50mK. No saturation was found in zero ﬁeld for T < 50mK.

(2) From the magnetoresistance measurements (symbols in Figure 4.21) and for
T > 0.1K, T.‘ for the KCumTisoCu sample are very close to those for the KCueoTiwCu
samples. However, for T < 0.1K, T.- for the CumTiw alloy saturates while 1',- for

CusoTiso alloy continues to decrease and may saturate for T < 20mK.

In the zero-ﬁeld ﬁts, saturation of T, was not considered. We will show that this
is not really a problem in the zero-ﬁeld ﬁts. For T > lOOmK, T,- in zero ﬁeld is
much smaller than its possible saturation value. Thus this possible saturation will
make a negligible change in the weak localization contribution to p. For T < lOOmK,
7, becomes comparable to its possible saturation value, but the total contribution
to resistivity related to r,- for T < lOOmK (weak localization) is much smaller than
electron-electron interaction (see Figure 4.10). Therefore the possible saturation of T,‘

in zero ﬁeld will not make any appreciable contribution to resistivity.

Next we will address the question of the relative size of the sample thickness and

the appropriate weak localization inelastic diffusion length L,, which is

L.- = \/D—r. , (4.10)

105

In other words, are we correct in assuming that our ribbons are three dimensional
for weak localization? To be correct, we require L.- < 11 (size of a sample). From
Figure 4.21, we know that the longest T.- is about 5x10‘ps. using D = 0.28cm2/s, we
have L.- = 2pm, which is indeed much less than d (> 20pm).

Table 4.7: The Fitting Parameters for Comparisons

m—

 

 

 

 

 

 

 

 

 

 

 

 

Bm“ Samples T 1 p no 1,, Tc F, ref
(T) 110 (pew) (pa) M)
0 CusoTiso 0052-27 1.7—2.5 30-200 0.06-1.7 9—l.0 0—0.3 Our results
1.7—20 2.78 16000 5.3 0.58 Schulte ‘
0.1—30 2(F 2) 410 1.0 0.37 Hickey b
CueoTiqo 0.015—2.7 1.8-2.5 30—100 0.0—0.3 3—0.2 0-0.37 Our results
1.7—20 2.99 11000 1.4 0.72 Schulte "
Cu55Ti35 0.1-30 2(F 2) 190 0.18 0.45 Hickey b
0.3-18 2.0—2.5 0.08—0.25 0-0.4 Lindqvist c
0.15 CusoTiso 0.02—4.0 1.6-1.8 680—750 13—20 5 0—O.2 Our results
6 4.2—20 3-4 4.4 0.37(F) Howson °
0.15 CusoTiw 0.02—4.2 1.6-1.8 870-950 7—15 0.1 0.0-0.3 Our results
4 0.12-4.2 11 0.6 Lindqvist d
4 Cu55Ti35 0045-92 2.0 5.5 0.4 Lindqvist c

 

1) The ﬁtting temperature range.
2) The ﬁxed parameters.

I‘) from reference [Sch 86].

b) from reference [Hic 86].

c) from reference [Lin 88].

d) from reference [Lin 89].

c) from reference [How 86].

Finally we want to compare our parameters with those of other people. Table 4.7
lists the ﬁtting parameters obtained by other researchers at both zero and ﬁnite ﬁelds.

In theoretical expressions given in Chapter 2, the phase-breaking time 73,, is commonly

deﬁned by

i

W

1 X
=—+— , (4.11)
Ti

T80

where different researchers use different values of X: 4 / 3, 2, or 4. Unfortunately the

1,0 value will depend upon the values of X. In our ﬁtting, we used X = 4. For other

106
X values, the equivalent values of 1,0 to that of X = 4 is given by

4
(Tao)equ = —X—(Tao)x - (4.12)

The values of 7.0 from other people in the Table 4.7 have been converted to our

equivalent values.

From the table, ﬁrst we see that our data were taken down to about 15mK and
the minimum temperatures for our ﬁts are 52mK and 15mK, respectively, for the
CusoTiso and CusoTim alloys. These are the lowest-temperature and highest-precision
measurements ever taken and ﬁtted to so far. We found that both CusoTiso and
CusoTim alloys are superconductors, so we have also considered the effects of Tc on

our ﬁts which most people have neglected.

Second, at zero magnetic ﬁeld, our parameters are consistent with Hickey et a1 . and
Lindqvist et al., whose data were taken and ﬁtted at very low starting temperature.
In fact, Lindqvist et aI took their data down to about 20mK, but they just ﬁtted their
data above 0.3K probably because their data were saturated below 0.3K (as shown
in Figure 4.9). The parameters of Schulte et aI., especially “no and 7,0, are different
from the others. The p value of from Howson et a1 is much larger than those from

others. The common feature is that their data were taken above 1.7K.

Third, the 7,0 and 7,0 parameters from ﬁnite ﬁelds are much bigger than those from
the zero ﬁeld. This may be due to an inconsistency in the theoretical formulas since
at the lowest temperatures zero ﬁeld resistance and magnetoresistance are dominated

by the electron-electron interaction and weak localization, respectively.

Chapter 5

Summaries and Conclusions

In this study, we have investigated the electrical and magneto-resistances of CusoTiso
and CumTiw alloys at very low temperatures and magnetic ﬁelds. The zero ﬁeld
resistances have been measured from 15mK to 6K and magnetoresistance measure-
ments were carried out in ﬁelds up to 0.2T at 20mK, 52mK, 0.1K, 0.25K, 0.5K, 1.0K
and 4.2K.

To measure the resistances and temperatures more accurately and reliably than
ever before, we have developed a novel method: measuring the electrical resistances
of the CuTi ribbons with the current perpendicular to the ribbons and using K as a
non-superconducting glue to achieve excellent electrical and thermal contact between
the CuTi ribbons and refrigerator with its attached thermometers. Our temperature
measurements, which can be quite difﬁcult at such low temperatures, are reliable at
least down to 15mK with an error much less than lmK. Using a SQUID null detector
and a high precision current comparator, we measured the resistances to a relative
precision of about 10'7—10’8. Even though this method causes some complications in
analyzing the resistances due to background contributions by the Cu electrodes, and
the K layers, and their boundaries, these problems have been overcome. In principle,

our method is applicable to any ribbon or ﬁlm-like high-resistivity sample.

We observed that resistances of the CuTi alloys decrease with increasing temper-

107

108

atures and that magnetoresistances increases with increasing magnetic ﬁelds. At zero
magnetic ﬁeld and for T < 0.15K, the electron-electron interaction contribution dom-
inates the total resistance, while the weak localization contribution is signiﬁcant for
T > 0.15K. On the other hand, at a ﬁnite magnetic ﬁeld and at the lowest tempera-
tures, the magnetoresistances are dominated by weak localization, Zeeman splitting,
and Maki-Thompson superconducting ﬂuctuations. We found that both Cu5oTi50 and
CusoTiw alloys are superconductors with the transition temperatures of less 15mK

and 5mK, respectively, which have never been quantiﬁed before.

The inelastic scattering time tends to saturate at T < 0.1K, especially for CuGOTim
alloys, which is possibly due to the presence of dilute magnetic impurities. We con-
clude that p = 1.8 - 2.5, where n = noT‘P, and If}, 2 0 — 0.3 for both CusoTiso and
CumTiw alloys. The lifetimes no and no that were obtained from the zero magnetic
ﬁeld measurements are about a order of magnitude smaller than those obtained from
ﬁnite ﬁeld measurements. The exact cause of the disagreement is not known, but one
possibility is that the theories are incomplete in the strong spin-orbit scattering limit

for zero and low ﬁelds and at very low temperatures.

Now that this novel technique has been perfected, a number future experiments

come to mind.

(1) In principle, thermoelectric power measurements could be made that would

provide a ﬁrst test of existing theoretical predictions [Hsu 89].

(2) The transport properties of CuTi/crystalline-metal multilayers could be stud-
ied in which the layer thickness spans the important length scales for electron-electron

interaction and weak localization, respectively: L1}, and L.- (see Equation 2.13 and

4.10).

Appendix A

Calibration of X ol d

As discussed in Section 3.3, for T < 50mK, a CMN thermometer called xold was used
to measure the temperatures of the sample. Xold varies linearly with 1 / T (Curie’s law).
However, its slope slightly changes with time. By comparing Xold with two germanium
resistors R6 and R7, which were at the same temperature, we could correct this change
as described in Section 3.3.1. The values of xom, R6, and R7 were measured directly

and are shown in Table A.l. The temperatures in the table are calculated from R6

Table A.1: xold vs T

 

 

RG/R7 T X011!
(9) (K)
1915 4.2010 0.0443
95.2 1.5190 0.0440
540 0.2765 0.0405
2530 0.1265 0.0356
9160 0.0786 0.0300

17500 0.0636 0.0266

50600 0.0465 0.0200

 

 

 

 

 

 

and R7. Fitting the data with Curie’s law, we obtained

0.001148

T (A.1)

Xold = 0.04466 —

For T < 50mK, we use this equation to calculate the temperature from the measured
ond- The error is estimated to be less than 0.5%.

109

Appendix B

Temperature Dependence of of
the Reference Resistance

As discussed in Section 3.3.2, for T > 0.4K, the resistance of our reference resistor
(Roof) slowly increases with increasing temperature. This temperature dependence,
even if is is very small in magnitude, may have a signiﬁcant effect on apparent tem-
perature dependence of the resistances of our samples. To correct this effect, we
measured R", vs Cu (indicating the reference temperature) as shown in Table 3.1
and plotted in Figure 3.7. Using the Equation 3.1, the data were ﬁtted very well as

shown in Figure 3.7. The ﬁtting parameters are shown in the following equation:

(3.1)

 

R, __ 7.263017, Cngm
*1 - 7.263001+(oofog‘iogfjffoo, Cll >1.4 1

where units of C11 and R"; are m0“ and p0, respectively. Once Cu is measured,

the value of Roof can be corrected according the above formula.

110

111

Table B.1: R", vs Cu

011
(mﬂ'l)

Rte]
(#9)

OR";
(#9)

 

 

1.0720
1.2123
1.4080
1.6040
1.8030
1.9030
2.0080
2.1330
2.2050
2.3130
2.4150
2.5270
2.6360
2.7620
2.8720
‘ 2.9490
2.9700
3.0990
3.2000
3.2850
3.3550
3.4130
3.4510
3.4900
3.5250
3.5480
3.5690
3.5800
3.6085
3.6270
3.6540
3.6720
3.6910
3.7035
3.7175
3.7290
3.7385

 

7.263016
7.263017
7.263018
7.263020
7.263022
7.263023
7.263026
7.263030
7.263033
7.263038
7.263044
7.263052
7.263064
7.263082
7.263102
7.263121
7.263126
7.263173
7.263222
7.263283
7.263351
7.263430
7.263498
7.263584
7.263677
7.263753
7.263832
7.263882
7.264024
7.264140
7.264336
7.264507
7.264712
7.264873
7.265081
7.265270
7.265453

 

0.000001
0.000001
0.000001
0.000001
0.000001
0.000002
0.000001
0.000002
0.000001
0.000002
0.000002
0.000002
0.000002
0.000002
0.000002
0.000002
0.000005
0.000005
0.000005
0.000005
0.000005
0.000005
0.000007
0.000007
0.000007
0.000007
0.000004
0.000004
0.000004
0.000004
0.000004
0.000004
0.000004
0.000005
0.000005
0.000005
0.000005

 

 

Bibliography

[Abr 79] E Abrhams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan,
Phys. Rev. Lett. 42, (1979)673

[Alt 81] B. L. Altshuler, A. G. Aronov, A. G. Larkin, and D. E. Khmel’nitski, Sov.
Phys. JETP 54, (1981)411

[Alt 85] B. L. Altshuler and A. G. Aronov, Electron-electron Interaction in Disor-
dered Systems, Efros, A. L. Pollak, M. (eds). Amsterdam: North—Holland
(1985)p1

[And 58] P. W. Anderson, Phys. Rev. 109, (1958)1492
[A31 68] L. C. Aslamazov and A. I. Larkin, Phy. Lett. 26A, (1968)238

[Bax 89] D. V. Baxter, R. Richter, M. L. Trudeau, R. W. Cochrane, and J. O. Strom-
Olsen, .1. Phys. France 50, (1989)1673

[Ber 69] G. Bergmann, Z. Phy 225, (1969)430

[Ber 82] G. Bergamann. Solid State Communication 42, (1982)815
[Ber 83] G. Bergmann, Phy. Rev. B 28, (1983)2914

[Ber 84] G. Bergmann, Phys. Rep. 107, (1984)]

[Ber 87] G. Bergmann, Phy. Rev. B 35, (1987)4205

[Bla 68] F. J. Blatt, Physics of Electronic Conduction in Solids, (1968)
112

113

[Dug 77] J. S. Dugdale, Electrical Properties of Metals and Alloys, (1977)
[Dug 87] J. S. Dugdale, Contemp. Phys. 28, (1987)547

[Edm 80] D. Edmunds, W.pratt Jr. and J. A. Rowlands, Rev. Sci. Inst. 51,
(1980)1516

[Fuk 81] H. Fukuyama and K. Hoshino, J. of Phy. Soc. of Japan 50, (l981)2l31

[Fuk 85] H. Fukuyama, Electron-electron Interaction in Disordered Systems, Efros,

A. L. Pollak, M. (eds). Amsterdam: North-Holland (1985)p155
[Hae 83] M. L. Haerle, Ph.D. Thesis, Michigan State University, (1983)
[Hei 83] V. O. Heinen, Ph.D. Thesis, Michigan State University, (1983)

[Hic 86] B. J. Hickey, D. Greig, and M. A. Howson, J. Phys. F: Met. Phys. 16,
(1986)L13

[Hic 87] B. J. Hickey, D. Greig, and M. A. Howson, Phy. Rev. B 36, (1987)3074
[Hik 80] S. Hikame, A. I. Larkin, and Y. Nagaoka, Prog. Theor. Phys. 63, (1980)707
[How 86] M. A. Howson and D. Greig, J. Phy. F 16, (1986)989

[How 88] M. A. Howson and B. L. Gallagher, Phys. Rep. 170, (1988)265

[Hsu 89] J. W. P. Hsu, A. Kapitulnik, and M. Yu. Reizer, Phys. Rev. B 40, (1989)7513
[Ime 74] J. Imes, Ph.D thesis, Michigan state University, (1974)

[Ime 75] J. Imes, G.Neiseihel and W.Pratt Jr., J. of Low Temp. Phys. 21, (1975)1

[Imi ] Imperial Metal Industries, PO. box 216, Witton Birmingham, B6 7ba, Eng-
land

[Joh 78] W. L. Johnson, C. C. Tsuei, and P. Chaudhari, Phy. Rev. B 17, (1978)2884

114

[Kaw 80a] A. Kawabada, Solid State Communication 34, (1980)431

[Kaw 80b] A. Kawabada, J. Phys. Soc. Japan 49, (l980)628

[Kaw 81] A. Kawabada, J. Phys. Soc. Japan 50, (1981)2461

[Kon 64] J. Kondo, Prog. Theor. Phys. 32, (1964)37

[Lar 80] A. I. Larkin, JETP Lett. 31, (1980)219

[Lee 82] P. A. Lee and T. V. Ramakrishnan, Phys. Rev. B 26, (1982)4009
[Lin 88] P. Lindqvist and o. Rapp, J. Phy. F 18, (1988)1979

[Lin 89] P. Lindqvist and G. Fritsch, Phys. Rev. B 40, (1989)5792

[Lin 90] P. Lindqvist, C. Rapp, A. Sahnoune, and J. O. Stréim-Olsen, Phys. Rev. B
41, (1990)3841

[Lou 74] O. Lounasmaa, Experimental Principles and Methods below 1K. Academic
Press, Inc., New York (1974)

[Mac 68] w. L. MacMillan, Phys. Rev. 167, (1968)331

[Mae 81] S. Maekawa and H. Fukuyama. J. Phys. Soc. Japan 50, (1981)2516
[Mak 68a] K. Maki, Prog. Theor. Phys. 39, (1968)897

[Mak 68b] K. Maki, Prog. Theor. Phys. 40, (1968)193

[Mcl 84] W. L. Mclean and T. Tsuzuki, Phy. Rev. B 29, (1984)503

[Miz 83] U. Mizutani, N. Akutsu and T. Mizoguchi, J. Phys. F: Metal Phys. 13,
(1983)2127

[M00 84] D. E. Moody and T. K. Ng, Proc. 17th Int. Conf. on Low Temperature
Physics, (1984)371

115

[Mot 89] A. C. Mota, P. Visani, and A. Pollini, J. Low Temp. Phys. 76, (1989)465

[0118 85] J. C. Ousset, S. A. Askenazy, H. Rakoto and J. M. Broto, J. Physique 46,
(1985)2145

[Pat 74] B. R. Patton, Proceeding of 13th International Confer. on Low Temp. Phys.,
Boulder, 172 (Plenum, New York, 1974) Vol. III.

[Pip 89] A. B. Pippard, Magnetoresistance in Metals, (1989)

[Ric 88] R. Richter, D. V. Baxter, and J. O. Strom-olsen, Phy. Rev. B 38, (1988)10421
[Ros 65] H. M. Rosenberg, Low Temerature Solid State Physics, (1965)

[Sam 89] A. Sahnoune and J. O. Str6m-Olsen, Phys. Rev. B 39, (1989)7561

[Sam 82] K. Samwer and H. V. L6hneysen, Phys. Rev. B 26, (1982)107

[San 85] J. M. B. LOpes dos Santos and E. Abrahams, Phy. Rev. B 31, (1985)172
[Sch 86] A. Schulte and G. Fritch, J. Phy. F 16, (1986)L55

[Sch 90] A. Schulte and G. Fritch, and E. Liischer, Z. Phys. B—Condensed Matter
78, (1990)457

[Sko 75] W. J. Skocpol and M. Tinkham, Rep. Prog. Phys. 38, (1975)1049

[Ste 81] S. D. Steenwyk, J. A. Rowland, and P. A. Schroeder, J. Phys. F: Metal Phys.
11,(1981)1623

[Tho 70] R. S. Thompson, Phys. Rev. B l, (1970)327

[Toy 84] N. Toyota, A. Inoue, K. Matsuzaki, T. Fukase, and T. Masumoto, J. Phys.
Soc. Japan 53, (1984)924

[Usa 69] K. D. Usadel, Z. Phy. 227, (1969)260

116

[Yin 87] Shi Yin, Ph.D. Thesis, Michigan State University, (1987)

[Zim 60] J. M. Ziman, Electrons and Phonons, (1960)

nrcurcsN STATE UNIV. LIBRnRIEs
111]] [1|] lll l]]|]|]]|] |||]]]||] [WW]
31293010296550