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IIZBIZY“)

IIIII III III IIII III III III III |III IIII III III III III III

This is to certify that the

thesis entitled

THE MAGNETOTHERMOELECTRIC EFFECT IN SINGLE
CRYSTAL ALUMINJM AND INDIUM AT LIQUID
HELIUM TEMPERATURES

presented by

Barry Jay Thaler

has been accepted towards fulfillment
of the requirements for

Ph.D . degree in _ED¥SJ$L5__'

 

/ Major professor

Date 7 ,- i ‘1 "-7- 7

 

0-7639

 

LIBRARY

Michigan State
University

 

THE MAGNETOTHERMOELECTRIC EFFECT
IN SINGLE CRYSTAL ALUMINUM AND
INDIUM AT LIQUID HELIUM TEMPERATURES

BY

Barry Jay Thaler

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Physics

1977

ABSTRACT
THE MAGNETOTHERMOELECTRIC EFFECT

_ IN SINGLE CRYSTAL ALUMINUM AND
INDIUM AT LIQUID HELIUM TEMPERATURES

BY
Barry Jay Thaler

We have studied the effect of a transverse mag-
netic field H upon the low temperature absolute thermo-
power S of very dilute aluminum and indium based alloys.
Measurements were carried out on single crystal samples
at temperatures T between 1.5 and SK and with magnetic
field strengths up to 20kG. The aluminum measurements
were performed on a sample grown along the [001] axis
and containing 50 ppm gallium The indium measurements
were performed on several samples containing from O to
500 ppm added impurities, which were grown along several
crystallographic axes. S of both metals consisted of
both a component which was oscillatory with increasing
magnetic field and a non-oscillatory component.

The non-oscillatory component of the aluminum
sample was obtained for H along both the {010} and
{110} families of axes. The data were consistent with
the equation s<n,r) = A(H)T + B(H)T3, allowing a

separation of the electron diffusion component,

Barry Jay Thaler

Sd = A(H)T, from the phonon drag component, Sg(H) = B(H)T3.
This separation allowed determination of the quantity

Asd = Sd(H + 0°) - Sdlfl = 0). The average value of ASd
for H along the crystallographically equivalent [100]
and [010] axes was ASd = (2.23 i .25)T x lO-BV/K, in
good agreement with both previous measurements on poly-
crystalline samples and with a calculation by Opsal and
Wagner, provided that a 45% electron-phonon mass enhance-
ment factor was included. For H along the {110}
family of axes, a value of ASd = (2.0 i,°2)T x 10-8V/K
was obtained. This value is slightly lower than the
average value of ASd along the {010} axes, in agree-
ment with the calculation of Opsal and Wagner.

The non-oscillatory component of the indium data
could not be unambigously separated into electron dif-
fusion and phonon drag components. It is concluded that
this was principally due to the fact that our available
temperature range was too close to the phonon drag peak.
For T between 3 and 5K, S becomes more negative with
increasing H. This is the antithesis of the aluminum
data in this T range, and is due to S of indium being
dominated by 59. For T between 1.5 and 2.5K, the high
field limit of S either gives strong evidence that it
will become more positive than the lower field values or
actually does become more positive. This is interpreted
as showing that Sd is increasing relative to So with

d

decreasing T.

Barry Jay Thaler

The size of the oscillatory component of S,
Sosc' due to oscillations in the density of states at the
Fermi energy is estimated for H along the [010] axis
for both the B-orbit in aluminum and the neck orbit of
the B-arms in indium. This estimate is several orders
of magnitude smaller than the measured oscillation
amplitudes of all large amplitude oscillations. We
interpret thistx>mean that these oscillations are caused
by magnetic breakdown, the only other mechanism expected
to cause this oscillatory behavior.

For aluminum, the above interpretation is sup-
ported by the fact that independent magnetoresistance
measurements strongly suggest that magnetic breakdown is
present for H directed along the [010] axis. In

addition, s°SC

was largest for H along this axis, being
an order of magnitude larger than the non-oscillatory
component at 20kG. As H was rotated away from the [010]
axis the oscillations decreased in magnitude, but per-
sisted throughout the (001) plane. The oscillations had
periods corresponding to those of the B-orbit. The

field and temperature variations of the [010] oscilla-
tions were dominated by a term exponential in the ratio
m*(T + TD)/H. Analysis using the de Haas-van Alphen pro-
cedure for determining effective masses and Dingle
temperatures yields m* = (0.093 : 0.01)me and T 2 l-4K.

D
*
This value of m agrees with the value obtained from

Barry Jay Thaler

de Haas-van Alphen measurements for the same orbit, and
corresponds to an electron-phonon mass enhancement of
approximately 40%.

Large amplitude, low frequency oscillations in
S were observed in indium below 2.5K for H between 10
and 20kG and directed along either the [010], [I01], or
[lIO] axes. The periods of these oscillations correspond
with those obtained from de Haas-van Alphen measurements
on the neck orbit of the B-arms of indium. Arguing by
analogy, comparison of the [010] indium data with similar
data for the [010] breakdown orbit in aluminum leads to
the conclusion that these oscillations also arise from
magnetic breakdown. This is the first evidence that there

may be magnetic breakdown in indium.

TO MY PARENTS

ii

ACKNOWLEDGEMENTS

It is a great pleasure to acknowledge my thesis
advisor Professor Jack Bass, whose suggestions and
criticisms throughout the course of this research were
invaluable. I would also like to thank Dr. Jon L. Opsal
and Professor Robin Fletcher for numerous helpful dis-
cussions. The time donated by Dr. E.L. "Terry" Stone
III and C.W. Lee in helping to make measurements is also
gratefully acknowledged.

I am indebted to z. Katsiapis for growing the
Al(Ga) single crystal. Specific thanks go to Mr. B.
Schumaker for preparing the alloys, and to D.A. Cady for
helping with the many aspects of sample preparation. I
would like to thank all the guys of the machine shop for
all their design tips and help in constructing the various
pieces of apparatus. I would also like to thank all the
secretaries who have been associated with the Physics
Department during my stay, especially the two SSMS
secretaries Jean Strachan and Delores Sullivan for all
their help. And last, but certainly not least, I'd like
to thank Diane Spero for helping to type this thesis.

I would also like to acknowledge the financial
support of the National Science Foundation.

iii

List

List

II.

III.

IV.

TABLE OF CONTENTS

of Tables

of Figures
INTRODUCTION
A. The Experimental Problem
B. Previous Work on the Magnetothermopower
of Aluminum
C. The Present Thesis

EXPERIMENTAL TECHNIQUE

A.

"1310001

Voltage Measurement

1. Nulling Technique

2. Field Sweep Technique

Thermometry

Sample Preparation

Alloys

Crystal Growth

Experimental Procedure and Data

Analysis

1. Nulling Technique Measurements
of S, K, and R

2. Field Sweep Technique Measurements
of S, K, and R

NON-OSCILLATORY THERMOELECTRIC PHENOMENA -
THEORY

A.

B.

Electron Diffusion

1. General Relations

2. Electron-Phonon Mass Enhancement
in S.

Phonon Drag

NON-OSCILLATORY THERMOELECTRIC PHENOMENA -
EXPERIMENT

A.

Aluminum

1. Introduction
2. Sample
3. Data and Analysis

iv

Page
vi

vii

33

33
33

40
48

56

56
56
59
59

Determination of the Non-

Oscillatory Results

AA for H Parallel to [010]

and [001]

AA for H Parallel to [011]

Effects of Magnetic Breakdown

Phonon Drag

4. Summary and Conclusions for
Aluminum

Indium

1. Introduction

2. Data and Analysis

3. Umkehreffect

4. Summary and Conclusions for Indium

u
C
N H

uww U)
U100.)

V. OSCILLATORY THERMOELECTRIC PHENOMENA -
THEORY

A.

B.

C.

Variation of the Density of States at
SF with H

Estimate of S Oscillations due to
Density of States Oscillations for

Aluminum and Indium
Magnetic Breakdown

VI. OSCILLATORY THERMOELECTRIC PHENOMENA -
EXPERIMENT

A.

B.

The Fermi Surfaces of Aluminum and
Indium
Aluminum .
1. Introduction
2. Experimental Procedure
3. Data and Analysis

3.1 H Parallel to [010]

3.2 H Along other Directions
in the (100) plane
Indium -
1. Introduction
2. Experimental Procedure
3. Data and Analysis

List of References

Appendix I :

Appendix II:

Calculation of Anisotropy of r(k)
for Al(Ga)

30(8)

Determination of as

Page

59

62
64
65
66

69
70
70
74

9O

92

92

95
102

112

112
114
114
116
116
116

125
132
132
133
133

143

148

150

Table

II-1

IV-1

VI-l

AI-l

LIST OF TABLES

Some characteristics of the samples
used in this thesis

Values of B(H) for Al(Ga)

*
Values of m /me and TD for two

orthogonal directions of the magnetic
field H and for different assumed
values of n for Al(Ga)

Average reciprocal relaxation times

for Al(Ga) obtained from Sorbello's
approximate phase shift mode1(7)

vi

Page

27
67

124

149

Figure

11.1

11.2

II.3

II.4

III.1

III.2

III.3

LIST OF FIGURES

The Superconducting Chopper-Amplifier

System: a) The superconducting chopper-
amplifier circuit; b) Schematic (actual
size) of the device (after Averback(3)).

The variation with magnetic field of the
gain of an unshielded chopper-amplifier
system

The variation with magnetic field of the
gain of a shielded chopper-amplifier
system

The Vacuum Chamber: A) lucite block,

B) heat sink for resistor leads,

C) sample, D) Glastic backing, E) car-
bon resistor, F) heater, G) cold sink
(O.F.H.C. copper), H) binding post,

I) flange for Pb O-ring, J) vacuum line,
K) epoxy seal, L) support rod, M) vise
clamp, N) sample support (brass) which
is attached to the cold sink

e vs. k relations for free electrons
(dashed curve) and for quasi-particles
(solid curve)

The change of electron wavevector
accompanying the annihilation of a
phonon by a (a) Normal process,

(b) Umklapp process (after Barnard(23)

)

Schematic diagrams of sections through

the 2nd (upper) and 3rd (lower) zones of
Al showing electron-phonon scattering
events (after Gripshover, et. al. (2 ))

vii

Page

13

15

18

46

51

54

Figure

IV.1

IV.2

IV.3

IV.4

IV.5

IV.6

IV.7

IV.8

IV.9

The variation of S with magnetic field
at 3.55K for the Al(Ga) sample. The
magnetic field is directed along the
[010] axis. The maxima and minima of
the quantum oscillations are repre-
sented by circles, and the non-
oscillatory component of S is repre-
sented by the crosses

The temperature dependence of the non-
oscillatory component of S of the
Al(Ga) sample for: H = lSkG and
parallel to [010] (squares); H = lSkG
and parallel to [100] (triangles);

H = lSkG and parallel to [110] (crosses),

H = OkG (circles). The full symbols

indicate data obtained with H reversed.

The temperature dependence of the non-
oscillatory component of S for sample
In(Ga) with H directed along [010]

The temperature dependence of the non-
oscillatory component of S for sample
In(Ga) with H directed along [001]

The temperature dependence of the non-
oscillatory component of S for sample
In(Sn)-II with H directed along [001]

The temperature dependence of the non-
oscillatory component of S for sample
In(Sn)-II with H directed along [110]

The temperature dependence of the non-
oscillatory component of S for sample
In(Sn)-I with H rotated 12° from the
[101] in the (101) plane

The temperature dependence of the non-
oscillatory component of S for sample
In(Sn)-III.

The magnetic field dependence of S for

sample In(Sn)-II with H directed along
[110]

viii

Page

61

63

75

76

77

78

79

80

81

Figure Page

IV.10 The temperature dependence of the
Lorenz ratio L divided by the ideal
Lorenz ratio L0 for samples:

A) In(Sn)-III, with H directed along
[001];
B) In(Sn)-II, with H directed along
[001];
C) In(Ga), with H directed along [010] 84

IV.ll Variation of S with rotation angle 6
at H = lOkG and T = 4.7K for
sample In-I. H was rotated about the
[101] axis. 6 = 213° corresponds to
H directed along [010] 87

IV.12 Variation of R (upper curve) and 8
(lower curve) with rotation angle 6
at H = lOkG for sample In(Sn)-I. H
was rotated about the [101] axis.
6 = 202° corresponds to H directed
along [010]. The R curve was measured
at 4.2K and the 5 curve at 4.3K 88

V.l Variation of the Fermi surface with
field, showing the increasingly
spherical shape as the field becomes
weaker. The field decreases going (a)
to (d) (after Adams and Holstein(43)) 93

v.2 (a) Free electron orbit in a magnetic
field. (b) In the periodic potential
of the lattice, the orbits are recon-
nected at the zone boundary; but in a
strong magnetic field may jump back to
the free electron path (after Ziman(51)) 104

v.3 Ashcroft's 4-OPW pseudopotential model

for the electron-like 3rd zone of the
Fermi surface of aluminum (after Abele
and BlatthZI) 106

v.4 Central [100] section through the (63)
aluminum Fermi surface (after Ashcroft )107

V1.1 (a) The Brillouin zone for face-centered
tetragonal indium; (b) the indium free-
electron model hole surface; (c) the
indium third zone electron B-arm sur-
face (after Hughes and Shepherd(52)) 113

ix

Figure

VI.2

V1.3

V1.4

V1.5

VI.6

The variation of S with magnetic field
at T = 3.55K for the Al(Ga) sample.
H is along the [010] axis. The solid
line labeled S = 0 represents the
x-y recorder output when there is no

' applied temperature gradient. The

broken line indicates the non-
oscillatory component of S. The
oscillating curve represents raw data,
uncorrected for a field dependent
amplification of the measuring system.
The magnitude 9f this correction is in-
dicated by the manner in which the

S = 0 curve and the broken line approach
each other as H increases. When the data
are corrected, these two curves are equi-

distant at all fields above about llkG
to within experimental uncertainty

The variation of the quantity

3/2 osc . .
H S /T (plotted on a logarithmic
scale) with H”1

along [010]

The variation of the quantity SOSC/T

(plotted on a logarithmic scale) with H-
for a series of temperatures for sample

Al(Ga). H was directed along [010]

The variation of the quantity

H-B/ZSOSC/T (plotted on a logarithmic

scale) with H"1

along [010]

The variation with temperature of the

slopes of the lines shown in Figure V1.4
for H parallel to [010] (circles) and of

similar slopes of similar lines for H

for a series of temper-
atures for sample Al(Ga). H was directed

for a series of tempera-
tures for sample Al(Ga). H was directed

Page

117

120

121

122

parallel to [100] (triangles). The upright

triangles represent initial measurements
with H parallel to [100]. To evaluate the
effects of changes in sample treatment and
alignment, after these initial measure-
ments were completed the sample was re-
moved from the holder, its potential leads

X

Figure

V1.7

V1.8

V1.9

V1.10

V1.11

Page

were removed, it was reannealed, the

potential leads were reattached and the

sample was returned to the holder. The
resulting data are indicated by inverted
triangles 123

The variation of S with magnetic field

at T = 2.4K for the Al(Ga) sample. H

is rotated 20° from the [010] axis in

the (001) plane. The solid line labeled

S = 0 represents the x-y recorder output

when there is no applied temperature

gradient. The oscillatory curve repre-

sents raw data, uncorrected for the field
dependent amplification of the measuring

system 125

The variation of S with magnetic field at

T = 2.8K for the Al(Ga) sample. H is

along the [110] axis. The solid line

labeled S = 0 represents the x-y recorder
output when there is no applied tempera-

ture gradient. The oscillating curve
represents raw data, uncorrected for the

field dependent amplification of the

measuring system 127

The variation of S with magnetic field at

T = 2.3K for H along a series of directions

in the (001) plane of the Al(Ga) sample.

0° corresponds to H directed along [100] 129

Variation of the oscillation period of the
third zone B-orbit with rotation angle in

the (001) plane. This figure displays

our present magnetothermopower data, the

de Haas-van Alphen data of Larson and
Gordon,(32) and the results of a pseudo-
potential calculation by Ashcroft(53) 130

The magnetic field dependence of S for H
aligned along the indicated crystallo-
graphic axes. The lines labeled S = 0
represent the recorder output when there
is no applied temperature gradient. The
oscillating curves represent raw data,
uncorrected for the field dependent

xi

Figure

V1.12

V1.13

amplification of the measuring system.
The amplification decreases by about 30%
over the field range shown. The curves
for H aligned along [101] and [010] are
from sample In(Sn)-I and the curve for H
along [110] is from sample In(Sn)-II

The variation of the quantity HSOSC/T
(plotted on a logarithmic scale) with
l/H for sample In(Sn)-I at T = 1.67K.
H was directed along [010]

Variation of S with rotation angle
measured from the [010] axis, at H = lSkG
for aluminum and indium single crystals.
The aluminum data are from sample Al(Ga).
The indium data for H rotated about the
[001] axis are from sample In-II and

the data for H rotated about the [lOl]
axis are from sample In(Sn)-I

xii

Page

134

136

140

I. INTRODUCTION

A. The Experimental Problem

 

Thermoelectric effects are usually classified in
one of three categories: the Seebeck, Peltier; or Thomson
effects. However, all three effects are related by the

(1)

Kelvin-Onsager Relations, and therefore provide equi-
valent information. The experiments forming the basis of
this thesis are measurements of the Seebeck effect. This
effect is characterized by the generation of an electric
field E by means of a temperature gradient VT which is
applied along an electrically isolated sample. Using

(2)

dyadic notation, the thermopower tensor 4S is defined

as

‘E = E (’v’T)'1 . (1.1)

. + + .

Since E = -VV, we may write

‘3 = -VV(VT)-1 . (1.2)
This is equivalent to

H _ -4 g! as

S - 1 dT 3 (1.3)

A

where i and j are unit vectors in the directions of
the voltage and temperature differences respectively.

1

 

 

Since measurements are performed over paths of

finite length, the experimentally measured thermopower,

‘Séxp, is defined as

where AV and AT are measured voltage and temperature
differences. If AT 5 dT then S 5 Sexp; this may be
attained when AT << T. Since the present measurements
were performed at temperatures between 1.5 and SK, temper—
ature differences of approximately 0.1 to 0.3K were used.
A typical value of S for aluminum or indium is lO-BV/K,

8 9

which implies that 10- - 10- V must be measured. To

make measurements with 1% accuracy required a voltage

10 - lO-llv. This is not a trivial

sensitivity of 10-
task in the presence of a large magnetic field (20kG.).
However this problem can be solved by using a supercon-

ducting chopper-amplifier.

B. Previous Work on the Magnetothermopower of

 

Aluminum
The thermopower is the most difficult transport
property of a metal to properly calculate. A major reason
for this is that thermopower is the transport property
which is most sensitive to the details of the scattering
processes. However, Averback found a property of the

electron-diffusion component Sd of the thermopower of

 

 

aluminum which was insensitive to the scattering details

and depended only on the host meta1.(3’4) Using a number

of polycrystalline samples of very dilute aluminum alloys,
he showed that although both the zero field and high field
values of S (note that S saturates in the high field
limit for aluminum or any other uncompensated metal with

no open orbits) varied from sample to sample, the difference
between these two values, ASd = Sd(H + w).- Sd(H = 0), was
approximately constant -- for all samples ASd/T varied
between 2.1 - 2.6 x 10-8V/K2.

Averback and Wagner(s)

calculated ASd using no
adjustable parameters and obtained the value ASd/T =
1.6 x 10-8V/K2, about 30% smaller than experiment. Some
possible sources of this 30% discrepancy are:
(l) The measurements were performed on poly-
crystalline samples, while the calculation
is appropriate for H directed along a
four-fold symmetric axis of a single crystal.
(2) The impurities in the samples might have
been anisotropic scatters in aluminum, while
the calculation assumed isotropic scattering.
(3) The 1-OPW Fermi surface used by Averback and
Wagner in their calculation might be in-
apprOpriate for comparison with experiment.
(4) The effects of mass enhancement were not in-

cluded in Averback and Wagner's calculation.

(Averback and Wagner neglected effects of
electron-phonon mass enhancement because
Prange and Kadahoff‘G) had claimed that such
effects were not present in the electronic

-properties of metals.)

C. The Present Thesis

 

This thesis began as an attempt to measure the
transverse magnetothermopower of an aluminum sample which
satisfied the requirements of the calculation of ASd made
by Averback and Wagner. We hoped that such measurements
would isolate the source of the discrepancy between theory
and experiment.

A single crystal foil, oriented so that the mag-
netic field could be directed along either a four-fold
symmetric [010] crystallographic axis or a two-fold
symmetric [110] axis, was used. This crystal contained
SOppm Ga, an impurity which should be a nearly isotropic
scatterer in aluminum.(7) The data obtained for ASd
with this single crystal are compared below with the data

from polycrystalline samples reported by Averback,

(4) (5)

et. al.; with the calculation of Averback and Wagner;

with an improved, 4-OPW version of this same calculation

(8)

by Opsal and Wagner which was stimulated by the experi-

mental results to be described; and, finally, with the
Opsal and Wagner calculation including the effects of mass

(8)

enhancement.

The results obtained with this aluminum single
crystal stimulated us to extend magnetothermopower measure-
ments to indium single crystals to see whether ASd be-
haved the same as in aluminum. It was anticipated that
such measurements would be interesting because although
aluminum and indium are both Group III metals, they
crystallize with different structures; aluminum in a face
centered cubic (FCC) structure, and indium in a face
centered tetragonal (FCT) structure. The slight tetragonal
distortion of indium causes some axes which are four-fold
symmetric in the FCC structure to be only two-fold symmetric
in the FCT structure. Thus indium would allow a study of
the effects of this small distortion on ASd. Unfortunately,
this study yielded results which did not allow the electron-
diffusion component of the magnetothermopower to be un-
ambiguously separated from the phonon drag component. The
indium data will be presented and will be analyzed to de-
termine the source of this lack of separation. Some
indium samples exhibited a significant magnetothermopower
umkehreffect; this will also be discussed.

In the process of making the aforementioned studies,
giant quantum oscillations were discovered in both metals
for H along a number of crystallographic directions. For
H along the [010] axis in aluminum these oscillations

could easily be distinguished from the noise for T between

1.7 and 4.8 K; therefore the temperature dependence of the

oscillation amplitudes was systematically studied. In
indium the oscillations could only be distinguished from
the noise for T between 1.5 and 2.5 K; therefore an
oscillation amplitude study similar to that performed for
the [010] direction in aluminum was not attempted. The
experimental results will be compared with theories which
predict oscillatory behavior. Also, the periods of the
magnetothermopower oscillations will be compared with the
same periods obtained from independent magnetoresistance
or de Haas-van Alphen measurements.

The remainder of the thesis is organized as
follows:

Section 11 provides a description of the experi-
mental apparatus and measurement techniques.

Section 111 furnishes the theoretical background
'for the calculation of magnetothermopower. Particular
emphasis is placed on the role of electron-phonon mass en—
hancement in thermoelectricity. In section IV the experi-
mental results for the non-oscillatory component of the
magnetothermopower for both aluminum and indium, are pre-
sented.

Section V presents the theory of the oscillatory
component of the magnetothermopower. These oscillations
can be caused by either oscillations in the density of
states at the Fermi level, or by magnetic breakdown. Both

of these pictures are developed. A simplistic calculation

is used to estimate the magnetothermopower oscillation
amplitude expected from oscillations in the density of
states. In section VI the experimental data for the
oscillatory component of the magnetothermopower of both
aluminum and indium are presented and compared with the

simple calculation.

II. EXPERIMENTAL TECHNIQUE

A. VOltage Measurement
The voltage measuring system is shown in Figure
11.1. This chopper-amplifier system was designed and de-

SCIiDEdeldetail by R.S. Averback.(3)

In view of this,
only the techniques of voltage measurement will be de-
scribed hithis thesis. The system was operated in two
modes: (1) At steady magnetic fields the signal gen-
erated by the sample was nulled out, using the chopper-
amplifier as null detector, (2) In continuously varying
magnetic fields the signal generated by the sample was
directly measured using the chopper-amplifier as a linear
amplifier. Operation in the first mode will be described
in the next section entitled "Nulling Technique". Opera-

tion in the second mode will be described in the section

entitled "Field-Sweep Technique".

1. Nulling Technique

 

As can be seen from Figure 11.1, the chopper-
amplifier system contains a reference resistor which is
in the liquid helium bath. This reference resistor was
a small segment of copper wire having a resistance of
about BuQ, which remained constant to within our
ability to measure it (< 2%) at all temperatures and

fields employed in these measurements.

8

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Figure 11.1: The Superconducting Chopper-Amplifier System:
a) The superconducting chopper-amplifier circuit; (b; Sche-
matic (actual size) of the device (after Averback( ).

10

When a known current (produced by a series of
12V dry cell batteries) is passed through the reference
resistor, a known voltage is generated in the chopper-
amplifier circuit. This voltage is used to null out the
voltage which is being generated by the sample. At null
condition there will be no net current flowing in the
circuit, so the panel meter (output) of the phase sensitive
detector will show no deflection. The accuracy of the
technique is limited by the circuit noise, which appears
as a small oscillation of the needle of the panel meter.
Using a time constant of 3 seconds on the phase sensitive
detector (PAR model JB-4), this noise was usually

2 - 4 x io'llv in zero field and increased to

10

1 - 3 x 10' v at 20kG.

2. Field Sweep Technigge

 

At all magnetic fields used in these experiments
the output of the chopper-amplifier system was quite linear
(to within 2%). So after independently calibrating the
panel meter, the voltage generated by the sample was
measured by reading the deflection of the needle of the
panel meter. The panel meter was calibrated by passing a
known current through the reference resistor under con-
ditions where the sample was not generating a voltage, and
then measuring the resulting deflection of the needle.

In addition to monitoring the output of the phase

sensitive detector on the panel meter, the analog output

11

was fed directly into the y-axis input of an x-y recorder
(HP model 7004B). Sweeping the magnetic field at a con-
stant rate and feeding the analog ouput of the Hall probe
gaussmeter (F.W. Bell model 660) into the x-axis, resulted
in a continuous recording of the voltage generated by the
sample as a function of H. Since a changing magnetic field
induces a voltage within our measuring system, the magnetic
field had to be swept up twice - once with the sample gen-
erating a voltage, and once with it generating zero voltage
(to define a zero voltage baseline). The difference be-
tween these two lines gives the voltage generated by the
sample. Data taken using this field sweep method agreed
with data taken independently using the nulling technique.
The field sweep technique is advantageous for recording
data when the voltage generated by the sample varies rapidly
with field (e.g. giant quantum oscillations).

There are two reasons why it is important not to

sweep the field too rapidly:

(1) If the field is swept at a rate such that the
thermally induced voltages are changing faster
than the time constants allow the Circuit to
respond, then the voltage variations will be
smoothed out (i.e. reduction of amplitude of
the quantum oscillations).

(2) If the field is swept too fast, the voltage

induced by the changing magnetic field may

12

cause a component of the voltage measuring

system to saturate, thus giving "false" data.

The field sweep technique has some disadvantages

compared to the nulling technique:

1)

2)

The field sweep technique offers less accuracy
because the quantity of interest must be ob-
tained by taking the difference between two
lines on a recording, each of which contains
noise. Additionally there may be slow varia-
tions in both the signal and the zero baseline
which lead to systematic errors. Comparisons
between equivalent data suggest that the field
sweep technique is about a factor of three less
accurate than the nulling technique.

A second problem with the field sweep technique
is that the gain of the chopper—amplifier
system is field dependent (at 20kG the gain

can be as much as 50% less than at lOkG).
Figure 11.2 shows a typical field dependence

of the amplification of the system. The raw
data must be corrected for this field depen-

dent gain.

Toward the end of the present thesis research,

the effect of shielding the chopper-amplifier with 5 mil

thick lead foil was investigated. When the shielding ex-

tended from above the chopper-amplifier to about 8cm below

13

 

 

(N I-

t:

Z I“

:3

a3 ..

(r 3

<1 I

Z I--

<i

(D l--—.
O t- l I J

IO (5 20
H(KG)

Figure 11.2: The variation with magnetic field of the gain
Of an unshielded chopper-amplifier system.

14

it, the amplification factor only decreased about 10% be-
tween lOkG and 20kG, rather than the 50% obtained pre-
viously. Figure 11.3 shows the field dependence of the
amplification factor of a shielded chopper-amplifier
system. Note that this field dependence is linear over a
larger portion of the region between 10 and 20kG than it

was for unshielded chopper-amplifier (see Figure 11.2).

B. Thermometry

 

The temperatures of the thermocouple junctions
were obtained using standard carbon resistance thermo-
metry techniques. Nominal 100 ohm, 1/8 watt Ohmite re-
sistors were used. These resistors are highly tempera-
ture sensitive between 1.5 and 5K, and they also have a
low magnetoresistance for H < 20kG. This combination of
properties makes these resistors quite satisfactory for
use as thermometers.

The resistance of the carbon resistors was

(9)

measured with an a.c. Wheatstone bridge. The off
balance signal was detected by a phase sensitive detector
(PAR model HR-8). This technique had a resolution of
about 0.1 ohms (~0.0004K at 5K and less than 0.0001K at
2K). I

The circuit was arranged so that RC (the resis-
tance of the resistor at the cold end of the sample) and

AR (AR==RC - RH' where RH 13 the re31stance of the re-

sistor at the hot end of the sample) could be alternately

15

GAIN (ARB. UNITS)
T

 

IO l5 7 20
H(KG)

Figure 11.3: The variation with magnetic field of the gain
of a shielded chopper-amplifier system.

16

measured. From these two measurements, R.C and RH could
be deduced. As a check against self-heating, the power
level of the phase sensitive detector (current source
of the bridge) was varied, and.the level which gave the
same measured resistance as the next lower level was used.
The carbon resistors were calibrated below 4.2K
against a pumped helium bath. The vapor pressure was
measured by a mercury manometer, which was mounted with a
silvered scale backing. The pressure could be read with
an accuracy of 0.2 - 0.3 mm Hg. These pressures were
corrected for hydrostatic head pressure and for the tem-
perature dependence of the mercury density. The pressure
vs. temperature relationship and the aforementioned
pressure corrections were obtained from the National Bureau
of Standards 1958 tables. The helium bath temperature was
controlled by a Walker Manostat, which controlled to
better than 0.001K above 2.5K and also between 2.0K and
the A-point of the liquid helium. Between the A-point
and 2.5K the system was somewhat less stable, presumably

(3)

due to bubbling on the vacuum can surface. Below 2K
the pressure difference across the diaphragm of the
Walker Manostat was not large enough to completely close
off the pumping line; thus the temperature normally de-
creased slowly while the Walker Manostat was trying to
control the bath temperature. I

The carbon resistors were calibrated every time

the cryostat was cooled, by fitting their resistances to

17

(10)

the equation:

lnR

*5
T ) + b . (II.1)

1n R = m (

For 1009 Ohmite resistors, typical values of m and b were
2.31 and 4.25 respectively. The temperature could be ob-
tained for any R by simply inverting eq. (11.1). The
calibrations were usually limited to a range of about 1K.
The "goodness of fit" was checked by substituting the
calibration resistances into the inverse of eq. (11.1) to
obtain the temperature corresponding to that resistance.
The calculated temperature rarely deviated from the
calibration temperature by more than 0.002K, and quite
often deviated by less than 0.001K. To obtain tempera-
tures between 4.2K and 5K, the calibration for the ranges
3.4 to 4.2K or 3.6 to 4.2K was simply extrapolated to
higher temperatures. Earlier measurements using a
calibrated germanium thermometer demonstrated that this
procedure introduced no significant error up to about
5K.(4’

It is well known that at a given temperature the
resistance of a carbon resistor slowly drifts with time.
A simple method was used to correct for this drift. Be-
fore this correction is described, it would be beneficial
to review the placement of heaters on the sample. Figure
11.4 shows that two heaters are attached to the sample -

one at either end. Aside from the connection of one end

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19

of the sample to a cold sink, which is in direct contact
with the helium bath, the sample is thermally isolated.
The end of the sample attached to the cold sink will be
called the "cold" end and the other end will be called the
"hot" end. When the heater at the "hot" end of the sample
(the "hot" heater) is turned on, a temperature gradient
is produced along the sample; this also causes the tem-
perature of the sample at the position of the "cold"
resistor to be higher than the bath temperature. After
measuring RC and AR the "hot" heater is turned off and
the "cold" heater is turned on. The "cold" heater heats
the sample to a uniform temperature. The heating power
of the "cold" heater is adjusted so that Rc has the same
value as when the sample was heated only by the "hot"
heater. Rc and AR are measured. The inverse of eq (11.1)
is used to calculate the temperature at either end of the
sample, and therefore T is determined. If the resistance
of the two resistors drifts with time this drift should
show up as a false temperature difference when only the
"cold" heater is on. By substracting this false tempera-
ture difference from the temperature difference obtained
using only the "hot” heater, the correct AT is determined.
The false temperature difference was usually less than 5%
for temperature differences greater than 0.03K.

If the sample is not completely isolated, the

"cold" heater will produce a real temperature difference,

20

which should be indicated by a thermally induced voltage.
In our measurements, such voltages where typically about
1% of those produced by the "hot" heater. It was there-
fore concluded that the real temperature difference pro-
duced by the "cold" heater was about 1% of that produced

by the "hot" heater.

C. Sample Preparation

 

Figure 11.4 shows how the measuring probes and
other apparatus were mounted on a sample. In order to
have enough surface area to accommodate all these elements
and also to maintain a sufficiently large temperature
difference, the samples were about 7cm long, and 2 to 5mm
wide. The following procedure was utilized to quickly
mount the elements on the sample while introducing a
minimum of cold work.

First the potential and current leads were
attached. These leads were 3 mil copper-clad NbTi wire,
chosen because NbTi remains superconducting at all tempera-
tures and magnetic fields employed in these experiments.
For aluminum samples, the leads were spotwelded on. In
preparation for spotwelding, the sample was glued to a
template using Duco Cement. The insulation and copper-
cladding were removed from both ends of each NbTi wire.
The leads were then varnished, using G.E.-703l varnish,
to the sample, so that one end of each lead was in posi-

tion for attachment. The leads ran from these points

21

to the center of the sample from where each pair of leads
was twisted tightly together as it led away from the
sample.. The potential and current leads were then spot-
welded to the aluminum sample. The sample was removed front
the template by soaking in an acetone bath, and then glued
to a Glastic backing, again using Duco Cement. Glastic

was chosen as a backing material because it is an in-
sulator whose thermal expansion is similar to that of a
metal.

Since indium is a much softer material than
aluminum, the potential and current leads were either
"acid-welded" or soldered to the indium samples; this
avoided the damage to the sample inherent in spotwelding.
Indium has the property that when the surface is well
cleaned with hydrochloric acid, it will bond to another
indium surface which has been similarly prepared. This
permitted NbTi leads which had been previously "wetted"
with indium to be "acid-welded" to the sample. Alter-
natively the leads could easily be soldered to the in-
dium samples using Rose's metal and Superior #30 liquid
soldering flux. Tests showed that the data were the same
for both methods of lead attachment. For both of these
methods the copper-cladding was not removed from the lead
ends which were to be attached to the sample. This was
done because it was much easier to "wet" the copper-

cladding with either indium or solder than it was to "wet"

22

NbTi. Since tests showed that the thermOpower of indium
with either method of lead attachment was zero (to within
experimental uncertainty), when the indium was in the
superconducting state, it is concluded that the copper-
cladding did not introduce any measurable spurious vol-
tages. Since the potential and current leads were not
spotwelded to the indium samples, these samples could be
glued to the Glastic backing before the leads were attached
to the sample.

Once the leads were attached and the sample
mounted on the Glastic, the sample was placed in a brass
sample holder which held the sample parallel (to within
1°) to the magnet pole pieces. The sample was isolated
from the brass holder by 2 small lucite blocks, one at
either end of the sample. These blocks were varnished
between the Glastic and the brass holder using G.E.-7031
varnish. In the case of aluminum, one end of the sample
was then mechanically clamped to the cold sink. 1n the
case of indium, a small piece of very pure indium with
about the same cross-sectional area as the sample was
clamped to the cold sink, and the end extending from the
clamp was melted and attached to one end of the sample.
This was done to avoid the strains which would be intro-
duced if the indium single crystal were directly clamped

to the cold sink.

23

The other ends of the potential and current leads
were then spotwelded to permanent leads which ran out of
the sample chamber through an epoxy seal (which was con—
structed in accordance with the procedure outlined by

Anderson‘ll)

). Since the insulation and copper cladding
had already been removed from the last centimeter or so
of these ends, only the superconductor, NbTi, remained.
Superconductors are poor thermal conductors, and therefore
these leads shouldn't provide a significant heat leak.
This was verified by the fact that the thermal emf gen-
erated when the sample was heated only with the "cold"
heater, was typically 1% or less of that generated by the
"hot” heater.

Next the carbon resistors and heaters were mounted.
The insulation on the carbon resistors was ground off on
one side. Cigarette paper saturated with G.E.-7031 varnish
provided electrical insulation between the resistor and
sample, and the varnish held the resistor in place. The
leads to the resistors were 3 mil. manganin wire. To pre-
vent large heat leaks through these leads, the manganin
wires were attached toarsmall heat sink at a temperature
within 0.01 to 0.02K of each carbon resistor before being
brought out to connections external to the sample. These
small heat sinks were made by varnishing several turns of
the manganin wire to copper posts which were either var-

nished to the aluminum samples or soldered (using Cerrolow

24

#117 solder) to the indium samples, about 0.5cm from the
carbon resistors.

The heaters were constructed by wrapping about
5000 of Evanohm wire (~130/cm) around a copper post and
varnishing the wire to the post. The copper post was then
attached to the sample in the same manner as the heat
sinks. The heaters were operated using a d.c. current
of l to lOmA.

The several leads from the heaters and resistors
were soldered to permanent leads on a vector board which
was mounted on the brass sample holder. The permanent
resistor leads were 3 mil. managanin wire and the per-
manent heater leads were 38 awg. copper wire. All these
leads entered the sample chamber via the vacuum line and
were heat sunk by varnishing them to a copper post which
was tightly screwed into the main frame of the cryostat
(which was in contact with the helium bath).

Finally, the brass vacuum can was attached. The
seal between the can and the sample chamber was made by
squashing a Pb + 0.05%As O-ring between the can and the
sample chamber with 12, 2-56 x 3/8" stainless steel Allen
head screws. The O-ring was made by wrapping Pb + 0.05%As
wire, which had already been greased with Apiezon M
grease, around the flange (see Figure 11.4) and twisting
it tight. Not a single O-ring prepared in this manner

leaked. A few times tempered steel (black) screws were

25

substituted for the stainless steel screws. In almost
every instance these seals leaked at 4.2K.

This entire sample preparation procedure could
be completed in 1-2 days. This procedure was conceived
to introduce a minimal amount of cold work to the rather

delicate samples.

D. Alloys

Both the aluminum and indium alloys were mixed
by Mr. B. Shumaker, using the procedure described by
R.S. Averback.(3) The only difference in procedure was
that the indium master alloys were prepared with a con-
centration of about 1% solute material. The "pure"
aluminum was 69 grade aluminum supplied by Cominco Inc.
The "pure" indium was either 69 grade indium supplied by
Cominco Inc. or 59 grade indium supplied by the Indium
Corporation of America. The solute materials, gallium
and tin, were 69 grade. The aluminum alloys were rolled
into 20 mil. thick foils, while the indium alloys were

rolled into rods with a 2mm square crosssection.

E. Crystal Growth

The Al(Ga) single crystal was grown by 2.

(12'13) Before

Katsiapis using a strain anneal technique.
mounting, the single crystal was annealed in air for 1

hour at 400C.

26

The indium crystals were grown in an atmosphere
consisting of 90% argon and 10% hydrogen using a simple
zone refiner and seeds of known crystallographic orienta-
tion. First the indium alloy and seed were placed in a
crucible made from Spectroscopically pure carbon. A heat
lamp housed in an elliptical reflector was used to pro-
duce a molten zone about 1.5 to 2.5cm. long. The molten
zone was first produced at the gnseeded end of the indium
alloy. This zone was then slowly moved along the entire
length of the alloy until a small portion of the seed
melted. The zone was then moved slowly away from the seed
back along the entire length of the alloy. This two-pass
procedure was adOpted with the intention of achieving a
homogeneous impurity distribution, since an odd number of
passes is expected to produce an impurity concentration
gradient along the sample. Once grown, the indium single
crystals were not further annealed. During initial trials
the indium sometimes separated into two pieces within the
molten zone. This problem was eliminated by completing
the etching (using concentrated HCl) and rinsing (using
double distilled H20 followed by methyl alcohol) procedure
between 5 and 10 minutes before the indium was placed in
the crucible. The crucible was then immediately placed
in a tube which was evacuated so that the argon-hydrogen

atmosphere could be introduced.

27

All samples were characterized by x-ray (Laue)
pattern analysis. Table 11—1 shows the intended impurity
concentration, the measured resistance ratio, RRR
(R (300K)/R(4.2K)), and the crystallographic axis along
which the crystal was grown, for all crystals used in this
thesis.

TABLE 11-1: Some characteristics of the samples used in
this thesis

Nominal Impurity Orienta- Misori-
SAMPLE Conc. RRR t1on entae
t1on
Al(Ga) 50 ppm 1,400 [001] < 1°
In-I 0 15,400 [101] 2°
In-II 0 4,350 [001] 3°
In(Ga) 50 ppm 7,400 [100] -
In(Sn)-1 50 ppm 2,300 [101] 3°
In(Sn)-II 100 ppm 1,650 [110] 2°
In(Sn)-III 500 ppm 390 [110] 2°

The rather high RRR of the In(Ga) sample indicates
that considerably less than 50 ppm gallium was in solution.
Single crystals were considered satisfactorily
oriented when the crystallographic axis along which the
sample was grown lay within a few degrees of the sample
axis. The misorientation was determined from X-ray
pattern analysis. The misorientation for each sample is

also given in Table 11-1. Since the sample holder aligned

28

the sample axis (not the crystallographic axis) trans-
verse to the magnetic field, the magnetic field could not
be aligned exactly along the crystallographic axes. How-
ever, in the interest of simplicity, the magnetic field
will be referred to as being aligned along a given
crystallographic axis when it was aligned as closely as

possible to that axis.

F. Experimental Procedure and Data Analysis
1. Nulling Technique Measurements of S, K, and R

Thermopower measurements using the chopper-
amplifier as a null detector were performed in the follow—
ing manner. The measurements were always begun with the
helium bath at 4.2K. The "hot" heater was turned on,
producing a temperature gradient along the sample, and the
power was recorded. Within a few seconds the sample
reached a steady state condition, and the thermally gen-
erated emf was then nulled by passing current through the
standard resistor. The current necessary for nulling was
recorded. RC and AR were also recorded. The "hot"
heater was then turned off and the "cold" heater turned
on. The power input to this heater was adjusted so that
Rc had approximately the same resistance as it did when
only the "hot" heater was on; this power was then re-
corded. RC and AR were again recorded. Sometimes a
relatively small thermal emf was generated by the "cold"

heater, this is probably indicative of a small heat leak.

29

When this occurred, this voltage was nulled and the current
necessary for the nulling was recorded. When analyzing
the data, this latter voltage was subtracted from the
thermal emf generated by the "hot" heater. This series
of measurements was repeated for a few different power
inputs to the "hot" heater and at several magnetic fields.

The electrical resistance of the sample could be
measured by sending an electrical current through the
sample and nulling the voltage which was generated (both
heaters were off when making this measurement). The
electrical current and the nulling current were both re-
corded. To correct for "thermals" the electrical current
was then reversed, the voltage was again nulled and both
currents were again recorded.

The bath was pumped down in 0.2K intervals and
the carbon resistors calibrated. Complete sets of the
types of measurements just described were normally made
at bath temperatures of 4.2, 3.0, 2.1, and 1.4K.

From the above measurements the thermally gen-
erated voltage V the temperature difference across the
sample AT (including the corrections described in the
thermometry section), and the average temperature of the
sample could be determined. With this information we

could compute

ILR

V
V H,T

30

where 2 is the distance between the 2 carbon resistors

R
and 2V is the distance between the two voltage probes.
2R
The correction factor I— was needed because the tempera-
V

ture difference was measured over a larger distance (10% -
30%) than the voltage difference. This correction factor
assumes that the temperature gradient is uniform between
the carbon resistors. As a by-product of the 8 measure-

ments, the thermal conductance K is also obtained:

(11.3)

”I
so

where Q is the power input to the "hot" heater.

Having measured the resistance of the sample R,
the thermal conductance K, and the average temperature T,
at which the thermal conductance was measured, the

Lorenz. number L can be computed from the relation

2
L = 1%; (11.4)
V
1R
The correction factor E—-is included for the same reason
V

as when computing S. At both H = 0, and in the high field
limit L should be equal to the ideal Lorentz number,
2.443x 10-3 watt-Q/K2 in the limit where elastic impurity

scattering is dominant.

2. Field Sweep Technique Measurements of S, K, and R
Measurements utilizing the field sweep method

were performed using the following procedure. The analog

31

output of the monitor of the lock-in amplifier in the
chopper-amplifier circuit was connected to the y-axis

of an x-y recorder, and the analog output of the gauss-
meter was connected to the x-axis. The field was then
swept up at a rate of 7G/sec. After the field had in-
creased approximatley lkG, the "hot" heater was turned on
for the remainder of the sweep. Upon completion of the
sweep, the "hot" heater was turned off and the field re-
duced to its initial value. The field was swept up again,
this time with the heater off. If initially the output
on the x-y recorder had shifted from the output of the
previous sweep when the heater was off, the y—axis off-
set was adjusted to align the two recordings.

Following completion of the second sweep, two
sets of calibration measurements were made with station-
ary magnetic fields. First, the output signal was
measured at several magnetic fields for a constant input
current to the reference resistor. A smooth curve was
drawn through these data points to determine the
amplifications factor at all fields of interest (see
Figures 11.2 or 11.3); this determined the correction of
the raw data for the field dependent gain of the measuring
system. Second, Rc and AR were measured in the usual
manner, with the "hot" heater adjusted to the same power
as during the first sweep. Since the temperature dif-

ference increased only about 10% over the field range

32

between 10 and 20 k6, these resistances were measured
only every 1 or 2kG. These two sets of calibration mea-
surements were made after each thermopower recording.

Eq. (11.2) was again used to calculate S. At
each magnetic field, V was determined by measuring the
difference between the two lines on the x-y recording and
then multiplying this difference by the appropriate
amplification factor. AT was found by first converting
the Rc and R values to temperatures and temperature dif-
ferences appropriate to the calibration fields and then

linearly interpolating to obtain AT at intermediate fields.

III. NON-OSCILLATORY THERMOELECTRIC PHENOMENA - THEORY

A. Electron Diffusion

 

Electron diffusion Sd refers to the thermopower
that would be measured if the phonon spectrum remained
in local thermal equilibrium. In section III.A.l a gen-
eral expression for ASd of any system of non-interacting
fermions (e.g. free electrons or quasi-particles) will be
derived. In section III.A.2 both Averback and Wagner's(5)

(8)

and Opsal and Wagner's results for the evaluation of
this general expression will be presented. Both of these
calculations neglected the effect of the electron-phonon
mass enhancement. Also in section III.A.2 we shall follow

1(14) to show how this enhance-

the work of Bass and Opsa
ment should be included in the evalution of the general

expression for ASd.

1. General Relations

 

The following derivation of Sd is primarily based

(5)

on the work of Averback and Wagner. In this section it

is advantageous to define the transport coefficients
(tensors) according to the following scheme:
‘3 ‘E - (-'v"r) (111.1)

+ +
J= ~E+

33

34
-+ 4+ -+ H -)

where 3 and 6 are the electrical and heat currents/unit
1

area respectively. IS is defined as E(VT)- 3:0, so that
‘3 and‘z are related by the equation

4+ ++-]_ H

S = 0 ° 8 . (III.3)

We assume a system of non-interacting fermions (electrons
or quasi-particles) whose motion in the presence of electric

and magnetic fields is governed by the rule:

_)
n§E=EE+§3xfi (111.4)
+ I
where v is g1ven by
n i; = i7}; a]; (111.5)

with ”h

Plank's constant divided by 2n
e = electronic charge
c = speed of light

ER = energy of an electron in state k

_ 3 3 3
I - ( 3k ' 3k ’ 8k )
x y 2
It can then be readily shown within the framework of

the linearized Boltzmann equation for E and 5 along the

 

x axis, and H along the z axis that:

e2 3+
0 = ——§ f d k v f'
x

(III.6a)
xx
4n

+

_ B 3+ r
xx - ——3-4 I d k v fE,T (III-6b)
n
ek T
_ B 3+ _ +
"XX '— IF I d k (5E 8F) Vx fl-E'E (1:11.76)
- kgT f d3? ( - ) I f' (111 7b)
Kxx - 3 5k EF vx K,T ' °

Here kB is Boltzmann's constant, e is the Fermi energy,

F

and [éE f'+ ([-kB$T f'E T]) is the non-equilibrium por-
I I

k E]
tion of the distribution function for an electron in state
k when only an electric field (temperature gradient) is

present. The above integrals are over k space, but we

can just as easily integrate over surfaces of constant

energy and then integrate over all energies. Performing
this change of variable gives:15
fd3k = f d f dzs (111 8)
E 457' -

where S is a surface of constant energy. For convenience

deS will be written as de, with the understanding that

de is a 2-dimensional surface integral. It is now con-

venient to define the function WE by

(-3f°§)

f]; = TWK (111.9)

1

where f3 = [l.+ exp(ei - eF)/kBT] .’

k
In the ensuing analysis the subscript R will be dropped,
except where essential for clarity. Combining equations

(111.6a, 8, 9) we see that

af°
OXX = I de (" 5?) Oxx(€) (III.10)
where
e2 dS
Oxx(€) = 1;? I‘D; waE (111.11)

Combining equations (111.7a) and (111.11) we see that

k T o
- B _ 3f _
"xx - e I de ( as ) (e 8F)oxx(e)
(111.12)
k T
_ 2 2 d B 3
- e LO T [3; OXX(€)]€=€F + 0(E;—)
nzk:
where L = -——7 (the ideal Lorenz number). From the
0 3e
(16) «+ MI-fi)
Onsager relations we know that 5(H) = E-T———q so
we may write
4+ _ (_i_4++ _
8(fi) - eLOT [de 0 ( fin€=€ (111.13)

F

where?+ is the transpose of the tensor‘E. Using the

Onsager relation oij(H) = oji(-H) it follows from
equation (III.3) that Sd is given by:
+ _ 4++ _c_'i_++
Sd(H,T) — eLoT p(H) do o(fi) (111.14)

where43(H) is the resistivity tensor,‘3(H) =‘3-1(H). It

is worth noting that eq. (111.14) holds for arbitrary

scattering mechanisms.

In the experimental arrangement described in

section 11, Se is measured under adiabatic boundary

XP
conditions (i.e., 6 = (Q,0,0)), with H along the z-axis.

37

Sexp can then be related to the components of‘S by the

relation

8 = s + s —1— . (111.15)

provided that H is along a 2-fold or higher symmetry axis.
Neglecting thermoelectric contributions we see from

eq. (111.2) that the quantity VyT/VXT is related to the
components of the thermal conductivity tensor,4z, by the

relation

V T K
V11 = - —X§ . (111.16)
K
x yy

Combining equations (111.15) and (111.16) we obtain

K
s = s - s —¥§ . (111.17)
exp xx xy Kyy

Evaluating the first term on the right hand side of eq.

(111.17) in the high field limit gives:

 

_ d
Sxle + a0) - eLoT [pxx(H + ) de Oxx(H + )I€=€F
+ p (n + m) 9— ln 0 (H + m)|
XY _ d6 YX €=eF
1 1 d
= eL T [0( ) - m '- O (H + 00)I _
o H: oxy(H + ) d8 yx E-EF
= eL T 9— 1n 0 (H + 00)I . (III 18)
0 de xy €=€F '

For an uncompensated metal in the high field limit

38

[ne(e) - nh(e)] ec

_ 1..
oxy(e) - H + 0(H3) (111.19)

 

where ne(e) and nh(e) are the number of electrons and
holes contained within the surface at e. Noting that the

electronic specific heat is just

2
yT — e LOT

ID.-

[ne(8) - nhIEIJIE=EF (III-2°)

D.)

e
we see that eq. (111.18) reduces to

m = YT
Sxx(H + ) (“e _ “n’e (111.21)

 

If we now assume that the scattering is predominantly

elastic, then we can use the relation

(111.22)

to evaluate the second term on the right hand side of eq.

(111.17) in the high field limit and obtain

 

 

 

xy yy
= eLoT [p x(H + m)::i:: : :I %; oxyiH + 0°)I€=€F
+ pxy(H + w) I:::: : :I g; oyy(H + m)|€=EFI
= e101 [g— in oxy(H + m) - gg.oyy<n + will€=€F
= (n YE nh) - eLOT g; In OYY(H + 0°)|e=eF' (111.23)

39

Therefore we obtain

ZYT - eL T
(ne - nh)e o

 

Sd(H+oo) = 1110' (H+oo)l€=€ ,

yy
F(111.24)

DIG
m

For H = 0, the tensors‘E and‘E are both diagonal. There-

fore from eq. (111.14) we find

_ _ L =
Sd(H — 0) — e101 d8 1n 0 (H 0)|€=€ . (111.25)

XX F

From equations (111.24) and (111.25) we find that the dif-

ference between Sexp(H + co) and Sexp(H = 0), ASd, 1s

2yT
AS =
d (ne nh)e

_ eLOT (g; ln oyy(H + w)oxx(n = 0)]|€=€F.(III.26)
The first term on the right hand side of eq. (111.26) is
independent of the details of the scattering and is com-
pletely determined by the host metal. If the scattering
can be characterized by a relaxation time, 1, whose energy
dependence is the same at both high and low fields, then
r cancels out of the second term, in which case this term
can be evaluated simply from a knowledge of the Fermi sur-
face of the host metal. This means that although both
Sd(H + co) and Sd(H = 0) may depend sensitively on the de-
tails of the scattering, their difference (ASd) should be
insensitive to the nature of the scattering. Such be-

havior has been observed in aluminum polycrystals.(4)

40

2. Effects of Electron-Phonon Mass Enhancement on Sd

Equation (111.26) was first evaluated for

(5)

aluminum by Averback and Wagner. In accordance with
the accepted theory of the time (due to Prange and
Kadanoff(6)) they used a value for y which was appropriate
to a free electron gas having the 8F of aluminum. This

is not the electron-phonon enhanced y which is directly
obtained from specific heat measurements. Averback and
Wagner also calculated the second term on the right hand
side of eq. (111.26) within a l-OPW approximation. The
resulting value of ASd was about 30% lower than the value
obtained from measurements on polycrystalline aluminum

samples.(3’4)

However, Averback and Wagner's calculation
was more appropriate for comparison with measurements made
on a single crystal with the magnetic field directed along
a four-fold symmetric axis (as measured in this thesis),
rather than with polycrystalline measurements. The re-
sults for H directed along a four-fold symmetric axis of
a single crystal aluminum sample, which will be presented
in section IV.a, are in good agreement with the poly-
crystalline results; thus the 30% discrepancy between
Averback and Wagner's calculation and experiment remains.
The fact that Averback and Wagner's calculation
did not correctly predict the single crystal results led

(8)

Opsal and Wagner to perform an improved calculation.

Within the more exact 4-OPW approximation, but still

41

neglecting the effect of electron-phonon mass enhancement,
they evaluated eq. (111.26) for H along the four-fold

symmetric [100] axis. This improved calculation increased

 

the discrepancy between theory and experiment. To deter-
mine how much ASd might change when the axis along which
H is directed is changed, they also estimated ASd for H
directed along the two-fold symmetric [110] axis. They
found ASd for the two-fold symmetric axis to be about 7%
smaller than for the four-fold symmetric axis.

Stimulated by the continuing disagreement between
theory and experiment, Opsal(l4'l7) then reinvestigated
the question of whether mass enhancement should appear
in thermoelectricity. He showed that contrary to the

(6) the effects of

statements made by Prange and Kadanoff,
the electron-phonon mass enhancement should appear in the
thermoelectricity of metals. Subsequent to Opsal's

initial work,(l7) (18)

Lyo confirmed, from a microscopic
viewpoint, that the effects of mass enhancement should
appear in thermoelectricity.

To see how Sd is affected by the electron-phonon
mass enhancement (which is a consequence of the electron-
phonon interaction), the work of Bass and Opsal(l4) will
be followed. It will be shown that whereas the electrical
conductivity is unenhanced, the thermoelectric power is
enhanced. This will be shown for the case of elastic

impurity scattering, by consistently renormalizing all

42

properties of the electrons to include the electron-phonon
mass enhancement.

Crudely speaking, the electron-phonon interaction
may be pictured as follows: An electron, by virtue of the
Coulomb interaction, attracts nearby positive ions; as the
electron moves through the crystal lattice it tries to drag
these positive ions along with it. This is equivalent to
an increase in the electron's effective mass. By dragging
these positive ions along, the electron distorts the crystal.
lattice in its immediate neighborhood; by virtue of the
long range nature of the Coulomb interaction, this dis-
tortion is "felt" by other electrons, and thus electrons
which otherwise would not have been coupled are coupled
together by the electron-phonon interaction. For weakly
coupled electrons we get a system of non-interacting quasi-

particles, described by the same wavevectors, E, but with

energy,(19) 5?, given by
51 = 5% + 2(51' E) . (111.27)

5% is simply the unperturbed band energy of the electron
and Z(e+, K) is the electron self energy due to electron-
phonon interactions.

Now we will consider an ideal system of these
quasi-particles which at low temperatures are elastically
scattered by a random distribution of fixed impurities.

The transport properties of these quasi-particles may be

43

described in terms of a mean free path-Ii which is re-

lated to the renormalized relaxation time 1+, by

k k
state k, and is defined by v+ =‘%-V k 8k' We wish to re-

1+ = viri. V+ is the velocity of a quasi-particle in
+

late the velocity of a quasi-particle to that of a free

electron, v2 = V Rafi. From eq. (111.27) we see that

 

 

1%
l + _ l + u l 32 l a;
6 vkek ‘ 6 Vkek 6 Mi vkek “K k
or
1 $ 3 1 32
1 + 11' kek 6 “I
is- Vkeiz (1 - 32 + (1 - 32 ) . (111.28)
38? 38+
32(ei,i)
The mass enhancement parameter A} is defined as - -———————u
38E
So by making use of the appropriate definitions, eq.
(111.28) may be rewritten as
'6“ 'fi
_ k __i?
Vk - (1 + xi) + (1 + AK) (111.29)

with the definition

1 32 (€‘+r E)
{'1 3k ’

 

++
”k

ER is much smaller than 3% (by approximately the ratio

of the sound velocity to the Fermi velocity(18)), and in
most cases can be neglected. We also note that at SF,
IE is always positive, so the quasi-particles will have

lower velocities than the free electrons, which is what

44

we expected from our elementary picture of the electron-
phonon interaction. Additionally, a perturbation treat—

ment of the impurity scattering between quasi-particle

(19)

states shows that Tk = 1E(l + 1E). Thus

u
+;

II:

_+u + u . + +11 -
IR - 1k + “ETK' S1nce ”k << vi, we find that I;

i.e. the mean free path is essentially unchanged by the

w

inclusion of the electron-phonon interaction. In the
interest of notational simplicity from here on the sub-
script R and vector symbols will be dropped from I, T,
3, A, and 3: except where essential for clarity.

Upon application of a small E field, the system
of quasi-particles may be described by the linearized

Boltzmann equation‘zo)

%%-+ eE - 6 = I (111.30)

with w given by eq. (111.9). Rewriting eq. (111.30) in
terms of unrenormalized (free-electron) variables and
neglecting n compared to vu gives

(1 + 1) 31 + eE - I“ = $— (111.30)

at u
1

For the steady state case, 3%

the same for both the renormalized (quasi-particle) and

0, so w is essentially

unrenormalized (free electron) cases; namely

w = eE . $1 5 eE - vuru. When eq. (111.10) is written in

dyadic notation we see

45

2
4+ e
0(8) = --— 1 ds
4nlfi 5(8)

82 Vin u-ru
—'—3—' f S — T V
4n'fi

<l<i~lr

III

S(e)d v

(III.3la)

(III.31b)

where the integral is over the renormalized constant

energy surface. As can be seen in Figure 111.1, the

e vs k relations are different for renormalized and un-

renormalized cases, except at the Fermi energy.

For the

temperature range examined in this thesis, the electrical

conductivity is simply 0(8 = EF

), and therefore the

electrical conductivity of a metal is not affected by re-

normalization.

We recall from eq. (111.14) that

_ d
sd — eLOT [dE 1n o(€)]l€=€

 

F
= eLoT do(€)l
o(e = 6F) de e=eF
In appendix II it is shown that
QELELI = 23——— Q§ V ° (TV)
d6 e=€F 12n3fi_ S(€F)‘fiv k

Substituting in unrenormalized variables yields

2 +
e dS V .

d0(€) f ___
12n3h S(EF)"hvu k

| _ = (1 + A)

u+u u+
(T v + T n) .

(111.14)

(AII.5)

(111.32)

46

 

 

 

 

 

 

7(- F--—-—-—-—-

FP—._._.

I
ku F

Figure 111.1: e vs. k relations for free electrons
(dashed curve) and for quasi-particles (solid curve).

 

47

u u 2 +
Identifying 92512.1I = §——§—-IS( ) EEG vk - ITUGu),
de su=eF 1211 n 8F div
u u
30(e)| _E = (1 + l) dou(e )I u
e E- F de 6 =eF

2 ' +

+ (1 + A) 6—7- fS(€ ) -d—_S; Vk ' (I: Tu).(III.33)
1211 ’h F ‘hv

Combining equations (111.14) and 111.33) and noting that

u u
SE = eLOT [QEEAE_1]| u we obtain
de 6 =8
F
_ u
Sd — (l + Msd + s (111.34)

where

e2 dS + +

S: (1+A) —3—-.I'——fi-Vk ° (leTu).
121r’h hv

Up to this point we have not considered the effect of a
magnetic field on the quasi-particles. Using the semi-
classical equation of motion for Bloch electrons in a
magnetic field, eq. (111.4), it can be shown that the
Boltzmann equation for steady state transport is not

(6)

affected by renormalization, so we may write in gen-

eral
Sd(H) = (1 + 1)33(H) + s(H) . (111.35)

Lyo(18) points out that although n/vu is very

small, the quantity [V - (firu)/V - (vuru)] is not nec-

essarily small;therefore one cannot justify neglecting

48

s(H) compared with Sd(H). However, it is expected that

(18)

s(H) is rather insensitive to H. Therefore, we see

for the quantity of experimental interest, ASd

(= SdIH + w) - Sd(H = 0)) that
~ 11
Sd - (l + MASd (111.36)

Equation (111.36) indicates that Sd will be enhanced by

the factor (1 + l).

B. Phonon Drag

 

In the preceeding sections we tacitly assumed
that only the electrons experienced a non-equilibrium con-
dition, while the phonons remained in local thermal equil-
ibrium. However, in actuality, this situation is not
realized. Since there is a temperature difference across
the sample, there will be a higher density of phonons at
the hot end than at the cold end of the sample; therefore,
a phonon flux will be created. These phonons can scatter
off electrons and thereby impart a net momentum to the
electron distribution. If no electrical current is per-
mitted to flow, then an electric field acting on the
electrons will be required to balance the force due to the
phonon flux. Thus, a thermal emf will develop across the
sample. This contribution to the thermopower is known
as phonon drag, 89.

Considering only phonon-electron scattering pro—

cesses for an isotropic phonon gas, MacDona1d(21) used a

49

simple kinetic theory approach to derive the relationship

1
9 3Ne g

. (111.37)
In eq. (111.37) C9 is the lattice specific heat per unit
volume, and N is the number of electrons per unit volume.
At low temperatures a Debye spectrum of phonons has a C

which exhibits a T3 behavior, so that we expect from eq.

(111.37)
S a T T << 6D , (111.38)

where an is the Debye temperature of the metal. Equa-

tion (111.38) was also obtained by Hanna and Sondheimer,(22)
using a more sophisticated approach involving the solution
of the coupled Boltzmann equations for electrons and
phonons. In deriving eq. (111.37), (and consequently eq.
(111.38)) all phonon scattering processes other than
phonon-electron scattering were neglected. To the extent
that these other scattering events do not produce a thermal
emf, eq. (111.38) overestimates 39. To a first approxima-
tion Sg would then be reduced by a factor Tpx/(Tpe + r )

px
yielding

T
s = 50 Tpx . (111.39)
9 pe + Tpx

 

In eq. (111.39), S0 = 1/3 (Cg/Ne), Tpe is the relaxation

time for the scattering of phonons by electrons, and 1

is the relaxation time for the scattering of phonons by

50

D' Tpx is

both C9 (and con-

everything else. At high temperatures, T >> 9

expected to be much shorter than Tpe'

sequently So) and Tpe are independent of temperature, and

Tpx is inversely proportional to T (for T >> GD, the number

of phonons is proportional to T). We thus see that the
phonon drag thermopower, as given by eq. (111.39), should
vary inversely with temperature. At low temperatures,

T << GD, 1 is much longer than 1 and therefore eq.

px
(111.39) reduces to eq. (111.38).

pe

We will now briefly examine the effects upon S9
of some departures from the ideal free electron gas be-
havior which may be expected in a real metal. A real
metal has Bragg reflection planes. Consequently, Umklapp
scattering processes are allowed, and these can have a pro-
found effect on 89. To illustrate this point we consider
both Normal and Umklapp scattering processes for a spherical
Fermi surface.

Umklapp scattering is distinguished from Normal
scattering by the appearance of a reciprocal lattice

+
vector G in the momentum conservation relation, i.e.,

E - 1' = a + 6 (111.40)

where q is the phonon wavevector. For a Normal scattering
.+

event G = 0. Figure 111.2(a) shows a phonon-electron

Normal scattering event. From this figure we see that

the change in the electron's momentum Ak is in the same

51

 

 

 

 

\
I
\
\
~
I .‘
5 ‘-
‘s ‘ H
‘
\—_ o’ ‘-

 

 

 

 

 

 

 

(a) (b)

Figure 111.2: The change of electron wavevector accom-
panying the annihilation of a phonon by a I?) Normal pro-
cess, (b) Umklapp process (after Barnard‘2 ).

 

52

direction as 3. Since the electron's velocity is directed
normal to the Fermi surface, i.e., in the direction of k
for the cases considered in Figure 111.2 16 is in the same
direction as a, so the electrons will move in the same
direction as the phonon flux, (i.e. down the temperature
gradient), thus contributing a negative 89. Figure
111.2(b) shows a phonon-electron Umklapp scattering event.
From this figure we see that after transforming the
momentum of the electron after the interaction k' by a
reciprocal lattice vector, Ak is now approximately anti-
parallel to 3; thus the electron will move up the temper-
ature gradient producing a positive 89. So for a spherical
Fermi surface,Norma1 scattering events produce a negative
89’ while Umklapp scattering events usually produce a
positive 89.

However, most metals, including aluminum and

indium, have complex Fermi surfaces. Bailyn(24)

and
Ziman(25) both employed variational techniques to cal-
culate S9 for a complex Fermi surface taking both Normal
and Umklapp scattering into account. They each obtained
quite general expressions for 59. A discussion of their
expressions would take us far afield from the major topics
of this thesis, so only the results concerning the sign

of S9 will be considered. Ziman's result leads to the
approximate rule that "the contribution to S will be

9
negative or positive according to whether the chord q

53

between the points k and k' passes through occupied or un-

occupied regions of the Fermi surface."(25)

Bailyn's
analysis leads to a similar rule.

To illustrate the significance of Ziman's rule
for a complex Fermi surface, we consider aluminum, a
nearly free electron-like metal. The Fermi surface areas
of aluminum lie only within the second and third Brillouin
zones. Figure 111.3 shows a schematic representation of
two dimensional slices through the second and third zones
showing the occupied regions. A number of phonon-electron
scattering events (both Normal and Umklapp) are indicated,
and their contributions to S9 can be discussed within the
framework of Ziman's criterion. In Figure III.3 all A
type scattering events should contribute negatively while
the B type should give positive contributions.

All the results stated so far concerning Sg have
been confined to the H = 0 situation. Almost all the
work in the presence of a magnetic field has concentrated
on the single component (a ) . Blewer, et. al.(27) have

XY 9
shown for an isotropic phonon gas that

%c , (111.40)

an expression which is quite similar to eq. (111.37).

Contrary to the electron diffusion terms, Opsal‘zg)

has
shown that none of the phonon drag terms of the thermo-

electric tensor are enhanced by the electron-phonon

54

 

Filled region

 

.. ._ ._ ._ _, N process

 

*— U process

Figure III.3: Schematic diagrams of sections throu h th

e
2nd (upper) and 3rd (lower) zones of Al showing elegE n-
phonon scatter1ng events (after Gripshover, et. a1.CZ g?).

 

55

interaction. Currently, calculations of the phonon
drag terms of the thermoelectric tensor in the presence
of a magnetic field have not been performed for a single

metal.

IV. NON-OSCILLATORY THERMOELECTRIC PHENOMENA - EXPERIMENT

A. Aluminum

1. Introduction

 

Measurements of the magnetothermopower of poly-
crystalline samples of "pure" aluminum and very dilute
aluminum based alloys (25 ppm to 350 ppm impurity concentra-
tion) are reported by Averback‘3) and Averback, et. al.(4)
Their results span a temperature range between 2 and 6K,
and a magnetic field range up to 20kG. The results may
be summarized as follows:

(1) For all samples the data were consistent

with the equation S(H,T) = A(H)T + B(H)T3,

allowing a separation of the electron diffusion

thermopower component Sd(H,T) A(H)T from the
phonon drag component Sg(H,T) = B(H)T3. This
separation assumes that in the presence of a
magnetic field Sg retains its zero field form
i.e. Sg a T3.
(2) Upon application of a transverse magnetic
field A(H) (i.e. Sd) first became more positive
and then appeared to saturate in high fields.

(3) Although A(H = 0) and A(H + co) varied from

impurity to impurity, the quantity
56

57

AA = A(H + 00) - A(H = 0) was nearly impurity in-
dependent; AA varied only from 2.1 - 2.6 x lO-BV/KZ.
This insensitivity to the nature of the impurity
suggests that ASd is determined primarily by

the properties of the host metal aluminum.

Using no adjustable parameters, Averback and

(5)

Wagner calculated AA for alumiunum. They obtained
AA = 1.6 x 10-8V/K2, which is approximately 30% smaller
than the experimental value. However, Averback and Wagner's
calculation assumed that the magnetothermopower was got
influenced by the electron-phonon mass enhancement, that
a l—OPW Fermi surface would adequately describe the
electronic properties of aluminum, and that the applied
magnetic field was directed along an axis of four-fold
symmetry in a single crystal containing an impurity which
scatters electrons isotropically. The experimental re-
sults, on the other hand, had been obtained with poly-
crystalline samples containing impurities for which no
independent information concerning scattering anisotropy
is yet available. It thus seemed possible that the 30%
discrepancy might arise from any of three sources:

(1) Incompatibility between theory and experi-

ment, in that the measurements were not per-

formed on appropriate samples under proper con-

ditions for comparison with the calculation.

58

(2) Inadequacy of a l-OPW Fermi surface for

calculation of the thermopower in aluminum.

(3) Neglect of the electron-phonon mass en-

hancement.

In an effort to experimentally test the via-
bility of source (1), measurements were performed on an
oriented single crystal foil of aluminum containing 50

(7)

ppm gallium. Recent calculations by Sorbello indicate
that gallium in aluminum should scatter electrons nearly
isotropically.* The single crystal was oriented so that
the magnetic field could be directed along both the four-
fold symmetric [010] axis and the two-fold [110] axis.
The data obtained from this crystal will be compared with
the polycrystalline data of Averback, et. al.,(4) and
also with the theory presented in the previous section.
As already indicated in section III.A.2, Opsal

(8)

and Wagner recently ruled out source (2), by showing
that a 4-OPW Fermi surface still leaves a significant
discrepancy between theory and experiment. It will be

shown that source (3) is the major cause of the dis-

crepancy between theory and experiment. The single

 

*

Sorbello showed explicitly that the scattering of the
third zone electrons by gallium was isotropic to within a
few percent, and indicated that the scattering of second
zone electrons was also nearly isotropic. Using his phase
shifts and amplitude factors, it was found that the
scattering of second and third zone electrons is the same
to within a few percent (see Appendix I).

59

crystal data can be explained by the theory of section
III.A.2 provided an electron-phonon mass enhancement of

45% is used.

2. Sample

The aluminum foil was 0.6mm thick, 1.6mm wide,
and the superconducting NbTi potential leads were 30mm
apart. The crystal was oriented so that the [001] axis
lay within 1° of the longitudinal axis of the foil and
the [010] axis was rotated 20° from the normal to the foil.
Once the sample was mounted in the sample holder, the heat
current flowed along the longitudinal axis, and the mag-
netic field H was directed transverse to the heat
current. The axis of rotation of the magnet coincided
with the longitudinal axis of the sample. Therefore the
magnetic field could be aligned along either the four-
fold symmetric {010} family of axes or the two-fold
symmetric {110} family. Thus, it was possible to deter-
mine whether AS was sensitive to the direction of H'

d
relative to the crystallographic axes of aluminum.

3. Data and Analysis

 

3.1 Determination of the non-oscillatory results
In the absence of any applied magnetic field,
the thermally generated voltages were measured using the
nulling technique described in sec. II.A.2. However, for
H > 10 k6, the presence of giant quantum oscillations when

H was directed along a four-fold symmetric axis‘29-3l)

60

(see section VI.A.3) made the field sweep measurement
technique (see sec. 1.A.3) advantageous. Figure 1V.l
shows the maxima, minima, and the derived values of the
non-oscillatory component of S for a typical field sweep
after making the corrections outlined in sec. II.G. The
non-oscillatory component of the thermopower was obtained
by averaging each minimum with the average of its two
adjacent maxima, and each maximum with the average of its
two adjacent minima. From figures similar to Figure IV.1
the non-oscillatory of S could be extracted as a function
of H at a constant temperature. As indicated in Figure
IV.1, the non-oscillatory component saturated in value,
to within experimental uncertainty, by 15kG*. 1n the
ensuing analysis the lSkG value will always be used as
the high field limiting value of S (above lSkG the measur-
ing uncertainties increased rapidly with increasing field).
Since the thermopower oscillations are so large,
small uncertainties in their magnitudes are magnified into
large uncertainties in the magnitude of the non-oscillatory
component. With care and patience the non-oscillatory
component was extracted with an uncertainty of about 10%
(at a 70% confidence level). The bulk of this uncertainty

arises from slow variations of the zero baseline and from

 

*

In some cases S had not quite saturated by lSkG, but the
resulting small increases do not significantly change any
of the conclusions reached in this thesis.

61

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62

electrical noise created by vibration of the sample holder
in the magnetic field.
On the basis of the polycrystalline aluminum

(4)

measurements by Averback, et. a1. it is expected that

the single crystal data should be of the form
S(H,T) = A(H)T + B(H)T3 (1v.1)

Therefore if we plot S/T vs T2, we should obtain a straight
line with slope B(H) and intercept (with the T2 = 0 axis)

A(H). We are interested in the quantity

ASd

‘1'— : AA = A(H + co) — A(H = 0) (1v.2)

3.2 AA for ngarallel to [100] and [010] axes

 

In Figure IV.2, the ratio of the non-oscillatory
component of S to the absolute temperature T is plotted as
a function of T2 for H = 0 and H = lSkG. The error bars
reflect the 10% uncertainty which was previously assigned
to each point. In each case the data are consistent with
straight lines. If there is any misalignment of the
potential leads on the sample, the value of S should change
when the magnetic field is reversed. To test for such an
effect, measurements were made with the magnetic field both
forward (open symbols) and reversed (filled symbols). To
within experimental uncertainty, no systematic differences

were observed. Straight lines were therefore drawn through

63

 

 

 

’-~ 10 . A. ‘3

N A . D
x ‘A‘
‘\~

> Ib—

~_a

00 X
O

_ o

><

h- ——

‘\~

(I)

 

 

T2IKZI

Figure IV.2: The temperature dependence of the non-
oscillatory component of S of the Al(Ga) sample for:

H = lSkG and parallel to [010] (squares); H = lSkG and
parallel to [100] (triangles); H = lSkG and parallel to
[110] (crosses); H = 0kG (circles). The full symbols in-
dicate data obtained with H reversed.

 

64

all data for a given field direction. The quantity AA
can be obtained directly from Figure IV.2; it is simply
the difference between the intercepts for H + w (here
taken to be 15kG) and H = 0.

To test whether the results were dependent upon
the orientation of H relative to the foil axes, ASd was
measured with the magnetic field along the two crystallo-
graphically equivalent [010] and [100] axes. Figure IV.2
shows that the data for these two directions are
systematically different, but the differences are compar-
able to the uncertainties in the data. The average of the
[010] and [100] values gives a "best value" of
AA = (2.23 i 0.25) X 10-8V/K2. This average value agrees
to within experimental uncertainty with the AA values
obtained for each direction individually.

The average value of AA is in good agreement with

(4)

the polycrystalline data of Averback, et. al. It is

also in agreement with the calculation of Opsal and Wagner,(8)
provided that a 45% electron-phonon mass enhancement is

included. A 45% electron-phonon mass enhancement is con-

sistent with values inferred from both de Haas-van Alphen‘32)

(33)

and low temperature specific heat data on aluminum.

3.3 AA for H parallel to {110} axes

We have noted in section III.A.2 that Opsal and
(8)

Wagner predicted that ASd should be somewhat smaller

for H directed along a two-fold axis than a four-fold axis

65

in aluminum. In this section results will be presented
for H directed along each of the four, two-fold symmetric
axes (the {110} family) lying in the (001) plane.

The data for H along any of the {110} axes are
shown in Figure IV.2. Since the data for all four equi-
valent axes were the same to within the specified un-
certainties, we will not distinguish between these axes
in Figure IV.2. Also, for these axes the magnetothermo-
power oscillations were an order of magnitude smaller than
for the {010} axes. This accounts for the smaller error
bars in Figure IV.2. From this figure we see that
AA = (2.0 i 0.2) X 10-8V/K2 for H along these axes. This
value is about 10% smaller than that for H along the {010}
axes, in agreement with the calculation by Opsal and

(8)

Wagner.

3.4 Effects of Magnetic Breakdown

 

The previously mentioned giant quantum oscilla-
tions in S for H along the {010} axes (Figure IV.l) are

(29,30) How_

believed to arise from magnetic breakdown.
ever, Opsal and Wagner's calculation of the non-oscillatory
component of S completely neglects breakdown. The follow-
ing argument is made against the presence of significant
effects of breakdown on the non-oscillatory component.

(34) effects

According to Balcombe and Parker,
of magnetic breakdown should appear primarily for H in the

(001) plane and should be most pronounced for H along the

66

[010] axis. In such a case the largest effect of break~
down on 8 would be expected for H along [010], a smaller
effect for H along [110], and the smallest effect for
polycrystalline samples. Indeed this is just what is ob-
served for the oscillatory component: there are giant
quantum oscillations for H along [010]; oscillations an
order of magnitude smaller for H along [110]; and no
evidence of quantum oscillations in comparable purity
polycrystalline samples. On the other hand, as regards
the non-oscillatory component little difference is found
between the values of AA for H along [010] and along [110]
and both values are consistent with those obtained with
polycrystalline samples by Averback, et. al.(4) There-
fore, it is concluded that magnetic breakdown does not
significantly affect this component, and thus the values
for AA should be amenable to the calculation of Opsal and

(8)

Wagner, provided that the electron-phonon mass enhance-

ment is included.

3.5 Phonon Drag

 

Besides measuring the electron diffusion contribu-
tion to S, our experiments also provide information con-
cerning the phonon drag contribution. Recalling from eq.
(1V.l) that S9 is simply B(H)T3, it follows that the slopes
of the lines in Figure IV.2 give B(H). The values of B(H)

for the aluminum single crystal are shown in Table IV-l.

67

Table IV—l: Values of B(H) for Al(Ga) Sample

B(x 10'1OV/K4) H
-3.2 i 1.5 lSkG, H ||{110}
-2.1 :_o.9 lSkG, H ||[0101
-o.5 : 1.5 lSkG, H III1001
-1.0 i 0.6 0.0kG.

The variance of B with magnetic field and also the measured
values of B are similar to the results reported by

(4)

Averback, et. al. The fact that B(H) is negative may
indicate that the A type scattering events of Figure III.3
predominate over the B type events (see section III.B).
Averback, et. a1. noted that they were unable to find any
quantitative feature of the variation of B with H which
was impurity independent. They therefore concluded that
the magnitude of the observed variation of B with H is
not an intrinsic property of the host metal aluminum, but
is determined primarily by the details of the scattering
of electrons by impurities. No detailed explanation of
the observed variation of B with H has been offered to
date.

One must be cautious when interpreting B as
solely a measure of phonon drag. If the scattering is
partially inelastic, then the electron diffusion magneto-
thermopower, in the presence of a magnetic field may also

(35)

contain terms which are non-linear in T. If these

68

contributions are small compared to the linear term, then
an S/T vs. T2 plot will incorporate these contributions
into B (the slope). The magnitude of inelastic scattering
may be estimated by examination of the variation of the
Lorentz ratio with temperature. Lorentz ratio data from

(36’shows

an aluminum sample of comparable purity by Willot
that this ratio changes about 10% between 0 and 4K. This
could cause Sd/T to vary by at least several percent over
this temperature range; moreover, the variation is expected
to be field dependent. Since S/T varies by only 25% over
this temperature range, we conclude that inelastic scatter-
ing could make a non-negligible contribution to B.

Nielson and Taylor(37) have shown that second
order multiphonon scattering processes may also give a
T3 contribution to Sd; the so-called "phony phonon drag."
Bourassa has estimated the size of the "phony phonon
drag" for aluminum in the absence of a magnetic field and
finds it to be orders of magnitude smaller than the ob-

(4)

served values of B. Currently the effect of a mag-
netic field on "phony phonon drag" is unknown. In con-
clusion, although the coefficient A(H) gives a quantita-
tive measure of 8d in the zero temperature limit, the
experimental value of B(H) does not necessarily measure
only 89. We therefore believe that a detailed theoretical
analysis of B(H) solely ins terms of phonon drag, may be

a bit premature at this time.

69

4. Summapy_and Conclusions for Aluminum

 

Transverse magnetdthermopower measurements of a
single crystal aluminum sample (containing 50 ppm Ga) have
been made with H along both the {010} and {110} families
of axes. In each case S was consistent with the form
S(H) = A(H)T + B(H)T3, allowing a separation of Sd from
$9, and therefore permitting a determination of the quantity
AA. For H along the four-fold symmetric {010} an average
value of AA = (2.23 i 0.25) x 10-8V/K2, was obtained” and
for H along the two-fold symmetric {110} axes a value of
AA = (2.0 i 0.2) X lO-BV/K2 was obtained. These values
differ only slightly, and they agree with the values pre-
viously reported by Averback, et. a1.(4) for polycrystalline
samples. We have argued above that these values are
appropriate for comparison with the calculations of Opsal

(8)

and Wagner. This comparison leads to good agreement,
provided that a 45% electron-phonon mass enhancement is
included. A 45% enhancement factor is consistent with
that independently obtained from other types of measure-
ments. The fact that AA is somewhat smaller for H along
the two-fold symmetric axis than for H along the four-fold
symmetric axis is also in agreement with Opsal and Wagner's
calculations.

Data was also obtained for S9 = B(H)T3. The

values of B(H) are shown In Table IV-l and agree with the

values previously reported by Averback, et. al.(4) for

70

polycrystalline samples. We have noted that these values

of B(H) may not be solely a measure of S .

B. Indium

1. Introduction

 

Indium and aluminum are both Group III metals.
Aluminum crystallizes in a face centered cubic structure
for which the {100} family of axes are all four-fold
symmetric. In contrast, indium crystallizes in a face
centered tetragonal structure (c/a = 1.08) for which the
[001] axis is four-fold symmetric but the [100] and [010]
axes are only two-fold symmetric. Thus, it seemed quite
natural to extend the magnetothermopower measurements made
on aluminum, to indium single crystals for H aligned along
both the [010] and [001] directions, in order to try to
observe the effect of the slight tetragonal distortion on
ASd.

Although the Fermi surfaces of aluminum and indium
are quite similar, previous measurements of their magneto-
thermopowers by Caplin, et. a1.(38) gave apparently dif-
ferent results. They measured the magnetothermopowers of
aluminum and indium polycrystals over the temperature range
4.2 to 60K in magnetic fields up to 50kG. The significant
difference between the results of the two metals showed up
in the quantity AS(H) (AS(H) E S(H) - S(H = 0)). Only in

aluminum did AS(H) change sign from positive to negative

71

as the temperature was increased. In indium, AS(H) re-
mained negative at all temperatures investigated.

Using magnetic fields less than 50kG., Caplin,
et. al. also noticed that S tended to saturate at high
fields for T.< 21K for aluminum and for T < 9K for indium.
Saturation indicates that the high field limit (wcr >> 1)
has been reached. They also measured the thermopower of
indium single crystals over the temperature range 4.2 to
300K in zero magnetic field. They observed that S was
always more positive for VT along [100] than for VT along
[001].

Averback and Bass(39)

performed measurements on
polycrystalline indium samples in the temperature range 2
to 5K in magnetic fields up to 20kG. Comparison of the
data of Averback and Bass with that of Caplin, et. a1.
shows a discrepancy in the sign of S. Measurements per-
formed during the course of this thesis indicate that the
sign given by Averback and Bass is incorrect.(4o)
Averback and Bass also observed that S saturated in high
magnetic fields. In addition, they noted that S did pg;
fit the simple form: S(H,T) = A(H)T + B(H)T3. Two

possible reasons are:

(l) The low temperature approximation of S9
(= BT3) may not be adequate. However, if elastic
impurity scattering is still the major scattering

mechanism then

72
S(H,T) = A(H)T + Sg(H,T) . (IV.3)

For this case, if a term linear in T can be
extracted from the data then Sd can be separated
from S9 (assuming Sg does not contain a term
linear in T).

(2) If there is a significant amount of in-

elastic scattering, then S will no longer be

d

linear in T. In this case Sd cannot be un-

ambiguously separated from S .

9
(41) measured the electrical

Blatt, et. a1.
resistivity of polycrystalline indium wires. The RRRs
of Blatt, et. al.'s samples were comparable to the RRRs
of Averback and Bass' samples. Blatt, et. al.'s data show
that the electrical resistivity is strongly temperature
dependent between 2.0 and 4.2K. This indicates that there
is a significant amount of electron—phonon (inelastic)
scattering at 4K. Thus, when the present study began it
seemed possible that reason (2) could be the cause of de-
viation of S from the simple form of eq. (IV.1).

In this thesis, measurements similar to those of
Averback and Bass were performed on oriented single
crystals of dilute indium alloys. It was anticipated that
added impurities would increase the electron-impurity
scattering to where it would finally dominate the in-
elastic scattering, causing Sd to become linear in T for

T < 5K, and thereby allowing a separation of S from 89.

d

73

On the other hand, the alloys must be very dilute in order
to reach the high field limit with H < 20kG. In addition
to measuring S, both the electrical and thermal con-
ductivities were measured as functions of H and T. This
allowed calculation of the Lorenz ratio L as a function
of H and T. The divergence of L from L6 (= 2.443 x 10'-8
watt-ohms/Kz) gives a measure of the amount of inelastic
scattering present at a given T. The variation of L with
T gives a more sensitive measure of the amount of inelastic
scattering than does the electrical resistivity alone.
To summarize, the aims of the magnetothermopower
measurements on indium were:
(1) To see the effect of the slight tetragonal
distortion on 48d by measuring S with H aligned
along both the four-fold symmetric [001] axis and
the two-fold symmetric [010] axis. '
(2) In order to achieve goal (1) it is necessary
to separate Sd from 89. We hOped to do this by
adding impurities to increase the electron-
impurity scattering to a point where it finally
dominates the inelastic scattering, causing Sd to
become linear in T for T < 5K.
(3) By extending the magnetothermopower measure-
ments on indium down to 1.5K we wanted to see
whether AS(H) would change sign from negative to

positive as the temperature was decreased, as it

does in aluminum.

74

2. Data and Analysis

 

Figures IV.3 - 1V.8 show magnetothermopower data
for a variety of indium crystals. The data are plotted
in the form S/T vs. T2 at constant H. Although in some
cases for a particular H the data lie on straight lines,
not a single sample produced data which lay on straight
lines for 211 H. This demonstrates that the magneto-
thermopower of indium, in contrast to that of aluminum,
does not obey the simple form: S(H,T) = A(H)T + B(H)T3
throughout the temperature range 1.5 to 5K.

Figure IV.9 shows a plot of S vs. H at both
4.7lK and 3.50K for a sample containing 100 ppm tin in
indium (In(Sn)-II). The data appear to be approaching
saturation, but have not reached saturation at H = 20kG.
Presumably this non-saturation by 20kG stems from the fact
that the sample contains 100 ppm tin; this places an
upper limit of about 100 ppm on the impurity content of
the indium alloys if the high field limit is to be attained
with the present measuring system. The fact that the data
appear to be approaching saturation, is in agreement with

the behavior found by both Caplin et. a1.(39)

(40)

and Averback
and Bass.

A goal of this research was to test the hypothesis
that S would fit the simple form of eq. (1V.l) throughout
the temperature range 1.5 to 5K for sufficiently high

impurity concentrations. Therefore measurements were

75

 

 

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Figpre IV.3: The temperature dependence of the non-
oscillatory component of S for sample In(Ga) with H
directed along [010].

 

76

I
<1x

S/TxIOBCV/KZ)
I
xc><0

X
' 0002 KG.

0 0.50 KB.
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Figure IV.4: The temperature dependence of the non-
oscilIatory component of S for sample In(Ga) with H
directed along [001].

 

77

 

 

 

 

 

F.
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0 d3
063 D 0
xx CD
953
l x
2_ 9A. 0 (CD) 0
G "x
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I I I I
0 IO 20
TZCKZ)

Figure IV.5: The temperature dependence of the non-
oscillatory component of s for sample In(Sn)-II with H
directed along [001].

78

 

 

 

 

 

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I— O
I o o
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A O V— _—
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r.AIOOO G
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I I I I
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T2(K2)

Figure IV.6: The temperature dependence of the non-

oscillatory compopent of S for sample In(Sn)-II with H
directed along [110].

79

 

 

 

 

55'—‘
I . 0.02 KG.
7 II? . o 0.50 KG.
_’ §<1C> <3 0
_- Xx x20.00 KG.
0 _ ______
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(1)-5—
7“ v
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l l I l l
0 IO 20
T2(K2)

Figure IV.7: The temperature dependence of the non-
oscillatory component gf S for sample In(Sn)-I with H
rotated 12° from the [101] in the (101) plane.

 

80

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81

 

 

 

 

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C).—
r\ n
X:
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PS? “I h— 5
x T=5.5K
a) a
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‘3‘— T:4e7K
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I I I J
0 IO 20
H(KG)

Figure IV.9: The magnetic field dependence of S for
sample In(Sn)-II with H directed along [110].

 

82

performed on a sample containing 500 ppm tin in indium
(1n(Sn)-III). Figure IV.8 shows that the data for

H

15kG and 20kG with H directed along the [001] axis
lie on straight lines. This tempts one to make the
hypothesis that all the data for H > lSkG will also lie
on straight lines. However, one must also note that for
this sample H = 20kG is a relatively low field (mcr < l);
and as illustrated by Figure IV.6 the form of an inter-
mediate field (H = 3kG or SkG) curve over a limited tem-

2 < 25K2) can be quite different

perature range (10 < T
from the high field curves (H > lSkG in Figure IV.6). In
fact these intermediate field curves over this limited
temperature range are quite similar to the curves for

H = 15 and 20kG with H directed along the [001] axis of
Figure IV.8. Therefore, since intermediate field curves
are not reliable indicators of the high field curves, and
also because the H = 0.50kG data do not lie on a straight
line, we believe that S will not fit the simple form of
eq. (IV.l) at higher fields over our available temperature
range. Figure IV.8 shows that the data for H directed
along the [110] axis deviate strongly from a straight line
already at 20kG. So it is concluded that for an arbitrary
H, S will not fit the simple form of eq. (IV.l) throughout

the temperature range 1.5 to 5k for indium samples with

impurity concentrations less than 500 ppm.

83

The nonlinearity of Figures IV.3 - IV.8 is most

2

pronounced in the temperature region above 3K (T = 9K2).

The experimental data of Caplin, et. a1. show a strong
negative peak in the vicinity of 10K. They ascribe this
peak to phonon drag. It seems likely that the available
temperature range is too close to the phonon drag peak and
therefore Sg cannot be adequately described by the low
temperature approximation (S = BT3). As a rule of thumb,‘
the phonon drag peak occurs between BD/S and 00/7 (0D is
the Debye temperature). Averback and Bass tentatively
attributed the inadequacy of the low temperature descrip-

tion in the temperature region above 3K, to the low Debye

temperature of indium (SD = 110K);aluminum which has a

much higher Debye temperature (0D = 425K), begins to
deviate from the simple form: S(T) = AT + BT3 above about
6K. ‘42)

The other apparent cause of the non-linear S/T
vs. T2 plots for indium concerns the effects of the in-

elastic scattering on S For this case, it would be

d'
expected that a sample with a significant amount of in-
elastic scattering would produce a different S/T vs. T2
plot when compared with that of a sample with a smaller
amount of inelastic scattering. Figure IV.10 shows the
Lorenz ratio as a function of T for a few samples. The

deviation of L/Lo from 1 gives an indication of the amount

of inelastic scattering present. This deviation is

84

 

 

 

 

x‘,,xx
100’— 88 "xx
C”) ”x I
O X
(3 no 00) ‘ A
" Q00 00
"99,I 00:0 09 I
x“ (la
.8)—- O xlxo D l
X0 8
O ‘0
_x___I
_ O
D
O
3'6I— V U o
x O
v D 0
)-— x D 00
D
V O
X on 00
V DD
.4?- xv DOD
XV
o 0.50 KG. ”V C
D I.OO KG. V
‘, Vvv
V 5.00 “6' X‘
.21 x20.00 KG. ""
l l l I l
O 5

T(K)

Figure IV.10: The temperature dependence of the Lorenz
ratio L divided by the ideal Lorenz ratio L for samples:
A) 1n(Sn)-III, with H directed along [001];

B) In(Sn)-II, with H directed along [001];

C) In(Ga), with H directed along [010].

85

largest in our purest sample, In(Ga) (RRR = 7400). The
fact that L deviates most from L0 is expected since this
sample should have the least electron-impurity (elastic)
scattering. Figure IV.10 also shows that sample In(Ga)
exhibits significantly more inelastic scattering than
sample In(Sn)-II. However, Figures IV.3 - IV.6 show that

these two samples produce similar S/T vs. T2

plots. Thus
it is concluded that the major cause of the nonlinearity
of Figures IV.3 - IV.8 in the region T > 3K is the break-
down of the low temperature description of 89, rather than
the influence of inelastic scattering on Sd'

Probably the most interesting region in Figures
IV.3 - IV.8 is below 2.5K (T2 < 6K).‘ For all samples ex-
cept In(Sn)-III, the high field set of data (H > lOkG)
either becomes more positive than the 0.50kG set (essen-
tially H = O) or it appears that this will happen at a
temperature somewhat below 1.5K (the lowest temperature
attainable with the present system). This behavior was
not suggested by any previously published data. However,
such behavior would be expected when Sd is the dominant
component of S. Moreover, the only abnormal case,
In(Sn)-III, can be understood by noting that this sample
has not reached the high field limit by 20kG, and we pre-
viously argued that the intermediate field curves over a
limited T range do not reliably predict the high field

S/T vs. T2 curves. The general low temperature behavior

86

of these samples is qualitatively similar to that of
aluminum, which is only slightly affected by phonon drag.
Perhaps this low temperature region is far enough removed
d is
comparable to $9, in which case S might be described by

from the large negative phonon drag peak, so that S

the simple form of eq. (IV.1). However, this is only a
conjecture concerning the scattering mechanisms responsible
for S. The true test will come when magnetothermopower
measurements are performed at still lower temperatures

(0.3 < T < 1.5K). At such low temperatures elastic,
electron-impurity scattering should be dominant, so Sd
should be linear in T, and moreover, Sg should be small
compared with Sd'

3. Umkehreffect

 

Samples In-I and In(Sn)-I demonstrated an inter-
esting umkehreffect. The umkehreffect is characterized
by the magnetothermopower changing drastically upon re-
versal of H.‘ Figures IVJLland IV.12 show this effect.
For H near [100] the magnetothermopower changes by a
factor of two upon field reversal, Figure IV.12 also shows
the angular variation of the electrical resistance. It
is clear that the resistance, in contrast to the magneto-
thermopower, is well behaved, i.e. not affected by field
reversal.

Such a result can be understood (at least

qualitatively) by considering the properties of the‘E and

87

 

9“ A+I0.00 KG.
_ “\. ’“.-IOOOO KGO
.yf' \‘ ‘LA \
h'/ \ A ‘f
\ /
n 6" \
k: \ _ /
s _ "‘ ‘ /
c \ 1
N9 ’— I \ / .\
€13 \ /
| A \ /O
3—
- - ‘(A \_.__'_—/ A
" [0:0] U01]
0 l l l l l I L I L Ill 1 J I I] l L I

 

O ICC 200 300
9(DEG)

Figure IV.11: Variation of S with rotation angle 0 at
H = lOkG and T = 4.7K for sample In-I. H was rotated about

the [101] axis. 6 = 213° corresponds to H directed along
[010]. '

88

8.5r-— A+l0¢00 KG.
5.. .-'0000 KC.

 

g t.
”9 8.0——
; _.
)—-—

(—

7.5%—

6—

-sno7(vn<)

 

 

 

[0:0] [10:]
lllJJllllJLlJlllllel
O ICC '200 300
B(DEG)

Figure IV.12: Variation of R (upper curve) and S (lower
curve) with rotation angle 0 at H = lOkG for sample
In(Sn)-I. H was rotated about the [101] axis. 0 = 202°
corresponds to H directed along [010]. The R curve was
measured at 4.2K and the S curve at 4.3K.

89

the‘S tensors. For either tensor, symmetry requires that
there be no umkehreffect for H directed along a three-
fold or higher symmetry axis. As a consequence of the

:13) ‘16)

Onsager relation pik(H) there can be no

= pki(
magnetoresistance umkehreffect for H directed along a two-
fold symmetry axis either. However, since there is no
such Onsager relation connecting components of the‘S
tensor, H must be directed along a three-fold or higher
symmetry axis for the magnetothermopower umkehreffect to
vanish. Since the [100] axis is only a two-fold symmetric
axis, it follows that the magnetothermopower may exhibit
an umkehreffect. Although such symmetry arguments allow
the existence of an umkehreffect, they alone do not pre-
dict its magnitude. Therefore, one cannot on theoretical
grounds alone unambiguously attribute the observed
magnetothermopower umkehreffect to an intrinsic property
of indium as opposed to an experimental artifact (e.g.
lead misalignment or sample preparation). To see if the
observed umkehreffect was due to a small misalignment of
the voltage leads attached to the sample, the leads were
removed and reattached, aligned as accurately as possible.
This had no significant effect on the measured value of S.
Therefore, it is concluded that lead misalignment was
not the principal cause of this irregular behavior.
Samples In-I and In(Sn)-I were both grown from

the same seed. Figures IV.11 and IV.12 show that both

90

samples exhibit the largest umkehreffect when H is in the
vicinity of [100]. H could also be aligned along [100]
with sample In(Sn)-III. The umkehreffect exhibited by
In(Sn)-III, as well as all other samples, was less than
20%; therefore, it is concluded that the very large
umkehreffect exhibited by In-I and In(Sn)-I is not
intrinsic to H along [100]. Although it seems unlikely,
one cannot rule out the hypothesis that this large effect
is a consequence of 5 being directed along [011]. There-
fore, further measurements on independently grown single
crystals are necessary to determine whether the afore-
mentioned hypothesis is correct or whether the large
umkehreffect was an experimental artifact due to sample
preparation, which did not show up in the X-ray (Laue)
patterns.* It should be noted that Figure IV.7 shows
data for a direction in which the umkehreffect was less

than 10%.

4. Summary and Conclusions for Indium

We briefly summarize the qualitative results for
indium: (1) Magnetothermopower measurements performed on
indium single crystals containing as much as 500 ppm

impurity showed that S was not consistent with the simple

 

*

It is unclear whether or not higher quality X-ray photo-
graphs would have shown a difference between these two
samples and the other indium samples.

91

form: S(H,T) = A(H)T + B(H)T3, over the temperature range
1.5 to 5K. It is concluded that the deviation from this
simple form is not primarily due to the effects of in-

elastic scattering on S but rather to the influence of

d’
the large negative phonon drag peak (near T = 10K), which
causes S9 to substantially deviate from the form S9 = B(H)T3.
(2) The magnetothermopower of samples with impurity con-
centrations of 100 ppm or less appear to be approaching
saturation for H = 20kG and T < 5K.

(3) At high temperatures, 3 < T < 5K, S becomes more
negative with increasing H. This is presumably due to S9
dominating S.

(4) A low temperatures, 1.5 < T < 2.5K, the high field
limit of S is more positive than the low field values.
This can be understood by assuming that-Sd is increasing
relative to S9 with decreasing T. The temperature at
which S starts becoming more positive instead of more

negative with increasing field varies somewhat from sample

to sample.

V. OSCILLATORY THERMOELECTRIC PHENOMENA - THEORY

A. Variation of the Density of States at SF with H

The state of a free non-interacting electron gas

 

can be described by a sphere in single particle momentum
space (k-space). At T = 0 the interior of the sphere has
an electron occupying every single particle state, while
the exterior states are all empty. The Fermi surface
separates the occupied and unoccupied regions.

In the absence of a magnetic field the single
particle states of the electrons are evenly distributed
throughout all of k-space. However, the application of
a magnetic field redistributes these states on to the
surfaces of cylinders aligned parallel to the magnetic
field. Figure V.l shows such a redistribution of states.
This figure also shows the H dependence of this redis-
tribution. An electron, in any particular state, orbits
around the circumference of the cylinder ("Landau
cylinder") with an angular frequency wc (the so-called

cyclotron frequency), given by
O.) = ___ 0 (Vol)
where me is the mass of the electron and c is the speed

92

93

     

(a) (b)

        

flu.
. .--..o‘ -

 

 

.u..‘..V-.-..-..

   

o .3'.J'. ' _
’- '.. ' . o
' .O..\‘......O...
. ' '. a
O .

a"

(C) (d)

Figure V.I: Variation of the Fermi surface with field,
showing the increasingly spherical shape as the field be-

comes weaker. The field decreases going (a) to (d) (after
Adams and Holstein(43)).

94

of light (in a vacuum). Thus, the effect of the magnetic
field is to quantize the electron orbits in k-space.

The radii of these cylinders increase with H.
As the outer cylinder passes through the Fermi surface,
the electrons on this cylinder "condense" back onto avail-
able states on lower energy (smaller radius) cylinders.
This process repeats as each Landau cylinder passes through
the Fermi surface, giving rise to an oscillatory magnetic
field dependence of the density of states at the Fermi
level. This dependence has been observed in measurements
of a number of high field (wCT >> 1) electronic properties
such as magnetization (de Haas-van Alphen effect) or
.electrical resistivity (de Haas-Shubnikov effect).

The exact solution of the free electron gas in a
uniform magnetic field is given in many solid state physics

texts.(44)

One finds that the oscillations in the density
of states at the Fermi energy are periodic in l/H, with

a period

A(l/H) {2‘23 A (v.2)

where A is the cross-sectional area enclosed by the
electron's orbit.

What happens when we are dealing with electrons
in real metals? In some metals, such as indium and
aluminum, the valence electrons can be treated as nearly

free electrons (NFE). The NFE approach treats the crystal

95

lattice potential as a perturbation. The important changes
introduced by NFE treatment are:
(l) The electron orbits (and consequently the
Fermi surface) are no longer spherical.
(2) The Fermi surface can be made up of pieces
from more than one band.
(3) The dynamics of the electrons in their orbits
are described in terms of an "effective mass"

*
m where

*
m =

lg

I (v.3)

.1.
2“ e e=€F

Q)

Using the Correspondence Principle, it can be
shown that electrons within a single band will still ex-
hibit an oscillatory density of states at the Fermi level

(44) The period is still given

which is periodic in l/H.
by eq. (v.2), but now A is the extremal cross-sectional
area of the Fermi surface in the plane perpendicular to

the magnetic field.

B. Estimate of S Oscillations Due to Density

of States Oscillations for Aluminum and

 

Indium
There do not exist any rigorous calculations of
the oscillatory component of the thermopower of a metal
which are suitable for quantitative comparison with ex-
periment. Only simplified calculations of limited appli-

(45) (46) al.,(47)

cability by Zil'berman, Horton, Grenier, et.

96

(48) are available. All of these cal-

and Long, et. a1.
culations predict for the particular orbit of present

interest within our experimental range of temperature and
magnetic fields, that the thermopower oscillation ampli-

tudes should consist of one or more terms of the form:

1 n 2
A1,nT H exp[-2n kB(T + TD)/BH] (v.4)

where A1 n is independent of H and T. Here 1 and n are
I
eh
* O
m c
In this section we will estimate the size of the

constants 2‘: 1‘: -1 and 3/2 Z.“ 1 -3/2; and B =

 

quantum oscillations of Sd induced by the oscillatory com-
ponent of the density of states, under a series of
assumptions which are necessary to allow the estimation

to be performed with data that currently exist in the
literature. The assumptions are indicated at the points
they are made. Only one of the assumptions, if incorrect,
might be expected to have a dramatic effect on the esti-
mate; the limits of this assumption are indicated in a
footnote following the assumption.

From equations (111.20) and (111.24) we see that

 

2eL T e
o d

Sd(H + 00)
e h F YY

where

97

If we assume that the electrical conductivity can be

described by a two band Sondheimer-Wilson model(49) then

 

en H en H
o =—_2————7ee +T‘7hh (v.6)
yy H+H H+H
e h
where
*
m.c
Hg= 1 .
1 e T.
1
3Hi 3Ti
If we further assume that 3E— (i.e. 52-) gives a

negligible contribution to the oscillatory component of

 

 

 

 

 

* os
Eyy' eyyc, then
H 3n H an
83:6 3 -e2LoT[%2 e2 (aee)osc + 2h 2 (36h)osc] . (V 7)
+He H +Hh e=eF
If we still further assume that He = Hh = Ho then
H an an
osc 2 o [ e osc h osc
e = -e L T ——- (-—) - (-—-0 I] (v.8)
2
YY o H 38 36 8:6
and
t (50) .
Recent measurements by Thaler, et. al. on a high

purity (RRR = 7000) aluminum polycrystal revealed that
under conditions where S displayed substantial quantum
oscillations, the Nernst-Ettingshausen coefficient (i.e.
exy) revealed none. Since exy is not dependent on

31/38, and eyy i§_dependent on 31/38, any significant

oscillatory component in 3T/3e would cause oscillations

in S but not in the Nernst-Ettinsghausen coefficient.
Therefore, Thaler et. al.'s result suggests that the
assumption that 31/38 contributes a negligible oscillatoryr
component may not be completely valid.

98

eHo
Oyy = ;7_ (ne + nh) . (v.9)

Dividing eq. (v.8) by eq. (v.9) gives

a eLoT 3ne osc 3n

£0 = " ___—n + n (38 I ' (3e ' . (v.10)
e h -

Measurements of the electrical resistivity of

the single crystal aluminum and indium samples used in

the experiments comprising this thesis show that for

osc
H : 20kG., g < .01. From these measurements it may

 

be inferred that ngsc and nfisc are negligible compared

with ne and n at all fields of interest. De Haas-van

h
Alphen results show that both a1uminumI32'51)

and

. . (52,53) . .
indium have very small third zone electron-like
orbits and much larger second zone hole-like orbits.
This means that the third zone orbits will produce

5 _ 107

relatively low frequency (~ 10 G.) oscillations
compared to the higher frequency (~ 108G.) oscillations
due to the second zone. From an experimental viewpoint,
the highest frequency of oscillation which may be ob-
served is limited by the homogeneity of the magnetic
field over the entire length of the sample. In an in-
homogeneous magnetic field, different portions of the
sample will oscillate out of phase with other portions.

This will wash out the oscillations in the thermal emf

generated over the length of the sample. The highest

99

frequencies observed with present apparatus were about
5 x 106G. Therefore, only oscillations due to density
of state variations of the third zone orbits can be ob-
served. In view of this, the estimate of the quantum
oscillations in 8d will be confined to these electron-
like third zone orbits. We therefore write

 

g eL T 3n
(_yy)osc = _ o ( eI _ )osc . (V 11)
Oyy ne + nh 36 5 -EF

In both aluminum and indium the volume of the
second zone hole-like portion is much larger than the
volume of the third zone electron-like portion. This

means that
n + n 5 n 5 n - n = l/atom , (v.12)
and consequently

(Eyy)osc _ I
Oyy ne - nh de e=eF

 

) . (v.13)

Combining equations (v.5) and (v.13) gives

osc eL T dne

_ o osc
sd (H,T) — n _ n ( ) , w T >> 1. (v.14)

e h d8 l€=€F c

 

dn

We now must evaluate the quantity (522' )osc

s=eF

Within the framework of the Lifshitz-KosevichI54) theory

for a NFE gas, GoldISS) obtains an expression for nose.

For the fundamental frequency his expression (in MKS

100

units) reduces to

 

1 _.
nosc = ——-l—-§-(e“hH)3/2 2" 4 IlKl sin(§% - 2wy+ %) (v.15)
n

where y is an undetermined phase factor,

Q)
3’

is the curvature of the extremal cross-sectional

muo

area of the Fermi surface in the direction of H,
X

I1 = m' kBT << fiwc'

*

2n2m k

X’ = TB T

H I

*
2n2m kB
K1 = exPI' ehH TD”
. . (56)
TD 18 the Dingle Temperature.
The 1 factor gives the temperature dependence of the

l
amplitude of the oscillations in the carrier density.

Since I1 decreases as T increases, the effect of a non—
zero temperature is to decrease the oscillation amplitude.
The oscillation amplitude will be further reduced if the
electrons suffer collisions. This means that the electrons
will possess a finite (not infinite) relaxation time. The

K1 factor gives this correction because T is inversely

D
proportional to the relaxation time.

Grenier, Zebouni, and ReynoldsI47) claim that
“ ‘ anosc
the most important term in 3? ls-s arises from the
‘ F

derivative of the phase (recall that A = A(e)). This

101

assumption gives

 

osc * l/2 _
e E—EF w'h '3 Al
2 (v.16)
BkH

osc
d I
the effect of a finite relaxation time will be neglected,

In an effort to estimate an upper bound for S

i.e. set K1 = l. The calculation will be performed for
H aligned along the [010] axis in both aluminum and in-

dium. For the B-orbit in aluminum the values:

m* = 0.095 m (32)
8

32A _ 7(57)

'7?" ‘

a H

-1 _ _ -29 3

n = 15.0kG.
T = 1.72K

give an oscillation amplitude.

SESC(A1) = 0.28 x lO-llV/K. (v.17)

For the neck orbit of the B-arms in indium the

values:
* 58
m = 0.2 m I )
e
32A ,_ 5(57)
'7' - 0b
3kH

102

15.5kG.

H

T = 1.67K

give an oscillation amplitude

535°(1n) = 1.0 x 10-11V/K . (v.18)

It is hoped that eqs. (v.17) and (v.18) give
order of magnitude estimates of the amplitude of mag-
netothermopower oscillations arising from oscillations
in the density of states at 8F of aluminum and indium.

These estimates suggest that such S oscillations should

d
be roughly the size of the signal noise in our measure-
ments and much smaller than the non-oscillatory component

of S.

C. Magnetic Breakdown

 

Up until now we have assumed that the electrons
can make transitions only between states within the same
band, i.e. we have ignored interband transitions. How-
ever, in large magnetic fields electrons can tunnel be-
tween bands, giving rise to the phenomenon of magnetic
breakdown. When this happens, the semi-classical
quantization scheme described in section V.A breaks

down. The review article by Stark and Falicov‘Go)

gives
a more comprehensive study of this phenomenon than will
be presented here.

To understand magnetic breakdown we will first

consider a free electron gas in a magnetic field.

103

Neglecting interactions among the electrons, the electron
wavefunctions are those of a particle in a circular orbit
in the (kx, ky) plane, which is normal to H. We then
introduce a lattice perturbation of the form V(x) =

Z VGelcx. When an orbit passes through a zone boundary

G
(i.e. its wavevector in the x direction, kx, is equal to

1 1/2 G), there is the possibility of Bragg reflection.
Using Figure V.2(a) as an example, this means that in-
stead of continuing along AB, the orbit may switch to the
direction AC, and so on.

If the strength of the perturbation is increased,
then the trajectories at A will be split apart in energy,
and the route AC will be favored. The electron is now
moving on an ordinary (open) orbit in the repeated zone
scheme. This is shown in Figure V.2(b). The part B of
the circle in Figure V.2(a) has now been joined up into
a separate branch of the Fermi surface, and is traversed
separately.

As the magnetic field is increased, the effect
of the lattice perturbation will decrease, so the
circular orbit will become more and more favored. In-
stead of the electron going along AC, it may break through
the energy gap, ("tunnel through" the region separating
the two orbits in k-space) and wind up going around

orbit B.

104

 

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105

In practice such behavior occurs at portions of
the Fermi surface where the band splittings are very

small (10'2 - 10‘3

eV.). This usually happens at symmetry
points of the Brillouin zone. The best known cases
correspond to splittings due to spin-orbit coupling.

The description of magnetic breakdown as illus-
trated by Figure V.2(b) can be extended to real metals
like aluminum and indium. Figure v.3 shows the

third zone pieces of the Fermi surface of aluminum
and Figure v.4 shows a central (001) section of this
Fermi surface. With a magnetic field along the [001]
axis, electrons travel clockwise on orbits like FABH...
(of Figure v.4), thus they are hole-like. In the absence
of magnetic breakdown, a hole arriving at B from A will
continue to H: otherwise it could break through the
energy gap and "jump" on to the small electron—like
B-orbit and proceed to C. Here it may "jump" to the
hole-like orbit and proceed to D, or remain on the
B-orbit and continue to B, then to H or C, etc. The net
result of this, ignoring complications due to the pos-
sibility of scattering while on the B-orbit, is that
either the hole remains on its original (second zone)
trajectory, or it moves across the zone boundary on to
another second zone trajectory and moves in the opposite
direction. When the breakdown probability is large, the

hole is "reflected" near the corner W' and thus travels

106

v x oasrr ‘3‘ ”‘8"
, N
I ’I \
I, ’ I)
I / \ I I
I / \ I /
§ ,’ \ , 4 l
\\ / \ a ’ f \ I
\ a
\ \\ )’ ‘\ I
\ \ ’ \l
\ ’ I
\ \ ’ /
\ /
\ /
‘\ \” ‘/
\.\ I!
\ \ , .—
\\ .—- .o- ’

Figure v.3: Ashcroft's 4-OPW pseudopotential model for the
electron-like 3rd zone of the Fermi surface of aluminum
(after Abele and Blatt(52)).

 

107

+v—“’. ‘ - .

 

 

 

 

 

 

 

 

Figure V.4: Central [100] section through the aluminum
Fermi’surface (after Ashcroft(53)).

 

108

over the lens shaped, electronélike orbit. Similarly
a hole arriving at C from G will move to either D or B:
if it reaches B, it then goes toward H or returns to C.
When the breakdown probability is large the resultant
orbit GCBH... is circular and electron-like.

Pippard(64) has derived an expression for the
breakdown probability for a junction between orbits of
the same topology as in Figure V.4. If R is the prob-
ability of a particle proceeding along CDE after arriving
at B from A, or proceeding along BH after arriving at

C from G, then

 

2
R = P 2 (v.19)
l - 2cose + Q
*
where P = exp(-HO/H), Ho =_egm c/effie, 69 being the

energy gap between the bands at the point in k-space
where the break through occurs, Q = l-P, and e is defined

as
6 = 6 + -—— (v.20)

where 60 is a constant and A is the area (in k-space)
through which breakdown occurs (the B-orbit in our case).
Thus, the breakdown probability has an oscillatory com-
ponent with a period (in l/H) of Zne/dfiA. This is the
same period as would be found in the ordinary (i.e. in
absence of magnetic breakdown) de Haas-van Alphen

oscillations associated with the small connecting orbit.

109

As the breakdown probability becomes alternately stronger
and weaker with increasing field, the orbits of the
central (001) section of the Fermi surface become
alternately more and less electron-like and both the equil-
ibrium and transport properties of the metal fluctuate
accordingly.

So far we have only described a simple micro-
scopic picture of magnetic breakdown, and have not con-
sidered the effect of this phenomenon on the macroscopic
properties of the material. The equilibrium properties
(e.g. de Haas-van Alphen effect) are the easiest to treat,
sovuawill consider them first. Once the electron systemis
free energy is known as a function of H, all the equili-
brium properties may be calculated from it via the
appropriate thermodynamic relations. To calculate the
free energy requires a knowledge of the magnetic field
dependent density of states p(e,H). Falicov and

(65)

Stachowiak have proposed a Green's function method
for calculating the oscillatory part of p(e,H). The
Green's function has been exactly calculated for the free
electron case, and it is believed to provide an accurate
description of the NFE case (after making the appropriate

changes(60)).

The important result of this procedure is
that oscillations, periodic in l/H, will occur in the
equilibrium properties. These oscillations will have a

frequency corresponding to the area of the breakdown

110

orbits: this frequency will be the same as that observed
in the ordinary de Haas-van Alphen type oscillations.
However, when the breakdown nears completion new fre-
quencies will appear: these frequencies reflect the new
and larger orbits formed by breakdown (similar to the
free electron circle shown in Figure V.3(a)). Such giant
orbits have been seen in a number of metals, e.g.
magnesium.(66)
Currently the description of transport phenomena
in the presence of magnetic breakdown is far from com?
plete. The effects on electrical conductivity have been

considered by a number of authors, some being Pippard,(67)

(68) (60)

Young, and Stark and Falicov.

They all agree that
magnetic breakdown will cause oscillations, periodic in
l/H, which will have the same frequencies as those in
the equilibrium properties. Stark and Falicov have per-
formed a calculation for an idealized case, which in-
dicates that the oscillation amplitude can be comparable
to the non-oscillatory component of the transverse
magnetoresistance.(60)
At the present time very little work has been
done on the effects of magnetic breakdown on magneto-
thermopower. YoungIGg) has considered this problem for
a different junction topology than that found in aluminum

and indium. He obtains an oscillatory component of the

magnetothermopower, which again exhibits the same

111

frequencies as those of the equilibrium properties. We
might note’in passing, that if parameters appropriate to
the particular orbits of present interest are substituted
into Young's expression, then within the experimental
range of temperatures and magnetic fields, this expression
may be approximated by one or more terms of the form of
eq. (v.4). Young also notes that for an "opening" net-
work (similar to the junction topology of aluminum and
indium) no actual calculations have been made, because a
simple expression does not arise. However, there should
be an oscillatory component.

The present state of theory of the effect of
magnetic breakdown on magnetothermopower oscillations is
not far enough advanced to allow calculation of the
amplitude of these oscillations. However, independent
experimental evidence obtained prior to this thesis in-
dicates that magnetic breakdown is present in all metals
that have exhibited giant magnetothermopower oscillations*
(this will be discussed in more detail in section VI.C.1).
These giant oscillations have frequencies that correspond
.to the small connecting orbits which facilitate the mag-

netic breakdown.

 

*

The fact that the oscillatory component is comparable to,
or much larger than the non-oscillatory component en-
genders the designation "giant oscillations".

VI. OSCILLATORY THERMOELECTRIC PHENOMENA - EXPERIMENT

A. The Fermi Surfaces of Aluminum and Indium

'The Fermi surfaces of aluminum and indium have
been extensively studied; and consequently the Fermi sur-
face dimensions are known in detail. The Fermi surfaces
of both these metals consist of a large second zone hole
surface and a small third zone surface which is made of
"arms" of electrons. Figure V1.1 shows the Fermi surface
of indium. The second zone hole surface (Figure V1.1(b))
is very adequately represented by the free-electron
model, and is quite similar for both aluminum and indium.
The third zone electron surface of indium shown in
Figure V1.1(c), consists only of B-arms, and these 8 arms
are joined together in four-sided rings in the (001) (but
not the (100) plane) by small junctions near point T.(59)
The third zone surface of aluminum is similar to that of
indium, but with a few significant differences:

1. The third zone of aluminum consists of both

a and 8 arms (see Figure v.3).
2. The third zone of aluminum consists of four

sided rings of "arms" in both the (001) and

(100) planes (a consequence of cubic symmetry).

112

113

EM!
23

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“mUMMHSm «do: H0008 couuomamlmmum sawmcm

 

How macs swooHHHHm one Ame "H.H> musmfim

 

114

3. In aluminum there is a swelling at the junc—

tion where the B-arms connect: in indium this

swelling is either smallISZ)

(59)

or completely
absent. This is a consequence of the

much stronger spin-orbit interaction present

in indium.

The free electron model predicts that the second

and third zones would be degenerate at certain high symmetry
points of the Brillouin zone. However, a non-zero spin-
orbit interaction breaks this degeneracy, thereby closing
both surfaces (i.e. there are no open orbits). The
oscillations observed in this thesis have frequencies which
are associated with the third zone B and y-orbits in

aluminum, and the neck and belly orbits of theE3"arm" in

indium.

B. Aluminum

1. Introduction

 

Since the spin-orbit interaction is quite small
in aluminum, the second and third zones are almost but
not quite degenerate at certain points in the Brillouin
zone. Therefore, aluminum has a Fermi surface which is
favorable to magnetic breakdown. Balcombe and Parker‘34)
have observed oscillations in the magnetoresistance of
aluminum for H along the [010] axis. These oscillations

had the same frequency as the electron-like 8 orbits.

Using a simple magnetic breakdown model, which allowed

115

holes (second zone) to "jump" from one orbit to another

in an adjacent Brillouin zone using a third zone electron-
like B-orbit as a bridge (see section V.C), their calculated.
values for pxx and pxy agreed "tolerably well with the
experimental results".

Recently, Bozhko and VOl'skiiI7o) have observed
oscillations in the magnetoresistance of aluminum for H
along the [110] axis, when the temperature was below 1K.
They also attributed these oscillations to magnetic break-
down.

Concurrent with the research constituting this
thesis, two other research groups: Kesternich and

Papastaikoudis,(29) (34)

and Sirota, et. al., independently
discovered the existence of giant quantum oscillations in
the magnetothermopower of aluminum for H along the [010]
direction. The oscillation frequency was the same as that

observed by Balcombe and Parker.(34)

Both of these groups
attributed these giant oscillations to magnetic breakdown.
Additionally, Kesternich and Papastaikoudis noted that
the amplitude of the.thermopower oscillations exhibited
a simple exponential decay in l/H over the entire field
range (H < 48kG).

Both of the aforementioned groups constrained
their thermopower measurements to having the cold end of

the sample remaining at about 4.2K, while producing

temperature differences along the sample of about 1K.

116

The great sensitivity of the chopper-amplifier system
allowed the measurements of this thesis to be performed
using relatively small temperature differences (typically
0.1 - 0.2K). Therefore, the temperature variation of the
oscillations could be studied. The thermopower measure-

ments were extended down to 1.7K.

2. Experimental Procedure

 

All measurements were performed on the Al(Ga)
sample (see Table I-l). The measurements were performed
using the chopper-amplifier in the field sweep mode (de-
scribed hlsections II.A.2 and II.F). This procedure pro-
vided a continuous record of the thermopower as a func-
tion of the magnitude of the magnetic field, while the

sample temperature remained essentially fixed.

3. Data and Analysis

 

3.1. H parallel to {010}

 

Figure V1.2 displays an example of the raw data;
it shows the relative magnitudes of the oscillatory and
non-oscillatory (broken line) components, and indicates
the precision of the data. In this figure the solid line
labeled 8 = 0 represents the recorder output when there
is no temperature gradient along the sample; the
oscillating curve is uncorrected for the field dependent
amplification of the voltage measuring system and also

for the slight increase in AT along the sample as H is

117

 

 

 

 

 

 

 

 

 

A

m

.2:

c v I
3 __———-— 1-..---‘Dd-(--.---.~-s o
D ., .- ,_~

‘5 S...
U

0)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I l 1 I 1 I l I
'2 '4 '6 I8 20

fiIkG)

Fi ure V1.2: The variation of S with magnetic field at

T = 3.55K for the Al(Ga) sample. H is along the [010]

axis. The solid line labeled S = 0 represents the x-y re-
corder output when there is no applied temperature gradient.
The broken line indicates the non-oscillatory component of
S. The oscillating curve represents raw data, uncorrected
for a field dependent amplification of the measuring system.
The magnitude of this correction is indicated by the manner
in which the S = 0 curve and the broken line approach each
other as H increases. When the data are corrected, these
two curves are equidistant at all fields above about llkG

to within experimental uncertainty.

118

increased. One can get a feeling for the size of the
corrections by noting that after correction, the S = 0
curve and the broken line remained equidistant at all
fields above llkG to within experimental accuracy. The
oscillations in Figure IV.2 are described by a single
frequency, characteristic of the third zone B-orbit of
aluminum. This is consistent with all previous measure-
ments of oscillatory transport properties.(29'30’34)
The oscillation pattern was insensitive to small mis-
alignments (~ 1°) of H relative to [010].

In section V.B the amplitude of oscillation in S
due to variation of the density of states at the Fermi
energy was estimated, and it was concluded that these
oscillations should be smaller than the signal noise in-
herent in the voltage measuring system. The variation
in the S = 0 curve in Figure V1.2 gives a measure of
this noise. It is quite clear from this figure that the
oscillation amplitude estimate of section V.B is much
too small to explain the data. In fact at 15.0kG, the
experimental oscillation amplitude was more the four
orders of magnitude larger than the estimate given by
eq. (v.17).* However, previous oscillatory magneto-

(34)

resistance data suggest that magnetic breakdown is

 

*
For comparison with experiment the value of S (Al)

given in eq. (v.17) must be multiplied by four, because
there are four independent 8 orbits which contribute to 8.

08¢

119

present in this direction in aluminum. If we assume that
breakdown is also the cause of the magnetothermopower
oscillations then we should expect a discrepancy between
the calculation of section V.B and experiment; and indeed
the discrepancy is large.

In section V.A it was noted that all calculations
predict for the orbit of interest within the experimental
range of temperatures and magnetic fields, that the
thermopower oscillation amplitudes should consist of one
or more terms of the form of eq. (v.4). If only one term
is present or dominant, then the data can be directly
analyzed in terms of eq. (v.4). According to this ex-
pression, if ln(SH-n) is plotted against l/H for a series
of fixed temperatures, straight lines with slopes
ankB(T + TD)/B should result. Figures V1.3 - V1.5 show
such plots for n = -3/2, 0, +3/2 respectively. The best
straight lines are for n = 0 and n = -3/2. Continuing
in the spirit of eq. (v.4), plotting these slopes
against T should also yield a straight line with slope
2n2kBm*c/efi and intercept at T = OK, ankBTDm*c/eh.

Thus, from knowledge of the slope and intercept both m*
and TD can be experimentally derived. Figure VI.6 shows
such a plot for n = 0; similar plots are obtained for

n = t 3/2. Table VI-l contains the values obtained for
m* and TD for two perpendicular (but crystallographically

equivalent) directions of H.

120

6
ITTIW

I

H3’ZSOSC/ T x10 (03’? V/Kz)

 

 

 

II-—
__ T=5.55K
T=4.57 K
I I I I I I I
0.!
4 6 8 IO
H" x105 (0")
Figure V1.3: The variation of the quantity H3/ZSOSC/T

 

(plottedcnia logarithmic scale) with H‘1 for a series of
temperatures for sample Al(Ga). H was directed along [010].

121

    

 

 

IO --- -
it
\
:>
”9 I Tzlo72 K
C:
c>‘ ‘y
can 7:2.43 K

T=5.55K
T34057K
0,“. 1 1 I I I II
4 6 8 IO
H"' x 105 CG")

Figure V1.4: The variation of the quantity Sosc/T (plotted

on a logarithmic scale) with H'1 for a series of tempera-
tures for sample Al(Ga). H was directed along [010].

 

122

H-3/2 soSC/T X {0'2 (6-3/2 V/KZ)

 

T=3.55 K

 

0"4 6 8 IO

H" x10“3 (0")

Figure V1.5: The variation of the quantity H-B/Zsosc/T
(pIOtted on a logarithmic scale) with H'1 for a series of
temperatures for sample Al(Ga). H was directed along [010].

123

1) x 10'“ (G)

11‘H2

 

 

 

IOQISOSCIHII/SOSCIH2IJ/(H

7"(K)

Figure VI.6: The variation with temperature of the slopes
of the lines shown in Figure V1.4 for H parallel to [010]
(circles) and of similar lepes of similar lines for H
parallel to [100] (triangles). The upright triangles rep-
resent initial measurements with H parallel to [100]. To.
evaluate the effects of changes in sample treatment and
alignment, after these initial measurements were completed
the sample was removed from the holder, its potential leads
were removed, it was reannealed, the potential leads were
reattached and the sample was returned to the holder. .The
resulting data are indicated by inverted triangles.

124

*
Table V121: Values of m /me and TD

directions of the magnetic field H and for
different assumed values of n for Al(Ga)

for two orthogonal

*

n m /me TD(K)
H parallel to [010] -3/2 0.096 1 0.01 2.5 i 0.6
0 0.093 1 0.01 1.1 :_0.6
+3/2 0.086 _t 0.02 ~ 0 i 0.6
H parallel to [lOO] -3/2 0.093 : 0.01 4.2 i 0.6
0 0.092 :_0.01 2.4 :_0.6
+3/2 0.095 1 0.01 0.7 i 0.6

We see that the values of m* are the same to within the
specified uncertainties for both directions of H and for
all values of n in the predicted range, whereas the values
of TD are not in agreement. This demonstrates that TD
can only be determined for an assumed value of n, but
reliable values of m* may be determined without such an
assumption. Additional studies showed that m* did not
change significantly when the magnetic field was reversed
or when the sample was removed from the holder, re-annealed
and returned.

From Table IV—l it is concluded that m* =
(0.93 : .01)me. The only other measurements of m: for
the B—orbit of aluminum are de Haas-van Alphen measure-

(32)

*
ments. These give m = (0.102 : .006)me, in agree-

ment to within the specified uncertainties with the value

125

derived from thermopower measurements. We might note in
*

passing that theory predicts the m derived from de Haas-

van Alphen data to be phonon-enhanced by about 50% over

*
(71'72) Therefore, the m derived

the bare band mass.
from thermopower measurements is similarly enhanced.
Finally, the thermopower data shows 1 i 1 5 2,

again in accordance with available predictions.

3.2. H along other directions in the (001) plane

Magnetic breakdown in aluminum is expected to be
greatest for H directed along [010]. In this section
data will be presented for H in directions other than
[010].

As previously noted, the S oscillation pattern
was insensitive to small rotations (~1°) of H from [010].
However, the oscillation amplitude did decrease as H was
rotated away from [010]. Figure V1.7 shows that the
oscillation amplitude has been reduced to about the size
of the non-oscillatory component (at T = 2.4K, H ~ l7kG)
when H is rotated 20° from [010] axis in the (001) plane.
Figure V1.8 shows that the oscillation amplitude has be-
come much smaller than the non-oscillatory component of
S when H is directed along [110]. Thus, it may be con-
cluded that although magnetic breakdown via the B-orbit
exists for H aligned along any direction in (001), the
probability of magnetic breakdown decreases as H is

rotated away from [010] (i.e. the energy gap between the

126

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8

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128

second and third Brillouin zones increases as H is
rotated away from [010]).

For a particular orbit, a magnetic field directed
in an arbitrary direction in the (010) plane will in gen-
eral intersect different extremal cross-sectional areas
(normal to H) on the individual third zone "arms" which
form the ring of arms. Since the cross-sectional area
determines the oscillation frequency, the general
oscillation pattern due to a particular orbit will be
composed of oscillations of several frequencies. Also
the oscillation amplitude is expected to be frequency
dependent. Thus, the oscillation pattern for the magnetic
field in an arbitrary direction may be quite complicated,
as shown in Figure V1.9. Only for H along either [010]
or [110] will the frequencies of all B-orbits be equal,
so that the oscillation pattern may be described by a
single frequency.

Figure V1.10 shows frequencies of the major low
frequency oscillations obtained from S data. Also shown
in Figure V1.10 are the frequencies obtained from de Haas-

(32)

van Alphen measurements and Ashcroft's predicted fre-

quencies for the B-orbits based on a pseudopotential cal-

(63) which was fit to de Haas-van Alphen data.

culation
This figure shows that the frequencies for a particular
orbit obtained from either de Haas-van Alphen or magneto-

thermopower oscillation measurements agree to within their

129

(ARB. UNITS)

S
.b
(N

o

 

 

34° Iv,
Awfuwe/V”n/\/
J J I I I I L I I I I
ICD l5 2()

H(KG)

Figure 21.9: The variation of S with magnetic field at
T = 2.3K for H along a series of directions in the (001)

plane of the Al(Ga) sample. 0° corresponds to H directed
along [100].

130

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131

mutual uncertainties. It is interesting to note that the
magnetothermopower oscillations for 0 between 0° and 20°
appear to be due to 81 orbits, in contrast to the de Haas-
van Alphen measurements which are sensitive to the 82
orbits in this range of angles.* Thus, magnetothermopower
oscillations apparently provide information regarding the
shape of the Fermi surface which is complementary to that
obtained from de Haas-van Alphen measurements.

If one carefully looks at the 46° and 49°
oscillation patterns of Figure V1.9, one discerns a small,

high frequency component above l7kG. The period of this

high frequency component is (3.4 i .3) X 10-76-1. This
_ _ **
is in agreement with the value 3.5 x 10 7G 1 obtained
(32)

from de Haas-van Alphen measurements of the larger
y-orbits of the third zone Fermi surface of aluminum (see
Figure v.3). We might note in passing that Bozhko and
Vol'skiiI7o) have observed magnetoresistance oscillations

with a period characteristic of the y orbits of aluminum

for H along the [110] axis.

 

* .

The geometrical arrangement of the 81 orbit with respect
to the 82 orbit is shown in Figure 5 of C.O. Larson and
W. L. Gordon, Phys. Rev. 156, 703 (1967).

**This frequency is actually due to the y orbit. The
geometrical arrangement of the y with reépect to the
other orbits is shown in Figure 5 of C.O. Larson and
W.L. Gordon, Phys. Rev. 156, 703 (1967).

132

C. Indium

1. Introduction

 

It is apparent from pseudopotential models of

the Fermi surface of indium,(52'53'59'73'74)

that magnetic
breakdown could occur for a sufficiently large magnetic
field along the [010] axis. However, until now, no ex-
perimental evidence of such breakdown has been obtained.
In this section experimental data showing giant, low fre-
quency magnetothermopower oscillations are presented.

The oscillations are attributed to magnetic breakdown.
This attribution is supported by a detailed comparison
between the indium data and similar data for the [010]
breakdown orbit in aluminum (presented in section VI.B).

Aluminum is one of only four metals (aluminum,(29-3l)

(75) (69’76) and zinc(47)) for which giant

magnesium, tin,
magnetothermopower oscillations have been observed. In

all four metals, these oscillations have been attributed
to magnetic breakdown. In each case, this attribution is

supported by independent magnetoresistance(34'77'78)

de Haas-van Alphen(66,79)

measurements and also by

analysis of the known Fermi surface of the metal. Aluminum
was chosen for the comparison because its electronic
structure and Fermi surface are most nearly similar to
those of indium.

In addition to the giant oscillations for H along

[010], large, low frequency magnetothermopower oscillations

133

for H along [101] and [110], and small, medium frequency
oscillations for H along [110] were also observed. These
low frequency oscillations are also tentatively attributed
to magnetic breakdown, but the medium frequency ones are

not.

2. Experimental Procedure

 

Oscillatory behavior was found in samples In-II,
In(Sn)-I, In(Sn)-II, and In(Sn)—III. As the temperature
was raised the oscillation amplitude fell off rapidly,
becoming comparable to the measuring uncertainty at about
2.5K. Searches for oscillatory magnetothermopower were
not performed on samples In-I and In(Ga); presumably
these samples would have exhibited this behavior when H
was aligned along the proper crystallOgraphic axis.

The measurements were performed using the chopper-
amplifier in the field sweep mode (see sections II.A.2

and II.F).

3. Data and Analysis

 

Figure V1.11 shows the output of the x-y recorder
(essentially a plot of S vs H) for samples In(Sn)-I and
In(Sn)-II for H along [101], [110], and [010]. No
oscillations were observed for H along [001]. In Figure

V1.11 four sets of oscillations are present, each periodic

in l/H. The oscillation frequencies are 1.4 i 0.1 x 105G

for H parallel to [101]; 1.55 i 0.1 x 1056 and

6

4.6 i 0.1 x 10 G for H parallel to [110]; and

134

 

 

HI) [1011
T=1.68K
{ Hutfl'OJ
>:
v T=1.98K
Q
2
03
HIIEOlOJ
T=1.67 K
I I I'

 

 

10 15 20
H (kGJ

Figure V1.11: The magnetic field dependence of S for H
aligned along the indicated crystallographic axes. The
lines labeled S = 0 represent the recorder output when
there is no applied temperature gradient. The oscillating
curves represent raw data, uncorrected for the field de-
pendent amplification of the measuring system. The ampli-
fication decreases by about 30% gver the field range shown.
The curves for H aligned along [101] and [019] are from
sample In(Sn)-I and the curve for H along [110] is from
sample In(Sn)-II.

 

135

1.04 i .04 x 105G for H parallel to [010]. All four
frequencies agree to within mutual uncertainties with

values obtained for indium from de Haas-van AlphenISZ'Bo)

and ultrasonic absorptionIBl)

measurements; measurements
which can be understood without invoking magnetic break-
down.

In an effort to see whether the oscillation
amplitudes were consistent with eq. (v.4), graphs of
ln(Hn SOSC/T) vs l/H with n = 0,1 were plotted for H along
the [010] axis. Figure V1.12 shows a typical graph. The
data clearly diverge from a straight line. Thus, in
contrast to aluminum, the oscillation amplitudes in indium
are not consistent with eq. (V.4). The relatively small
signal to noise ratio for H along either [101] or [110]
precluded any similar oscillation amplitude analysis for
these directions.

Samples In-II and 1n(Sn)-III also exhibited
oscillations for H directed along the [010] and [110]
axes respectively. For H along the [010] direction,
the data for the purer sample In-II were similar to those
for 1n(Sn)-1, but the oscillation amplitude was about
25% smaller for a field of about lSkG. at 1.7K. One
also obtains a similar comparison with the data of the
purer In(Sn)-II with those of In(Sn)-III for a magnetic
field of about lSkG. directed along the [110] axis at

1.9K. Therefore, it is concluded that the oscillation

136

 

 

50M
(gt 9
\x o
>
9 I0—
¢ )—
9 +
x r-
F— I-
6‘
‘8 5—
w
I _
' I I I I I
4 6 8 IO
l/H x no5 (0")

Figure V1.12: The variation of the quantity HSosc/T
(plotted on a logarithmic scale) with l/H for sample
In(Sn)-I at T = l.67K. H was directed along [010].

'— fl,“ .m. 1.4

137

amplitudes were limited by extended crystal lattice
imperfections (mosaic structure, lattice dislocations,
etc.) rather than by impurity scattering. Such a result
is reasonable since indium is a very soft material which
can be easily damaged.

Like aluminum, the oscillation pattern for these
indium crystals were not significantly affected by 1°
rotations of H relative to the major symmetry directions;
and similarly the oscillation amplitudes gradually de-
creased as H was rotated away from the major symmetry
directions.

The remainder of this section will concentrate on
the giant oscillations for H along [010], both because
these are the largest oscillations and because they can
be compared directly with breakdown oscillations in aluminum
for the same field orientation.

As noted earlier, electronic transport oscilla-
tions are expected to arise from either magnetic break-
down or oscillations in the density of states at SF.
Clearly, the most convincing proof that the giant thermo-
power oscillations which are observed are a result of
magnetic breakdown would be a demonstration that their
amplitude could be predicted from theory. Unfortunately,
the theory of the effect of magnetic breakdown on the
thermopower of metals is not sufficiently far advanced

to allow direct quantitative comparison with experiment.

138

Therefore, the breakdown hypothesis was indirectly tested

in two ways involving comparison of the indium data with

similar data for the [010] breakdown orbit in aluminum.
First, the estimation of the oscillation amplitude

due to the other mechanism which might cause oscillations:

the variation with H of the density of states at the

Fermi energy in the absence of magnetic breakdown given

by eqs. (V.l7) and (v.18), was compared with the experi-

mentally measured amplitudes for both aluminum and indium.

L”fl:.-’rl€.";m 5'
. . ' .4 -..

For aluminum the calculated oscillation amplitude for H

——-

along [010] was four orders of magnitude smaller than the
experimental amplitude, and for indium it was three

orders of magnitude smaller. Since the oscillations in
aluminum result from breakdown, a discrepancy between this
calculation and experiment is expected and indeed it is
very large. The fact that indium also exhibits a very
large discrepancy is taken as evidence in favor of magnetic
breakdown in indium. The large oscillations for H parallel
to [101] and [110] are also two or more orders of

magnitude larger than the calculated value of eq. (v.18),*

 

2
* *
In eq. v.18, the values of |§—%| and m appropriate to H
3k
H

along [010] was used. Proper calculations would use the
* - _.
3—%| and m appropriate to H along either [101] or [110].
3k
H

However, it seems unlikely that these changes will signif-
icantly reduce the discrepancy between calculation and ex-
periment.

139

making them candidates for breakdown too. On the other

hand, the small medium frequency oscillations for H

parallel to [110] are within one order of magnitude of

the value predicted by this simple model, and it is there-

fore unnecessary to invoke breakdown to explain them.
Second, the angular variation of the thermo-

powers of the two indium samples was compared with that

of the aluminum single crystal as H was rotated away

from the [010] direction. Agreement between the forms

of these two angular variations might provide additional

 

evidence for magnetic breakdown in indium. The angular
variations are shown in Figure V1.13. The field was
rotated around the [001] axis of the aluminum sample and
of sample In-II, and around the [101] axis of sample
1n(Sn)-I. Indeed, the forms of the rotation patterns
are nearly the same for both aluminum and indium.
Similar patterns were also obtained for rotations with

H near [101] and [110].

Having shown that the low frequency data can be
plausibly attributed to magnetic breakdown, this section
will be concluded by considering the ramifications of
the presence of such breakdown for the most recent
pseudopotential model of the Fermi surface of indium by

(59)

Holtham. For H along [010], this model yields a

breakdown field H0 in the range of 200-400kG,(S7) more

than an order of magnitude larger than the largest field

 

 

8 ——
6 - AI.
T: 2.26K
4 — ROT. AXIS-[001]
2 __
O _
4...
_ In
A 2 T21.76 K
x O __ ROT.AXIS-[001]
\
3
‘6 -2r-
2'
0)
6h In
4 1.. 131.71 K
ROT. AXIS-[101]
2 __
0 _—
_2 _—
I I I I I I I
-60-40-20 0 20 4O 60
£010]
9 (DEG)

Figure V1.13: Variation of S with rotation angle measured
from the [010] axis, at H = lSkG for aluminum and indium
single crystals. The aluminum data are from sample Al(Ga).
The indium data for H rotated about the [001] axis are fronI
sample In-II andtfluadata for H rotated about the [101] axis
are from sample In(Sn)-I.

 

141

used here. In view of the exponential dependence of break-

down probability on the ratio H/Ho,(64)

it seems unlikely
that the experiments performed in this thesis would show
significant effects of breakdown for such a large HO. The
experiments suggest that H0 is roughly an order of

magnitude smaller. As Holtham points out,(59)

this would
imply that the second and third zones are closer together
than in his model.
In conclusion, large, low frequency oscillations
have been found for H along the [010], [101], and [110]
axes, while small, medium frequency oscillations are
found for H along the [110] axis. All frequencies agree
to within mutual uncertainties with the values obtained by
de Haas-van Alphen and ultrasonic absorption measure-
ments on indium. Arguing by analogy with aluminum, a
metal for which there is independent evidence that magnetic
breakdown exists in directions where there are giant
magnetothermopower oscillations, the large, low fre-
quency oscillations are attributed to magnetic breakdown.
This attribution implies that the energy gap between the
second and third zones is smaller than predicted by the
latest pseudopotential model of indium.(59)
To confirm the existence of magnetic breakdown
in indium, it is suggested that a search for its effects

on other properties be begun. Because of the apparent

sensitivity of magnetothermopower oscillations to

 

142

(29)

breakdown, other properties are likely to require

both magnetic fields well above 20kG. and temperatures
below 2.5K. This dual requirement is the probable rea-
son for the lack of prior evidence of magnetic breakdown
in indium, since measurements of less sensitive properties

such as magnetoresistence have only been made at either

high fields or comparably low temperatures, but not both

at once.

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10.
11.
12.

13.

14.

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APPENDICES

APPENDIX I

CALCULATION or ANISOTROPY OF 1(1) FOR Al(Ga)

We will use Sorbello's approximate phase shift
model(7) to calculate the anisotropy of 1(k) of Al(Ga).
Within the Born approximation (52 << n) the average value
of l/To(k) around an orbit on the Fermi surface,

<l/To(k)>,may be written

(D

 

<l/TO(E)> = gficg z (22 + l)(6:A)2<F£(k)> . (AI.1)
‘F 0 i=0
SEA are the phase shifts obtained from the Born approxima-

tion, and depend only on the impurity scattering potential.
The quantity F2(k), called the "amplitude factor"’measures
the square of the magnitude of the 1 component of
¢K(;)(¢k(;) is the pseudowavefunction associated with the
Bloch state k) when ¢k(;) is expanded in spherical harmonics
about a lattice site. In the free electron (P.E.)

l for all 3.

III

approximation F£(k)
Using the 63A obtained from the Heine-Abarenkov-

Animalu form factors in the Born approximation, which are

given in Sorbello's table 1; and the average amplitude

factors <F£(k)> for orbits on the Fermi surface of

148

149

aluminum, which are given in Sorbello's table 3, we can

calculate the ratio <1/Io(k)>/(1/To) for Al(Ga).

P.E.
Table AI-l shows the results for the third zone y, a,
B and orbits and second zone [110] orbit on the Fermi

surface of aluminum.

Table AI-l: Average reciprocal relaxation times for
Al(Ga) obtained from Sorbello's approx-
imate phase shift model 7

ORBIT
<l/To>/(l/TO)F.E. 1.10 1.10 1.19 1.08

Table AI—l shows that the average reciprocal relaxation
times of the third zone of Al(Ga) are isotrOpic to
within 10%. Sorbello also claims that the average
reciprocal relaxation times of the second zone should

be more isotropic than the third zone times.(82)

There-
fore from table A1-1 it is concluded that the average
relaxation time is nearly isotropic (to within an
accuracy of about 10%) over the entire Fermi surface

(which only consists of second and third zone pieces) of

dilute Al(Ga) alloys.

APPENDIX II

30(8)

DETERMINATION OF 38

From eq. (III.3la) we see that

2 v.
e f dS —$ T v

0"“) S(e) v j'

13 4flzfi
which may be rewritten as

2

_ e E
Oij‘e’ ' ggzg [S(e) as v

- a. T v., (AII.1)

A . . . .th
where ei is a unit vector in the l

(83)

direction. Apply-

ing the divergence theorem to eq. (AII.1) gives

 

 

_ e 3+ . A
e 8 ds A

= d ' ' -—— . . . 0 AI 02
4fl2h f0 8 15,8 ) hv 9k (e1 1 v3) ( I )

So it follows that

30..(€) 2

1 _ e dS . A
38 _ 4fl3fi fS(e) hv 6k (ei T vj)’ (AII‘3)
We recognize that V - (ei T vj) = [$(t v)]ij so
30..(e) 2
13 = e dS + +

150

151

For cubic symmetry eq. (AII.4) becomes

2
30(8) e dS +

38 12655