F>Ipl , 5 .1‘1

             
     

 

.. 2.... .r r.

   

      

 

          

       

         
                     

:f. ............:4 ;..Vq~
.5... . . I 2 . .2 . m2... 2...??? «diam .
I . 2 2.. .2... 5.: 4...: 5...... v .

    

         

                   
 

a.

a

         

  

$.45 .o- p

. an “Mummy”

  

       

 

    

 

        

         

 

 

    

    

 

 

.
z: ..
A I 3 Z 1. .3 t
. 31.9.?- 1 . s.
I 2.12:... $.w...-.fln 1 nib
.:.:... .. u..:1f..;a:»b~.. ..r’
.331... 1:. .1. $1212.... :— _ ‘
22.1.3... .32» Z :1? ». .Tq.
.,..ZI_.:1..__.£\...1 $5.3... ‘ . .
. .1. 1.41.322?
0 .

               

. 3’1”.”
.. . .
1_finw..,r£
..

 

          

. .1.»
.. Tr Jury“

1rd .21.. .
w .1?» 212.55.23.22. .
, aw... xii...
., I, _ .Wt.1..x_rurx

   

 

3.. .

1: ._ .
1 A Yaw...”
. —n all 9., a
’— :1. cl a 1
554.... J
.

1.

V
.

 

 

          

    

 

            

      

 

    

 

u4’d

..a' L.

1—t‘ ..:J...
2.

 

     

 

       

L.
... i 5.
3.1.2..

Tia. r f» v

f... 2.....x
,,

                     

             

      

   

  

..w,
r.

min
1... M...

 

a

,1 2? hr.

9.,H..e.r.fi..,..£.w

.3 ”2.1.9.3. x:
.

    

w
1 a. .

any .7 br. 1“.
”mafia.” any?

..17 ... .1 , 1 1 .
name szwfl Wfiyfi
. 3.21.1. a .r ..r‘ .
2?..x...¢rmimgw1wdr
r .

583er .4 .
1 if?

M.

                   

'I

            

 

 

 

UM

           

 

 

 

IND]

 

  

S

r

STATE ,

    

‘2'!
N}

y

1. (.L .z

ELF
[LOY

[0.
2A

 

 

 

   

 

s

 

v
i

 

 

HERM
r
HIGA

      

     

' ,

"HOT

1
gummy, . 4%
. 4v v. P '4‘?” a.

 

I 1

       

(I: F in.
I

 

 

  

 

”WW
93
MIC

 

  

 

      

  

I}! . (VI
.7): lift—5mm”
1 f

 

   

 

 

 
 
 

     

an 3.2

              

 
 

 

 

           

      

    

 

 

 

      

  
 

 

   

. 1y 1 1 ,1 . 1 , . . .. 5511311.,1 1.12 f: . . .. 1A, .1 _..,1. 1.5.2 2 , 5:51.»! 2.1x.\.52mrr..;
1.1.. . .1 . _ , .. .2 , . 1 ..L..........a..._b:; . .1 . éfifiea Hi3? .12
12.1.... . .. . . 1.1.. .. ,. . Raw ......,:....1.. . . . . _ a .21 x&
2.2912. 12 .12.... e1. .1 .. .. p 1. . _..1.:..2.:“... 1 1 . . .1 1. . 1 .1 .3. .11“.an mufiwmrfiifiu

 

This is to certify that the

thesis entitled
THE MAGNETOTHERMOELECTRIC EFFECT
IN ALUMINUM, ALUMINUM ALLOYS,
INDIUM, AND LEAD
presented by

ROBERT SCOTT AV ERBACK

has been accepted towards fulfillment
of the requirements for

PH-D . degree in _BHXS_LCS_

[Ls/a “an,

( Major professor

Date Dec.‘7l, l 77/

G7 639

LIBRARY

Michigan State
University

 

ABSTRACT

THE MAGNETOTHERMOELECTRIC EFFECT IN ALUMINUM,

ALUMINUM ALLOYS, INDIUM, AND LEAD

BY

Robert Scott Averback

We have studied the effect of a transverse magnetic
field upon the low temperature absolute thermopowers of Al,
Al-alloys, In, and Pb, with primary emphasis upon the Al and
Al-alloys. Measurements were carried out on polycrystalline
specimens over a temperature range from 2.5K to 4.5K and with
magnetic field strengths up to 20 k-Gauss. The thermopowers
of the pure Al and Al-alloys were negative in zero magnetic
field, became more positive with increasing magnetic field,
and then saturated in the high field limit. In most cases,
this resulted in a magnetically induced sign change. The
thermopower of pure In also became more positive with increasing
magnetic field and saturated in high fields. The thermopower
of Pb, on the other hand, became more negative with increasing
magnetic field and showed no clear tendency toward saturation
in high fields. We associate this difference in behavior with
the different valences of these metals; Al and In each have
three valence electrons, while Pb is a "compensated" metal
having four valence electrons. The behavior of Pb is consistent
with a prediction for "compensated" metals by Azbel, Kaganov,

and Lifshitz.

Robert Scott Averback

In order to interpret the behavior of the Al and Al-alloys,
it was necessary to separate the total thermOpower into electron-
diffusion and phonon-drag components. The electron—diffusion
component was then found to be initially negative, to become
more positive with increasing magnetic field, and to saturate
in high fields. Although in zero field the electron-diffusion
thermopower varied from specimen to specimen, its total
variation from zero to high magnetic field was found to be
nearly the same for all of the specimens studied. A "two
band" model of the thermopower of Al is presented and shown
to adequately account for this general behavior, yielding a
predicted magnitude for the change with magnetic field within
a factor of 2.5 of the experimental value. The phonon-drag
thermOpower was found to be negative in zero field and, for
all but one alloy studied (Al-340 ppm Cu), to become initially
more negative with increasing magnetic field. In a number of
cases, the phonon-drag component reached a maximum value and
then decreased at higher fields. No satisfactory explanation
for this behavior is available. The phonon—drag thermopower of
the Al-340 ppm Cu sample was found to decrease slightly with
increasing field. We also have no explanation for the difference
in behavior of this alloy.

It was not possible to unambiguously separate the thermo-
power of the In samples into electron-diffusion and phonon-drag
components. But unlike in Al, the zero field phonon—drag
component was positive (this fact was confirmed by measurements

at higher temperatures), and most probably increased substantially

 

Robert Scott Averback

in magnitude with increasing magnetic field. The difference
in sign and magnetic field dependence between the phonon-drag
components of Al and In is tentatively associated with
differences in the forms of the Fermi surfaces in the third
Brillouin zones of the two metals.

During the course of these studies, we also obtained
information concerning the zero-field electron-diffusion
and phonon-drag components of the thermopowers of Al and A17
alloys. The electron-diffusion components were found to be
consistent with Kohler's theory for the effect of the concen-
tration of an impurity on the thermopower. The characteristic

electron-diffusion thermopowers of the impurities Cu, Th, and

Cd in A1 were found to be, respectively, SCu-Al = 0.56 i 0.02
-8 __ -8 . _.

x 10 V/K, STh-Al — 3.2 i 0.5 x 10 V/K, and SCd-Al — 4.0 i

0.6 x 10-.8 V/K. The phonon-drag component of the thermopower

was observed to be different for different impurities, differing

in the most extreme cases by more than a factor of three.

 

THE MAGNETOTHERMOELECTRIC EFFECT IN ALUMINUM,

ALUMINUM ALLOYS, INDIUM, AND LEAD

by

Robert Scott Averback

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1971

To Marian and my Daughters, Leslie and Jennifer

ii

Acknowledgements

It is a pleasure to acknowledge that this thesis was
carried out under the guidance of Professor Jack Bass, whose
advice and criticism at all stages of this study were
invaluable. I wish also to acknowledge Professor Frank
Blatt as the original source of the idea that the thermopower
of aluminum might change sign in a magnetic field. In addition
I wish to thank Professor Carl Foiles and Dr. Craig Stephan
for many helpful discussions.

Specific thanks go to Dr. G. J. Edwards for his help
in constructing the superconducting chopper amplifier, to
Mr. B. Schumaker for preparing the alloys, and to Professor
W. P. Pratt for the suggestion of using a COpper shield
around the chamber.

I would also like to acknowledge the financial support

of the U.S. Atomic Energy Commission.

iii

TABLE OF CONTENTS

List of Tables

List of Figures

I.

II.

III.

Introduction

A.

C.

Sign Changes in the Hall and Righi-Leduc
Coefficients of Al and In

Previous Work on the Magnetothermoelectric
Powers of Al and In

The Present Thesis

Experimental Technique

A.

G.

Introduction: The Experimental Problem
I. Seebeck Effect
2. Peltier Effect
3. Thomson Effect
Voltage Measurement
1. Magnetic Amplifier
2. Superconducting Chopper Amplifier
Thermometry and Temperature Control
Magnetic Field
Specimen Preparation
Alloys

Experimental Procedure and Analysis

Theory of Thermoelectricity

A.

Electron Diffusion

iv

Page
vi

vii

10
12
13
18
27
30
32
38
4O
44

44

Table of Contents (Continued)

IV.

1. Introduction
2. Effect of Two Groups of Carriers
3. Effect Of Two Types of Scatterers
B. Phonon-Drag
Theory of the Magnetothermoelectric Effect
A. Electron Diffusion
1. Introduction
2. Formal Theory for Aluminum and Indium
B. Phonon-Drag: Theory
Experimental Results and Discussion
A. Zero Magnetic Field
1. Introduction
2. Aluminum
Electron-Diffusion
Phonon-Drag
3. Lead
4. Indium
B. Magnetothermoelectric Power
1. Aluminum
Electron—Diffusion
Phonon-Drag
2. Indium.and Lead
3. Conclusions
References

Appendix 1: Thermal Conductivity of Aluminum

V

Page

44
47
53
55
61
61
61
65
78
80
80
80
83
83
89
93
95
97
99
99
110
113
116
120

123

Table

II-l:

LIST OF TABLES

The intended concentration (in P.P.M.)
and measured resistance ratio for the
various aluminum alloy specimens.

Values of a(H), b(H), and residual
resistance ratios for aluminum and
aluminum alloys.

The Lorentz number (in 10—8 watt-ohms/Kz)
of AlCu for various magnetic fields

1
at 4.5K.

vi

Page

39

84

125

LIST OF FIGURES

Figure Page
II-l: The Conditions for Measuring the Thermo-

electric Effects 9
II—2: The Magnetic Amplifier System:

(a) Schematic of magnetic amplifier;

(b) Idealized hysteresis curve for mumetal;

(c) Magnetic amplifier circuit 14
II—3: The Superconducting Chopper-Amplifier System:

(a) The Superconducting Chopper—amplifier

circuit;
(b) Schematic (actual size) of the device 19
II—4: The specimen holder: a, styrofoam support;

b, heat sink for resistor leads; c, specimen;

d, germanium resistor (c0pper housing);

e, carbon resistor; f, heater; g, aluminum

brace; h, copper clamp; i, binding post;

j, flange for lead O-ring; k, vacuum line;

1, epoxy seal; m, support rod; n, heat

reservoir 25
III—l: Cross-section of the one O.P.W. Fermi Surface

of aluminum in the (100) plane superimposed

upon the Brillouin zone structure. The dashed

line indicates a constant energy surface

above the Fermi level 49

vii

List of Figures (Continued)

Figure

IV-1:

IV-2:

IV-3:

The one O.P.W. Fermi surface of aluminum in
the reduced zone scheme. Cyclotron orbits in
the second and third zones are indicated in
the (001) plane. (After Ashcroft, Ref. 9)
Experimental Condition for Measuring the
Adiabatic Magnetothermoelectric Power

The velocities, VX(0), of electrons executing
the orbits shown in Figure IV—l. The dashed
lines represent the velocities for a Fermi
surface based upon the free electron sphere.
The dotted curves represent the function
-cose, normalized at 0=w to the maximum value
of VX(0). (After Ashcroft, Ref. 9)

The thermopowers of Al, In, and Pb vs. the
reduced temperature, T/GD.

The variation with temperature of the thermo-
powers of Al, In, and Pb below 6K. A straight
line indicates a thermopower of the form
S=aT+bT3.

The coefficient of the electron diffusion
component of the thermopower of Al alloys vs.
the inverse resistivities of the alloys.

The intercept at l/p is the "characteristic

thermopower" of the impurity in Al.

viii

Page

63

66

7O

81

82

88

 

List of Figures (Continued)

Figure

V—4:

A schematic illustration of electron-phonon
scattering events in the second and third
zone portions of the Fermi surface of Al
The thermopowers of Al, In, and Pb versus
magnetic field at 4.5K

The temperature dependences of the thermo-
(0) and A1'(x) for a series

1 1

of magnetic fields. A straight line

powers of Al

indicates a thermopower of the form
S=a(H)T+b(H)T3. The magnetic fields are
denoted by: a)0, b)0.5, c)l.0, d)l.5, e)
and f)5.0, g)12.0 k—Gauss

The temperature dependence of the thermo-
power of a thin (thickness~electronic mean-

free-path) Al specimen, Al for a series

2,
of magnetic fields. A straight line
indicates a thermopower of the form
S=a(H)T+b(H)T3. The symbol ¢ indicates

two indistinguishable data points. The
magnetic fields are denoted by: a)0, b)l.0,
c)3.0, d)5.0, e)l0.0 k—Gauss

The temperature dependence of the thermo—

power of specimen Al for a series of magnetic

3

fields. A straight line indicates a thermo-

ix

Page

90

98

100

101

   

List of Figures

Figure

(Continued)

power of the form S=a(H)T+b(H)T3. The

magnetic fields are denoted by: a)0, b)0.5,

C)1.5, d)3.0, e)5.

0, f)10.0

k—Gauss

The temperature dependence of the thermo—

power of specimen AlCu

magnetic fields.

1

for a series of

A straight line

indicates a thermopower of the form

S=a(H)T+b(H)T3.

The symbol é indicates

two indistinguishable data points. The

magnetic fields are denoted by: a)0, b)3.0,

c)5.0, d)10.0, e)

15.0,

f)20.0

k—Gauss

The temperature dependence of the thermo-

power of specimen AlCu

magnetic fields.

2

for a series of

A straight line

indicates a thermopower of the form

S=a(H)T+b(H)T3.

The symbol 6 indicates

two indistinguishable data points. The

magnetic fields are denoted by: a)0, b)l.0,

C)3.0, d)5.0, e)1

0.0, f)20.0

k—Gauss

The temperature dependence of the thermo-

power of specimen AlCd

magnetic fields.

1

for a series of

A straight line

indicates a thermopower of the form

S=a(H)T+b(H)T3.

denoted by:

a)0,

The magnetic fields are

b)0.5,

c)l.

0,

d)3.0 k-Gauss

Page

102

103

104

105

 

List of Figures (Continued)

Figure

V-12:

The temperature dependence of the thermo-
power of specimen AlTll for a series of
magnetic fields. A straight line indicates

a thermopower of the form S=a(H)T+b(H)T3.

The symbol § indicates two indistinguishable
data points. The magnetic fields are denoted
by: a)0, b)l.0, c)3.0, d)5.0, e)l0.0 k-Gauss
The temperature dependence of the thermo—
power of specimen AlSn for a series of
magnetic fields. A straight line indicates

a thermOpower of the form S=a(H)T+b(H)T3.

The symbol 0 indicates two indistinguishable
data points. The magnetic fields are

denoted by: a)0, b)1.0, c)3.0, d)5.0,
e)10.0, f)15.0 k-Gauss

The electron diffusion coefficient a(H) versus

wCT for representative specimens: a)AlCul;

 

b)A1Cu2; C)A13; d)A1Tll; e)A18n; f)A1Cdl.
__ R(300)/R(4.2)
Here wcr — 0(300) nec where R(300)/R(4.2)

is the measured resistance ratio of the
specimen and 0(300) is the resistivity of

A1 at 300K.

xi

Page

106

107

108

 

List of Figures (Continued)

Figure

V-15:

The temperature dependence of the thermo-
power of Pb for a series of magnetic
fields. A straight line indicates a
thermopower of the form S=a(H)T+b(H)T3.
The magnetic fields are denoted by:

o = 0.8,A= 1.5, U: 5, ¢ =10, A= 20
k—Gauss

The temperature dependence of the thermo-
power of two In specimens for a series

of magnetic fields. A straight line
indicates a thermopower of the form
S=a(H)T+b(H)T3. Symbols A! V 0
indicate the less pure sample.

The crossing field, HC (i.e. the field at
which S(H)=0), for specimen A1l as a
function of angle. The magnetic field

is being rotated in the horizontal

plane, so that it always remains trans-
verse to the temperature gradient. All
is polycrystalline; and preferential
alignment of crystallites is suspected.
The temperature dependence of the
thermal resistivity of specimen A13.
A straight line indicates a thermal

resistivity of the form W=A/T+BT2.

xii

Page

114

115

118

124

 

I. Introduction

A. Sign Changes in the Hall and Righi—Leduc Coefficients

 

of Al and In:
In the free electron theory of metals, the signs of the

Hall coefficient R

'H’ the Righi—Leduc coefficient, and the

absolute thermopower S, are all the same as the sign of the

charge carrier in the metal. It was therefore intriguing

when recent observations showed that the low temperature Hall(l’2)

(3,4)

and Righi—Leduc coefficients of aluminum and indium

change sign upon application of a sufficiently strong transverse

 

magnetic field. This is especially so, since both aluminum
and indium have nearly spherical Fermi surfaces(5), and the
electronic transport properties of aluminum had previously

been observed to be in reasonable agreement with the free

(6)

electron theory. The free electron theory is, however,
inadequate to account for these sign changes. For the reader
unfamiliar with transport in metals, we briefly review the
free electron theory of the Hall effect and the extension to
the "two band" model necessary to explain the observed sign
changes.

In the free electron theory, the Hall field is generated
as follows. Electrons drift in a metal with velocity vd
antiparallel to an applied electric field. Upon application
of a magnetic field transverse to the motion of the electrons,

the electrons experience a Lorentz force, are deflected in

the direction vdx H and thus accumulate on one side of the

specimen. The accumulation of negative charge creates an

electric field, EH. This charge continues to grow until EH

is sufficient to balance the Lorentz force, i.e.

(y x H) = eEH. (I-l)
Since the current density is
J = nev (1‘2)

»- ~d'

where n is the number density of electrons, the Hall field is

E =__].'_...

.~H nec (g X E)' (IT3)

The Hall coefficient is defined as,
R = EH/(Q x H) (I-4)

and, in this model, is given by

R = l/nec. (I—5)

Since R does not depend upon the magnetic field, a magnetically
induced sign change is inexplicable.

The simplest addition to the theory which can be made
to account for a change in sign is the allowance for a second
group of carriers, holes. These carriers would have, effectively,
a positive charge. The holes would therefore establish a Hall
field of sign opposite to that produced by the electrons and
would compete with the electrons in determining the resultant
sign of the Hall coefficient. The possibility for a magnetically
induced sign change will then exist if the result of this
competition is dependent upon the magnetic field.

(7) (8)

Sondheimer and Wilson and Sondheimer originally

developed the two carrier model for metallic conduction.

In this model the Hall coefficient is given by the expression
2‘ R.O’.2
l l

1 1 + '\wcr)i2

 

I (1’6)
Oi(wcT)i 2

i 2
[g 111w01)i ] + [ g 1+(wCT)i ]

where Ri is the Hall coefficient for the ith carrier acting
alone (i e. R. = ~34——) and (w T). = €E—‘T. where m? is the

' l niec c l mic 1 ’ 1
effective mass of the ith carrier and Ti is the relaxation time
for the ith carrier. The key to the possibility for a sign
reversal lies in the numerator of Eq. (I-6). At low fields

(wcTi<<l)’ the sign of RH is determined by the quantity

RO+RO. (I-7)

At high fields (wcri>>l), on the other hand, the sign of RH

is determined by the quantity

2 2
R101 R202
—————— —————— (I—8)
(w )2 (w )2

cT 1 cT 2

In going from the zero field to the high field limit, the
relative importance of the two carriers has been changed by

. 2
the square of the ratio (wcr)2/(wcr)l. If R1 and R2 have

opposite signs, this change in relative importance can lead

to a change in sign of R Thus Eq. (I-6) provides a means

H'
for understanding how the sign of the Hall coefficient can change

(9)

upon application of a magnetic field. Ashcroft has used

this model to explain the sign reversals observed in aluminum

and indium. Alternative calculations have been made by Feder

(10) (11)

and Lothe and by Van der Mark, Ott, Rasmussen and Sargent.

 

 

B. Previous Work on the Magnetothermoelectric Powers of

 

A1 and In:

 

In 1948 Sondheimer(8)

developed a general two band model
to treat the effect of a transverse magnetic field on the
thermopower of metals. Unfortunately the form of his results
made them difficult to apply to real metals. Since then, a few
theoretical papers concerning this effect have appeared in

the literature, but these did not apply to either aluminum or
indium. The reason is twofold: l) the thermopower is a
difficult property to calculate properly; according to Ziman(l6)
it is "the most sensitive transport property of a metal"; and
2) there have been no experimental data to provide a basis

for a calculation.

This situation is however not surprising. Few laboratories
have had voltage measuring devices sufficiently sensitive to
measure the thermopower of these metals in a magnetic field.
Those devices which had adequate voltage sensitivity were also
quite sensitive to magnetic flux and were often difficult to

(12, 13, 14)

keep in operation. To perform these experiments

we have used a recently designed superconducting chopper
amplifier which is both insensitive to magnetic flux and

reliable.(15)

 

 

C. The Present Thesis:

 

As noted above(and shown in section III.A.1), in the free
electron theory, the thermopower, like the Hall coefficient,
is of the same sign as the charge carrier. It was therefore

(17) that the magnetically induced sign change in

suggested
the Hall coefficients of Al and In might be reflected in a
similar magnetically induced sign change in the thermOpowers

of these metals.

This thesis began as a search for a sign change in the
thermopower of aluminum at low temperatures. When a sign
change was observed, the thesis was extended to include a
systematic study of the effect of a magnetic field on the
thermOpower of very dilute aluminum alloys and of two other
metals, indium and lead. By studying a pure metal of similar
electronic structure (indium) and pure metal of dissimilar
electronic structure (lead), in addition to the dilute aluminum
alloys, we hoped to ascertain whether the sign change was linked
more closely to the electronic structure or to the details of
electron scattering in aluminum.

In the process of making this study we have also obtained
information concerning the thermopower of dilute Al alloys in
the absence of a magnetic field which is relevant to two
questions raised by previous experiments.

It will be shown below that the thermopower of a metal
may be sensitive to the type of impurity atom present, but
for a given impurity it should not be sensitive to the concen—

tration of that impurity once the impurity concentration

 

exceeds a certain level. This "rule" is obeyed by several

(18) However, the available data makes it unclear

(l9)

metals.

whether it is obeyed by Al. DeVroomen et a1. did not

find the rule to be obeyed for AlMg alloys. Boato and Vig<20)
obtained data which led them to remark, "The observed concentra-
tion dependence appears to be random, and is not understood".
We therefore decided to investigate whether this rule was
obeyed in our alloys.

Additionally, the phonon-drag thermopower of many metals

(21)

(e.g. the noble metals ) varies with the type of impurity

present. Whether this is also true for Al is still unclear.

(19) (20) were able to

Both DeVroomen et al. and Boato and Vig
describe their data in terms of a constant phonon-drag contribu-
tion. And the values they obtained agreed to within 15%.

(22), on the other hand, reported a difference

Holwech and Sollien
between the phonon-drag contributions for the two alloys Al + Cu
and Al + Fe. But the uncertainty in their data made their

claim inconclusive. We have also investigated this phenomenon.

The remainder of the thesis is organized as follows:

Section II is a description of the experimental technique,
with particular emphasis on the problem of measuring low level
D.C. signals at liquid helium temperatures.

Section III provides the theoretical background for the
calculation of the thermopower in the absence of a magnetic
field. In this section we develop the two band model for the
thermopower of aluminum to provide the framework for the theory

of thermopower in a magnetic field.

 

 

Section IV contains the theory for the thermopower in
a magnetic field. In Section V we report our experimental

results and compare them with theory.

II. Experimental Technique:

A. INTRODUCTION: The Experimental Problem:

 

There are three thermoelectric effects; the Seebeck,
Peltier, and Thomson effects. The conditions under which
they are observed, are as follows:

1. Seebeck Effect:

 

When a temperature gradient is applied to a specimen
which is electrically insulated, an electric field is
established. The ratio of the electric field to the tempera-
ture gradient is called the thermopower, S, of the specimen.
To measure the thermopower of a conductor, potential leads
are attached to the specimen as illustrated in Figure (II-la)

and the voltage at the terminals is measured for AT # 0.

The Seebeck effect refers to this voltage, which is obtained

by integrating the electric field around the circuit, i.e.

B To T T+AT To
VAB = IA E°d£ = f(E/AT)dT = f S(A)dT + f S(B)dT + f S(A)dT
T T T T+AT
O O
T+AT
= f [S(B)-S(A)]dT (II-l)
T

Thus the measured voltage is determined by the difference in the
thermOpowers of materials A and B.

2. Peltier Effect:

 

When an electric current is passed from one metal to
another under isothermal conditions, heat is reversibly

generated or absorbed at the junction (Fig. II-lb). The

9

 

 

L

/ \
A. A://///
j E
- +
VAB

0. SEEBEC K EFFECT

J K .1 2
i A ("v/7 ’8 E
b. PELTIER HEAT

E :ilmx A E

c. THOMSON HEAT

.T < p- T+AT

 

 

 

 

 

 

 

 

 

 

 

 

V

Figure II-l: The Conditions for Measuring the

Thermoelectric Effects.

 

 

10

Peltier coefficient HA is defined as the heat evolved per

B

second when a unit current passes through the junction,

 

H = H - H = . (II-2)

3. Thomson Effect:

 

The Thomson effect refers to the reversible heat generated
by a conductor when a temperature gradient is maintained in
the specimen and an electric current flows in it, (Figure II-lC).
The measure of the reversible heat generated per unit volume

per second is the Thomson coefficient,

[00

_ REV _

 

The Thomson coefficient is the only thermoelectric coefficient
which can be measured without reference to another material.
These coefficients are not independent. They are related
through the Kelvin—Onsager Relations:
1) n = TS (II-4)

2) u = T é§ (II-5)

It is therefore necessary to measure only one of the three.
Of the three, the conditions under which the Seebeck effect
is measured are most easily obtained experimentally.

The difference between the thermopowers of materials A
and B is proportional to the temperature derivative of the
generated thermal emf. If AT is kept below a few tenths of
a Kelvin (see page 42) the measurement becomes quasi-differen—

tial, and Eq.(II-l) becomes SB - SA = AV/AT. If material A

 

 

11

is a superconductor (which has zero thermOpower in the super—

(23) AV . .
), measurement of AT yields directly

conducting state
the thermopower of material B. The experimental problem is
therefore reduced to the measurement of AV and AT.

The electron densities in metals are in excess of 1022/cm3.
Therefore, the electron system, which is a Fermi system, is in
the extreme degenerate limit, i.e. the Fermi energy, sf,
greatly exceeds the thermal energy KT. The degeneracy prevents
all but a small fraction of the conduction electrons, roughly
KT/Ef, from being thermally excited from the ground state.

. (24)
For this reason

the magnitude of the thermopower of metals
at low temperatures is small, approximately 0.1 uvolt/K at

5K. The combination of a small thermopower and small tempera-
ture difference, AT, gives rise to a small thermal emf.
Therefore, special voltage measuring devices are necessary

for measuring the low temperature thermopowers of metals.

The following two sections, entitled voltage measurement

 

and thermometry and temperature control describe how these two

 

aspects of the experiments were handled. In order to clarify
the problems involved, this material is presented in

terms of the general problem of dealing with low level
signals at low temperatures. The next sections, magnetic

field, specimen preparation, alloys, and data taking and

 

 

 

analysis, are particular to these experiments, and provide
a detailed description of how the experiments were set up

and performed, and how the data were analyzed.

 

12

B. Voltage Measurement:

 

The problem of accurately measuring low level D.C.
signals is two-fold: l) unwanted thermoelectric emfs are
developed in the leads which carry the signal to the measuring
device, and 2) the measuring device must be sufficiently
sensitive to detect the low level signal. The thermal emfs
which are develOped in leads carrying a signal from 4.2K
to 300K can be kept below about l uvolt, and fluctuations
in these "thermals" over a short period of time can be limited
to about 1 x lo"9 volts. If the constant "thermals" can be
eliminated, then the smallest signal which can be distinguished
from noise is about 1 nanovolt. Reversing switches (for
resistance measurements) and superconducting shorting switches<25)
(for thermopower measurements) have been used successfully
to eliminate the constant "thermals". Potentiometers used
in connection with sensitive photocell galvanometer amplifiers
have been developed to detect nanovolt D.C. signals. For the
present study it was necessary to detect signals considerably
smaller than lO_9 volts; beyond the limit of purely D.C.
measuring systems.

To measure signals smaller than 10-9

volts, two possibilities—
exist. One method is to do away with the D.C. signal altogether,
and form the signal which carries the desired information by
A.C. techniques. This method fails for precision resistance

measurements of pure metals at low temperatures because of

the high reactive part of the impedance relative to the

Iii TEN. Elwin” gt

1..
..T... vthrfl
.. . n..... . -...-... .u . .

 

 

13

(12)

resistive part. For thermopower measurements this technique-
has been successfully applied at high temperatures where a
chopped light signal was used to heat the specimen and thereby

(26) At low temperatures this procedure

produce an A.C. signal.
is also possible, although difficult. Such measurements have
not yet been performed.

An alternative procedure is to convert the D.C. signal
to A.C. in the helium bath, where "thermals" are small, and
to bring the resulting A.C. signal out to room temperature.
In these experiments two devices were used to affect the D.C.
to A.C. conversion. The first device was a magnetic amplifier
(or saturable reactor) and the second was a superconducting

chopper amplifier.

1. Magnetic Amplifier:

 

A magnetic amplifier was used in the earliest experiments.
As the device is adequately described in detail elsewhere<27),
the description presented here is limited to a brief analysis
of the physical principle involved and a description of the
particular model used in these experiments. Figures (II-2a)
and (II-2b) illustrate the principle of the device. Figure
(II—2c) is a schematic of the amplifier circuit. The window
frame shaped core is comprised of several laminated plates of
mumetal with dimensions 1" x 3/4" x l/2". Two, one-hundred
turn coils of copper wire, coils "a" and "b" in Figure (II—2a),

are wound in series opposition on the two side posts of the

transformer so that the net flux they produce in the center

 

 

14

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
       

 

 

 

 

 

    

 

 

 

 

 

 

 

 

    
 

 

 

 

I ° I
r—-)' ——
o c —o
c_D EFT—ID ”‘1: b
I d I
(a) (b)
”ASE LOW NOISEA * ’
SENSITIVE ~ CURRENT
DETECTOR AMPLIFIER SUPPLY
FREQUENCY
DOUBLER '—"" — ————— r—_._.._._.i
I I
4 AUDIO I- MAGNETIO REFERENCE l
OSCILLATOR | AMPLIFIER RESISTOR I
I

| A.C. CHOKE Rc-RK

—_ -———_--——-——J

CURRENT I

 

 

 

SUPPLY

 

 

(c)

Figure II-2: The Magnetic Amplifier System:
a) Schematic of magnetic amplifier;
b) Idealized hysteresis curve for mumetal;

c) Magnetic amplifier circuit.

 

 

 

15

post is zero. An A.C. current is applied to these coils; at
its peak this current drives the point of operation of the
core on the B-H curve almost to saturation. By applying a
D.C. current to coil "c" (100 turns of tantalum) the symmetry
is altered, in that the D.C. flux adds to the flux on one side
post and diminishes the flux on the other. As can be seen

in Figure (II—2b) one side of the core is driven to saturation
but the other is not. This produces an unbalanced flux in

the center post. When the phase of the A.C. current has
changed by l80°, the roles of the side posts are reversed.

The flux in the center post thus fluctuates at twice the
frequency of the current in the side post windings. The
unbalanced A.C. flux produces an A.C. voltage in the 10,000
turn coil wound on the center post, coil "d". THis voltage

is then led out of the helium bath, amplified and phase sensitive
detected. This is shown schematically in Figure (II-2c).

The success of this device arises, in part, from the fact
that the D.C. signal is converted to A.C. at twice the frequency
of the current in the side post windings. This makes it possible
to discriminate against any undesired signal caused by imperfectly
balanced side fluxes. (In practice these signals are often
considerably larger than the signal of interest.)

The magnetic amplifier is a current sensing instrument,
whereas the information of interest is in the form of a voltage.
The conversion from current to voltage can be affected in two

ways. By inserting a known resistor, Rk' into the primary

 

 

16

loop (see Figure (II-2c)), the D.C. signal can be nulled by
introducing a known current, ik’ into the circuit as shown.

When a null in the output occurs, all the current, passes

ik’

through Rk and the unknown voltage is i A second method

kRk .
makes use of a second coil on the center post, wound identically
to the coil in the primary loop. This second coil is used to
calibrate the phase sensitive detector output with respect to
the current flowing in the primary loop. The output is directly
calibrated by passing a known current through the second coil.
The unknown voltage is then the product of primary resistance
and the current measured in the magnetic amplifier. The

primary resistance is obtained by supplying a known current,

ik' to the circuit as shown in Figure (II-20) and measuring
the current flowing through the primary coil in the magnetic
amplifier. Elementary analysis shows that the primary resistance

is given by Rp = Rklk/Im, where im 18 the measured current

ri
in the magnetic amplifier.

Each technique had a difficulty; the calibration of the
device was not strictly linear for currents in the primary
larger than about 10”5 amps; the nulling was hampered by the
long time constant (3 sec.) of the device. By using a combina-
tion of the two methods these problems were overcome. A guess
at the correct nulling current was made and this current was
applied to the circuit. The remaining output was accounted for
by the calibration.

The device used in these experiments had a current sensiti-

vity of about 0.1 uamps which did not depend upon the temperature

 

 

17

of the magnetic amplifier to above 80K. The circuit resistance

5 ohms at liquid helium

could have easily been kept below 10-
temperatures, and we originally believed that voltages of
the order 10_12 volts could be resolved. However, as the
primarly impedance is reduced, the flux change in the center
post is also reduced. (By Lenz's Law, an induced current in
the primary circuit tends to impede flux change in the center
post.) This reduced the sensitivity of the device. To elimi—
nate the problem, an A.C. choke was installed in the primary
loop. The inductance of the choke was about 1 mh. corresponding
to an impedance of 1.5 ohms at the operating frequency, 260 Hz.
The time constant for the device is given by the inductance
divided by the resistance and hence the minimum primary
resistance commensurate with a reasonable time constant was
about 10_3 ohms. This yielded a voltage sensitivity of about
1 x 10—10 volts.

The relatively long time constant of the device made it
tedious to use. In addition,the experiments were carried out
in a magnetic field, which required the transformer to be well-
shielded and far-removed from the pole-faces of the magnet.
Despite the shielding, the device showed a considerable increase
in noise above 5 kG. Finally the voltage sensitivity, which
at best was 1 x 10"10 volts, limited the accuracy to about 2%.
Despite these drawbacks, the device operated reliably and
provided reproducible results. We do not present these results

as they were superceded by more accurate results obtained with

the superconducting chopper amplifier.

 

18

2. Superconducting Chopper Amplifier:

 

The superconducting chopper amplifier resolved the
difficulties of the magnetic amplifier. The specific advantages
of this device for the experiments carried out in this thesis
are as follows:

11 volts)

1) excellent voltage sensitivity (2 x 10—
2) the time constant is determined by the phase sensitive
detector's filtered output (1 sec. was usually used.)
3) eliminates the need for shielding since it operates
virtually independent of field to above several kilogauss
4) can be constructed quickly, cheaply and reliably (The
final version was built in four hours for under $1.00
and has lasted several monthslundergoing several
cyclings from 300K to 2K with little change in sensitivity)
The principle involved in converting a D.C. signal to A.C. is
straightforward. By oscillating a piece of superconducting
wire into and out of the normal state, resistance is switched
into and out of a circuit. This periodic variation in
resistance produces an oscillating signal from a constant
potential. This signal can be detected by ordinary A.C.
techniques.

(12) was

The first superconducting chopper amplifier
developed by Templeton who used tantalum wire as the chopping
element, and an oscillating magnetic field superimposed on a
D.C. magnetic field in order to break the superconductivity.

By using a D.C. magnetic field just below the critical field

of the superconducting element, the modulating field could

 

 
 
 
 

19

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PHASE Low NOISE
SENSITIVE ,
DETECTOR AMPLIFIER

I
FREQUENCY CURRENT
DOUBLER SUPPLY
1

AUDIO _POTENTI-

OSCILLATOR OMETER
”‘2.
L_______.__ 4.__

 

 

 

 

 

 

REFERENCE W

I

' I

' I
I

RESISTOR I

I Nwwwe '

I WM” %SPECIMEN |

' I
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CHOPHNG

 

,4 Current Leads

3 Potential Leads
Oscillator Lead
Ground

Secondary Leads
Sample Leads
Reference Resistor
Transformers

Chopping Element
Vector Board

—QG’U|N—
(0‘3

g
_
_

 

 

 

 

—‘

 

 

(O41)

 

 

 

 

Io—»-l I’<——II

(b)

Figure II-3: The Superconducting ChOpper—Amplifier System:
a) The superconducting chopper-amplifier
circuit;

b) Schematic (actual size) of the device.

 

20

be kept small, thereby minimizing pickup. Even with this
procedure and elaborate shielding, the pickup had to be

(13)

balanced with an A.C. bias. DeVroomen later constructed
a similar device, using thallium as the superconducting
chopping element. Both devices were reported to have a
sensitivity around 1 x 10”11 volts.

Switching a superconductor with a magnetic field was
rejected for use in the present experiments for two reasons.
First, the experiments were to be carried out in a magnetic
field; this would require the amplifier to be extremely well
shielded. Second, the problem of constructing the chopping
assembly and balancing the pickup with a bias signal was felt
to be exceedingly difficult. We therefore decided to switch
the superconductor into and out of the normal state by means

(15) By winding a heater wire

of an oscillating heat signal.
non—inductively on the superconductor, pickup problems can be
virtually eliminated.

The superconducting element was a length (1/2") of
.007" dia. niobium zirconium (NbZr) wire. The heating element
consisted of 6" of Evanohm wire (.0014" dia. 429 ohms/ft.)
wrapped non—inductively on the bare NbZr wire in one tightly
spaced layer. The chopping element was sealed into a 3/4"
length of glass capillary, (1/4" O.D., 1/32" I.D.) with
Apiezon grease. The capillary plus grease served as a thermal

resistance to the helium bath. The resistance of the segment

of superconductor when switched to the normal state was about

 

 

21

0.02 ohms, allowing the specimen to have a relatively high
impedance. The power required to raise the NbZr element to
above its transition temperature of 10.8K, and thereby achieve
chopping, was about 50 mwatts. The heater current was taken
directly from an audio-oscillator (Krone—Hite, distortion
below 0.1%, at 200 Hz). It was found that the element would
switch prOperly anywhere in the range 200Hz to 2000Hz, although
at the higher frequencies the power supplied to the chopping
element was quite critical for smooth chopping. No advantage
was found in operating near 2000 Hz and so 200 Hz was chosen
for the heating signal.

Two tests were made to determine whether the sensitivity
of the device was limited by pickup from the oscillator. At
power levels below and above that necessary to achieve chopping,
the output of the phase sensitive detector is comprised solely
of pickup and noise. Increasing the power in the heater should
increase the pickup. As the power to the heater was increased,
the output was monitored and found to remain zero until the
chopping frequency was reached at which time the output jumped
indicating a D.C. signal. As the power was further increased
the output quickly died away to zero, and remained zero when
the chOpping element remained in the normal state. Since, in
this test, the input impedance without chopping is not the
same as when chopping occurs, a second test was performed. A
second heater was wound onto the superconductor and D.C. power
was supplied by it to raise the temperature of the NbZr to

near its transition temperature. A small A.C. signal was

 

 

22

superimposed upon this signal to chOp the superconductor.

The noise level using this technique was the same as that
observed using A.C. power alone. We concluded that pickup

did not constitute a major problem. The reason for such
little pickup from the oscillator (aside from the careful non—
inductive winding of the heater wire on in the chopping
element) is that the power in the heater oscillates at twice

the frequency of the current in the heater wire.

2R + 1/2 i R cos(2wt).

Power = i2R = 12 c082(wt)R = 1/2 10

0 O

II—6
The signal is thus chopped at twice the oscillator frequency.
This allows the pickup from the oscillator at the primary
frequency, w, to be eliminated by using phase sensitive
detection and detecting at 2w.

The transformers in the primary circuit were constructed
with teflon cores, 1/4" in diameter and 0.1" in length. The
primaries of the transformers consisted of five turns of a
superconducting material, niobium—titanium (NbTi), and the
secondaries contained 2,500 turns of 46 awg. copper wire.

The primaries were varnished in place, but not the fine wire
secondaries. Teflon was used for the core material instead
of mumetal, since mumetal would have needed shielding from
the magnetic field. The two transformers were screwed onto

a square of vector board and positioned so as to cancel stray
flux.

The reference resistor in Figure II-3 was a short segment

of copper wire, 8 x 10"6 ohms at 4.2K, its resistance being

23

measured in place on various occasions. It was used for
nulling the unknown voltage.

The components of the chopper amplifier were positioned
as shown, actual size, in Figure (II-3). The components were
soldered to 0-80 brass screws which held them rigidly to the
vector board. In order to minimize the voltages generated
by the primary loops of the superconducting chopper amplifier
vibrating in a magnetic field, it was positioned 22" above
the pole faces of the magnet. A special Helium—Nitrogen dewar
with an extended tail section contained only 1/2 liter of
helium below the chOpper amplifier.

The signal was carried out of the dewar via a screened
twin cable to a shielded audio transformer. The transformer
had a balanced input and a turns ratio of 100. The signal
was then led to a low noise amplifier (PAR Model CR-4A),
using less than 6" of coaxial cable since the output impedance

of the transformer was in excess of l megohm. Finally the

signal was carried to a phase sensitive detector, (PAR Model
JB—4), and the output was monitored on either the panel meter
or an X-Y recorder. The frequency doubler shown in Figure (II-3)

served to provide a signal with even harmonics of the oscillator

frequency, to the reference of the phase sensitive detector.
Operation of the device is straightforward. A D.C.

signal of several nanovolts is applied across the known resistor

in the primary circuit, and the output is monitored. Initially

there is no signal. The power to the chopping element is

24

slowly increased until a deflection in the output meter is
observed. The power, frequency, and phase are then adjusted
to maximize the output. Once these adjustments are performed,
the chopper amplifier needs no additional adjustment until
the bath temperature is changed, at which time the power
needs adjusting. The total time for these adjustments is
under a minute.

Although the output of the device appeared to be quite
linear, the device was used only as a nulling instrument,
since the accuracy of the nulling technique is limited only
by the sensitivity of the device. Using a calibrated output
the accuracy is limited by either the sensitivity of the device
or the accuracy of the output of the phase sensitive detector,
whichever is worse.

On occasion a small standing voltage was observed, less
than 2 x 10_10 volts. This voltage remained constant for
several minutes and was subtracted from the voltage readings.
When this occurred the standing voltage was checked before and
after each measurement for changes. The voltage sensitivity
of several devices ranged between 2 x 10—ll and 5 x 10”11 volts
in zero field. The sensitivity deteriorated to no worse than
3 x 10—10 volts at 20 k-Gauss, although a longer time constant
in the filtered output of the phase sensitive detector was
necessary (3 seconds). A superconducting shorting switch was
used once in connection with the superconducting chopper amplifier.

This device allowed the signal from the thermocouple to be shorted

until equilibrium was reached. The measurements could then be

 

 

25

made instantaneously, which increased the sensitivity of the
device. Unfortunately, in the one attempt, the device
introduced a thermal into the circuit. The idea was

abandoned since the device was already sufficiently sensitive

for the present experiments.

.n .f' ..
. . III I
. I Io

.

 

26

.HHO>HmmOH been .c “cos psommsm LE “Hmom wxomm .H “mafia Essom> .x umcHHIO
pmoa How omgmHMILm “Umom mcflpceb .fl “QEOHU Hommoo L: “woman ESCHESHO .m numbed: .w
“poemflmmu conumo .o “Amcflmsos Hommoov Hoemflmmu Edflsmasom LU “coSHOOQm .o “momma

HOUmHmmH MOM Rafim pom: .Q “unommsm Emowoumpm .m "Hopaon cmfifloomm use “SIHH ossmflm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

27

C. Thermometry and Temperature Control:

 

The temperatures of the thermocouple junctions were
obtained by using carbon resistor sensors. Nominally 100 ohm,
1/8 watt Ohmite resistors were used. These resistors are
well-suited to the temperature range over which these experi—
ments extended (1 - 6K), since the ”knee" in their resistance
vs. temperature curve occurs at about 6K. In addition these

(28) Unless

resistors show only a small magnetoresistance.
fields over 20 k-Gauss are employed, this magnetoresistance
can usually be neglected. Where necessary, formulas exist to

(29) In early experiments,

correct for the magnetoresistance.
the resistance of each sensor was obtained by measuring the
voltage across the carbon resistor for a known current, about
one microamp. The voltage measurements were performed using

a Leeds and Northrup K-5 potentiometer in conjunction with

a galvanometer amplifier. The tedium involved in balancing

the potentiometer and averaging thermal voltages generated

in the leads prompted the abandonment of this technique in
favor of an A.C. Wheatstone bridge. The circuit was arranged
so that both the resistance of the cold end resistor and the
difference between the two resistances could be measured.

From these measurements the hot end resistor's resistance could
then be later deduced. The power input to the resistors was
kept at about 1 x 10_8 watts to prevent internal heating of

the carbon resistors. A phase sensitive detector, PAR model

HR-8 was used to detect the off balance signal in the bridge.

28

A difference of 0.1 ohms could be resolved; at the highest
temperatures (~5K) this corresponded to about 0.4 milliKelvins.
To check for internal heating, the power level was varied
during several measurements. At a power level of 1.5 x 10—7
watts nearly every resistor used showed an increase in tempera-
ture of about 1 milliKelvin.

The resistors were calibrated below 4.2 K against the
pumped helium bath for each run. The pressure over the bath
was obtained by inserting a tube just above the liquid helium
level which led directly to a mercury manometer which was
mounted with a silvered scale backing. The pressures could
be read to about 0.3mm of Hg and were corrected for hydrostatic
head pressure and for the variation of mercury density with
temperature. Below 25mm of Hg a McCleod gauge was used,
although rarely did the experiments extend this low in tempera—
ture. The National Bureau of Standards 1958 tables were used
both to obtain the temperature for a given pressure and to make
the pressure corrections.

The bath temperature was controlled by a Walker Manostat
which seemed to control to better than a millidegree. It was
found that as the bath was pumped to lower temperatures, the
specimen temperature became increasingly unstable. The
condition was seriously worsened when power was applied to
the specimen to provide a temperature gradient. If it were
not for the fact that the difference in sensor resistance

was being measured, which remained stable to milliKelvins,

29

rather than each junction temperature, measurements with the
bath temperature between 2.3K and 3.0K may well have been
impossible. This condition was attributed to bubbling on

the can surface, since below the lambda point, where no
bubbling occurs, no fluctuations were observed. In later
studies the bath temperature was maintained either above

3.4K or below the A—point. To reach temperatures above 4.2K,

a second heater was used to bodily raise the specimen tempera—
ture. It was mounted as shown in Figure (II—4). A germanium
resistor was positioned on the specimen between the two carbon
resistors and was used to calibrate the carbon resistors. To
check that the specimen was indeed at a uniform temperature,
the thermal voltage output of the specimen was monitored during
the calibration. For specimen temperatures up to 7K, no output
voltage was observed, indicating that the thermocouple junctions

were at the same temperature to within 0.1 milliKelvin.

 

 

30

D. Magnetic Field:

 

The magnetic field was generated by a Harvey Wells iron
core electromagnet. With the pole faces set three inches
apart, the maximum field intensity was 21 kilogauss. The
pole faces were six inches in diameter and homogeneity was
better than 0.5% over a four inch region. Although the size
limit imposed on the specimen chamber by the magnet was not
serious, it did limit the absolute accuracy of measurements
to about 2%. The problem arose from the finite width of the
carbon resistors, and from their placement relative to the
junctions between the specimen and the two superconducting
leads which completed the thermocouple. For the maximum
specimen length of 2 1/2", the width of the carbon resistors
(l/l6") gave rise to a maximum error of 2%. This was the
largest error involved in the measurements. Since it was a
constant factor for a given specimen, it was not important
in the determination of the temperature and field dependence
of the thermopower. The magnetic field strength was measured
with a Hall probe manufactured by the F.W. Bell Company.

Its accuracy was about 0.1%.

The most serious problem caused by the magnet was an
increase in noise upon application of a sufficiently strong
magnetic field. Before careful precautions were taken, the
sensitivity of both the magnetic amplifier and the super-
conducting chopper amplifier were reduced by an order of

magnitude at 5 k-Gauss, and they were often unusable above

 

31

10 k-Gauss. The problem arose from two factors: 1) noise and
instability in the magnet and 2) vibration of the primary

loop in the magnetic field. Since the signal was analyzed
using a phase sensitive detector, the noise should not have
significantly altered the sensitivity. (Most of the noise

was at a different frequency than the signal and had an

uncorrelated phase.) However, the induced voltages were so
large as to overload the amplifiers. To overcome this problem
several precautions were taken. Vibration transferred from the

forepump along the pumping line to the vacuum can was minimized
by a series of transverse hose connections. The diffusion
pump was mounted free of the dewar platform. All leads in the
primary loop were tightly twisted (see specimen preparation).
The tail of the dewar was wedged between the pole faces of the
magnet with clay. Finally, the specimen holder was placed
inside of a cylindrical c0pper pipe, 1/8" thick, 1 3/8" O.D.,
and 10" long, located in the helium bath. The attenuation

of 60 Hz noise by this pipe was computed to be a factor of
seven.(30) With these precautions, the sensitivity of the
superconducting chopper amplifier was not affected by fields
less than 10 k—Gauss and decreased only to 3 x 10'.10 volts

at 20 k-Gauss, although a longer time constant in the phase

sensitive detector was necessary as noted above.

32

E. Specimen Preparation:

 

The loaded specimen holder is shown in Figure (II-4).
To accommodate two carbon resistors, a germanium resistor,
two heaters, and two potential and two current leads, and
yet to have sufficient surface area available to attach one
end of the specimen to a heat reservoir, the specimens were
all rolled into thin strips, about 3" long and 3/16" wide.
The following procedure was adopted to mount these elements,
with the intention of making specimen changing easy, while
introducing as little cold work as possible.

The annealed specimen was first washed with alcohol.
The potential and current leads were then attached as follows.
(These leads were niobium-titanium (NbTi) wires, which were
superconducting below 10K and had a critical field HC2 of over
100 k-Gauss.) The insulation and c0pper plating was removed
from one end of each lead, one inch from the end. The leads
were taped tightly to the specimen, so that the prepared ends
were in position for attachment. The leads ran from these points
to the center of the specimen from where they were twisted
tightly together as they led from the specimen. The tape
provided electrical insulation between these leads and the
specimen. For aluminum and lead, the potential and current
leads were spotwelded to the specimens. The tape proved quite
valuable in reducing cold-work during the spotwelding, since
it held the resilient NbTi wires in place. The main purpose

of the tape, however, was to hold the potential leads close

 

33

to the specimen to reduce the area of the primary loop and
to prevent the wires from vibrating.

The resistance of the spotwelded joints was measured to
be under 5 x 10‘.5 ohms for one of the aluminum specimens.
This is well below the maximum resistance tolerable for the
superconducting chopper amplifier to work properly. A low
junction resistance was imperative for a second reason,
since the welded area has a thermopower different from that
of the specimen and the potential lead. With the specimen
temperature at 6K there is approximately a 2K temperature
difference between the specimen and the bath. The temperature
difference is divided between the junction region and the

potential leads according to:

W.
= _l. = _§E. _
ATj W AT and ATSC W AT (II 7)
where W : Wj + Wsc' "j" indicates junction material, and "sc"
indicates superconductor. Assuming the Wiedemann-Franz Law

to hold and using the normal state resistance for the super-
conductor, (approximately 2 ohms), the temperature drop across
the junction region is about 5 x lO—SK. Unless the junction
material has an anomalously large thermopower, the thermal emf
generated at the junction would not be detected. It was noted
previously, section (II—c), that when the base heater was used
to bodily raise the specimen temperature, no thermal emf was

observed, thereby confirming that the junctions did not introduce

spurious voltages into the measurements. In the case of lead

 

 

34

the thermal emf generated by the Pb versus NbTi thermocouple
for gradients of 2K was undetectable when both junctions
were below 7.8K, the transition temperature for lead. This
test was further evidence that the spotwelding technique
was suitable for thermopower measurements.

The same general procedure was used for the indium specimens,
except that the potential and current leads were "acid-welded"
to the specimens. Indium has the property that when the surface
is well cleaned with hydorchloric acid the surface will bond
to another indium surface similarly prepared; this procedure
is called "acid-welding". NbTi is not easily "wetted" with
indium and therefore it was necessary to have the copper bonding
(which could be wetted) on the NbTi wires at the tip. Indium
has a superconducting transition temperature of 3.4K, and it
was thus possible by measuring the thermopower below the
transition temperature to show that the copper plating introduced
no spurious voltages.

The specimen with all the leads attached was turned over
and the carbon resistors, heaters, and germanium resistor were
mounted. The carbon resistors were ground flat on one side
to increase the surface area in contact with the specimen. The
leads were 3 mil manganin wires. Cigarette paper saturated
with GE-703l varnish provided electrical insulation between
resistor and specimen, and the varnish held the resistor in
place. The maximum power which could be delivered to the carbon
resistor without causing a detectable change in resistor tempera—

ture, (AT~0.3 mk), was just under 0.1 uwatt. The power

 

35

necessary to adequately measure the resistance of the carbon
resistor was under 0.01 uwatts. To prevent a heat leak
through the resistor leads, the manganin leads were attached
to a heat sink at a temperature within 0.01K of each carbon
resistor before being attached to a heat sink at the bath
temperature. These additional heat sinks were provided by
varnishing several turns of the manganin wire to copper posts
which were glued to the specimen 0.25" from each junction.

The heaters were made by wrapping several feet of Evanohm
wire non-inductively around a copper post and varnishing the
wire to the post. The copper post was then varnished to the
specimen. The power needed to provide the temperature gradient
in the specimen ranged between 1 and 40 mWatts. The power
necessary to raise the average temperature of the specimen
by l or 2K was typically a few tenths of a watt. The close
proximity of the heater to the carbon resistor made it possible
for the heater (part of which may have reached a temperature as
high as 30K) to radiate energy to the resistor and thereby
cause incorrect specimen temperature readings. To prevent
the realization of this possibility, a radiation shield was
mounted between the heater and carbon resistor. Finally, the
copper housing for the germanium resistor was varnished into
place with cigarette paper providing electrical insulation.

The germanium resistor was inserted into the housing after

the varnish dried.

 

 

 

36

One end of the loaded specimen was inserted into the
copper clamp shown in Figure (II-4) and the other end was
taped to a styrofoam support. Although styrofoam provided
a heat leak from the gradient heater to the bath, it was
found that the resultant heat flow was insignificant compared
to that through the specimen to the bath. For thermopower
measurements the heat leak, even if large, has no significance.
However, the thermal conductivity of each specimen was being
measured simultaneously with the thermopower. For these
measurements any heat leak can be disasterous. To test for a
heat leak through the styrofoam, we used the base heater to
raise the average specimen temperature 2K above the bath tempera—
ture, and monitored the thermal emf generated by the specimen.
No emf was observed. This indicated that the ratio of the
gradient across the specimen to that across the styrofoam
support was less than 5 x 10-5. Hence, no appreciable heat
was being carried by the styrofoam when the gradient heater
was operating.

The many leads from the heaters and resistors were then
soldered to permanent leads on the back side of the aluminum
brace, Figure (II-4). The permanent leads were 36 awg. manganin
wire which entered the specimen chamber via the vacuum line.
These leads were wrapped around the COpper support and glued
to the aluminum brace before being soldered to the leads coming
from the specimen. The heater leads were 38 awg. copper wire

which also entered through the vacuum line. They were wrapped

 

37

around a copper post before being attached to the copper support
and aluminum brace with the other leads. The potential and
current leads were spotwelded to permanent NbTi leads which
entered the specimen chamber through an epoxy seal. The seal
was made from Stycast GT-2850 manufactured by the Emerson
Company, according to a recipe published by A.C. Anderson.(3l)
The seal was extremely reliable, never developing a leak in
several cyclings between room temperature and liquid helium
temperatures. It was found, however, that unless the copper
plating was removed from the NbTi wire at the seal, the seal
would invariably have a superfluid leak.

Finally, the brass can was attached. The seal between the
can and the specimen chamber was made by squashing a Pb +0.05%
As O-ring between the can and the specimen chamber with 12
screws. The O—ring was made by wrapping the Pb +0.05% As wire
around the flange, Figure (II-4), twisting it tight, and
greasing it. Only one such O—ring developed a leak in these
experiments, and that was a superfluid leak. From the annealing
process to the time the specimen chamber was sealed, it should
be noted that the only time the rather delicate specimens were
handled was during the taping of the potential leads and the

spot welding.

 

Q

 

 

38

F. Alloys:

The alloys were made according to the following procedure.
First a master alloy was prepared with a concentration of
about 0.1% solute material. This was done by heating the
aluminum and the solute material under vacuum in an induction
furnace. The aluminum and solute were contained in a carbon
crucible, which in turn was in a vycor collet. The crucible
had previously been cleaned and baked under vacuum. When the
aluminum and solute formed a molten solution in the crucible,
the solution was chill cast by pouring it into a cold aluminum
mold. The alloy was then etched to remove surface contaminents.
To provide better homogeneity, the alloy was remelted and chill
cast again. By diluting the master alloy with the appropriate
amount of pure aluminum, the desired concentration of the
solute could be obtained. The starting material was 69 grade
aluminum supplied by Cominco Inc. and had a bulk resistance
ratio, R(300)/R(4.2), of about 8,000. The solute materials,
copper, thallium, tin, and cadmium were also 69 grade.

The prepared alloys were first rolled in a stainless
steel mill to 50 mil diameter wires. The mill had been first
cleaned and a pure piece of aluminum rolled through it to
collect contaminants. The surfaces of the wires were probably
still dirty after rolling, and hence the specimens were etched.
The wires were then placed between clean sheets of mylar and
this arrangement was sandwiched between stainless steel plates.
The entire sandwich was then rolled in a mill, which in one pass,

reduced each aluminum alloy to a flat strip about 10 mils thick.

 

39

A pure aluminum speciman, Al prepared according to this

3,
procedure but with no intended impurities added, had a measured
resistance ratio of 4600. The high resistance ratio indicated
that the procedure introduced very few undesired impurities.
Table II-l is a list of the specimens used in this study
including for each alloy the intended concentration and the
measured resistance ratio. The fact that the resistance ratio
of AlTll is greater than AlTl

tion was greater for AlTl

2 although the intended concentra-

1 may indicate that the master alloy

was not sufficiently homogeneous.

Table II-l: The intended concentration (in P.P.M.) and
measured resistance ratio for the various

aluminum alloy specimens.

SpeCimen AlCul AlCu2 AlTll AlTl2 AlSn AlCdl AlCd2 Al3

 

Conc. 340 50 100 50 100 25 50 0

 

R(300)/R(4.2) 127 650 2600 1250 1200 2200 2000 4600

 

 

40

G. Experimental Procedure and Analysis:

 

The taking of data was carried out in the following manner.
The carbon resistors were first calibrated against the germanium
resistor using the base heater to bodily raise the specimen
temperature. After about ten points were measured, extending
up to 7K, the heater was turned off and measurements of the
thermopower, thermal conductivity, and electrical resistance
were begun. The gradient heater was switched on and the power
into it was recorded. The specimen came to equilibrium
instantaneously, and the resistors reached equilibrium before
the bridge could be balanced, a few seconds. Both the resistance
of the carbon resistor at the cold end and the difference

between the resistances of the two carbon resistors were

recorded. The thermal emf signal was then nulled and the
current necessary for nulling was recorded. If a standing
10

voltage greater than 1 x 10- volts had appeared before the
gradient heater was switched on, the heater was turned off
after the measurement to check if the "zero" had changed. If
not, as was usually the case, the heater power was increased
and the process repeated for about five points. When the gradient
was as large as could be tolerated for a differential measure-
ment, the base heater was switched on to increase the average
specimen temperature and more points were taken.

The base heater was useful to test for systematic errors.
It allowed the thermopower to be measured at the same average

temperature using both a large temperature gradient and a

small temperature gradient. We always found the thermopower

41

to have the same value, regardless of the gradient used.

The magnetic field was then turned on and the same
series of steps carried out for about five field strengths.
For the higher field strengths the measurements were repeated
with the polarity of the field reversed. Taking fifty or
so points consumed about 90 minutes. Usually at this point
liquid helium was retransferred into the dewar. The bath was
next pumped down in 0.1K intervals and the carbon resistors
calibrated. Complete thermopower measurements were usually
carried out using three bath temperatures, 4.2K, 3.5K, and
2.1K. Electrical resistance measurements were obtained at
4.2K for each field intensity. Finally the copper standard
resistor was checked on various runs. No change in its resist-
ance was ever observed.

The carbon resistors were calibrated by fitting their

resistances to the equation:

 

+ b. (II—8)

By inverting Eq.(II-8) the temperature could be obtained
for any resistance value. Upon plugging the calibration
resistances into the inverted form of Eq.(II-8), the temperature
obtained never deviated by more than 0.5mK from the calibration
temperatures. Since the temperatures obtained from the
inverted form of Eq. (II-8) for the calibration resistances
of the two resistors were within 0.2mK of each other, the

determination of temperature differences was probably better

 

42

than 0.5mK. Eq.(II—8) fit the calibration in the range 2.6K
to 4.2K so well that it was decided that extrapolation to
6K should not introduce significant errors. This assumption
was verified on two occasions using a germanium resistor.
Hence for most experiments the germanium resistor was not used.
The differential technique for measuring the thermopower
greatly simplifies the conversion from raw data to thermopower
data. By dividing the thermocouple voltage by the temperature
difference of the junctions, the thermopower is obtained. The
error introduced by using a finite temperature difference was
negligible for the conditions used in the present experiment.
This can be seen by the following argument. The voltage
generated by a material with its ends at different temperatures
is given by:
T2
v = f s dT. (II—9)
T1
Assuming a temperature dependence of the form aT + bT3 for the

thermopowers of the metals studied, one obtains:

_ , 4 _ 4 _
V — (aTavg AT) +(b/4)(T2 Tl) (II 9a)
where
Tavg = (T2 + Tl)/2 and AT = T2 - T1.
This reduces to:
_ 3 2 _
AV — aTanAT + [bTan+(b/4)(Tavg[AT] )]AT (II 9b)
and the calculated thermopower is:
s — AV A — + b 3 + b 4 A 2 (II 9 )
cal — / T — aTan Tavg ( / ) Tavg[ T] )' C

 

43

The "error" thermopower is therefore (b/4)(Tavg[AT]2). For

aluminum "b" is known to be about 1 x 10_10 volts/K4.(19)
For lead it is roughly three times that value549) For

indium, it will be Shown to be near these values. At 4K,
with a temperature difference of 0.5K, the voltage due to
the "error thermopower" is just detectable with our
measuring system. At lower temperatures or with smaller
gradients it becomes too small to be detected. Above 4K
it is detectable with a temperature difference of 0.5K,
however in this range, the other two terms are much greater.
The thermal conductivity was just as easily computed.
Having measured the resistance of the specimen, R, the temperature,T,
and the temperature difference, AT, for a known heat current,Q, the

Lorentz number was obtained from the equation

 

_ Q'R . -
L — T.AT (II 10)

The Lorentz number was the physical property of interest in
these experiments. It could be converted to the thermal

conductivity through the relation:

PE
0

where

II

p [o(300)/[R(300)/R(4.2K)I],
and the resistance ratio, R(300)/R(4.2K), was determined

experimentally.

 

 

III. Theory of Thermoelectricity.

 

A. Electron Diffusion:

 

1. Introduction:

 

In the relaxation time approximation, the Boltzmann
transport equation describing an assembly of non—interacting,
indistinguishable Fermi particles under the influence of an

applied electric field and a temperature gradient is given
by(l6):
E(5)~EF

8fO(E)
flUfi) = f(h)—fo(€) = —T(]§)V(}§){—(——T—*—~

) 88(5) VT +

8fO(E)

e 850;)

1
(g - e— yep} (III-1)

where: .E = applied electric field; Ek = electron energy;

’V

e = Fermi energy; T(k) = relaxation time; f(k) = electronic

f
distribution function; fO(€) = equilibrium electronic
distribution function; and fl(k) = f(k) - fo(€) = non-
equilibrium part of the electronic distribution function.

To obtain Eq.(III—l), the following assumptions were

invoked:

l) the distribution function, f(k), does not deviate
far from equilibrium, so that only its lowest order
non-vanishing term need be used;

2) a time of relaxation exists, and is independent of
the type of perturbation. (This assumption is not in
general exact.)

A discussion of the limitations imposed by these assumptions
and of the general use of the Boltzmann Equation can be found

in several texts.(l6’24)

44

 

45

Assumption (1) allows the electric and thermal currents to
be expressed as linear functions of the electric field and

temperature gradient

:1 = as E +2321: ° YT ”II-280
U = {11‘s. E + ifTT - yT (III-2b)

’v

The transport coefficients gij are in general tensors, but
if we limit the discussion to crystals of cubic symmetry,
they reduce to scalars. The thermopower is defined (page 8)
as the ratio of the electric field to the temperature gradient
which exists in the specimen when a heat current, but no
electric current, flows. From Eq. (III-2a) it follows that
the thermopower can be expressed in terms of the transport
coefficients as,

S=-L

L (III-3)

EE/ ET'
To determine S we need to calculate the ratio LET/LEE' To
do so, we proceed via the Boltzmann Equation. We begin by
calculating the electric current density and setting it equal

to zero.

The electric current density is given by:
g = f e y(k) f(k) dk (III-4)

Since the equilibrium distribution function, fo(s), is
symmetric in k—space it does not contribute to the current.
The solution for fl(k) is Eq. (III—l) is

8f
_. _ O ,_ E]; _ o —_]; °
151(k) .. { E—IIIIEIEI Ef)(T)( yr) g<g>1+eIEeyefI guy}

(III-5)

 

 

46

where £(k) = (k)T(k) is the mean free path.
Inserting Eq.(III-S) into Eq.(III—4), integrating, and
setting the result equal to zero, yields

K

_ E _ 1 _
S — VT " ETR— (III 6’
o
where the functions Kn are given by
' 2 2
__ _ .n 0(a) IL. 2 2 §___ _ 0(6)
Kn.-_ [(5 cf) ——jT—-+ 6 K T? 2 ((E Ef)-—jf- )-+ ...]€=€
e BE e f
(III-7)

To first order in the expansion of Kn the thermOpower is given

by

 

 

_ n K T { Sing

36 I6 } (III-8)

f

For a cubic material having an isotropic mean free path,

 

the conductivity is given by(l6),
2
O z e 1% (III-9)
12n %
Under these conditions the thermopower reduces to,
2 2
_ —n K T 1 32 1 3A _
S - “33—— IT SE + A elef (III 10)

Two factors determine the sign of the thermopower, the mean
free path of the electrons, and the phase space available
for excitations (i.e. the geometry of the Fermi surface).
The determination of the sign of 3% is in general complex,
but in a classical model more energetic electrons would be

less likely to be scattered than slower ones and would thus

have longer mean free paths. Thus the mean free path will be

 

47

an increasing function of energy at the Fermi energy. The term
gé-will then be positive and yield a corresponding negative
contribution to the thermopower. For free electron Fermi

surfaces, which are spherical, the surface area, A, is proportional
to the energy. Thus the contribution to the thermopower

from this term would also be negative.

2. Effect of Two Groups of Carriers:

 

Eq. (III-9) relies on the condition that Q be constant
on constant energy surfaces. Experimental evidence suggests
that for aluminum and indium, this assumption is not valid533’ll)

We therefore go beyond this assumption and break up the

conductivity integral into regions over which R is constant:
O = Zio. , (III-ll)

where oi is the contribution to the conductivity from a region

having constant mean free path ii. Substitution of Eq. (III-ll)

 

into Eq. (III—8) yields the well known result(34):
Ci
8 = Z. ——-S. (III-12)
i c i
where
TT2K2T afinoi
Si 2 3e { as lsf} (III-l3)

is the thermopower of region i.

In polyvalent metals, the Brillouin zones provide natural
boundaries for breaking up the electrical conductivity integral
in Eq. (III—ll). Consequently, as a first approximation, we

take the mean free path within a zone to be constant (for

 

 

48

aluminum and indium there is some justification for this in

(9)).

the literature Before applying Eq. (III-12) to the
specific cases of aluminum, indium and lead, we review briefly
how the Fermi surfaces of metals are mapped into various zones.

The periodic potential of the lattice serves to render
the energy contours in k-space discontinuous at a set of
planes. These planes, called Bragg planes, are determined
by the condition

2 pg - g = G2 (III—l4)
where G is a reciprocal lattice vector. They divide k-space
into regions called Brillouin zones. The first Brillouin zone
is defined as the smallest volume bordered by Bragg planes
which contain the origin, 5 = 0. Regions in k-space which
can be reached from the origin by crossing one Bragg plane
lie in the second zone; regions reached by crossing two planes
lie in the third zone; and so on for higher zones. Each zone
has the same volume as the first zone, and can be mapped
onto the first zone by translation of its pieces by a reciprocal
lattice vector.

To see how a Fermi surface is divided into zones, we begin
by drawing the free electron sphere in k-space, superimposed
upon the Brillouin zone structure. This corresponds to the
one O.P.W. Fermi surface in the extended zone scheme. In
Figure (III-l) the solid line represents a slice through the
Sphere for a metal with 3 electrons per atom. It resembles

a slice of the Fermi surfaces of aluminum and indium in the

 

49

 

 

 

 

 

Figure III-l: Cross—section of the one O.P.W. Fermi
surface of aluminum in the (100) plane
superimposed upon the Brillouin zone
structure. The dashed line indicates a

constant energy surface above the Fermi

level.

 

 

50

(100) plane. In aluminum and indium the first zone is full,
the second zone contains approximately one electron per atom,
the third zone contains approximately 0.03 electrons per atom,
and the fourth zone is empty. In lead, a four electron metal,
the first zone is full and the second, third, and fourth zones
are partially occupied. By translation of the pieces of the
Fermi surface lying in higher order zones back to first zone,
one obtains the Fermi surface in the reduced zone scheme. This
scheme is particularly useful for describing the electron system
in a magnetic field. We therefore defer further discussion of
it until we consider the effect of a magnetic field on the
thermopower.

Using the zone boundaries to define the regions 1 in
Eq. (III~ll), Eq. (III-12) for aluminum and indium can now
be written
S = :—2— 82 + 2}- S3. (III-15)
For lead, a contribution from the fourth zone would need to
be added. If, as we assumed above, the mean free path is

constant within each zone, Eq. (III—12) takes the form

2 2 3A 32 3A 3%
_ n K T 2 2 3 3
_ —3e ){Q2 86 + A2 38 + 23 38 + A3 BE }€

 

 

 

(III-l6)

3A.
where EA 2 2 A + Q A and is the rate of change with

1 1 2 2' 88 lef

energy of the area of that portion of the Fermi surface contained

in the ith zone.

 

 

 

51

 

 

 

3A
In the second zone gg—IE is negative (as indicated
. f
by the dashed line in Figure (III-1)) and yields a positive
3A
contribution to the thermopower. In the third zone EE—le
f
is positive and yields a negative contribution. Within our
3A 8A
one O.P.W. approximation, the terms gzg-and 5E" are not
independent, since
3A2 8A3 3A
——— + 2 __EE , (111-17)
8 3E 88
8A8
where 38 p is the energy derivative of the Fermi surface
of a free electron sphere.
If the mean free path is isotropic (i.e. £2 = 23),
Eq. (III—l6) reduces to the free electron result
2 2 8A
_ n K T l s l 32 _
S I 3e {A BE + I _E}€=€ (III 18)
sp f

If 12 ¢ 23, S will not have the free electron value, and
3A
may even be positive. For example, the term 5E4-was found

to be negative. Therefore, if I >>£ the net contribution

2 3'
from the terms containing energy derivatives of the Fermi
surface area will be positive.

The gé-terms are in general complex and to even obtain
their correct signs requires a detailed knowledge of the
scattering potentials seen by electrons in the metal. For
example, for lithium and the noble metals, which have nearly

(31)

free electron Fermi surfaces, Robinson has shown that a
suitable choice of the scattering potential can account for

the anomalous positive sign of the diffusion thermopower.

 

 

52

Because of the difficulty in determining gé-we will begin

by focusing our attention on the %% terms to see what aspects

of the thermopower can be understood without a detailed

knowledge of %& .
E 32 323
If Q = 2 , and if both ——— and ——— are zero, measurements
2 3 88 SE

of the thermopower would yield direct information about the
area of the Fermi surface, since then Eq. (III—l6) would

reduce to

2 2
S = n K T (

3e ——-)I . (III-19)

One way to approximate these conditions experimentally is

to make the specimen so thin (t<<£ where t is the specimen

bl
thickness and lb is the mean free path in the bulk material)

that all electrons with energy in the vicinity of 8f will be
scattered only by the Specimen surface. Then all these

electrons will have the same mean free path, 2 ~ t. Thus i

will be isotrOpic, and %%IE will be zero. On the basis of
f

our one O.P.W. model, we would expect S for aluminum to be

given by the value appropriate to the 3-electron sphere,

S = -2.1 x 10_9T (volts/K). Indeed, recent size-effect measure-
ments on dilute aluminum alloys (containing 5 ppm Fe and

20 ppm Cu, respectively), yielded just this value to within

(22)

eXperimental uncertainty. This result reinforces other

(35) and gives us confidence in the

experimental information
one O.P.W. model as a valid first approximation to the Fermi
surface of aluminum. (It also indicates that if the 3% term
can be accounted for, if not in fact solved for, then reasonable

agreement between theory and eXperiment is possible.)

53

3. Effect of Two Types of Scatterers:

 

We will now show that the effect of impurities on the
thermopower can, in part, be understood without explicit
knowledge of %%° We limit this discussion by considering
only a metal with a single group of carriers (the extension
to two groups of carriers is straightforward but algebraically

(36)

cumbersome). Kohler has shown that, subject to certain

conditions, the thermopower can be written as
Wi
S = Z W— S (III-20)

where Wi is the contribution to the thermal resistivity of
the ith scattering mechanism and
TT2K2T ainpi

Si = 3e 36 lef (III—21)

 

is the "characteristic thermopower" of that mechanism. The
conditions are:
(l) the electronic conduction is a result of a single
homogeneous group of charge carriers.
(2) the electron scattering mechanisms act independently,
(Matthiessen's rule).
(3) electrons are solely responsible for the heat
transport in the conductor.
An alternative derivation of Eq. (III-20) can be made by
assuming that the Wiedemann-Franz Law holds (an assumption
which is valid for the aluminum specimens studied in this thesis).

By noting that o = l/p and using condition (2) above, the

 

 

54

thermopower can be written as

2 2 BQan.
_ _ n K T i i _
S — 3e 38 IE (III 22)

 

or

pi
S. (III-23)

5:2—
10 l

which is the Gorter—Nordheim rule.(37)

Eq. (III-23) is identical to Eq. (III—20) when the
Wiedemann-Franz Law is valid. In addition it is also similar
in form to Eq. (III—12), the difference being that the former
describes the thermopower when two scattering mechanisms are

present, whereas the latter describes it when two groups of

carriers are present. Eq. (III-23) when rewritten in the form
0b
S = Simp + E— (Sb—Simp) (III-24)

(where "imp" signifies a particular impurity ion and "b"
signifies all other scattering mechanisms) provides a recipe

for obtaining the "characteristic thermopower", Simp' of a
given impurity. If the thermopower S is measured for a series
of alloys containing various concentrations of a known impurity,
then a plot of S vs. l/p will yield a straight line whose

slope is (Sb—Si )‘pb and whose intercept at 1/p=0, is Si

mp mp'

We see from Eq. (III—24) that S will be different for
different impurities, and that once pimp becomes much greater
than Oh, the thermopower becomes independent of the concentra-

tion of the impurity.

55

B. Phonon Drag:

 

In the preceding discussion it was tacitly assumed that
the phonon system remained at all times in thermal equilibrium
with the lattice. This is, however, not true when a tempera-
ture gradient exists within the specimen. In such a case, a
net flux of heat is carried down the gradient by a phonon

(38) this current can

current. As first noted by Gurevich
transfer momentum to the electron system via the electron-
phonon interaction, and thereby drag electrons down the gradient.

(34) has shown,

This phenomena is called phonon-drag. MacDonald
using a simple kinetic model, that the phonon drag contribution

to the thermopower is given by:

Sg = 1/3 [C/Ne] (III-25)

where C is the lattice specific heat and N is the number of
electrons per unit volume.

At low temperatures Eq. (III-25) reduces to
__ 3 _
Sg — 77.8 eNa [T/OD] (III 26)

where Na is the number of free electrons per atom and 6D is
the Debye temperature of the metal. The same result was

(39), using a more sophisticated

obtained by Sondheimer and Hanna
approach involving the solution of the coupled Boltzmann
Equations for electrons and phonons.

In obtaining Eq. (III—26) all phonon scattering events

other than electron-phonon scattering have been neglected.

56

Such scattering events can be taken into account by writing the

phonon-drag thermopower as(34)
T
sg = 80 ( ?——§§—— ) (III-27)
Pe PP
where S0 = l/3[C/Ne], Tpe is the relaxation time for scattering

of phonons by electrons, and Ipp is the relaxation time for
scattering of phonons by everything else (e.g. phonons). At
low temperatures T<<OD, Tpp is much longer than Tpe, and Eq.(III-27)

, T is
D PP

shorter than Tpe (Tpp is dominated by phonon-phonon scattering,

since at high temperatures the lattice vibrations are no longer

reduces to Eq.(III-26). At high temperatures, T30

governed by a harmonic potential) thus

5 = S —BE (III-28)
g o I
pe
At high temperatures both the specific heat and '%e(24) are
independent of temperature. Tpp is, however, inversely

proportional to T (at high temperatures the number of phonons
is prOportional to T). Thus in this regime, the phonon-drag
thermopower should vary inversely with temperature.

In the derivation of Eq. (III—26), Sondheimer and Hanna
neglect Umklapp scattering. Umklapp scattering is distinguished
from Normal scattering by the appearance of a reciprocal

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

k' — k = q + G (III—29)

1"

where q is the phonon wavenumber. For normal scattering the

reciprocal lattice vector is zero. For the case of a spherical

 

57

Fermi surface, which does not intersect the Brillouin zone
boundaries, the effect of Umklapp scattering is readily
seen. An electron in state k having a velocity v is scattered

by a phonon of momentum q to the state k' and obtains the

velocity 3'. For spherical Fermi surfaces, v is proportional
tofik and therefore from Eq. (III—29) y' u y is parallel to q,
the phonon momentum absorbed. The electron system gains

momentum in the direction of g, which is down the gradient,
and thus yields a negative (normal) contribution to the thermo-
power. For Umklapp scattering y' - y is parallel to fi,+ G,
which is antiparallel to q. Hence Umklapp scattering increases
the electron momentum up the gradient, yielding a positive
(anomalous) contribution to the thermOpower.

To calculate the phonon—drag thermopower in a real metal,
whose Fermi surface may contact the zone boundaries (as for
the metals studied in this thesis), a more realistic approach

(40) (41)

is necessary. Ziman and Bailyn have both made use

of the variational technique to calculate the phonon—drag

thermopower, taking Umklapp scattering into account. At low

(40)

temperatures, Ziman obtains
Sg = K/en [CL/3Nk][—PlL/PLL] (III-30)
where
-P1L = ET If! (Ck—gmb PE. dk d (135' (III—30a)
~ ~ 9 -2 ~
and ~

__1 215' . _
p _ KT fff (Oq) qu a5 a3 a5 (III 30b)

~__,

 

 

 

58

k

PEG is the probability for an electron to scatter from.k to k'
by absorbing a phonon of wavenumber q. Qk and Oq are functions

describing the deviation from equilibrium of the electron and
phonon systems respectively. It is not our intention to
delve deeply into these equations. However, one point is
essential to subsequent discussion in this thesis. Ziman

chose the trial functions

I

k
— 41.3 ° _
it — Te k YE E (III 30c)
and

T

S
@q = ?— q°u (III-30d)
.e L ”’~

where u is a unit vector in the direction of the temperature

gradient, Tk and Tq are the relaxation times for electrons in

~

the state k’and phonons in the state g respectively, and Te

«.1

and IL are defined through the kinetic formulae

nNe2 Te/m (III-30e)

0
II

_ l 2
K — 3 CLVLT

L (III-30f)

L

Te and TL are chosen to make these formulae yield the correct

ideal electric conductivity and lattice heat conductivity.
With these assumptions, Ziman finds PLL to be positive definite,
so that both the sign and magnitude of Sg are determined by

_P1L° The essential feature of P1L is the factor

) (III—31)

 

59

which shows that for a nearly free electron Fermi surface,

as noted above, the contribution of normal scattering to

the thermOpower tends to be negative and that of Umklapp
scattering, positive. These remarks lead to the approximate
rule that "the contribution to QL (Ziman's symbol for 89)
will be negative or positive according to whether the chord q
between the points k and k' passes through occupied or

(40)

unoccupied regions of the Fermi surface".

Bailyn arrives at the following expression for the phonon-

 

 

drag thermopower:(4l)
3N (qj) .
K2e22[——9—7——]Z(3q)<r(jq;kl,k'l) [v(k1)T(kl)-v(k'l')'r(k'l')].v(jq)
S = 3 OKT ~‘7” N ~ V ~ ~ ~ ,~ ~ V
g 3f
2 2 2 0
3e Zyjfil) T(]gl)[—88 I

(III-32)
in which a(jgiEl7E'l') is the relative probability that the
jg phonon will scatter an electron from state kl to 5'1'

(1 is the band index). There are two essential features in
this equation. First, the sum over k can be divided into

various regions of the Fermi surface such that

 

 

 

o.
9 O 91
where
2 2 2 8f0
Oi = -3- e XI (E1)\C(51) (I-‘gz- ) (III-33a)
and
3N (jq) . .
K-Z-QZX. {—3—]Z(j’q’l)oc(jq;kl;k'l')[v(kl)T(kl)-v(k'l')r(k'l')]
S = 3 3q BKT
gi 2 2 8fo
—2 V(kl) T(k1)[- ———J
3 ~ ~ . BE
. 35(qu
2 2 3f0
3'2 17,051) ”((351) [" —3_€— (III-33b)

 

60

This expression is particularly useful if each region
is characterized by a constant relaxation time, and if Sgi
is nearly constant throughout the region. If T is constant
within a band, then the T(kl)'s cancel out of Eq. (III-33b)
so that Sgi is independent of impurity. In such a case,
Sgi is affected by the presence of impurities only through
the weighting factors, Oi. If the Fermi surface consists

of only two different regions of different I which have

substantially different Sg's then measurements of the phonon-

 

drag thermopower for different alloys would yield direct
information regarding anisotropy in the scattering of electrons
by impurities (i.e. information about the quantities oi).(21)
In obtaining Eq. (III-33) from Eq. (III-32), it was
assumed that interregional scattering by phonons was small.
This assumption is best justified at low temperatures<4l),
where the present experiments were performed.
The second point concerns the determination of the sign
of Sgi' The sign of Sgi is the same as the sign of
Z E [ylkl)r(kl) - ZIE'l')TIE'1')]'XXjr§)' Bailyn's rule for
such a determination is "--When the shortest surface connection
between k and 5' (in the extended zone scheme, where 56 is the
equivalent state to 5' that is closest to k) lies on a convex
strip of surface (relative to the filled part) the contribution
tends to be normal (negative); but it the strip lies on a
.. (41)

concave part, the contribution tends to be anomalous (positive).

This is similar to Ziman's rule.

 

 

 

IV. Theory of the Magnetothermoelectric Effect

A. Electron Diffusion:

 

1. Introduction:

 

We begin by first describing electron dynamics in the
one O.P.W. model and offering a physical argument to explain
the observed variation of the thermopower with magnetic field.
The rate at which the state 5 of an electron changes

due to the Lorentz force is given semiclassically by

(W’

 

Since the force is perpendicular to the motion of the electron
in real space, the magnetic field does no work on the
electron, and the electron remains on a constant energy
surface in momentum space. This conclusion can also be
arrived at by noting that k is normal to y, and y is proportional
to the gradient of the energy. In addition the motion of

an electron in k-Space is confined to a plane perpendicular

to the magnetic field, and thus its trajectory in k—space

is the intersection of this plane with a surface of constant
energy. The Bragg planes are of fundamental importance in
determining these trajectories, since an electron moves on

a spherical arc of the Fermi surface, normal to the magnetic
field, until it either scatters or reaches a Bragg plane. At
a Bragg plane, barring magnetic breakdown, it is diffracted to
an equivalent site on the Fermi surface and hence remains in
the same band.

61

62

In Figure (IV—l) the Fermi surface of aluminum and
indium in the second and third zone are shown in the reduced
zone scheme.

Figure (IV—1) also displays various intersections
possible for a plane normal to a symmetry axis (also the
field direction) with the Fermi surface. The orbits shown
in Figure (IV-l) are obtained from the one O.P.W. representation
of the Fermi surface of aluminum. In this approximation, the

arcs in each orbit are portions of spheres and thus

e_k (IV—2)
m

V:

thereby allowing Eq. (IV-l) to be rewritten
k: 9— (k x H) (IV-3)
A. mc ~~ ,c'

The state k therefore moves with constant angular

velocity w: = 2% along the Fermi surface until (in the

extended zone scheme) it reaches a Bragg plane, at which
time it is diffracted to another portion of the Fermi
surface and continues its motion again with angular velocity
0

we. The time required to complete an orbit in this model

can be computed from the following rule for the frequency.

0
Zfiwc
we : ‘26—,— . (IV—'4)
i
0. = angle through which the electron moves on the ith arc.

l

Figure IV-l:

63

 

The one O.P.W. Fermi surface of aluminum
in the reduced zone scheme. Cyclotron
orbits in the second and third zones are
indicated in the (001) plane. (After

Ashcroft, Ref. 9).

64

From Figure (IV—l) it is clear that electrons in the second
zone move along orbits which, for the most part, have path
lengths comparable to those for orbits of free electrons;
whereas electrons in the third zone sample very little of
k-space.

Using Eq. (IV-4) and assuming I to be approximately
the same for both zones (or alternatively, using results

I

of cyclotron resonance experiments to estimate wet for

 

the second and third zones), one obtains the ratio,
(“CH3 - (9)
IZEIT—-_ 10.
c 2

With these results and the equation for the zero field

thermopower in the two band model,
5 r (III-15)

we can qualitatively understand the observed variation of the
electron diffusion thermopower with magnetic field. The
weighting factors Oi dictate the relative importance of the
contribution of each band to the total thermopower. For
electrons on a closed surface, in a strong magnetic field,
i.e. in the limit that the electrons complete several orbits
before scattering, the contribution to the conductivity tends

(42) that for such electrons

to zero. It can in fact be shown
the conductivity OXX+ A/(wcr)2 as (wcT) + w. Herein lies
the mechanism for the field variation of the thermopower of

aluminum and indium. As H + w, 02(H)/O3(H) + 10002(0)/O3(0)

65

and thus S + S . In Section III it was shown that S (due

2 3A 2

to the negative §E£|€f) was more positive than S3. Thus
the total thermopower becomes more positive than its zero
field value, and then saturates in high field. In making
this argument we have assumed that Si does not vary with
field and have used a scalar equation for S rather than
the apprOpriate tensor equation (See Eq. (IV-8) below).

We now turn to the formal solution for S(H). We will
show that the free electron model (i.e. no Bragg reflections)
is totally inadequate to explain the observed variations of
the thermopowers with magnetic field in the metals studied.
We will also show that, with certain approximations, the

formal solution for S(H) yields a result similar to that

obtained with the simple argument just presented.

2. Formal Theory for Aluminum and Indium:

 

In the presence of a magnetic field, the electric and

thermal current densities can be expressed as linear functions

of the electric field and thermal gradient.(l6)

g.=.£EE(H) ..E +,gET(H) °,ET (IV-5a)
and

.2 = gTE(H) "E + ~TT(H) ’ET (IV-5b)

In section II, the tensorsJLij reduced to scalars as a result
of the cubic symmetry of the crystals. However, the magnetic
field introduces a preferred direction into the problem and

in its presence the tensor nature of Lij must be retained.

 

 

 

 

66

We assume that the magnetic field is applied in the z—direction

and that the voltage and temperature differences are measured

between points on the x-axis (Figure IV-2)

IZ
I

 

   

 

 

Figure IV—2: Experimental Condition for Measuring the

Adiabatic Magnetothermoelectric Power.

The thermOpower is defined subject to the condition that

the electric current density vanishes. With this condition

Eq. (IV—5a) yields

_ _ -1
,g- [(LEE(H)) gETIHII 1T

.V

(IV-6)

Due to the nonvanishing RighiwLeduc coefficient in Al and
In(3’4), we must distinguish between isothermal and adiabatic

boundary conditions. The isothermal (in the y direction)

condition requires (ET)y = 0, whereas for adiabatic conditions,

U = 0. In the present experiment adiabatic conditions

Y

 

 

.I.I II! , . lililiilllllliiili'lt

67

apply, and it is the solution to Eq. (IV—6) under these
conditions that we will be concerned with. However, the
solution for isothermal conditions is also interesting and
will be discussed.

Under adiabatic conditions the thermopower, S, is

given by the relation

(VT)

 

 

 

_ _ _ -l _ -l y
S — Ex/(VT)X — [(‘I‘EEEIHH [LIET(H)] [(EEEWH éET(H)]xy(VT)X
. . _ (Iv—7)
and the Righi-Leduc Effect is determined by
LTELET
[LTT _ L 1 K
EE xy xy
AT /ATX = = K , (IV-8)
y L L xx
TE ET
7 [LTT L 1
EE XX
where Kij is the thermal conductivity tensor.
The thirty-six components of the four transport tensors
ETT’ gTE' EET'IEEE are not all independent, but must satisfy
the Onsager Relations,
Lij<§) = LjiI—g) (IV-9)

In addition, if consideration is limited to isotropic poly-
crystalline specimens, the number of independent components

reduces to nine.(l6)

To obtain these remaining coefficients
the transport equation must be solved. Under the prescribed

conditions (i.e. VT#0, H#0, E#0) the Boltzmann Equation in

the relaxation time approximation is given by(16)
af 3f (5)
O . O _l§_ —§L_o_a__
SE 3??“ .3T 3 TEE—"T — IE+CIX X.E) kak avlfIIE)

rd

(IV-10)

68

The relationship between L and L is therefore the same

EE ET

in a magnetic field as in absence of a magnetic field, i.e.

 

 

HZKZT 8
£ET(H) = 7 3e ( SE'EEE(H))e=ef . (IV-15)
Similarly
gTT(H) = - LOTgEE(H) (IV-16)
n2K2
where L0 is the ordinary Lorentz number, L0 = 2 .
3e

This argument, that the thermomagnetic properties can be

derived from the galvanomagnetic properties (in the relaxation

 

 

time approximation), was originally developed by Kohler<43)

and can be found in Electrons and Phonons.(l6) Using Eq.(IV-15),

Eq. (IV—7) can now be rewritten

2 2 (VT)
_ n K T -l BO(H) -l 30(H)
S — 3e {(G(H) BE '6 xx + [(O(H) 38 IE ]xy (VT) }
f f X

(IV—17)

Eq. (IV—l7) provides a prescription for calculating the
thermopower in a magnetic field similar to that prescribed in
zero magnetic field by Eq. (III—8). The complexity of such
a calculation can be appreciated from the difficulty confronted
in the zero field calculation. However, by making various
assumptions to simplify the problem, Eq. (IV—l7) can be used
to compute the change in the thermopower, S(H=W) - S(H=0),
semiquantitatively.

We begin by solving Eq. (IV—l7) for the case of a free
electron Fermi surface, to show that for this case, the thermo-
power should not be affected by the presence of a magnetic

field. According to Eq. (IV—l7)

69

2 2

_ n K T
ij — 3e {(0(H))

-l

 

80(H)
} l

~3E——- e (IV-18)

f 13
(Note the distinction between the tensor Sij and S).

The elements of the conductivity tensor are in general given

by the expression<44)
2 8f (8) m
Oij — Z;§;2f __§E—— dE [wdkzzwa f0 Vi(6)d6fevj(¢)d¢ exp{ f96;?;TE)}
(IV-l9)

where:

mg and ma are the effective mass and cyclotron frequency

 

of an electron making an orbit of type d;

Vi(8), Vj(¢) are the velocities at points 9 and ¢ on the orbit
(measured from an arbitrary origin);

and T(w) is the relaxation time appropriate to the orbit d at
position w. The sum is over all orbits defined by E and k2.
For free electrons, wCT is constant and vi(e) and Vj(¢) vary
sinusoidally around the Fermi surface. Under these conditions

Eq. (IV—l9) reduces to

 

/ l w T 0

0o

G = 2 —w T l O (IV—20)
c
l + (wcT) 2
0 O 1+(w T)
c
where
2
O = e RA
I
O 12n3h

is the zero field conductivity, and

eH
c mc

 

‘ TfIII. i), . ‘ \Nlfilly

 

Figure IV-3:

 

 

 

 

The velocities, VX(6), of electrons
executing the orbits shown in Figure IV-l.
The dashed lines represent the velocities
for a Fermi surface based upon the free-
electron sphere. The dotted curves
represent the function-c056, normalized
at e=n to the maximum value of VX(6).

(After Ashcroft, Ref. 9).

 

 

71

Substitution of Eq. (IV—20) into Eq. (IV-18) yields

 

 

Tr2K2T l 800 wCT a(wcr)
S = ————-—{——-—*——-- } (IV-21a)
xx 3e 0 36 2 38
o l+(wCT)
2 2 8w T ‘

Sx : Tr3: T { l 2 a: } (IV—21b)

y 1+(w T)
AT ny Oxy
JAT : 'K—-— = ;— : LDCT (IV‘ZlC)

x xx xx

so that from Eq. (IV-l7)

1T2K2T 30
3e

0

Be lef

s = }, (III-8)

 

 

l_
G

which is just the zero field value.

In evaluating Eq. (IV—l9) to obtain Eq. (IV-20), the
effect of zone boundaries was ignored (since we were treating
free electrons), so that the velocities vi(e) and Vj(¢) varied
sinusoidally around the Fermi surface. For aluminum and indium
we cannot neglect Bragg reflections. The correct velocities
for representative orbits are shown in Figure (IV—3). In the
one O.P.W. model we can solve Eq. (IV-l9) exactly, at least
in principle, since the velocities are known. This is, however,
difficult, and we use an alternative procedure which allows an
approximate solution which we believe retains the essential
physics of the problem.

We first expand the Vij(6) in a Fourier series,

\

v (e) = z Vix elne (IV-22)
n

where an are the Fourier transforms of Vi(6), and assume

 

”L

 

72

that (wCT)i is constant within band i. The Fourier transforms
can be obtained, since Vi(6) is known. Thus Eq. (IV—20)

could be integrated term by term. We note, however, that

the first term in Fourier series is clearly the most important
term (see Figure (IV-3)), and should therefore be a reasonable
first approximation to the correct velocity. We therefore

assume that it is the only term present. Physically this
approximation changes the orbits from those shown in Figure (IV—l)
to circular orbits. For the values of ma in Eq. (IV-l9)

we use the average cyclotron frequency of the band; the

variation within the band being much smaller than that between
the bands. The solution to Eq. (IV—l9) is now greatly simplified,

and in fact reduces to the tensor

1 w T O
. c
c l
o.(H) : o —wCT l 0 (IV-23)
3* l+(w r)2 2
c i O O 1+(w T).
c i
where i : 2,3 designate the second and third zones respectively.

The total conductivity is

C(H) = O (H) + 03(H). (IV-24)

2

Upon inserting Eq. (IV-24) into Eq. (IV—7) we obtain the
following results:

1) in the limit H+O,

 

 

73

in agreement with Eq. (III-l7).

2) in the high field limit, H + m,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' _ /o3(0)2 03(0)02(0)‘ S, 02(0)2 03(0)02(0)
— (0) + +
XX ((w T) (w T) (w T) 3 (w T)2 (w T) (w T)
c 2 c 3 c 2 c 2 0
03(0) 03(0)02(0) > a(wCT)3
{(wcT)§ (wcT)2(wc T)2 36
{03(0) 02(0) 2 {7% T) 330 (0) (wCT)202(0) 2
+
Um T)§ (w +i(:c) T) (NOT):
(0)2 o (0)0 (0) 2M 1)
_f’2 3 2 “’c 2
(wCT)§ (wCT)§(wCT)3 86
"03(0) 02(0)’ 2 i_(mCT)3o3(0) (wcT)202(0)-] 2
( >2 < )2 + ( )2 + ( )2
L0G T wcT 2 (HOT 3 wCT 3
(IV-25a)
2
03(0) 03(O)02(0) + 203(0)02(O) a(wcT)3
s' _ (wCT)3 (wcT)§(wCT)§ (wCT)§(wCT)2 , Se
xy — -- 2 ,_ 2
03(0) 02(0) (wCT)3O3(0) + (wCT)202(0)]
(m r>2 (w n2 (w T)2 (w T)2 J
- c 3 c 2 — c 3 c 2 _
(62(0)2 02(0 )03 (0) + 3(0)02 (0) amenz
[(wCT):O (wC T)3 c): (w cT)2 3(wc T)3 86
03(0) “(w CT) 330 ()0 + 3(wCT)202(fl2
(w T)2 (wc T) (wC T)? (w T)2 I
c 3 c 2 _

 

(IV-25b)

\BSZ(O) +
3

+

 

74

 

 

 

 

2 2 .
ny = 33X = (wCT)3(wCT)203(O) + (wcT)2(wCT)302(O) (IV-25c)
Kxx Tx (w T)20 (0) + 0 (O)(w r)2
c 2 3 2 c 3
where
S(H)
S'(H) =
TZKZT
3e

The parameters in these expressions are estimated as
follows:
1) The cyclotron frequencies of the two zones are

obtained according to the procedure described earlier. The

(wCT)3
ratio, 75—?7- , is taken to be 10.
c 2
2) The zero field conductivity of each zone is obtained
from Eq. (III-9) and hence the relative contribution to the

zero field conductivity is given by the percentage of the
Fermi surface within each zone (assuming an isotrOpic mean
free path). From a graphical construction using the one

(45) estimates that 76% of the surface

O.P.W. model, Pippard
area liea within the second zone while 24% lies in zone three.
(He notes also that rounding off of the monster arms in the
third zone reduces significantly both the area within that
zone, and the velocity; the latter reduces the mean free
paths. Pippard therefore estimates the third zone to contribute
only 12% of the total zero field conductivity.)

3) We estimate the ratio Sé(O)/S;(O) from the equations,
s'(0) = £—-353 3%
2 A e

|
2 a 6f 2

(IV—26a)

 

I 1 8A 82
83(0) = A_ as e

]E (Iv—26b)
3 f 3 f

We begin with the terms involving the logarithmic

derivatives of the areas. Since, as we noted earlier
8A2 3A3 BASp
35— + §E_ = _§E_ , we need calculate only EE— .
For a sphere intersecting a plane, the surface area of
the segment is 2Tr(r—d) where r is the radius and d is the

distance from the center of the sphere to the plane. Thus

22
Sr

in aluminum intersecting the Brillouin zone boundaries

~ 2nr for r~d. This situation resembles the Fermi sphere

separating the second and third zones. However the inter-
sections are much more numerous and complicated, in that the
spherical segments often contact more than one plane.(46)

As a rough estimate we approximate this situation with a

 

 

 

8A
sphere intersecting an octehedron and obtain 5E; = 28fir %% .
Since sp = 8Tr £3 , we get ——— = 20fir EE and
as 88 Be 36
3A3 8A2
(F)/(a—€—)=7/5-
If we again use our one O.P.W. estimate that A /A = l/3
3 2
8£nA3 aznAz 21
then the ratio of ( _§E_— ) 38 = E— ~ 4.
Unless the logarithmic derivatives of the mean free paths
82n£3 32n22
dominate the thermopower and have a ratio ( as )/( _§E__ )

greater than four, the ratio of the two thermopowers

S3(O)/SZ(0) will be less than or equal to four. In fact, we

82n£3 BQnQ

3e )/( 88
With these estimates, the leading term in the high field

) to be closer to unity.

 

 

would guess the ratio (

thermopower is

 

 

L

76

2 2 3A 31
S(H+m) = { Egg + %—
2 2

N

 

E }€f I (IV—27)

vra
0)

which is simply the thermopower of the second zone. With
S3(O)/SZ(O) = 4, the next term is approximately 12% of the
leading term. For smaller values of S3(O)/SZ(0) the correction
term decreases in magnitude relative to the leading term.

We therefore conclude, in agreement with our physical
model, that S(H)+SZ(O) as H+W. To obtain the variation in
the thermopower with magnetic field in the high field limit,

we write

 

 

 

 

 

2 2 8A 3% 82 3A
S(H+w) = W K T {i_ __g + i. __g _ _l_ __§B 1 __EE } .
3e A as 2 8 2 as A Be a
2 2 sp f
(IV—28)
8A2 3 5
Using the estimates A = 0.76 A and ——— = —2.5 p ,
2 sp as as
Eq. (IV—27) reduces to
AS =_n2K2T [4 2( 1 BAsp ) + £_ 322 _ 1 325p
3e L' A as 2 as 2 Be
sp 2 sp
(IV—29)

The first term in Eq. (IV-29) yields the value

AS = 8.7 x 10’9

area T V/K (IV-30)

We cannot calculate the terms involving logarithmic
derivatives of the mean free path. To do so we would need
detailed knowledge of the form of both the scattering
potentials seen by the electrons and the electronic wave

functions.

 

77

For free electrons the two terms involving logarithmic
derivatives would cancel. It therefore seems reasonable to
assume, as a first approximation, that only the first term
in Eq. (IV—29) is important. In this case the change in S
with H should be the same for all impurities.

If the logarithmic derivatives are significantly different
from each other, and comparable in magnitude to
(l/Asp)3ASp/8€, we would expect the total variation of S(H)
to depend upon the type of impurity present. Thus an
investigation of the dependence of the magnitude of the
change in S upon the nature of the impurities present should

provide information concerning the importance of these terms.

 

 

 

78

B. Phonon drag: Theory

 

The theory for the effect of a magnetic field upon the
phonon drag thermOpower is limited. In 1956 MacDonald and
Pearson, in theisztudy of the effect of a magnetic field
on the thermOpowers of the "one electron metals" sodium
and c0pper, had written "we assume, as appears reasonable
that the presumed "phonon-drag" component will be uninfluenced

by the magnetic field ...."(47) More recently (1969)

(48)

Sugihara showed that a sharp minimum in the thermOpower

 

of the semi-metal bismith, at low temperatures and high
magnetic fields, could be explained as a magnetically induced
change in the phonon drag component. There have been no
calculations for Al and In. We therefore suggest a possible
mechanism which can account for variations in phonon drag
with magnetic field in these metals.

Similar to the electron diffusion component, the phonon-

drag component of a two band metal in zero field is given by,

02 03
8g = 3— ng + E— Sg3 (III—33)
02
We have shown previously that the ratio 3— changes in
3
a magnetic field and will show below thatES is different

g2
from 593. If we assume that Eq. (III—33) also applies when
a magnetic field is present, then we would expect S9 to
change in a magnetic field. Since we cannot determine the
(section V) we cannot predict whether

93 03(H) 03(0)

Sg should decrease of increase as E;Tfi7’+ IEBEETBT-in high

sign or magnitude of S

79

magnetic field. Perhaps the observed variation may be viewed

S
as a measure of the ratio —32 .

ng

 

 

V. Experimental Results and Discussion:
A. Zero Magnetic Field:

1. Introduction

The thermopowers of pure specimens of aluminum, indium

and lead are shown in Figures (V-l) and (V—2). The data for
lead above 7.2K is due to Christian et al.(49), for aluminum
above 10K, to Gripshover et al.(50), and for indium between

10K and 300K to Averback and Greig.(51) The data below 10K,
Figure (V—2) have been plotted as S/T versus T2 in anticipation
of a thermopower of the form S=aT+bT3. The thermopowers

above 10K, Figure (V-l), have been plotted versus the reduced
temperature T/BD. The variation with temperature of the
thermopowers of these metals are seen, when plotted versus

T/B to be quite similar to each other in general form.

DI

We can understand this in the following way. The thermopower

of a metal is the sum of electron diffusion and phonon-drag

components(52), i.e

S = Sdiff + Sg . (V-l)
From the discussion of section III, we obtain

S = aT + bT3 for T<<6D (V-la)
and

S = cT + d/T for TieD . (V—lb)

80

 

 

 

 

 

8l

 

 

8x :06 VOLTS/K

 

\‘V

\
—-2.81—————-~~~----‘ 1 1 1 l J . n L 1

 

 

 

o .2 .4 .6 .8 LO I2 L4 |.6 LB 20
T/eD

Figure V-l: The thermopowers of Al, In, and Pb versus

the reduced temperature, T/GD.

 

 

82

 

2.8
2.6 r-

24 -- Pb
2.2 —-

2.0“—

AL

-.4‘- [TCCDOJZ

L J l \i/l l L 1 l L j
o 2 4 6 8 no :22 I42 I6 l8 20 22 24 24
T (K )

Figure V-2: The variation with temperature of the thermOpowers of Al,

 

In, and Pb below 6K. A straight line indicates a thermo-

power of the form S = aT + bT3.

 

83

The linear term in each equation is the electron diffusion

component; the coefficients 'a' and c' are determined by the
dominant scattering mechanism in the metal in the particular

temperature region, i.e. impurities at low temperatures,

and phonons at high temperatures. The other term in each

equation is the phonon—drag component. Thus at low tempera- ’
tures the magnitude of the thermOpower increases substantially

with increasing temperature due to phonon-drag. At approximately

GD/S, phonon—phonon scattering begins to dominate electron-

phonon scattering, and thereby reduces the effect of phonon—

 

drag (see Eq. (III-27)). The electron diffusion term is
responsible for the monatonic increase in magnitude near the
Debye temperature. Finally, for T>8D the thermOpower becomes
approximately linear in temperature, signaling that the phonon-
drag contribution has become small.

2. Aluminum:

We are concerned here only with the low temperature
thermopower of aluminum; the high temperature thermOpower
has been discussed in detail elsewhere.(50)

Electron Diffusion:

The coefficients "a" in Eq. (V-la) for the various
aluminum specimens are listed for zero magnetic field (and
also for a series of magnetic fields) in Table V-l. It is
clear that in zero magnetic field, the electron diffusion

thermopower is sensitively dependent upon the type of impurity

present. This is expected from both theory (Eq. (III-23)

 

 

 

84

Table V-l: Values of a(H), b(H), and residual resistance

ratios for aluminum and aluminum alloys.

 

 

85

 

.pmmum>muss mpawflm Am

H.vi\mu.Ho> aIOH cH n AN
NM\muHo> mIOH CH m HH
s.m ms.m s~.m OH.N mm.H we
I I I I I I I I s.HI Hm.H+ H.HI Ns5.+ oo.om
m.NI s.H+ I I I I o.mI omm.+ I I m.HI msm.+ oo.mH
N.NI mm.H+ I I H.mI ss.H+ m.mI omm.+ mm.HI mv.H+ mm.HI smm.+ oo.OH
m.HI mam.+ m.mI mm.H+ H.mI mm.H+ o.mI mos.+ s.HI mo.H+ mm.HI sHN.I oo.m
I I I I I I mm.HI mm.H+ o.HI os.o+ ms.HI mam.I oo.m
I I I I s.mI m.+ I I I I I I om.H
mm.HI no~.I m.mI va.+ I I m.HI mNB.I m.HI som.I I I oo.H
I I I I o.mI om.I I I I I I I com.
mm.HI mo.HI m.HI mm.I N.HI mm.I m.I no.HI v.HI mmm.I m.HI mm.I 0.0
a m mg mm mg mm a m n M Na Hm @HmHm
oomH comm comm oomH 0mm smH .umm .mmm

N H H N H

HHE m.o Ha HHE s .Hm HHE s H4 cmHm suH< soH< cmeHommm

 

 

 

 

86

o.m+ mN.I

omNH

NHBHd

.m.

9

o
H

mv.m

No.H+

oo.H+

mmw.+

mmN.+

hv.Hl

0mm

HEH<

m.NI mv.H+
m.mn ov.H+
H.v1 vm.H+
w.m| om.+

oomv

ad

m.m+

ooom

NcoH<

I HH.N
«.mI hov.l
m.m| mn.I
m.NI SN.HI
N.HI om.m|

Q m

comm

HUUH¢

m<
00.0w
00.mH
00.0H
00.m
00.0
0m.H
00.H
00m.
0.0
meHm_
.Pwm .mmm

swEfloQO

 

 

 

87

(19,20) We

and from observation by other experimenters.
examine the dependence of the thermopower upon the concentra—

tion of a given impurity in terms of the equation

0
_ b _ _
S — Simp + p (Sb Simp) (III 24)
or, Since S = aTé Slmp = aimpT’ and Sb = abT,
_ 2 _ _
a — aimp + p (ab aimp). (V 2)

For pb and S we use the measured values for a reference

b’
specimen prepared identically to the alloys, except that no
impurities were intentionally added. Figure (V-3) is a
plot of "a" versus 1/p for three aluminum alloys. For each
alloy, the intercept at l/p = 0 yields aimp' We find for

the characteristic thermopowers of these impurities the

following values:
8

1) Copper in aluminum, aCu—Al = 0.56i0.02 x 10 V/K.
. . . _ -8

2) Thallium in aluminum, aTh—Al — 3.2iO.5 x 10 V/K.
. . . _ -8

3) Cadmium in aluminum, aCd-Al — 4.0iO.6 x 10 V/K.

The quoted uncertainties represent the differences in
the values obtained for two specimens of each alloy. The
large uncertainties in Al + T1 and Al + Cd arise from the
low concentration of these impurities, coupled with the

difficulty of maintaining S and ob constant in the various

b
specimens during the alloying process. For example, the
open circle in Figure (V-3) represents a four mil aluminum

specimen which was rolled from the same starting material

as that which was used for the reference specimen, but which

 

 

 

 

 

Figure V-3:

88

oSTANDARD AIS
A AlCu

vAITI

IAICd

0 TEST Al,

. AICcI
AITI

 

,4 .6 ,8 LO I.2 L4 |.6 l.8

The coefficient of the electron diffusion
component of the thermOpower of A1 alloys vs.
the inverse resistivities of the alloys.

The intercept at l/p = O is the "characteristic

thermopower" of the impurity in Al.

 

 

 

 

 

89

did not go through the process used to produce the alloys.
We see that for very pure specimens (resistance ratios above

3,000) is very sensitive to small variations in treatment.

pb
We attempted to produce more concentrated alloys so that the
0
term —9 (S -S. ) would be less important, but, for unknown
0 b imp
reasons, only a fraction of the impurity atoms appear to have
gone into solution with the aluminum.
Phonon—Drag:
The values for the zero magnetic field phonon-drag coefficient
'b', are listed in Table V—l. These values do not appear to
vary in a systematic fashion with either the type or concen-
tration of the impurity atoms in the aluminum. To account

qualitatively for the fact that there are substantial variations

we make use of the equation

Oi
S9 = Z 3_ Sgi . (III-33)
Since Sgi was shown to be independent of the type of impurity
(see Section III), the variation in Sg due to impurities
must be a consequence of anistropic scattering (i.e. 02#03),

in combination with a difference between the values for

592 and S

93'

To understand why the scattering may be anistropic we
consider the effect of electron scattering through a small
angle in both the second and third zones. Over most of the

second zone, small angle scattering produces only a small

change in the electron velocity. Several such events are

 

2ND ZONE
Il— .— ‘_\.

 

/' \ ./' '\

399 ZONE

___—I NORMAL SCATTERING
..... . UMKLAPP SCATTERING
_ -— BPILLOUIN ZONE BOUNDARY

FILLED REGION

Figure V—4: A schematic illustration of electron—phonon
scattering events in the second and third zone

portions of the Fermi surface of Al.

 

 

 

91

therefore necessary to relax the electron to equilibrium
(i.e. randomize its velocity). In the third Zone, on the
other hand, small angle scattering can drastically alter
the electron velocity and hence few such events are necessary
to relax an electron in the third zone.

Exact determination of 892 and S93 is a formidable
task. However, it is not difficult to see that they are
probably not the same. Figure (V-4) is a cross-sectional
view of the Fermi surface in the second and third zones in
which several possible electron-phonon scattering events
are indicated. We consider both the relative probability
for each scattering event, and its relative effectiveness
in contributing to phonon—drag (see Section III). Transitions
across the open portion of the second zone are effective,
due to the large transfer of momentum (i.e. 2,. (yiji) is
large), but these transitions are not probable at 4.2K.
At low temperatures only phonon modes of wavenumber about
q = (T/GD)°kf are pOpulated and, except in corners, a transi-
tion across the Open portion would require a phonon corres-
ponding to temperature near 0D. In the corners, transitions
are possible but here the Fermi surface area is small, so
that the weighting factor Oi in Eq. (III-33) is small, and
the contribution from the corners to S9 should be small.

These transitions, according to Ziman's rule (q passing

through an open region) contribute a positive thermopower.

 

92

Transitions in the second zone in which k remains on the

same spherical segment are ineffective since q ° (v — vi)
N ~ ~

f
is small. However, these transitions are possible at all
temperatures, and the large surface area available probably
makes them important. They provide a negative contribution

to the phonon drag. This leaves, on the second zone, only
Umklapp scattering. These scattering events are effective

and have a large Fermi surface available. But they require
phonons corresponding to temperatures well above 4.2K.

These transitions yield a negative phonon-drag. On the third
zone there are also three types of transitions. Transitions
on the same spherical segment act the same as in the second
zone. In total they are perhaps 30% as important as those

in the second zone, due to the smaller amount of Fermi surface
available. Transitions across a monster arm are effective

and have a considerable amount of Fermi surface available.

But they require a phonon corresponding to approximately 40K.
They contribute a negative phonon-drag component. The last
type of transition is across the Open regions of the third
zone. These would be effective, and have a large Fermi
surface area available. The significant portion of that area
is near corners so that low phonon wavenumbers can affect
electron transitions. These transitions contribute a positive
phonon drag. We therefore surmise that l) the second zone

produces a net negative contribution to the phonon drag, and

2) the third zone's contribution is determined by a delicate

 

 

 

93

balance between positive and negative contributions. Without
a detailed calculation, it cannot be determined which contri-
bution will dominate in the third zone. But it is unlikely

that S will be exactly equal to S We can therefore

93 92'
understand qualitatively the fact that the phonon drag thermo—
power is different for different impurities, since different
regions in the Fermi surface have different Sgi and through
gi the relative importance of these Sgi will be dependent
upon the type of impurity. When detailed calculations of
Sgi are performed, these experiments should yield useful
information regarding the anisotrOpy of the scattering from
impurities.

3. Lead
From the low temperature data of Christian et a1.
between 7.2K and 10.0K, the coefficients "a” and "b" are

8 ~10 V/K4,

found to be -O.98 x 10" V/K2 and —3.5 x 10
respectively. The data below 5K were obtained with the
specimen in an 800 gauss transverse magnetic field. (The
field was necessary to switch lead out of the superconducting
state; its transition temperature is 7.2K and its critical
field is 800 gauss.) The data at these lower temperatures

are not well described by the simple form of the thermopower

of Eq. (V—la). Inspection of Eq. (III-20)
S = Z ~— S. (III-20)

shows that the electron diffusion component of lead within

 

 

   

94

the temperature range of these experiments should in fact
not be linear in temperature.

We now rewrite Eq. (III-20) as

imp _ _
Th w (Simp STh) (V 3)

where "Th" indicates a thermal scattering mechanism,

"imp" indicates an impurity scattering mechanism, and

 

W = Wimp + WTh.
At low temperatures<l6>(Appendix I),
Wimp = A/T (V-4)
and
w — BT2 (V-S)
Th" '
so that Eq. (V-3) becomes
A/T
S . = s + (s. — S ). (V-6)
diff Th (A/T) + 8T2 imp Th
. . 2
At suffiCiently low temperatures, (A/T)>>BT and Sdiff Simp'

At higher temperatures (this temperature depends upon sample
2

hpurity through A) BT >>(A/T) and Sdiff = STh' The lead
specimen of Christian et al. was sufficiently pure so that
BT2>>(A/T) at 7.2K; consequently their data is well described
by Eq. (V-la). For our specimen, which was measured below
SK, the ratio W. /W was about 0.18 at 4.8K and 0.4 at

imp Th
3.3K. Thus the diffusion thermopower should not be linear
in temperature. (These ratios were obtained by measuring

the thermal conductivity of the specimen simultaneously with

the thermopower.)

 

95

If we draw a line tangent to the highest temperature
data in Figure V~2, we can obtain an approximation for the
phonon—drag coefficient "b", since the non-linear portion
of the electron-diffusion term is becoming small. The value
obtained is 4.8 x 10”10 V/K4, about 25% higher than that
obtained by Christian et a1. From the shape of the curve in
Figure V-2 and the expected temperature dependence from
Eq.(V—6), it appears that the slope will decrease at higher
temperatures and thus be in better agreement with the Christian
et a1. value. In any case the agreement should not be exact,
since our data were obtained in an 800 gauss magnetic field.
Unfortunately, the curves in Figure V-l4 cannot be reliably
extrapolated to zero magnetic field.

4. Indium:

The low temperature data for Indium (see Figure V-2),
like those for lead, are not well described by Eq. (V-la).
The reason is believed to be the same as for lead, since
the ratio Wimp/WTh z 0.6 at 4K. This ratio was obtained by
measuring the resistance ratio of the indium specimen and

using the Wiedemann-Franz law to determine W. The value

imp'

(53,54)

for W was taken from the literature. Since Eq.(V-la)

Th
does not apply to the data, we cannot make an unambiguous
separation of S into electron diffusion and phonon-drag

components. But if we assume that the electron diffusion

term is approximately linear in this temperature range, then

 

 

 

 

96

we can infer that the phonon-drag component is positive,
in agreement with the results of measurements at higher tempera—
tures (Figure V-l).

In View of the similarity between the Fermi surfaces of
indium and aluminum, it is perhaps surprising that the two
metals have phonon-drag thermOpowers which differ in Sign.
Because aluminum is face—centered-cubic, whereas indium is
face-centered-tetragonal with c/a ratio of 1.08, their Fermi
surfaces are not identical. The second zone in indium is
believed to be very similar to that in aluminum. The most
significant difference between the two Fermi surfaces is
found in the third zone, where the so-called "d—arms" are

(11) In

present in aluminum, but apparently not in indium.
aluminum we found that the second zone should contribute a
negative phonon-drag component, but that the sign of the
contribution from the third zone was unclear without detailed
calculations. It appears from comparison between the aluminum
and indium data that the third zone is important in determining
the sign of the phonon-drag thermopower, which is quite sensitive
to the exact shape of the Fermi surface and to the scattering
potentials.

Finally, we note that above the primary phonon-drag peak
at 0D/5, the thermopower of our indium specimen decreased
rapidly and even became slightly negative (Figure V—l). It

may be that we are seeing here the effect of a second phonon—

drag peak, having negative sign.

 

 

 

 

97

B. Magnetothermoelectric Power:

 

The effects of a transverse magnetic field on the thermo-
powers of pure aluminum, indium, and lead at 4.5K are shown
in Figure (V—S). For each metal, similar results were obtained
with specimens of comparable purity obtained from different
sources. It is clear that, although the temperature dependences
of the thermopowers of these metals are similar, Figure (V-l),
the magnetic field dependences are not. The thermopowers of
aluminum and indium become more positive in a magnetic field
and saturate at high fields. The thermopower of lead, on the
other hand, becomes more negative with field, and shows no
strong tendency toward saturation. The significant difference
in the electronic structures of these metals is that aluminum
and indium are trivalent, whereas lead is quadrivalent.
Apparently the electronic structure is important in determining
the general form of S(H).

In order to compare experimental results with the theory
of section IV, it is first necessary to separate the electron
diffusion and phonon drag components. We do this by assuming

that the thermopower has the form

3

s = a(H)T + b(H)T (V-7)

in the presence of a magnetic field. In section IV we showed
that the electron diffusion component retains its linear
dependence on temperature in a magnetic field. For the phonon

drag component there exists no reliable theory to guide us.

-20

-3O

-40

Figure V-S:

98

 

 

 

 

 

 

 

 

Pb
I J L l J
4 8 l2 IS 20

H (kilogauss)

The thermOpowers of A1, In,

field at 4.5K.

and Pb vs. magnetic

 

99

1. Aluminum:

Figures (V-6) through (V-l3) are plots of S/T versus
T2 for a series of magnetic field strengths for our purest
aluminum specimens and for several dilute aluminum alloys.
Indeed, the data are generally consistent with Eq. (V-7).
The coefficients a(H) and b(H) are listed for the different
specimens, along with their respective resistance ratios, in
Table V-l. In this table, subscripts distinguish specimens
having different concentrations of the same impurity.
Electron Diffusion:

According to Eq. (IV-27) and (IV-30), a(H) should become

more positive with increasing H, should saturate in high field,

and its total variation should be given by

9

a(H=w) - a(H=O) = 8.7 x 10‘ Volt/K2. (v—a)

Figure (V-l4) is a plot of a(H) versus wCT for several

representative specimens. wCT is defined here according to

R(300K)/R(4.2K) H
p(300K) nec

 

the free electron model (i.e. wCT = ),
and is not to be confused with the quantity (wCT)i defined
above.

From Figure (V-l4) we see that, indeed, a(H) becomes
more positive with increasing H and appears to saturate in
high field. In more cases this behavior leads to a magnet-
ically produced change of sign in a(H). (In addition to

the sample shown in Figure (V-l4), two other samples —-A1Cd2

and AlTl2 -- also did not change sign in high magnetic field.)

 

 

100

 

-L6
-l.8 -

 

 

 

 

1 1 JO 1 1 1, ll

 

Figure V—6:

l l I 21
O 2 4 ES 8 IO l2 l4 l6

IS 20 22 24
22
T(K)
The temperature dependences of the thermopowers

of All(O) and Ali(x) for a series of magnetic

fields. A straight line indicates a thermOpower

of the form S=a(H)T+b(H)T3. The magnetic fields

are denoted by: a)0, b)0.5, c)1.0, d)l.5, e) and

f)5.0, g)12.0 k—Gauss.

.. S/T X '08 (VOHS/Kz)

Nb'm'oo'o

 

1111
02468

101.

 

—

 

 

1111111111
IO 12 14 l6 l8 202224262830

T (K2)

 

 

Figure V-7:

of a thin

The temperature dependence of the thermopower

(thickness~

electronic meannfree-path) Al specimen,

Al

for a series of magnetic fields.

A straight line

2!
indicates a thermopower of the form S=a(H)T+b(H)T3. The

symbol 0 indicates two indistinguishable data points. The

magnetic fields are denoted by: a)0, b)1,0, c)3.0, d)5.0,

e)10.0 k-Gauss.

 

 

 

102

 

 

 

 

 

I 1 1 J 1 1 L J 1 1 L L 1
2 4 6 8 IO I2 I4 l6 I8 20 22 24 26
T2(K2)

Figure V-8: The temperature dependence of the thermopower

of specimen A1 for a series of magnetic fields. A straight

3

line indicates a thermopower of the form S=a(H)T+b(H)T3.

The magnetic fields are denoted by: a)0, b)0.5, c)1.5,

d)3.0, e)5.0, f)l0.0 k-Gauss.

 

 

 

 

 

103

 

 

 

 

 

12*-
IO
.8
1: -6
4‘
g .4
§ ,c
:o" .2.
9
x O—
[.—
U>\ -.2—d
-4—e
-.6—
f
-.8-
4.0—I I I I I I I I I I I I 11
0 2 4 6 8 IO l2 I416 |8202224262830
T2(K2)

Figure V-9:
specimen AlCul for a series of magnetic fields. A straight
3 The

line indicates a thermopower of the form S=a(H)T+b(H)T

The temperature dependence of the thermopower of

The

b)3.0, c)5.0,

I

 

symbol 0 indicates two indistinguishable data points.
d)10.0,

magnetic fields are denoted by: a)0

e)15.0, f)20.0 k—Gauss.

 

 

104

 

 

 

 

_f
"511111111111

 

 

O ,4 8 I2 I6 20 24 28 32 364044
‘ T2(K2)

Figure V-lO: The temperature dependence of the thermopower of
specimen AlCu2 for a series of magnetic fields. A straight
line indicates a thermopower of the form S=a(H)T+b(H)T3. The
symbol 0 indicates two indistinguishable data points. The
magnetic fields are denoted by: a)O, b)1.0, c)3.0, d)5.0,

e)10.0, f)20.0 k-Gauss.

 

 

105

 

   

_

 

 

.6
.4
.2
O

1 1 1 l '1 1 J J l L J
O 2 46 8Io I2 I4 16182022
T2(K2)

Figure V-ll: The temperature dependence of the thermopower of

 

specimen AlCd for a series of magnetic fields. A straight

1
line indicates a thermopower of the form S=a(H)T+b(H)T3. The
magnetic fields are denoted by: a)0, b)0.5, c)1.0, d)3.0,

k—Gauss.

106

 

 

   

-l.0 -

 

111111 1 1 [11 [Li
0 2 4 6 8 1012 l4 l6 |82022 242628

T2 (K2)

Figure V-12: The temperature dependence of the thermopower of

 

 

specimen AlTll for a series of magnetic fields. A straight
line indicates a thermopower of the form S=a(H)T+b(H)T3. The
symbol § indicates two indistinguishable data points. The

magnetic fields are denoted by: a)0, b)l.0, c)3.0, d)5.0,

e)10.0 k-Gauss.

 

V '

 

107

 

 

 

 

I I 1, J 1 L, 1 I 1 1 1 l 1 l
O 2 4 6 8 IO I2 I4 I6 I820222426 2830
T2(K2)

J,

 

Figure V-13: The temperature dependence of the thermopower of
the specimen AlSn for a series of magnetic fields. A straight
line indicates a thermopower of the form S=a(H)T+b(H)T3. The
symbol 9 indicates two indistinguishable data points. The
magnetic fields are denoted by: a)0, b)1.0, c)3.0, d)5.0,

e)l0.0, f)15.0 k-Gauss.

 

 

 

 

 

P)
-b

N
0

LG

 

F

108

 

ITT

ITII

T

 

 

 

 

.01 .1 I 10
we T

igure V—14: The electron diffusion coefficient a(H) vs. wcT

for representative specimens: a)A1Cul; b)A1Cu2; c)Al3;

, , = R(300)/R(4.2) H

1’ e)AlSn, f)AlCdl. Here wCT 0(300) nec

R(300)/R(4.2) is the measured resistance ratio of the

d)AlTl where

specimen and 0(300) is the resistivity of A1 at 300K.

 

109

From Table V-l we see that the total change in a(H)

8 Volt/K2. These values are

ranges from 2.1 to 2.6 x 10-

approximately two and one-half times greater than predicted.

In View of the approximations made in obtaining Eq. (IV—30)

(e.g. using only the first term of the Fourier expansion

in Eq. (IV-22); estimating rather than calculating 8A3/86;

and assuming that the logarithmic derivatives of the mean—

free-path exactly canceled) this agreement is heartening,

and suggests that the basic mechanism is correct. We believe

that the observed differences in the total variation of a(H)

are real, and arise from anisotropy in the scattering of

electrons, a factor which was neglected in obtaining Eq.(IV-30).
With these results and assumptions, two interpretations

of the experimental data are possible: 1) that the two terms

(1/AS )BASp/BE and ((1/22)822/86 - (l/Qsp)8£Sp/86) are compar-

P
able in magnitude, and that the term involving logarithmic
derivatives of the mean—free-path varies only moderately

from impurity to impurity; or 2) that the term (l/Asp)8ASp/86
dominates the thermOpower change, in which case the other
term may vary strongly from impurity to impurity. Our data
do not allow us to choose between these two possibilities.

A more precise calculation of the two terms is necessary.

As an aside, we briefly consider the theoretical expectation

for the magnetothermoelectric power measured under isothermal

 

conditions. For these conditions

S(H) = SXX' (V-9)

 

 

 

110

and in the high field limit (see Eq. (IV-25a)

WZKZT BQDAZ 82n22 82nwcT
+ + —————— )1 (IV-25a)
8f

 

 

SXX(H+m) + 3e ( 86 86 86

Noting that £=VT, and assuming we to be independent of energy
(this is valid in the free electron model and is consistent
with the approximations invoked in section IV) Eq. (IV—25a)

reduces to

HZKZT ( 82nA2 8£nV

3e + )1 (V—lO)

8f

 

 

Sxx(H+m) + 86 86

This result is independent of the type of impurity present,
(so long as the impurity concentration is sufficiently small
so that the band structure of the specimen is not altered).
Thus the magnetothermoelectric power measured under isothermal
conditions should become more positive with magnetic field,
and should saturate at a constant value independent of the
nature of the impurities present. This is just the behavior

(1'2) We see

observed in measurements of the Hall effect.
that the isothermal magnetothermopower would be expected to
behave like the Hall coefficient, but that the adiabatic
magnetothermopower does not.
Phonon Drag:
It is clear from Figures (V-6) through (V-13), or from
Table V—l, that the phonon drag component of the thermopower

(i.e. the slopes of the lines in Figures (V-6) through (V-13))

varies with magnetic field strength. For all the specimens

 

 

 

 

 

111

except AlCu the phonon drag component initially increases

2,
substantially with magnetic field. At high fields the data
suggest the existence of a phonon drag "peak" in the magnetic
field dependence for specimens AlSn, AlCu2, A13 and AlTlZ.
Because of the difficulty in obtaining reliable high—field
phonon drag data (at 4K the phonon drag component is only
10% to 20% of the total thermopower, thus small systematic
errors which would not much affect the value of the electron
diffusion component may significantly alter the value of the
phonon drag component) we examine the possibility that this
peak is associated not with the thermopower of aluminum,
but with some breakdown in the measuring technique at high
field. The difficulty might arise for three reasons:
1) carbon resistors show a noticeable magnetoresistance
above 10 Kgauss.
2) probe effects (all spurious voltages not accounted
for by reversing the magnetic field) may be serious.
3) the superconductivity reference leads may acquire a
nonzero thermopower, either by becoming normal over
a small region or due to flux flow motion.
We treat these problems in turn.
The magnetoresistance in the carbon resistors was measured
at 4.2K and found to be small; at 20.0 K—gauss the change in
resistance from the zero field value corresponded to approxi-

mately 0.4% of the absolute temperature (i.e. 15 millidegrees

 

112

higher than the actual temperature). Since both the hot
and cold resistors show nearly the same magnetoresistance,
the percentage errors in AT should be comparable to 0.4%.
This is much too small an error to explain the "peak".
Confirmation that the error was small was obtained from
measurements which showed the Lorentz number to be constant
up to 20 k—Gauss. (see Appendix I) This is as expected
from theory, (Eq. (IV—16)), and indicates that the measure-
ments of T and AT were indeed correct to within a few percent.
Probe effects often cause spurious results in measure-
ments of magnetoresistance, and may be the cause of spurious
results in measurements of thermOpower in a magnetic field.
However, since the phonon—drag components all show a decrease
in magnitude at high fields, we do not think that probe effects
are responsible for the observed peaks. We would imagine
that probe effects would have been as likely to increase
as to decrease the phonon-drag component, depending upon which
junction was most in error. We should therefore have seen
some increases as well as decreases.
Finally, we tested whether the reference leads acquired
a nonzero thermOpower by constructing a thermocouple consisting
of NbZr and NbTi spotwelded to a pure aluminum intermediary
held at a uniform temperature. The aluminum intermediary was
used to simulate the experimental conditions involved in the
measurement of the thermopower of aluminum. Upon raising

the temperature of the aluminum piece in a field of 20 k—Gauss,

 

 

 

 

113

Ive found no voltage output (to within our measuring uncertainty
()f 2.3 x 10_lOV) until one of the superconductors turned
normal, at which point the superconducting chopper-amplifier
stopped chopping. So long as the system was chopping, no
signal appeared. We conclude that any effects of alloying

at the superconductor-aluminum junctions or of flux flow

were small; too small to explain the "peak" in the phonon-drag
thermopower with increasing magnetic field.

At present we have no explanation for these peaks.

2. Indium and Lead:

 

The data for indium and lead are shown in Figures (V-15)
and (V-l6) respectively. Since we cannot separate the electron
diffusion and phonon-drag components from each other (see
sections V-2 and V—3), it is difficult to compare the magneto-
thermoelectric powers of indium and lead with theory. We
therefore make only some general comments.

The data for indium are interesting in that at high
fields the two specimens had the same thermOpower to within
2% (the data for the less pure specimen has been omitted for
clarity), but at low fields their thermopowers were consider-
ably different. This behavior is quite different from that
of the thermopower of aluminum, which for different samples
shifted by approximately equal amounts at high and low fields.
From the fact that the zero field thermopower of indium does

3

not deviate much from the simple form S = AT + BT , we

conjecture that the phonon—drag term increases substantially

 

- S/T X I07(V01ts/K2)

114

 

 

 

 

1.0
.91- A
(9
.8—
.TL— 4’
.6—
o
A o
.5— '
’ a
.4— a
DC)
3*- . Do ‘ A ‘0
A 0°
2— o o‘°
05‘
1*- A
o
o
C)“. O
-|_.
[A 1 L 1 1 l l L 1 l L l l
0 2 4 6 8 1012 I416 18202224262830
T2(K2)

Figure V-lS: The temperature depende
Pb for a series of magnetic fields.

a thermopower of the form S=a(H)T+b(

nce of the thermopower of

A straight line indicates

H)T3. The magnetic fields

are denoted by: o = 0.8,.A = 1.5, C]: 5, 0 = 10, A.= 20 k-Gauss.

 

 

 

115

 

 

 

 

‘ A 306
I D 5006
' V I.OOkG
0 + 5.00kG-
0 I0.0kG
x |5.0kG
IOO-
90- x o,
80— 05+ *.
70— :°+ 3 '
1:? 60'- 030°.
m\¥ 2o. 0
E 50'- 2 o
v40— 00“
GE) 0 V V w V V
_ 30— W V
.. v v v
a): 20 as“, 'D'L'u U “3‘“
IO— Vs D”
u . I All.A
0" .- A A
- o— A A ‘ ‘ A
I L “A.
-20
-30—
l I I l l l l J l l l l J
O 2 4 6 8 IO l2 I4 l6 I8 20 22 24 26
T2(K2)
Figure V-l6: The temperature dependence of the thermopower of

two In specimens for a series of magnetic fields. A straight
line indicates a thermopower of the form S=a(H)T+b(H)T3.

Symbols: AI V 0 indicate the less pure sample.

116

with increasing magnetic field, and dominates the high field
thermopower.

Unlike aluminum and indium, lead is a compensated metal;
it has equal numbers of electrons and holes. Azbel, Kaganoff,

and Lifshitz<55)

calculated that for a compensated metal

the “xx component of the Thomson coefficient tensor should
vary linearly with magnetic field in the high field limit.
Since the Righi-Leduc effect vanishes in a compensated metal
in the high field limit, we can calculate S using only uxx'
Using Eq. (II—5) to relate the thermOpower to the Thomson
coefficient, we find that the thermopower should also be
linear in magnetic field in the high field limit. Our results
are in reasonable agreement with this prediction. However,
we do not wish to over-emphasize this agreement, both because
we have not gone to extremely high fields, and because the
theory applies to the electron-diffusion component of the
thermopower alone. Since our data include the phonon-drag

component, a strict comparison with theory may be misleading.

3. Conclusions:

 

We have investigated the effect of a magnetic field
upon the thermOpower of aluminum and aluminum alloys. We
find that the electron-diffusion component of the thermopower
becomes more positive with increasing magnetic field, and
saturates at high field. The change in this component from
zero field to high field is nearly the same for all samples

studied. We have shown that these results can be understood

 

 

 

 

ll7

semi-quantitatively in terms of a simple "two-band" model.

A comparison of data taken on In, Pb, and Al alloys with

data for pure Al suggeSts that the basic mechanism in the
field dependence of the thermOpower is associated with the
band structure of the metal, rather than with the details

of electron scattering. It would therefore be interesting

to study the effect in oriented single crystals of aluminum
and dilute aluminum alloys. Additional impetus for such a
study is given by data shown in Figure (V—l7), which contains
a plot of the crossing field, (i.e. the field at which the
thermopower is zero), versus rotation angle for sample All.
Here the sample is being rotated around a vertical axis,

with the magnetic field in the horizontal plane. Sample All

is not a single crystal, but a rolled and annealed foil.

As can be seen in Figure (V—l7), the crossing field varies

as much as 20% during the rotation, and has four—fold symmetry.
This data demonstrates that the sign change is not a size
effect —— for which we would expect two-fold symmetry.

However, it also demonstrates that the sample is not a collection
of randomly distributed crystals —- for which we would expect
no angular variation. Rather, we suSpect that preferential
alignment of crystallites is responsible for the four-fold
symmetry observed. The fact that there is an angular variation
suggests that studies of oriented single crystals will be

fruitfhl.

 

 

 

 

 

.UmuoQOSm mH mmuHHHmumxno mo ucmficmaam Hafiucoummmnm Ucm

umqflaamumwuowaom mH HH< .ucmflpmum musumuomEmu wnu Ou mmuo>mcmuu mchEmH mkdBHm us umcu om

.mcmHm HmucoNHuoc ms» CH pmumuou mCHmQ ma paoflm oflumcmmfi one .wamcm wo :ofluocsM m mm

Haa cmeflommm no“ .Aoulmcm goflgz pm efimflw or» .@.Hv om .wamflm mcflmmouo was "5H-> musmflm

33.39 m

own oom ooN oo. o
_. _ _ 00m

 

 

.118

1 000.

00:

 

 

 

(ssnom H

 

 

 

119

In addition to the information concerning the electron-
diffusion thermopower in aluminum in the presence of a magnetic
field we have also obtained information concerning the phonon-
drag thermopower in the presence of a magnetic field and
both the electron-diffusion and phonon-drag thermOpowers
in the absence of a magnetic field. In the presence of a
magnetic field, the phonon—drag component first becomes more
positive, and then appears to "peak" and become less positive
with increasing field. Neither phenomena is understood.

Our data indicate that Kohler‘s theory (Eq. III-24) is approxi-
mately valid for the electron-diffusion thermopower in dilute
aluminum alloys containing Cu, Th, and Cd. Our data also

show that the magnitude of the phonon-drag components of

these alloys is senSitive to the type of impurity present.

 

 

 

 

 

 

 

REFERENCES

 

10.

ll.

12.

13.

14.

15.

16.

17.

18.

 

 

References

R. Lueck, Physica Stat. 801. $9, 49 (1966).

J. N. Cooper, P. Cotti, and F. B. Rasmussen, Phys. Lett.
99, 560 (1965).

C. H. Stephan and B. W. Maxfield, Sol. St. Comm. 1,

1039 (1969).

T. Amundsen, Phil. Mag. 99, 687 (1969).

W. A. Harrison, Phys. Rev. llg, 1182 (1960).

E. S. Borovik, Zhur. Eksptl. i Teoret Fiz. 99, 83 (1952).

 

E. H. Sondheimer and A. H. Wilson, Proc. Roy. Soc. (London)
A999, 435 (1947).

E. H. Sondheimer, Proc. Roy. Soc. (London) A999, 484 (1948).
N. W. Ashcroft, Phys. Kond. Mat. 9, 45 (1969).

J. Feder and J. Lothe, Phil. Mag. £9, 785 (1966).

W. Van der Mark, H. R. Ott, F. B. Rasmussen, and D. Sargent,
Phys. Kond. Mat. 9, 63 (1969).

I. M. Templeton, J. Sci. Instr. 99, 314 (1955).

A. R. DeVroomen, C. Van Baarle, Physica 99, 785 (1957).

J. Clarke, Physics Today 93, 30 (1971).

G. J. Edwards, J. Sci. Instr. 4, 299 (1971).

J. M. Ziman, Electrons and Phonons. (Oxford University

 

Press, London, 1960).
F. J. Blatt. Private communication.
See e.g. A. V. Gold, D. K. C. MacDonald, W. B. Pearson,

and I. M. Templeton, Phil. Mag. 9, 765 (1960).

120

 

19.

20.
21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

121

A. R. DeVroomen, C. Van Baarle, and A. J. Cuelenaere,
Physica 99, 19 (1960).

G. Boato and J. Vig, Sol. St. Comm. 9, 649 (1967).

J. S. Dugdale and M. Bailyn, Phys. Rev. 999, 485 (1967).
I. Holwech and V. Sollien, Phys. Stat. Sol. 99, 403
(1969).

H. B. G. Casimir and A. Rademakers, Physica 99, 33 (1947).
F. J. Blatt, Physics of Electronic Conduction in Solids.
(McGraw-Hill, U.S.A., 1968).

I. M. Templeton, J. Sci. Instr. 99, 172 (1955).

R. H. Freeman and J. Bass, Rev. Sci. Instr. 99, 1171 (1970).
C. L. Foiles, Rev. Sci. Instr. _9, 731 (1967).

G. K. White, Experimental Technigues in Low Temperature
Physics. (Oxford University Press, London, 1968).

L. J. Neuringer and Y. Shapira, Rev. Sci. Instr. 99,

1314 (1969).

W. R. Smythe, Static and Dynamic Electricity. (McGraw-Hill,
New York, 1967).

A. C. Anderson, Rev. Sci. Instr. 99, 605 (1968).

J. E. Robinson, Phys. Rev. 999, 533 (1967).

R. Risnes and V. Sollien, Phil. Mag. 99, 895 (1969).

D. K. C. MacDonald, Thermoelectricity: an introduction to

the Principles. (Wiley, Inc., New York, 1962).

 

G. N. Kamm and H. V. Bohm, Phys. Rev. 131, 111 (1963).

M. Kohler, Z. Phys. 26, 481 (1949).

 

37.

38.

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

49.

50.

51.

52.

53.

54.

55.

122

L. Nordheim and C. J. Gorter, Physica 9, 383 (1935).

L. Gurevich, J. Phys. (Moscow) 9, 477 (1945).

I. I. Hanna and E. H. Sondheimer, Proc. Roy. Soc. (London)
9999, 247 (1957).

J. M. Ziman, Advan. Phys. 99, l (1961).

M. Bailyn, Phys. Rev. 999, 480 (1967).

E. Fawcett, Adv. Phys. 99, 139 (1964).

M. Kohler, Ann. Phys. 99, 601 (1941).

R. G. Chambers, Proc. Roy. Soc. (London) 9999, 344 (1956).

A. B. Pippard, Proc. Roy. Soc. (London) A282, 467 (1964).

W. A. Harrison, Solid State Theory. (Mc-Graw-Hill, U.S.A.,

 

1970).

D. K. C. MacDonald and W. B. Pearson, Proc. Roy. Soc.
(London) 9999, 257 (1957).

K. Sugihara, J. Phys. Soc. Japan 91, 356 (1969).

J. W. Christian, J. P. Jan, W. B. Pearson, and I. M.
Templeton, Proc. Roy. Soc. (London) 9999, 213 (1957).
R. S. Gripshover, J. B. Van Zytveld, and J. Bass, Phys.
Rev. 999, 598 (1967).

R. S. Averback and D. Greig (unpublished).

F. J. Blatt and R. H. Kropschott, Phys. Rev. 999, 480 (1960).
R. E. Jones and A. M. Toxen, Phys. Rev. 999, 1167 (1960).
C. H. Stephan, Thesis (unpublished), (1971), Cornell
University.

M. I. Azbel, M. I. Kaganov, and I. M. Lifshitz, Soviet

Phys. — JETP g, 967 (1957).

 

 

 

 

 

APPENDIX

 

 

Appendix I. Thermal Conductivity of Aluminum

The thermal resistivity of metals at low temperatures

is expected to have the form(l6)

w = A/T + BT2 . (A-l)
The first term on the right represents the contribution to
the thermal resistivity of impurity scattering, whereas the
other term is the contribution of phonon scattering. Plotting
WT versus T3 should yield a straight line with "A" given by
the intercept and "B" by the slope. Figure (A-l) is such

a plot for the purest aluminum specimen, A1 The coefficient

3.
"A" is a measure of impurity scattering ahd thus varies with
the impurity content. "B", on the other hand, is a measure
of phonon scattering and therefore should be insensitive

to impurities. We obtain B = 3.3 x 10-5 cm/K-Watt which is

5

in good agreement with the value, 3.6 x 10- cm/K-Watt found

in the literature.(19)
Although both the electrical and thermal resistance

resistivities of metals vary in a magnetic field, the Lorentz

number should not vary with field when the resistivity is

dominated by elastic scattering, Eq. (IV-16). Table A-1
lists the measured Lorentz number, L = %%T’ (Q is the heat

transported by the specimen and R is the resistanCe of the
specimen), of AlCu, for various magnetic field strengths.

It is observed that indeed L is independent of magnetic field.

123

124

 

 

 

 

 

2 5 I I I J I I I I I I I I
O 20 4O 60 8O IOO I20
T3IK3)
Figure A-l: The temperature dependence of the thermal resistivity
of specimen 913. A straight line indicates a thermal resistivity

of the form W = A/T + BT2.

 

125

Table A—1. The Lorentz number (in 10_8 watt-ohms/Kz) of

AlCul for various magnetic fields at 4.5K.

Field (kG) 0 l 3.0 5.0 10.0 15.0 20.0

Lorentz Number 2.43 2.42 2.42 2.43 2.48 2.43 2.43

as...

>.. Zr

II.

ERSITY LIBRARIES
3‘2 504

UNIV
W ‘I
‘I
U“
030

STATE
‘ I

L

. IL.

HAN
lII

H

l

HII

\sc.
.3... “Six- MM

2..» \ .MxrvnVikk

Y 3~1;~
1.11)....

. e
1 1.13.4.1
1

 

;...A
‘ 1 .
1 1...}.