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dissertation entitled

The Efiecznicaz Rebiazivity and the Thenmoe£ectnic Ratio
06 PotaAAium, Sodium, Lithium, Rubidium, and PotaAAium—
Rubidium AZZoyA 610m 0.07K to 4.2K

presented by

Zhao-Zhi Va

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THE ELECTRICAL RESISTIVITY AND THE THERMOELECTRIC RATIO
OF POTASSIUM, SODIUM, LITHIUM, RUBIDIUM, AND POTASSIUM-
RUBIDIUM ALLOYS FROM 0.07K TO 4.2K

BY

Zhao-Zhi Yu

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1984

© 1985

ZHAO-ZHI YU

All Rights Reserved

THE ELECTRICAL RESISTIVITY AND THE THERMOELECTRIC RATIO
OF POTASSIUM, SODIUM, LITHIUM, RUBIDIUM, AND POTASSIUM-
RUBIDIUM ALLOYS FROM 0.07K TO 4.2K

BY

Zhao-Zhi Yu

AN ABSTRACT OF A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1984

ABSTRACT
THE ELECTRICAL RESISTIVITY AND THE THERMOELECTRIC RATIO

OF POTASSIUM, SODIUM, LITHIUM, RUBIDIUM, AND POTASSIUM-
RUBIDIUM ALLOYS FROM 0.07K TO 4.2K

BY
Zhao-Zhi Yu

The electrical resistivities (P) of thick (diam. =
1.0 - 3.0 mm) samples of K, Na, Li, and Rb; of thin (diam. a
0.09 - 0.5 mm) samples of K and Na; of samples of K and Na
encased in polyethylene and teflon tubes; and of samples of
K-Rb alloys were measured from 4.2K down to 0.07K with a
precision of 0JL4L01 ppm. The thermoelectric ratios (G) of
the same samples were measured over the same temperature
range.

In free-hanging, bare, high-purity thick samples, we
found a T2 variation of the electrical resistivity for K,
from about 1.1K down to about 0.3K; for Li, from 4.2K down
to about 1.6K; and, perhaps, for Na from 1.9K down to 1.2K.
No temperature range over which a T2 variation was dominant
was found for thick Rb samples. Clear deviations from a T2
variation for K, Na, and Li below the temperature ranges
just listed were also found.

In thin samples of K cooled in He gas, we found a size

effect pattern in P leading to a negative dP/dT when the

Zhao-Zhi Yu
diameter became smaller than electron mean-fee-path. The
only model we know which might explain the negative dP/dT we
observed is an interaction between surface scattering and
normal electron-electron scattering first prOposed by Gurzhi
and then calculated by a Monte Carlo method by Black.

A Kondo-type anomaly was observed in the electrical
resistivity below 1K for K samples encased in polyethylene
tubes. The data can be well fitted by p= 90 + AT2 - BlnT,
where -BlnT is a typical Kondo effect term.

The resistivity of 9.4 at.% Rb alloy samples evidenced
a strong deviation from the expected T2 behavior for TglK.
The reasons for this deviation are not yet clear.

‘All G data of thick samples of pure K, Na, Li, and Rb
and K-Rb dilute alloy could be well fitted by c a G0 + 13*1'2
+ (C*/T)Exp(-6*/T), where the third term is negligible for
Na and Li. Both the normal and Umklapp phonon drag terms
were quenched by increased surface and impurity scattering.
G data of K samples encased in polyethylene tubes were well .
fitted by G = G0 + B*T2 + D*/T, (D*<0) where the last term

is attributed to the Kondo effect.

TO MY WI FE

ii

ACKNOWLEDGEMENTS

It is a great pleasure to thank my thesis advisor,
Professor Jack Bass, for his supervision, support, criti-
cism, and invaluable aid in all aspects of this research. I
would also like to thank Professors Pratt and Schroeder for
their precious advice, discussions, and help at many stages
of this study.

Thanks also are due to Professor Spence for his help in
testing the single-crystal potassium, and to Professor J. C.
Garland of Ohio State University for providing high-purity
sodium.

I wold like to extend thanks to Dr. Mark Haerle and Dr.
Vernon Heinen for their unselfish aid in various stages of
this research, to Mr. John Zwart for his help in design and
construction of the sample can, and to Zhao Jing and Qian
Yao-Jin for their help in taking some data. I wish also to
thank the good humored fellows in the machine shop for
their help in constructing the sample can, the presses and
other apparatus.

Finally, the financial support of the National Science

Foundation is gratefully acknowledged.

iii

LIST OF

LIST OF

CHAPTER

I.

II.

TABLE OF CONTENTS

TABLES O O O O O O O O O O O O O O O O O O

FIGURES O O O I O O O O O O O O O O O O 0

INTRODUCTION 0 O O O O O O O O O O O O O O

1.1 Alkali Metals and Basic Electronic
Transport PrOperties . . . . . . .

1.2 PreVious work O O O O O O O O O O I I
1.2.1 Previous Work on Resistivities
1.2.2 Previous Work on ThermOpower S

and Thermoelectronic Ratio G

1.3 Present Thesis . . . . . . . . . . .
1.3.1 Electrical Resistivity . . . .
1.3.2 Thermoelectric Ratio G and

Thermopower S . . . . . . .

THEORY O O O O O O O O O O O O O O O O O O

2.1 Standard Theory of Electrical Resistivity

for Alkali Metals . . . . . . . . .

2.1.1 Electron-Phonon Scattering . .
2.1.2 Electron-Electron Scattering .
2.1.3 The Residual Resistivity . . .

2.2 Beyond the Standard Model of Electrical

Resistivity for Alkali Metals . . .

2.2.1 Electron-Electron Scattering in the
Presence of Anisotropic Scat-

terers such as Dislocations

2.2.2 Inelastic Scattering of Electrons
Due to Low Energy Excitations

Associated with Dislocations
2.2.3 CDW Theory and Electron-Phason
Scattering . . . . . . . . .

iv

Page
vii

ix

13
l4
l4
17
18

19
19
24

25

25

29
32

III.

IV.

2.3

The Knudsen Flow Model . .
Size-Dependent DMR . . . .
The Gurzhi Theory . . . .
The Kondo Effect . . . . .
Inelastic Scattering of Electrons
by Impurities and Defects . .
Electron-Electron Interaction
Effect and Localization Effect .

o o e o o
N NNNNN
o o o o o

\D (DNO‘UIIh

N NNNNN

Thermoelectric Power S and Thermoelectric
Ratio G I O O O O I O O O O O O O I O 0

2.3.1 Thermoelectric Power S
2.3.2 Thermoelectric Ratio G . . . . . .

EXPERIMENTAL TECHNIQUES . . . . . . . . . . .

3.1

3.6

Introduction . . . . . . . . . . . . . .

3.1.1 The Main Equipment for Measurements.

3.1.2 Thermometers O O O O O O O O O O 0
3.1.3 The Main Equipment for Sample
Preparation . . . . . . . . . .
The Glove Boxes and Presses . . . . . . .
Sample Can . . . . . . . . . . . . . . .
Sample Preparation . . . . . . . . . . .
Measurement Method . . . . . . . . . . .
3.5.1 Resistivity . . . . . . . . . . .
3.5.2 Thermoelectric Ratio G and Thermo-
power S O O O O O O O O O O O O
Uncertainties . . . . . . . . . . . . . .
3.6.1 Uncertainties in G and S
Measurements . . . . . . . . . .

3.6.2 Uncertainties in P(4.2K), Po, an
dP/dT Measurements . . . . . . .

EXPERIMENTAL RESULTS AND ANALYSIS . . . . . .

4.1

Free Hanging, Bare, Thick (d=1.5-2.0 mm)
High-Purity K Samples 0 o o o o o o o 0

4.1.1 The Resistivity . . . . . . . .
4.1.2 The Thermoelectric Ratio G . . . .

V

51
51
54
55
55

56
57

59
S9
62
68
87
87
97
99

99
100

103
103

104
107

V.

4.2

4.3

4.4

4.5

4.6

4.7

Free Hanging, Bare, Thin (d=0.09-0.5 mm)

4.2.1 The
4.2.2 The
4.2.3 The

Resistivity . . . .
Thermoelectric Ratio G
Thermoelectric Power 8

Nominal Single-Crystal K Sample .

Resistivity . . . .
Thermoelectric Ratio G

High-Purity K Samples in Contact with

4.3.1 The
4.3.2 The

Plastics
4.4.1 The
4.4.2 The
K-Rb Alloy
4.5.1 The
4.5.2 The

Free Hanging, Bare, Thick, High-Purity Na,

or Oil 0 O O O O O O O

Resistivity . . .
Thermoelectric Ratio G

Samples . . . . . . .

Resistivity . . . .
Thermoelectric Ratio G

Li, and Rb Samples . . . . . .

4.6.1 The

Resistivity . . . .

4.6.2 The Thermoelectric Ratio G .

Free Hanging, Bare, Thin, High-Purity Na

Samples .
4.7.1 The
4.7.2 The
Na Samples
4.8.1 The
4.8.2 The

Resistivity . . . .
Thermoelectric Ratio G

Encased in Polyethylene Tubes

Resistivity . . . .
Thermoelectric Ratio G

SUMMARY AND CONCLUSIONS . . . . . . .

LIST OF REFERENCES . .

vi

Page

113
113
132
139
145
145
148
148

148
168

175
175
185
187
188
201
206

206
209

211
211
213
215

222

Table

1..

1

LIST OF TABLES

Previous work on potassium . . . . . . . . . . .

Characteristics of
samples . . . . .

Characteristics of
samples . . . . .

Characteristics of
samples . . . . .

Characteristics of
samples . . . . .

Characteristics of
or plastics . . .

Characteristics of
samples . . . . .

Characteristics of
ethylene tubes . .

bare, free-hanging, pure K

bare, free-hanging, pure Rb

bare, free-hanging, pure Li

bare, free-hanging, pure Na

K samples in contact with oil

bare, free-hanging, K-Rb

Na samples encased in poly-

The electrical resistivities of some alkali metals

Coefficient A and other parameters of thick K

samples . . . . .

L for isotrOpic
($3?. 38)) . . . .

scattering (after Black

Coefficient A' deduced from the fitting of Eqn.

(4'2*) 0 o o o e e

The coefficients from fits to the G data of thin
samples cooled in He gas . . . . . . . .

The coefficients from fits to the G data of thin
samples cooled in Ar gas . . . . . . . . . . .

L(T)/Lo values calculated from G*/G and from
Wiedemann-Franz Law for sample K-1/2H . . . . .

vii

Page

10

70

73

73

74

81

85

86
89

106

123

125

135

138

146

4-10'

Page

The coefficient B from fits to the P data of K
samples encased in polyethylene tubes . . . . . . 165

The coefficients from fits to the G data of K
samples encased in polyethylene tubes . . . . . . 174

The coefficients from the local phonon mode fits
to P data of Li samples . . . . . . . . . . . . . 200

The coefficients from the bound electron level
fits to 0 data of Li samples . . . . . . . . . . . 200

viii

\IO‘U'IuwaH

LIST OF FIGURES

Page
Normal and Umklapp process of e-ph scattering
(A) Phonon annihilation . . . . . . . . . . . . . 21
(B) Phonon creation 0 o o o o o o o o o o o o o o 21
The effect of a CDW perturbation on the Fermi
surface. The sphere is distored to a lemon
shape (AFter Overhauser (ref. 1) . . . . . . . . . 34

The Umklapp scattering for a distored Fermi
surface (After Overhauser (ref. 1) . . . . . . . . 34

Plot of resistivity vs T for the data of sample

K2C of Rowlands et a1. (ref. 16). The fitting

curve with J4 is shown. In the inset, extrapola-
tions of Tl's, J2, J , J5 fits to lower tempera-

tures are shown. (After BishOp and Overhauser

(ref. 17)) O O O O O O O O O O O O O O O I O I O O 37
Two kinds of stainless steel presses . . . . . . . 61
sample hOIder 1 O I O O O O O I O O O O O O O O O 63
Cover of the sample holder . . . . . . . . . . . . 64
Three different ways of mounting samples . . . . . 65
The way of sucking potassium into a tube . . . . . 75

The system for purifying paraffin oil . . . . . . 77

The system for pulling single crystal potassium
sample 0 O O O O O O O O O O O O O O O O O I O O O 7 9

The low temperature circuit. The components
inside the broken line are inside the sample can . 90

(P(4.2K)/PT) (AP/AT) vs T for free hanging, bare,
high-purity, thick K samples . . . . . . . . . . . 105

A vs PO for 5 thick K samples cooled in He . . . . 108

G vs T for free hanging, bare, high-purity, thick
K samples 0 O O O O O O O O O O O O O O O O I O O 109

ix

4-10

4-11

4-12

4-13
4-14
4-15
4-16

4-17

4-18
4-19

4-20

P(4.2K) (AlnP/PT) vs T for thin wires of K cooled
in a He atmosphere . . . . . . . . . . . . . . .

P(4.2K) (AlnP/AT) vs T for two K samples of
Rowlands et a1. (ref. 13) with d=0.08 mm cooled
in He atmosphere. For comparison, the solid
lines represent data from Fig. 4-4 for K samples
having the diameters indicated . . . . . . . . .

P(4.2K) (AlnP/AT) vs T for two K-0.08 at.% Rb
alloy samples with d=0.25 mm . . . . . . . . . .

P(4.2K) (AlnP/AT) vs T for samples indicated in
Fig. 4-4 With fitting Curves Of Eqno 402 o o o o

P(4.2K) (AlnP/AT) vs T for thin wires of K cooled
in an Ar atmosphere or a partial vacuum. For
comparison, the solid lines represent data from
Fig. 4-4 for K samples having the diameters
indicated . . . . . . . . . . . . . . . . . . .

90 vs 1/d for thin K samples . . . . . . . . . .

G vs T for thin wires of K cooled in a
atmosphere 0 O O O O O O O O O O O O O O O O O O

G vs T for thin wires of K cooled in a
atmosphere . . . . . . . . . . . . . . . . . . .

cooled in a partial

G vs T for thin wires of K
vacuum 0 O O O O O I O O O

S vs T for sample K-1/2H . . . . . . . . . . . .
G and G* (=S/LOT) vs T for sample K-l/ZH . . . .
L(T)/Lo vs T for sample K-l/ZH . . . . . . . . .

(P(4.2K)/P) (APAAT) vs T for the nominal single
crystal K sample . . . . . . . . . . . . . . . .

(P(4.2K)/PT) (AP/AT) vs T for the nominal single
crystal K sample C O O O O O O O O O O O O O O O

G vs T for the nominal single crystal K sample .

(P(4.2K)/T) (AlnP/AT) vs T for sample K-S and
five runs of sample K-PHl . . . . . . . . . . .

A version of Fig. 4-19 to a larger scale . . . .

Page

115

117

119

126

127
131

133

136

140
142
143
144

147

149
150

153
154

Figure

4-21

4-22
4-23

4-24
4-25

4-26
4-27

4-28
4-29
4-30

4-31
4-32
4-33
4-34

4-35
4-36

4-37
4-38

4-39
4-40

(P(4.2K)/T) (AlnP/AT) vs T for four runs of

sample K-PHZ. For comparison, the data of

samples 2a, 2b, and 2c of van Kempen et al. are
indicated 0 O O O O O O O O O O O O O I O O O O O
A version of Fig. 4-21 to a larger scale . . . . .

(9(4.2K)/T) (AlnP/AT) vs T for five runs of
sample K-PAl O O O O I O O O O O O O O O O O O O O

A version of Fig. 4-23 to a larger scale . . . . .

(P(4.2K)/T) (AlnP/AT) vs T for five runs of
sample K-PAZ o o o o o o o o o o o o o o o o o o o

A version of Fig. 4-25 to a larger scale . . . . .

(P(4.2K)/T) (AlnP/AT) vs T"2 for samples K-PHl
and K-PHZ o o o o o o o o o o o o o o o o o o o o

(P(4.2K)/T) (AlnP/AT) vs T"2 for sample K-PAl . .
(P(4.2K)/T) (AlnP/AT) vs T‘2 for sample K-PAZ . .

(P(4.2K)/T) (AlnP/AT) vs T for K samples in
contact with other plastics . . . . . . . . . . .

G VS T for sample K-PHl o o o o o o o o o o o o o
G vs T for sample K-PHZ . . . . . . . . . . . . .

G vs T for samples K-PHl, K-PHZ, K-PAl, and K-PA2
with fitting curves at temperature below 1K . . .

p0 VS C for K-Rb samples 0 o o o o o o o o o o o o

(P(4.2K)/T) (AlnP/AT) vs T for dilute K—Rb alloy
samples 0 O C O O O I O O I O O O O O O O O O O 0

Normalized AP/AT vs T for K-Rb alloy samples . . .

Three different trial fittings for (P(4.2K)/T)
(AlnP/AT) of two K-9.4 at.% Rb alloy samples . . .

C vs T for K-Rb samples 0 o o o o o o o o o o o o

(P /T) (AlnP/AT) vs T for free hanging, bare,
thick, high-purity Na samples . . . . . . . . . .

xi

Page

155
156

157
158

159
160

162
163
164

167
169
170
171

173
176

178
180

182
186

189

Figure Page

4-41 (P(4.2K)/T) (AlnP/AT) vs T for Rb, Li, and K
samples 0 O O O O I O C C C O C O O Q C C O O O O 191

4-42 P(4.2K) (AlnR/AT) vs T for Rb, Li, K, and Na

samples from 0.07K to 1.4K. The data are the

same as in Fig. 3-40 and Fig. 4-41. The dashed

lines indicate T resistivity variations

inferred from all of the available data . . . . . 194
4-43 Trial fittings for P(4.2K)A1nP/AT of Li samples . 196

4-44 Trial fittings for (9(4.2K)/T) (AlnP/AT) of Li
samples 0 O O O O O O O O O I O I '0 O I O O O O O 197

4-45 G vs T for thick Na samples . . . . . . . . . . . 202
4-46 G vs T for Li samples . . . . . . . . . . . . . . 204
4-47 G vs T for Rb samples . . . . . . . . . . . . . . 205
4-48 (Po/T) (AlnP/AT) vs T for thin Na samples . . . . 207

4-49 (ea/P) (AP/AT) vs T for selected thin Na
samples 0 O O O C C I O O O O I O O O O O O C O O 208

4-50 G VS T for thin Na samples 0 o o o o o o o o o o o 210

4-51 (t%/T) (AlnP/AT) vs T for Na samples encased
in polyethylene tubes . . . . . . . . . . . . . . 212

4-52 pbAlnP/AT vs T for Na samples encased in
polyethylene tubes . . . . . . . . . . . . . . . . 214

xii

CHAPTER 1

Introduction

This dissertation is a report of experimental studies
of electrical transport properties such as electrical
resistivity and thermo-electric ratio in the alkali metals
K, Li, Na, Rb, and in K—Rb alloys. One of the main purposes
of this study is to try to find the concealed reasons for
disagreements between experimental results from different
groups (see section 1.2.1). Particular attention is
concentrated on: 1) a size effect in pure K samples at
temperatures below 1K; 2) a contact effect of
polyethylene and oil on K at temperatures from above 1K down
to 0.1K; 3) anomalous deviations from the simple T2
variation expected for electron-electron scattering in
thick, pure, free hanging K, Li, Na, and Rb samples when the
temperature is lower than a certain limit for each; and
4) a deviation from the expected T2 behavior in the
resistivity of K-Rb alloys at temperatures below 1K.

In this introduction, we first explain why people are
interested in measurements on the alkali metals, especially
potassium, and then review previous work on K, Li, Na, Rb,

‘ and K-Rb and discuss some disagreements among work done

previously by different groups. In Chapter 2 we introduce

2
all related theories. In Chapter 3, the experimental
techniques of the present study are described. The
experimental results and interpretations are presented in

Chapter 4. Summary and conclusions are given in Chapter 5.

1.1 ;_1; V a, a P =‘ 3.8 _-0 RA _'0:. "0’ i .E.

The alkali metals have been attracting renewed

theoretical and experimental interest in the past few
decades. These metals are simple monovalent.metals which
have bec. lattice structures at room temperature, so that
they have nearly spherical Fermi surfaces (to within 0.1%
for K according to deA measurements). They have no
unfilled d- or f- shells to complicate calculations. No
superconducting phenomena and Kondo effects at low tempera-
tures have been previously seen in the alkali metals.
Potassium has been studied most often among the alkali
metals. The reasons are:

1) Unlike lithium and sodium, potassium does not
undergo a martensitic phase transformation at low
temperatures. It thus keeps its bxxc. structure
and spherical Fermi surface.

2) Potassium is less reactive than rubidium and cesium
and thus easier to work with.

3) Potassium is softer than lithium and sodium, and
can be easily extruded through small dies into thin

wires.

3
4) Overhauser has claimed that potassium is the best
material for testing the existence of charge
density waves (CDW) (ref. 1) in simple metals. If
found to exist in K, CDW theory would change our

basic understanding of electronic transport in

metals.

In all studies of electronic transport, the fundamental

bases are the microscopic transport equations,

3.. =’- Ell-E. + L12VT (1‘1)
2'5 = RAE + 322% (1-2)

where 3 is the electrical current density, 3 is the heat
flow current density, E is the electric field, and‘aT is the
temperature gradient. In general, all four 3 coefficients
are tensors, but in cubic metals, such as alkali metals

under zero magnetic field, the L's are scalars. Then we can

rewrite (l-l) and (1-2) as

E = p§+s‘v"r (1-1')
5 = 3p-S-E-K3T <1-2')

The electrical resistivity is defined as

a (1‘3)
' 'VT = 0

'b
n
'ullml

where E and 3 are in the same direction.
Considering electron-phonon, electron-electron, and

electron-defect scattering, the electrical resistivity at

4

temperatures well below the Debye temperature is expected to

have the form (ref. 2)

P(T) = Po + Pe_e +"§-ph +’Pg-ph (1-4)

Here!% is the residual resistivity caused by elastic scat-

tering of electrons by impurities or lattice defects in the
metal and is temperature independent in the first approxima-
tion; ‘E-e is attributed to electron-electron scattering and
is expected to have a temperature dependence of the form
ATZ, where A is basically independent of 05; p§_ph is due to
electron-phonon scattering normal processes and, according
to Bloch-Gruneisen theory, is expected to have the form
p§_ph = BTS; and the last term pg-ph' comes from electron-
phonon Umklapp scattering processes which, in a metal with a
spherical Fermi surface completely within the first

Brillioun zone, are expected to lead to the form (ref. 3,4)

U n -6*/T
pe_ph = CT e (1-5)

The thermal conductivity K is defined as

s
K = - ——- (1-6)
VT E-O.

We have the Wiedemann-Franz law (ref. 2)
K9 _ _
T — L(T) (l 7)

When elastic scattering of electrons is dominant,

n2 k2
0 3 e2

L(T) = L 2.45 x 10'3v2K'2 (1-8)

where L0 is the ideal Lorentz number.

The thermopower S is defined as

E

S = -—— a (1-9)
The thermoelectric ratio G is defined as
G = % a (1-10)
q B=0
In general,
, S
S = GLT 1.e., G = LT = xp (1-11)
Where L is usually temperature dependent.
When elastic scattering is dominant, L(T) = Lo and
S = GLOT (1—12)

In pure potassium this is true only when the temperature is
below 1K (according to our present data).
At low temperatures, when elastic scattering is

dominant, G is expected to have the form (ref. 5,6,7,8)

Be"e1"/T

- 2
G - G0 + AT + T

(1-13)

Where Go is attributed to electron diffusion, the second

term is due to normal phonon drag, and the last term is an

6
Umklapp phonon drag term appropriate to the alkali metals

with b.c.c. lattice structure.
1.2 EBEEIQH$_EQBK

1.2.1 WW

Earlier resistivity measurements on the alkali metals
have made the electron-phonon scattering contribution to the
electrical resistivities of the alkali metals, including the
contribution of normal and Umklapp processes, well
understood (ref. 9,10,11,12,13).

Since van Kempen et al. developed high-precision (1
ppm) measurements of the electrical resistivity at low
temperatures (1K and above) in 1976, (ref. 14) the electron-
electron scattering contribution to the electrical
resistivity of simple metals, and deviations from this
contribution under various conditions, have been a subject
of interest.

Four groups in the world have made high precision
measurements of the electrical resistivities of alkali
metals in the last few years. They are the Wyder, van
Kempen group in the Netherlands, the Greenfield group in
Israel, the Woods and Rowlands group in Canada, and our
group at Michigan State University (MSU). This previous

work is briefly described as follows:

7

1.2.1.1 W
W

In 1976 van Kempen et al. measured the resistivity of
high-purity potassium (purity 99.97%) samples each clad in a
l-m-long polyethylene tube with an inner diameter of 0.9 mm.
They reported that from 1.7SK down to 1.1K, their data could

be fit to the formula
p a po + ATS + BTn e-9*/T (1-14)

with n a l, 9* = 19.9 i 0.2K, and s = between 1 and 2.
Using 3 a 2, as expected for electron-electron scattering,
their coefficients, A varied from sample to sample by a
factor of as much as 3.6, in contradiction to the standard
theory. They also reported that the residual resistance
ratio [RRR = R(300K)/R(OK)] of the samples changed after
annealing the samples at room temperature. One of their
samples originally had RRR a 3000, then changed to 6300
after two days annealing at room temperature in a He
atmosphere, and finally changed to 8100 after 80 days
annealing at room temperature in vacuum. The coefficient A
decreased as the RRR increased (ref. 14).

Following on the heels of the results of van Kempen
et al., a theory of electron-electron scattering in the
presence of anisotropic scatterers such as dislocations was
prOposed by M. Kaveh and N. Wiser (ref. 15) to explain the
sample-dependence of A in P(T) = ATZ.

In 1978, Rowlands et al. (ref. 16) measured the

resistivity of bare, high-purity potassium samples, 1.8 m

8
long with diameter of 0.79 mm, which were wrapped into a
helical groove on a 3-cm-diam Teflon cylinder under a helium
atmosphere. They reported that the RRR of their samples
increased upon annealing at room temperature (e.g., from
1300 to 4800 in 3 weeks), and that the best fit of their
data from 0.5K to 1.3K was pm = 3133/2 with B = (86 1». 10) x
10'6 Po K'3/2. However, P(T)<x T2 could not be ruled out
because of the uncertainty in the measurements. They
proposed an explanation involving a size effect associated
with Knudsen flow of electrons. Alternatively, Bishop and
Overhauser (ref. 17) prOposed a mechanism of electron-phason
scattering (according to CDW theory), to explain the T3/2
behavior.

In 1979, Greenfield et a1. (ref. 18) measured high
purity potassium samples inside polyethylene tubes with
diameters of 1.0 mm. They reported that by heating and
cold working their samples, they sharply increased the
impurity content in their samples, and brought the RRR down
(e.g., from 14000 to 2000). They plotted their data as
(P-P’OVT2 versus T with an adjustable parameter pb, and
claimed that a term ATz'oio'l, attributed to electron-
electron scattering, was found from 1.4K down to 1.1K. They
also reported their coefficient A was sample-dependent, and
decreased as RRR decreased. They plotted A versus (Pa/Pb)2,
for a given sample, leaving Pd as a constant parameter, and

claimed their A<X(Pa/Pb)2, consistent with the theory of

9
Kaveh and Wiser mentioned above. Here Pd was an assumed
anisotropic resistivity due to dislocations.

In 1982, C. W. Lee et a1. (MSU group) (ref. 19) reported
0.1 ppm precision measurements of dP/dT for free-hanging,
bare, high-purity potassium samples, with diameters ranging
from 0.9 mm to 3.0 mm, down to 70 mK under Ar atmosphere.
They reported that dP/dT varied closely as T, (i.e., P(T) =
ATZ), from 1.3K to 0.35K. The RRR of their samples did not
change much upon annealing or contaminating when left at
room temperature, (e.g., the RRR of one of their samples
changed from 5100 to 4100 in 83 days, and another one
changed from 3600 to 3700 in 11 days). They found that the
coefficient A did not change much from sample to sample,
with a mean value of 0.24 i 0.02 chm/K2 consistent with
simple electron-electron scattering (ref. 20,21). They also
reported a deviation form T2 behavior below 0.35K.

The main disagreements among these various sets of
results and their different experimental conditions are
summarized in Table 1-1.

More recently, M. Haerle (ref. 22) reported that
deformation in K samples brought the coefficient A up,
enhanced the anomalous turn-up at low temperatures found by
Lee et a1. (ref. 19), and enhanced the electron-phonon
scattering term from 4.2K to about 1K. For the anomalous
turn-up at the low T end, he suggested an explanation based
on low energy excitations associated with dislocations (or

conceivably some other product of deformation) which would

10

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11
cause inelastic scattering of electrons. His deformed K

samples showed complete recovery upon annealing at tempera-

tures above 165K.

1.2.1.2 W
111.915.

In 1980, C. W. Lee et a1. (ref. 23) reported resistivity
measurements of bare, free-hanging K-Rb alloy samples with
nominal concentrations of 2.24, 0.83, 0.32, 0.13, and 0.05
At.% Rb from 180mK to 4K. The samples were about 4cm long
with d = 3.0 mm. They claimed that below 1K, dP/PdT =
(mo/(Do + 2Ai)T, (i.e., p= p0 + (A0 + Aipo)T2, with A1 =
(8.5 i 0.3) x 10"5/K2 and A0 = (2.2 i 0.31) x 10'13 52cm/K2.
The AC was consistent with the A they found in pure K, and
the Ai was comparable to the theoretical values of 13.7 x
10"5/x2 from P. L. Taylor (ref. 24) and 12.5 x 10"5/K2 from

Kus and D. W. Taylor (ref. 25).

1.2.1.3 W
In 1979, Greenfield et a1. (ref. 18) reported

resistivity measurements of high-purity Na samples clad in
polyethylene tubes with d = 1.0 mm. As they did for their K
samples, they plotted their Na data as (P-Po)/'r2 versus T
with an adjustable parameter Po, and claimed (KT) = AT2 was
found from 1.1K to 2.1K. The RRR of their samples increased
slightly with sample annealing time at room temperature; in
9 days Po changed from 0.916 n cm to 0.787 chm, while the
coefficient A changed form 0.195 to 0.180 chm/Kz. They

claimed that A was a linear function of (Pd/Po)2, in accord

12
with the theory of Kaveh and Wiser (ref. 15). The lowest
temperature they reached was 1.1K; the sensitivity they

achieved was one to few ppm.

1.2.1.4 Ere1i9us_work_on_resistixitx.9f_hi

In 1971, G. Krill (ref. 26) reported electrical
resistivity measurements of a high-purity Li sample with
RRR a 7000 and Po = 7.30 chm from 40K down to 1.3K. His
sample was 50 cm long with d = 0.5 mm. He found PCT) « T4
at 10K<T<40K, and P(T) = AT2 with A = 3.3 poem/K2 at
4.5K<T<10K. In 1981, Greenfield et a1. (ref. 27) reported
dR/dT measurements on a bare high-purity Li sample of 4.3 m
long with d = 3 mm and Po = 12.1 nncm. They got dP/dT =
2AT, (i.e., P(T) a ATZ), throughout the temperature range of
the experiment, 1.2 - 4.2 x, with A = 3.0 i 0.1 poem K‘Z
consistent with data from Krill, but an order of magnitude
larger than that predicted by MacDonald et a1. (ref. 20).
They proposed that this anomalously large value for A was
due to the anisotropy of the electron relaxation time for

Li, in accord with the theory of Kaveh and Wiser.

1.2.1.5 ErexiQus_nork_on_rssistixit¥_9f_hish:
Ruritx_Bb_at_low_tsmpsratures.

No high precision resistivity measurements on high-

purity Rb at low temperatures have been previously reported.

13

1.2.2WWW
I' El' 3

More than twenty years ago, MacDonald et a1. (ref.
5,6,7) measured the thermal e.m.f. E of pure K and reported
that their deduced thermopower S = dE/dT could be fitted,

from 3K down to 0.1K, to the formula
5 = A'T + B'T3 + c'e‘e’VT (1-15)

The first term, A'T, was attributed to electron diffusion
thermopower; B‘T3 was attributed to normal phonon drag
thermopower, and the exponential term was attributed to
electron-phonon Umklapp processes with 6* = 21K, where C'
could be T-dependent.

In 1980, C. W. Lee (ref. 8) reported G measurements on
pure K and K-Rb alloys from 4.2K down to 80 mK. Their
samples were the same as the ones for electrical resistivity
measurements described above. They claimed that for pure K,
G had the form 0 = Go + B*T2 + (c*/T)e‘9*/T, with co = -0.03
i 0.03 v’l, 3* = -o.30 i 0.01 v‘lx‘z, and 9* a 23.1 2K.
These results were consistent with the results of MacDonald
et al., (i.e., G0 was the diffusion term, B*T2 was the
normal phonon drag term, and the last one was the Umklapp
phonon drag term). For K—Rb alloys, they found roughly the
same three terms as above. The normal and Umklapp phonon
drag terms were quenched more and more as the impurity
concentration increased. They also found that the diffusion
term Go is positive and roughly obeys the Gorter-Nordheim

rule: co = oi + (pp/ponep - Gi) with oi = +0.48 i 0.01v'1,

14
where Pp = the residual resistivity of pure K, P0 = the
residual resistivity of K—Rb sample, Gp = the diffusion term
of G for pure K, and Gi = the diffusion term of G due to
impurity scattering.

In 1983, M. L. Haerle et a1. (ref. 22) reported G
measurements on deformed pure K samples from 4.2K to 0.08K.
They claimed that G = G0 + B*T2 + (C/T)*e'e*/T could fit
their data for T)z 0.2K, with 9* ~ 18 1 2K, Go<0, B*<0, and
C*>0. But below 0.2K, their data start turning down from
the fit, and can be fitted better by G = G0 + 8*T2 + A*T,

with Go<0, B*<0, and A*>0.

1.3 EBESENI_IHEELE

1.3.1 W

As noted in the introduction above, at low enough
temperatures, the electrical resistivity P(T) in metals is
predicted to vary as T2 due to electron-electron scattering.
It is predicted (ref. 21) that such behavior would be seen in
alkali metals such as K when the temperature is lower than
about l-2K. However, experimental results on potassium
below 1.5K by various groups showed discrepancies with this
simple theory, as well as disagreements with each other, as
described above in section 1.2.1. In order to find the
reasons for these discrepancies and disagreements, in this
thesis more than 100 measurements of dP/dT were made on
alkali metals such as K, Na, Li, Rb, and on K-Rb alloys,

from 4.2K to about 0.07K with 0.1 ppm - 0.01 ppm precision.

15

1. To study the deviation from T2 behavior found by
Lee et a1. below 0.35K for pure, thick K samples cooled in
Ar gas, as described above in section 1.2.1, we used im-
proved techniques to measure 14 free-hanging, bare, pure K
samples, with d = 1.5 and 2.0 mm, under Ar gas, He gas, or
partial vacuum, respectively. The improved techniques in-
cluded a better glove box, an improved sample can with an
external thermal radiation shield, and a computer averaging
technique for data taking.

2. To investigate the approximately T3/2 behavior
reported by Rowlands et al. for pure K samples of d = 0.79
mm cooled in the gas and measured from 1.4K to 0.5K, as
described in section 1.2.1, sets of pure samples with d
ranging from 1.5 down to 0.09 mm were cooled in He gas, Ar
gas, or partial vacuum and measured from 1.8K down to 0.07K.
An unusual size-effect was found, which lead to a negative
dP/dT in very thin samples. To test whether the electron
mean-free-path was an important parameter in this size-
effect, measurements of dR/dT were made on two K-0.08% Rb
samples with d a 0.25 mm, which had a much smaller electron
mean-free-path (A.z 0.04 mm) than the pure bulk K samples
(A.= 0.2 mm). In an attempt to get a longer electron mean-
free-path, so as to see whether the size-effect could be
observed in a thicker sample, we tried to make and measure a
single crystal K sample with d = 1.5 mm. To test for

effects of surface corrosion, two different thicknesses of

16
samples were allowed to thin by means of such corrosion,
which occurs naturally inside the sample can.

3. To investigate further the variations in magnitude
of the T2 coefficient reported by van Kempen et a1. and Levy
et al. for pure thick K samples clad in polyethylene tubes
and measured down to about 1K, and described in section
1.2.1., we measured 4 pure K samples clad in polyethylene
tubes with d a 1.6 or 0.9 mm down to 0.07K. To test for
annealing effects, after each measurement, each sample was
allowed to anneal for varying periods of time at room tem-
perature under Ar gas, He gas, or partial vacuum, respec-
tively. Samples were remeasured after being annealed for a
few days, a few weeks, and a few months. Anomalous behavior
in dP/dT which might be due to a Kondo effect was seen. To
test whether the contact with polyethylene is an important
condition for this anomalous behavior, we measured pure K
samples in contact with other materials such as Teflon, Kel-
F, and paraffin oil.

4. To investigate further the resistivity P of K-Rb
alloys, with improved techniques, ‘we made and measured K -
0.08 at.% Rb and K - 9.4 at.% Rb samples from 4.2K down to
about 0.07K.

5. Previous work on Li and Na at temperatures down to
about 1K by Greenfield group, as described in section 1.2.1,
showed T2 behavior in P(T) from 4.2K down to 1.1K for Li,
and from 2.1K down to 1.1K for Na. No previous work on p(T)

of Rb has been reported. In order to see whether there are,

17
as in K, anomalous deviations from T2 behavior of P at lower
temperatures in Li, Na, and Rb, we extended resistivity
measurements of all these three pure metals down to about
0.07K. We measured thin high-purity Na samples and high-
purity Na samples encased in polyethylene tubes to test
whether the size-effects and polyethylene effects seen in

pure K can also be seen in pure Na.

1.3.2 W
About twenty years ago, MacDonald et a1. (ref. 5,6,7)

measured the thermal e.m.f. of pure K from 3K down to 0.1K
and deduced the thermopower S from their data. More
recently, Lee et a1. made G measurements on pure K and K-Rb
alloys from 4.2K down to 0.07K, and M. Haerle et a1. made G
measurement on deformed pure K samples from 4.2K down to
0.07K, as described in section 1.2.2. We measured G for
most of the samples mentioned above concurrently with dP/dT;
(i.e., we extended G measurements to thin samples of K, Na,
and K-Rb, thick K samples in contact with plastic and oil,
and thick samples of Li, Rb, and Na from 4.2K down to
0.07K). In order to find the low temperature limit of L(T)
a Lo (equation 1-8) and thus 8 = GLOT (equation 1-12), and
to find the form of L(T) in K, the thermOpower S was
concurrently measured with G in one pure K sample with d =

0.5 mm.

CHAPTER 2

Theory

In this chapter the theory of electronic transport,
especially for alkali metals, is reviewed.

First, the basic "standard model" of the electrical
resistivity for alkali metals is described.

Second, since many measurements on many different
topics have been done, and various deviations from
the standard model have been seen, it is necessary to review
quite a few models which might be candidates for explaining
our data. Some of these models were proposed to explain
previous work on the resistivities of alkali metals. Some
were previously used for other metals. Some are candidates
for only one topic, some for more than one. The candidates
for explaining deviations from the standard theory in our
bulk pure samples are: 1) anisotropic electron-electron
scattering; 2) electron-phason scattering based on CDW
theory; 3) inelastic scattering of electrons due to low
energy excitations associated with dislocations; and 4)
inelastic scattering of electrons by impurities. The
candidates for explaining deviations in our thin samples

are: 1) the Knudsen flow model; 2) electron-phason

18

l9

scattering; 3) size-dependent DMR; and 4) the Gurzhi theory.
The candidates for explaining deviations in alloys and in
pure samples in contact with hydrocarbons are: l) the Kondo
effect; 2) inelastic scattering of electrons by impurities;
3) localization effect; and 4) electron-electron interaction
effects. These models are introduced and discussed
separately.

Third, the theory of thermopower S and thermoelectric

ratio G is introduced and discussed.

2.1 40.41. ..o: 0

METALS

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I!
p.
i
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O
.00
4
p
)0

When an electric potential is applied across a piece of
metal, the conduction electrons in the metal will drift in
a definite direction to form a current. Any mechanism of
electron scattering that reduces the total crystal momentum
of the electrons will cause resistance. The standard theory
considers electron and phonon systems inside a crystal with
impurities and defects. So the mechanisms contributing to
the resistivity include: electron-phonon scattering,
electron-electron scattering, electron-impurity scattering,

and electron-defect (including electron-surface) scattering.

2.1.1 W (ref. 2,28)

There are two different kinds of collisions between
electrons and phonons: normal and Umklapp. It is helpful
to understand these two processes first. When an electron

is in the periodic potential of a lattice, the free electron

20
model can no longer be used; hk, where k is the wave vector
nhvu) of the electron wave function, is not a real
momentum. hk acts, however, rather like a real momentum,
and is thus called the crystal momentum of the electron.
Strictly speaking, phonons do not carry momenta, because
they are associated with the vibrations of whole lattice.
But when a phonon interacts with another kind of particle,
such as a photon, a neutron, or an electron, it acts like it
possesses a momentum hq, called the crystal momentum of the
phonon. Here, a is the w.v. of the phonon. When an
electron is scattered from a phonon, the total crystal
momentum is conserved to within a reciprocal lattice vector
G, (i.e., hk+hq+h5=hk' or k'=k+§+§wherekis
the wave vector of the incoming electron and k“ the wwv. of
the scattered electronh This process is called phonon anni-
hilation. In phonon creation we have bi? + hG = hk' + THE or
k + G = k" + q. For a normal process, 5 is zero, and there
is no crystal momentum exchange between the lattice and the
electron. When G is not zero, it is called an Umklapp
process, and there is crystal momentum exchange between the
electron and the lattice. The alkali metals in the bcc
structure have nearly spherical Fermi surfaces, completely
inside the first Brillouin zone. Figure 2.1 shows schemati-
cally the normal and Umklapp processes of phonon annihila-
tion and creation in the alkali metals with bcc structure,

where the Brillouin zone has been simplified. Clearly, for

 

  

 

 

 

 

 

 

 

 

 

Fermi . .
surface\‘* Simplified
M Brillouin
zone
k" k

 

 

 

 

(A) Phonon

Annihilation

 

kF/OM =O.877 for the
alkali metals in bcc
structure

Normal process:

'k + a = k'

Umklapp process:

a.

k + a +'§ =‘F'

Normal process:

72

1.12.4.3

 

 

 

 

37L

 

Q

 

ll ..
k .
1;::::::::::>

 

 

 

 

(B)

Phonon

Creation

Umklapp process:

A A

k + G =‘k' +‘a

Fig.12-l. Normal and Umklapp processes of e-ph scattering.

22
an Umklapp process to take place, a minimum value of phonon
w.v. (qmin) is required.
For electron-phonon scattering, assuming: 1) thermal
equilibrium phonon distribution; 2) the relaxation time
approximation; 3) only normal processes; and 4) the Debye

model of phonons, Bloch (ref. 28) predicted that,
pe-ph = const (T/ea)5 j5(6d/T), (2-1)
where:
99/4
jn =J£ zn (ez-l)'1 (1-e'z)’1dz (2-2)

and.6D is the Debye temperature. At high temperatures,
T>>Gb, j5~(6b/T)4, so Pe_ph«(T/eb); at low temperatures,
T<<GD, j5 is constant, so pe_phoc(T/9D)5.

When the phonons are not in thermal equilibrium, a
phenomenon called phonon drag can occur. Consider the
system of all electrons and phonons. If there are only
collisions among themselves and none of these collisions are
Umklapp, (idh, this system has no crystal momentum exchange
with the whole lattice or anything else such as impurities
or defects), then the total crystal momentum of this system
should be conserved. The only effect of the collisions
between electrons and phonons is then to drag each other
along so that they drift together. This is called phonon
drag for electrons, or electron drag for phonons. Phonon
drag reduces the resistivity by reducing the Bloch T5 term.

When phonon drag occurs, the electron-phonon scattering can

23
contribute to electrical resistivity only through Umklapp
processes.
For electron-phonon Umklapp processes in the alkali
metals with bcc structure, we have already mentioned that a

minimum value of phonon wrv. is required. The minimum

qmin
vuv. is associated with a minimum energy humin' When kBT is
well below humin' the number of phonons available for such
events should become proportional to exp(-hwmin/kBT), i.e.,
exp(-6*/T). That is why Ziman (ref. 2) concluded that the

Umklapp process contribution to the resistivity is

-8*/T

pe_ph or e (2-3)

For the resistivity of potassium at low temperature,
Kaveh, Leavens and Wiser (ref. 3) proposed an empirical

expression,
pe-ph = CTn exp(-6*/T) (2-4)

where the value of n was related to that of 6*; increasing n

by one would reduce 6* by about 2.8K.

2.1.2 Warring. (ref. 4)

As in electron-phonon scattering, the total crystal
momentum in electron-electron scattering is conserved to
within a reciprocal lattice vector 5, (i.e., k1 + k2 = iii +
kg + G where El and k2 are the w.v.s of the incoming
electrons, kl' and R3 are the w.v.s of the scattering
electronsh Since electrons obey the Pauli exclusion

principle, k1, k2, kg, and k} are all within kBT of the

24
Fermi Surface. So, the numbers of possible vuv.s k1 and k2
are both proportional to T. Thus, the scattering rate,
l/Té_e, is prOportional to T2. But, notlall electron-
electron scattering events contribute to electrical
resistivity; For normal processes, G = 0, the total crystal
momentum of the electrons is strictly conserved. Such scat-
tering thus makes no contribution to the resistivity, assum-
ing an isotropic relaxation time. For Umklapp processes,
é’go, crystal momentum is exchanged between the electrons
and the lattice, thus there is a contribution to the resis-
tivity, pe-e = (m/ne2) (l/fe-e)diT2. Lawrence and Wilkins
(ref. 21) developed a theory of screened Coulomb interaction
between electrons for potassium. MacDonald et a1. (ref. 20)
refined this theory. They found that the contribution of
screened Coulomb interaction was much smaller than that
calculated by Lawrence and Wilkins, and proposed a dominant
term due to phonon-exchange scattering instead. Both
theories predicted an AT2 term which accidentally had the

same values A = 0.17 pflcm/KZ.

2.1-3 Wadi—325.131.111.131

In the first order, the scattering of electrons by
static imperfections in the lattice, such as impurities,
vacancies, dislocations, surface and so on is elastic, and
makes a temperature-independent contribution to the

resistivity, called the residual resistivity Po.

25
However, inelastic scattering of electrons by lattice
imperfections can also cause a temperature-dependent

resistivity, as will be discussed in section 2.2.8.

According to Matthiessen's rule (ref. 2), the standard

theory for the alkali metals described above should give
P =.Po + AT2 + 3T5 + CTne'e*/T (2-5)

At a very low temperature, when the last two terms

become negligible, we expect
p a pa + AT2 (2-6)

and

dP/dT = 2m, or (71;) dP/dT 2A (2-7)
If dP/dT is plotted versus T, a straight line with slope =
2A should result. If (l/T0(dPVdT) is plotted versus T, a

horizontal straight line should result.

2.2 : .\' . :t-::- won 0 _ - a 4 a 0:

2.2.1 Electron:B1sctr9n_Ssattarins_in_the_Rrssence_Qf
Aaégotfgpic_Scatterers_snch_as_nislocatigns.

As mentioned in section 1.2J1, Kaveh and Wiser proposed

an anisotropic electron-electron scattering model to explain

the variations in magnitude of the T2 coefficient reported

by van Kempen et al. for pure K sample clad in polyethylene

26
tubes. Their model was also used to explain some later
experimental results from others. In this section, the
details of this model are described in what follows.

Under the following conditions: a) steady state, b) no
temperature gradient, c) no magnetic field, and d) relaxation
time approximation, the Boltzmann equation, which lies at
the heart of the theory of electronic transport in metals,

is of the form

 

[vkeuzn [$51) = hvuz) figs-(1‘11.

Siflce' §kf(E)

 

thus, f(k) - fo(k) = e7(k)V(k)'B (Be ) (2-7)
where E is the external electric field, f(k) is the
electronic distribution function, fo(k) is the equilibrium
distribution function, and f(k) is the relaxation time.
Kaveh and Wiser defined ¢(k) as f(k) - fo(k) =
-¢(E)(—a-§é-k~)), and argued that in terms of Mk), the inte-
grand of the multiple integral for Pe_e(T) contains the

product (Ziman 1960) (ref. 2)

mil) + f(kz) - f(R3) - f(k4)]2 [E1 + £2 — E3 - i541
«(Nip + 45062) - ¢(f<’3) - ¢(k‘4)12 [£1 + E2 - E3 - E41
(2‘8)
where El and k2 are the initial states, 12:; and E4 final

states. They argued that for one-OPW electron wavefunctions

27

(which are primarily apprOpriate for alkali metals) ¢Uba

whiz-E. Thus,

MEI) + 4022) — ¢<E3) — ¢(R’4)12 blkl + :22 - E3 - E41
«Huang + 7(E2)k’2 - 7023))2’3 - «r(i€4) E4)-E12
alk’l + k2 - E3 - E4]
(2‘9)

The 6 function in equations (2-8) and (2-9) implies that
there is no contribution from Umklapp scattering. So, only
normal scattering needs to be considered. Moreover, they
pointed out that the 71k) is determined by all electron
scattering mechanisms, not only by electron-electron
scattering. At low enough temperatures, electron-impurity
and electron-dislocation scattering are dominant. There are
thus two different limits for the determination of f(k). In
the first limit, the dominant scattering mechanism is
electron-impurity scattering. They argued that electron-
impurity scattering is approximately isotropic [71k) is
constant]. This is termed the 'isotropic limitJ In this
limit, integrand (2-9) is equal to zero, so P§_e(T)
vanishes. In the other limit, the dominant scattering
mechanism is electron-dislocation scattering. They argued
that electron-dislocation scattering is highly anisotropic,
so that the resulting relaxation time 11k) will be
anisotropic over the Fermi surface, (idh, not equal for
different directions). This is called the anisotropic

limit. In this limit, the 71k)s in equation (3) are

28
different for El, k2, E3, and k4, p§_e(T) does not vanish,
and can assume a relatively large value.

Next, they generalized the discussion to include
multiple-OPW electron wavefunctions, and argued that the
Umklapp scattering term.pU_e(T) no longer vanishes but is
still considerably smaller than.pg_e(T) in the anisotropic
limit for alkali metals, because the wavefunctions for the
alkali metals have primarily one-OPW character.

In summary, they had:

Pe_e(T) --> Pg-e(T) = AUT2 isotropic limit
. (2-10)
-e(T) __>‘,N,ani(T) +'PUe- e(T) anisotropic limit
~ pNLani(T) = ANTZ (2-11)

Finally, they predicted the coefficient of the e-e T2
term to be A = AU + AN with

A“ a: (pod/po)2 = (1 + Poi/Paal'2 (2-12)

where P61 and pod are the residual resistivity contributions
due to impurities and dislocations, respectively.

This model was able to describe the sample dependence
of the coefficient A for K data of van Kempen et al. of Levy
et al., and of Rowlands et al.

The main difficulty with this model is that the re-
quired number of dislocations in the K samples are at least
two orders of magnitude more than is believed could easily
exist in strain free K samples. Moreover, there was no way

to determine poi' pod' and their ratio, so that one

29
adjustable parameter was needed to make each set of data fit

the theory.

2.2.2 W
E E 'II' E 'Iill'IlE'] I‘

.As mentioned in section lu2.1, M. Haerle (ref. 22) used
theories of inelastic scattering of electrons due to low
energy excitations associated with dislocations to explain
anomalous deviations at the low T end in bulk potassium
samples. In this section, this model is introduced. Two
kinds of low energy excitations are discussed; one is asso-
ciated with local electronic states, the other with local
phonon modes.

Theoretical and experimental work on understanding the
contribution of dislocations and grain boundaries to the
residual resistivity in metals was summarized by Brown in
1981 (ref. 29). He suggested that additional bound states
for electrons with energy slightly larger than the Fermi
energy exist near the cores of dislocations. He estimated
the energy of these additional levels to be above the Fermi
level by e which is a few meV for common metals such as
c0pper, gold, aluminum, silver, molybdenum, tungsten and
zinc, and about 0.1 meV for potassium. Earlier, Gantmakher
and Kulesko (ref. 30) had derived an equation for the addi-
tional resistivity due to changes in the elastic scattering
cross-section because of filling of electron levels local-

ized at dislocations as

pe-dis = all + flexp(e/kT)1'1 (2-13)

30

where p is the spin degeneracy of the additional level andc:
is a pr0portionality constant. Fulde and Peschel (ref. 31)
calculated the resistivity contribution of inelastic scat-
tering of electrons off of a localized energy level. Their
theory could be applied to scattering off local electronic
states caused by dislocations such as the ones predicted by
Brown. They obtained

yzé/T

Sinh(6/T)] (2'14)

0(T) = ao[1 -
They compared the above result with the corresponding
expression one would obtain by applying Matthiessen's rule,
2

aM(T) = ao[l + 7 1'1 (2-15)
1 + 2/3 Sinh2 6/2T

 

where do is the residual conductivity, 6 is the energy
separation between the associated two states, and )’is a
constant. These two functions have the same limiting values
for T<<6 and T>>6.

In another mechanism, inelastic scattering of electrons
off of dislocations is associated with local phonon modes.
Anderson and O'Hara worked on the lattice thermal
conductivity in undeformed superconducting Al and deformed
superconducting Pb and Ta, and claimed that their data
showed resonant phonon absorption at certain energies.

Their results were in good agreement with calculations based
on the elastic-string model of localized phonon modes

associated with dislocations (ref. 32). The elastic-string

31
model of dislocations considers a dislocation which is
anchored at both ends. Since there is an elastic energy
associated with unit length of dislocation, there will
always be a tendency for a dislocation to make itself as
short as possible so as to minimize this energy, like an
elastic string does. In this model, additional local phonon
modes can come from the oscillation of the elastic disloca-
tion string; the longest wave length, and thus the lowest
energy state, are associated with the length of the disloca-
tion. The anchoring sources can be impurities, nodes in a
dislocation network, or some other crystal defect. Adding
more impurities and dislocations will shorten the length of
dislocations, and thus increase the.lowest energy of this
system. ‘The random placement of the dislocations and im-
purities would give a random distribution of lengths of the
anchored dislocations.

The scattering of electrons from these local modes
should be inelastic. Approximating the local modes by a
single frequency oscillator, Gantmakher and Kulesko (ref.
30) calculated the contribution of such inelastic scattering

to the resistivity as of the form
Pe-dis = (C/4T) Sinh’2(hw/2kT) (2-16)

where a2is the frequency of the ground state and C is a

constant.

32

2.2.3 Wm

An electron-phason scattering mechanism from CDW
theory, first proposed by Boriack and Overhauser (ref. 33)
and later improved by Bishop and Overhauser (ref. 34), was
proposed to explain the T3/2 variation in the resistivity of
potassium, reported by Rowlands et a1. (ref 16).

According to Overhauser (ref. 1), in a metal for which
the positive-ion lattice closely approximates the deformable
jellium model, the electronic ground state is a charge
density wave (CDW) state. The spatial density of conductive
electrons is not uniform throughout the crystal, but is of

the form
P(F) = -en[1-P cos(5°r +¢)] (2-17)

where n is the density of the electron gas, and P, 6, and.¢
are the amplitude, wave vector, and phase of the electronic
CDW. In potassium Q is claimed to be slightly greater than
2kF (ref. 1).

In order to neutralize this charge modulation, the
lattice tends to be deformed. Potassium might undergo this
deformation easily because of its softness. This deforma-

tion of the ionic lattice is given by

-.

6(E) = A sin(6°fi +95) (2-18)

with parameters defined above, and [AI = P/IGI.
In general, the CDW wave vector 6 is incommensurate

with the reciprocal lattice vector G. Thus the lattice

33
should have an extra one dimensional periodicity along the
direction of 6. This extra periodicity introduces addi-
tional Bragg planes, and thus, extra Brillouin zone bounda-
ries. In potassium, Q is presumed to be slightly greater
than 2kF, so the first extra zone boundary is very close to
the Fermi surface, and the spherical Fermi surface of potas-
sium is changed to a lemon shaped surface that just touches
the extra zone boundary (Figure 2-2).

One of the consequences of this modification is that in
the region of contact points between two Fermi surfaces in
neighboring new Brillouin zones, the minimum wave vector of
phonons joining electron-phonon Umklapp scattering is
reduced (Figure 2-3), thus the scattering is enhanced, and
its contribution to the electrical resistivity increases.

In addition, electron-electron Umklapp scattering can occur
not only through E1 + k2 = k} + k3 + 5, but also through
k1 + k2 = ki + k} + 5, (i.e., the CDW enhances this
scattering and its contribution to the resistivity).

According to Overhauser, a CDW crystal is typically
divided into 5 domains. The direction of 6, along with the
domain boundaries, is very unstable at high temperature, but
is frozen at low temperatures. Overhauser predicted that
the enhanced electron-phonon Umklapp scattering along the
direction of 5 would cause an anisotropy of the electrical
resistivity of potassium at low temperature as large as four
to one (ref. 34). For wire samples, the orientation of the

5 domains is sample-dependent, so the electron-electron

Fig: 2-2.

 

 

ii "
0
ll

  
 

.—-.——.-’--

New Fermi
;//surface
\ due to CDW

z' perturbation

7\_.- _ -._._._

————-—-—

/

New zone boundaries ”/r
due to CDW perturbation

The effect of a CDW perturbation on the Fermi

surface. The Sphere is distored to a lemon shape.

(After Overhauser (ref.l))

 

 

The Umklapp scattering for a distored Fermi

surface. (After Overhauser (ref.l))

35
scattering contribution to the resistivity should also be
sample dependent. This can be used to explain the sample
dependencies of Po and of A, the coefficient of the T2 term
in the resistivity of potassium, reported by van Kempen et
al., Levy et a1. and Rowlands et al.

0 Another consequence of a CDW, is phonon-like
excitations, called phasons. Without a CDW, phonons arise
from thermal vibration of the lattice ions because of the
lattice periodicity; Similarly, phasons arise from the new
ionic periodicity of CDW, since 6 is incommensurate with G.

Boriack and Overhauser (ref. 33) suggested an electron-

phason interaction of the form
Ve¢ = 1/2 G§¢q {cos[Q + q)°r - wqt] -
cos[(5 - ii)? - <0th (2-19)

where G cos(6ۤ is the total self-consistent potential, and
¢H*‘“§* a are magnitude, frequency, and wave vector of the
phason, respectively. With this potential, and assuming the
Fermi surface to be rigidly displaced in k-space by the
external electric field, Bishop and Overhauser (ref. 34)
obtained the electrical resistivity by using the variational

method as

Pe¢(T) = A(\/‘2' T/6¢)5 J5(6¢/\/‘2' T) + 3(T/9¢)4
J4(eh/T) + C(T/6¢)2 J2(e5/T)
(2-20)

36

where

 

x n
Jnix) 8] 2 dz
0 (ez-l)(l-e“z)

is the Bloch-Gruneisen function. Each one of the three
terms given above can fit well to Rowlands' data with
different parameters 6¢, but their extensions below 0.5K
differ from each other (Figure 2-4).

It is important to note that this model cannot lead to
a negative dP/dT, since all three terms given above increase
monotonically with temperature. This means that this model
of electron-phason scattering cannot explain the appearance
of a negative dP/dT in our thinnest K samples, which will be

analyzed in Chapter 4.

2-2-4 Ih£_KnndS£n_£lQE_MQdfil

By analogy with the Knudsen flow (ref. 35) of rarefied
gases in a cylindrical tube, Rowlands et al. (ref. 16)
-proposed a size effect model associated with the Knudsen
flow of electrons to explain their experimental results for
K (section 1.2.1).

They noted that the relaxation time (T) for electron-
electron normal scattering in potassium at 1K from Lawrence
and Wilkins' calculation (ref. 21) is about 1.09 x 10-7 sec,
and the Fermi velocity (VP) of potassium is about 8.6 x 105
m/sec. Thus, the mean free path (A) of this scattering is
about 9 cm. Comparing A with the radius (r) of their

samples (r = 0.4 mm), they argued that most collisions of

250 l

zoo—r

RESISTIVITY (|O"50hm-cm)

Fig. Z-Q,

(50-

'00" 0.2 0.4

37

 

 

\T|.5

 

 

 

 

 

 

I
i l
I l

l
0.0 0.5 LO (.5
TEMPERATURE

Plot of resistivity vs T for the data of sample KZC

 

of Rowlands et al. (ref.l3). The fitting curve with
Jh is shown. In the inset, extrapolations of TI'S,
J2, J4, J5 fits to lower temperatures are shown.

(After BishOp and Overhauser (ref.l7))

38
electrons below 1K were with boundaries, and thus the
Knudsen flow model can be used. Assuming diffusely
scattering walls they proposed an appropriate formula for

the resistance to the electron "Knudsen flow" as
p = po (1 + kr/A) (2-21)

where k is about 1 for gasses (found by Dushman) (ref. 36),
for phonons in liquid helium (found by Whitworth) (ref. 37),
and for electrons (found later by Black in a Monte Carlo
calculation (ref. 38)),

For electron-electron scattering, AaT'Z, so the

temperature-dependent resistivity should be
_ _ 2 -
with
Borpor

Black (ref. 38) pointed out later that in the Knudsen
case, Po is proportional to 1/r, and hence, the coefficient
B is independent of the residual resistivity Po. Black also
argued that if P0 is something other than the size—limited
resistivity, as suggested by Rowlands et al., then B a pot
could be valid.

We noticed that r/A a 0.005 at 1K in Rowlands case, and
if k ~ 1, as calculated by Black, the PT(1K)/Po is about 5 x
10'3. But the data of van Kempen et al., of C. W. Lee et

ale, and even of Rowlands et al. themselves, showed

39

P'T(1K)/Po to be about 2 x 10'4. Even considering a possible
smaller effective radius due to additional scattering from
other scatterers, as suggested by Rowlands et al. for their
data, the predicted ratio PT(1K)/Po is still about an order
of magnitude too big.

Alternatively, according to Kukkonen and Smith (ref.
39) 7'could be an order of magnitude smaller. In such a
case, Rowlands et al. proposed that a term in (r/A)2 ought

to be taken into account to give
9T = BT2 - CT4 (2-23)

In this case, the ratio PT(1K)/Po is even two orders of
magnitude too big compared to experimental results-
Starting from a force-force correlation function
formula for the residual resistivity, March and Woods
provided a possible alternative justification for equation

(2-21) above (ref. 40). In their form

p.-.p (1+£+ ) (2-24)
0 A C O C

where A? is found to reflect long-range correlations in the
electronic motions and not necessarily to be size limited.
A.is electron-electron scattering mean-free-path in low
temperature resistivity study of potassium. In explaining
data of the potassium wire samples from Rowlands et al.,
they suggested A" ~ 1/2 mm. So, the PT(1K)/PO in their

model is still an order of magnitude too big.

40

As mentioned above, we need a model to explain the
negative dP/dT in our thinnest K samples. In the two models
described in this section, only the second term in equation
(2-23) seems to be able to generate a negative dP/dT. But
when sample radius r is smaller, huQJz is less important,
so -CT4 term is less important. So if for bigger r we do
not see a negative dP/dT, (idh, with our thick samples),
then for thinner samples this equation cannot lead to a

negative dP/dT.

2.2.5 W

In this section we first describe two size-dependent
DMRLmodels, which were previously used for explaining size-
effects in other metals. The possibility for using each of
these models to explain the negative dP/dT found in our
thinnest K samples is discussed.

The well known Matthiessenfs rule (MR) can be written

in the form
Pa(c.T) = Pp(T) + 90(c) (2-25)

where Pa(c,T) is the resistivity of a dilute alloy contain-
ing a concentration c of impurity, Pp(T) is the resistivity
of an ideally pure metal, and Po(c) is the impurity produced
resistivity at zero K.

In fact, there are always deviations from MR. Pa(c,T)

can be written exactly as

Pa(C.T) = Pp(T) + 90(0) + A(C,T) (2-26)

41

For 'normal' size dependent DMR, induced by a spatial
variation in the distribution function fk(f) due to the
existence of boundaries, Dingle (1950) (ref. 41) developed a
theory for metal wires of circular cross section. Dingle
assumed that the electrons in the metal were free-electron-
like with a spherical Fermi surface and that the surface
scattering might be characterized by a single specularity
parameter, p (defined as the probability that an electron
incident upon the surface is reflected specularly as though
from a mirror) which was assumed to be independent of the
incident angle. A few years later, Ziman (ref. 2) (1960)
formulated an angle-dependent specularity parameter Ps(0) =
expl-(4nh/Ae)2coszel, by assuming that all electrons have
the same wavelength Xe and that the surface can be charac-
terized by a root mean square surface roughness h. More
recently this 95(0) was incorporated into Dinglefls model by
Sambles et a1. (ref. 42), obtaining an expression for the
ratio of bulk to thin wire resistivity,

pa, 12 ”/2

2 2 ”72
- l — -—' cos 95in OdOJf sin¢d¢
P flk o o

(l—p)[1-exp(-ksinw/sin9)]
1-p.exp(-ksinw/sin0)

 

(2-27)

where k d/Awand p = expl-(llrrh/Ae)2 sin20 sinzw]. Since
Rn/P is a function of only k, Sambles defined f(k) =
(fa—poo )k/Poo = (p/poo - l)k. Thus P-Poo= f(k)P°° /k = f(k)°

{LAW/d. To determine the sign of 6P/6T, Sambles (ref. 43)

42

argued that

23:3. - 6p
6T 6T (’0 pm) +5?

BmQ-f-é—kl- + % (since Mis a constant)

P A 6f(k) 6k 6p . . .
_emaa_____.. __ m
d 6k 8T + 6T (Since f(k) is a function of

 

only k)

- if. he, - , 9. = an...

— (l + 6k) 6T (Since k A“, Ankw thus
Bk/d‘l‘ = (ti/Pack...) (GAO/6T)

(2-28)
Finally, Sambles argued that in their model, 6f/ak cannot be
smaller than -1, dP/OT cannot be negative unless ago/am is
negative (ref. 43).

Boughton and Neighbor (1972) (ref. 44) calculated size-
dependent DMR under several different situations. For wires
of circular cross section, their calculation led to two
different results: a) For metals with cylindrical Fermi
surface, they obtained the ratio of thin wire to bulk
resistivity
.312 [
16k

P .. _ iii-1 _
P..' 0.590 ln3fl] (2 29)

as a function of only k, with k = d/Aq” This gave

f(k) - (——p 1) k
- 16 [0.590 1n3 ] k (2-30)

and

43

6f

ar’l
with the same argument for equation (2-28), one can conclude
that aP/BT cannot be negative unless dam/6T is negative. in
,For metals with spherical Fermi surface, they obtained
p 1 3 3n
-—-= —-+ — -—-- - - 1
D; it 8 [1n 8 lnk IE] (2 3 )
where I; is a constant, for wires of circular cross-section

I; < -1 can be obtained from an equation given by them (ref.

44). This gave

3k 3n

 

f(k) = 1 +-§- [ln§—-- lnk - [g] - k (2-32)
Thus,

62:“ >
and,

g; > 0 (as long as ggé > 0).

2.2.6 W (ref. 45)

None of the theories associated with size-effect
discussed above can provide an explanation of negative dP/dT
in our data of thinnest potassium. In this section, the
only published candidate which might provide such an expla-

nation is introduced.

44

In 1962, Gurzhi predicted that normal electron-electron
scattering (NEES) should affect the path length between two
sequential electron-surface collisions in impurity-free
conductors and thus cause a negative temperature derivative
of resistivity (dP/dT < 0).

Gurzhi argued that a) The frequency of electron-
electron scattering decreases with temperature more slowly
than the frequency of the electron-phonon scattering and
will be dominant at low temperatures. b) In a bulk
conductor, electron-electron scattering gives rise by itself
to electrical resistivity only through Umklapp processes,
NEES makes no contribution. c) When 1V>>d, where 1V is the
mean free path connected with the scattering by the
inhomogeneities such as phonons, impurities and internal
defects, and d is the sample diameter, the effective mean
free path with diffusely scattering walls should be of the
form leff ~ d if the effect of NEES is not taken into
account. But, when d>>1e-e (le-e = the mean free path of
NEES), before an electron situated deep in the sample
reaches the walls, it would be scattered by other electrons
many times and thus would move like a Brownian particle.
According to the formula of Brownian motion, the length of
the path length between two collisions with the wall is of

order d2/1e_e, i.e., leff ~ d2/1e_e. But 1/1 or l/‘re_eat

e-e
T2, so Pal/leffale_e/d2aT'2. This gives us dP/ch1-T‘3, a
negative derivative!

According to the Gurzhi theory, a negative temperature

45
derivative of resistivity would be expected in a high-purity

thin sample with le_e<<d.

2.2-7 Ih£_KQndQ_Bf£s£t (ref 4,46,47)

In the standard theory (section 1n3.1) nonmagnetic
scatterers, such as impurities and lattice imperfections,
led to a temperature-independent term toward which the
resistivity monotonically drops with decreasing temperature.
But if magnetic impurities are the predominant type of
scatterers, the scattering from them will be the primary
source of electrical resistance at low enough temperature.
Then, instead of dropping monotonically, the resistivity can
rise with decreasing temperature (ref. 4,47).

In 1963, Kondo (ref. 46) argued that when the scatter-
ing center has a magnetic moment, the exchange interaction
between the conduction electrons and the local moment leads
to flipping of the electronic spin with a compensating
change of spin on the local moment. Kondo discovered that
in all higher orders of perturbation theory the magnetic
scattering cross-section is divergent, yielding an infinite
resistivity as in a term of -1og T. The balancing of this
term against the electron-phonon scattering term results in
a resistivity minimum.

According to subsequent analysis by Kondo and others,
the thermal rounding of the electron distribution removes
the divergence. Experiments showed that the logarithmic
Kondo behavior is followed by the resistivity only down to

a certain temperature known as Kondo temperature, below

46
which the resistivity becomes approximately constant (ref.
47).
No Kondo effect in the alkali metals has been reported
in previous work. In section 4.4 we will report a possible

Kondo effect in the present study.

2-2o8 W
W

Considering inelastic scattering of electrons by the
thermal motion of impurity ions, Koshino (ref. 48) in 1960
first obtained a T2 behavior of the resistivity at low

temperature of the form

finei ctJ2Z2T2 (2-33)

where
2.4 for T-¢0
Jz=.(
GD/T for T>>6D
and Z is the excess charge of the impurity ion.
Considering that the displacement of the impurity
potential induced by the thermal motion of the impurity
ion leads to a reduction of elastic as well as inelastic
scattering of electrons, Taylor (ref. 24) pointed out that
this reduced the additional resistivity due to inelastic
scattering. In 1963, Koshino (ref. 48) argued that this
reduction cancels out the additional resistivity due to
inelastic scattering at high temperatures, but brings no
essential change in the additional resistivity at low

temperatures. Taylor (1964) (ref. 24) recalculated the

47
additional resistivity at low temperatures and obtained an

expression without z as

”ZhZKZ
_ av 2 _ -5 2 -
‘Hnel - -——-§-POT - 1.37 x 10 PCT (2 34)
2MkBGD

where Kav is an averaged wave vector close to 2kF (kg =
Fermi wave vector), M the host ionic mass, 6D the Debye
temperature, and k8 the Boltzmann's constant.

Kus and Taylor (ref. 25) obtained for K-Rb
_ -5 2 _
pinel - 1.25 x 10 POT (2 35)

Frobose and Jackle (ref. 49) derived the same formula
as equation (2-34) for structure defect inelastic

scattering, and calculated pinel for K and obtained
p. - 1 8 x 10"5 p T2 (2-36)
inel ‘ ' o

C. W. Lee et al. (ref. 23) reported that in K-Rb

alloy resistivity measurements they obtained
p = p0 + (8.510.3l x 10‘6pr2 + (2-2i0-3) x 10’13T2:

and claimed the second term, (8510.3) x 10‘5po'r2, as an

unambiguous Koshino—Taylor term.

2.2.9 WWW
Effects

In this section, two different theories of the

temperature dependence of the resistivity in impure metals

with nonmagnetic impurities are given.

48

The first is the theory of electron-electron interac-
tion effects.

A square-root temperature dependence of resistivity due
to the electron-electron interaction effects was first ob-
tained by Altshuler et a1. (ref 50). They argued that
electron-electron interaction accompanied by the impurity
scattering leads to a singularity in the density of states
near the Fermi surface, and thus leads to a zero bias anoma-
ly in the tunneling current-voltage characteristic. They
showed that at low temperatures, the resistivity decreases
as -x/"l" with increasing T.

In 1980, Fleurov et a1. (ref. 51) obtained a more clear
picture. They argued that when T<ceb, (where c denotes the
concentration of the impurities, 6D denotes Debye
temperature), the electron-phonon interaction is negligibly
small compared with Coulomb interaction, thus resistivity
decreases with increasing temperature according to APKTl~
-c5/2(T/£F)1/2, due to the Coulomb interaction between the
electrons scattered by the static impurities. Moreover,
they argued that the quasiclassical mechanism of the
scattering from vibrating defects compensates the electron-
phonon contribution over all this temperature region. At
T~ceb only this quasiclassical mechanism remains, giving an
increase in the resistivity term of about cTz/eDeF, which
exceeds the Coulomb term at T~cEK6VeF)l/3. The temperature

of a resistivity minimum lies near this value.

49

The second theory given in this section is localization
theory.

Scaling theories of localization have been discussed by
Thouless et al. (ref. 52), by Wegner (ref. 53), by Schuster
(ref. 54), and by Abrahams et al. (ref. 55). Abrahams et
al. (1979) (ref. 55) using a scaling approach to localiza-
tion, have obtained a correction of the conductivity to the
Boltzmann result in three-dimensional disordered metallic

state:

5., 2 __e2 (1 -_1_) (2-37)
23/2 flzh L L
where L is the length of a cubic system, or, at finite
temperatures, the inelastic diffusion length, Li : LE a 1/2
leli’ l/Lo is a constant of integration. le is the elastic
mean free path and Li is the inelastic mean free path.
Using this result along with that of electron-electron in-
teraction model, Howson (ref. 56) recently analyzed the

temperature-dependent part of the scaling result of conduc-

tivity in the form

 

 

1 e2 1 (kFle)2 1
0(T) ‘-‘-'--— ( - —— ) (2-38)
h Li(T) 3 11m

where the first term will always give a non-metallic posi-

tive contribution to dq/dT while the second term will give a
metallic negative contribution. Howson argued that in high-
resistivity metallic glasses ( >150 pacm), for which 16 may

be as small as an atomic spacing, the second term in Eqn.

50

(2-38) can be neglected, then,

_3 1

1
0(T) :- ._.._..
n2 h Li(T)

 

At low temperature,

1101-) or T‘2 T<6D

(2-39)

for both electron-electron and electron-phonon inelastic

scattering. At high temperature, the electron-phonon

scattering leads to

-1
11(T) at T T>6D

but
2 -
Li(T) - 1/2 leli(T)
Thus, Howson obtained

0(T) « T T<6D

and 0(T) «VT T>9D

(2-40)

when ab>>a«T), where do is temperature-independent part of

conductivity, then one has

p = p0 "' CT T<6D

and p 00 - CJT T>9D

in first approximations, where C ang.

(2-41)

51

2.3 W

As described in section 1.1, generally there are three
terms in S (or G) at low temperatures. The first is the
diffusion term. The other two terms are due to normal and
Umklapp phonon drag, respectively. In normal phonon drag
(described in section 2Jul), electrons tend to flow along
with phonons from the hot to the cold end, and thus
contribute a negative term in S (or G). In Umklapp phonon
drag, electrons tend to flow opposite to the phonon flow,
and thus contribute a positive term to S and G.

Details of the theory of S and G for Alkali metals have
been described by C. W. Lee (ref. 8) and M. Haerle (ref. 22)
in their theses. Only the main points of this theory are

briefly reviewed in this section.

2.3.1 W
At low temperatures, the thermopower S of alkali metals
with bec. lattice structure, can be written in the general

form
8 = A'T + B'T3 + C'(T) exp(-9*/T). (2-42)

The first term (A'T) is the diffusion term, where A'
can be written as the generally used Mottls expression (ref.

2):

_ fizkz 61na(E)
3e 88 E=Ef .

 

A' = (2-43)

52
Here Ef = the Fermi energy, 0(E) = energy dependent

conductivity. Simplifying this general form, Guenault and

MacDonald (ref. 5) obtained

 

 

. nzkz aln[n(E)v2(E)l aln‘r(E)
A = " 3 i + -- (2-44)
er alnE alnE E=Ef
where 7(8) = the relaxation time, n(E) = the density of

states, and V(E) = the average electron velocity at energy
B. When temperature independent mechanisms such as impurity
scattering, dislocation scattering, etc. dominate, a
temperature-independent A' is expected from either equation
(2-43) or (2-44). And for spherical Fermi surface, as in
alkali metals with bxnc. structure, we expect to have a
negative A' (ref. 8).

Applying Matthiessen's rule P= Pi + Pd to Mott's

formula, one gets the Gorter-Nordheim relation

where Pi and Si are the electrical resistivity and
thermopower, respectively, due to impurity scattering, and
Pd and 3d are the electrical resistivity and thermopower,
respectively, due to another temperature independent
scattering (such as dislocation or surface scattering).

A slight change in the concentration of impurities, or
in the density of dislocations, can easily change the

dominant scattering mechanism and thus change the magnitude

of A'.

53

The second term B'T3 is the normal phonon drag term,
where B' is a simple constant. At very low temperatures in a
sufficiently pure metal, the boundary scattering of phonons
will dominate,the contribution of normal phonon drag to S
will then go roughly as the lattice specific heat, which has
T3 temperature dependence (ref. 8).

The third term (C'(T)exp(-6*/T) is the Umklapp phonon
drag term. For Umklapp scattering, as described in section
2.1.1, a minimum phonon wave vector (qmin) is needed, and
thus the number of phonons available for such events should
be pr0portional to exp (-6*/T).

Guenault and MacDonald (ref. 7) obtained an

approximation for C'(T) at low temperatures as

 

l/r ..
c'(T) = [ 9“ e 1 (0.200) [5) [9:1
1/"'ph--e + 1/ Tph-i e an
9* T
[5" + 3 + 65;] (2-46)

where Tph-e = phonon-electron scattering relaxation time,
Tph-i = phonon-impurity scattering relaxatin time, 9D =
Debye temperature, and EN = a constant.

Ziman (ref. 57) and Bailyn (ref. 58) developed more
detailed theories, but so far only the simplest forms for
the Umklapp scattering have been used for fitting
experimental data.

Guenault and MacDonald (ref. 7) claimed that their

potassium data could be fitted by

54

s = A'T + B'T3 + c'exp(-e*/T)

with C' = a constant and 6*~21K. They found, as expected,

that A' and B' were negative while C' was positive.

2.3.2 Wm
‘Very little theory has been worked out for G. ‘But, as

described in section 1.1, G may be related to S by
S = GL(T)T (1-11)
and when elastic scattering of electrons is dominant

L(T) = L = 2.45 x 10‘8 V2 K'2 (1-8)

0

According to our present data, L/Lo ~ 0.96 in K at 1K,
the ratio L/Lo becoming closer to l at lower temperatures.

At very low T, G is thus expected to be

I
G = G0 + B'T2 + $~exp(-6*/T) (1-13)
or even
G = G0 + B'T2 (2-47)

where G0 = A} is the diffusion term, B'T2 is the normal
phonon drag term, and C'/T exp(-6*/T) is the Umklapp phonon

drag term.

CHAPTER 3

Experimental Techniques

3.1 W

In this chapter the basic experimental techniques and
equipment, and our samples, are briefly described. Since
most of the techniques and equipment have already been
described elsewhere (ref. 8,22), only some modifications are
given in detail.

As mentioned above, one of the main purposes of this
study is to try to find the concealed reasons for disagree-
ments between experimental results from different groups
(section 1.2JJ, and to study various deviations from
electron-electron scattering when according to standard
theory electron-electron scattering should dominate over
electron-phonon scattering at low enough temperature. Thus,
it was necessary to get the temperature below 1K. Second,
there are two ways to study the temperature dependence of
resistivity at low temperature» One is to measure P'direct-
1y. But P = P0 +P (T), and at 1K, P(T)/Po is about 200 ppm
for pure potassium of typical RRR~5000 (ref. 14,59). Since
Po cannot be directly measured, it must then be left as an
adjustable parameter, which produces an inevitable uncer-

tainty in P(T). The other way, used in this study, is to

55

56
measure dP/dT. Since PO is a constant, this must be exactly
equal to dP(T)/dT with no adjustable parameters. To deter-
mine the measured quantity AP/PAT with a precision of 1%,
extremely high precision (g 0.1 ppm) of AC/C (=AP/P) was
needed, because for our highest purity potassium AP/P z 10"5

with a AT = 0.1K near 1K.

3.1.1 W

1) WWW with good
vibration isolation, giving a temperature range from 4.2K to
60 mK. The whole system could be cooled down to liquid
nitrogen temperature (~77.4K) in about 12 hours by adding
liquid nitrogen to the outer dewar with appropriate exchange
gasses, and then cooling further to liquid helium
temperature UL2K) by transferring liquid helium into the
inner dewar. This transfer took about 2 hours. 1.3K could
be obtained at the 1K pot by pumping on the liquid helium.
The lowest temperature of 70 mK was reached at the mixing
chamber and the attached sample can about 3~5 hours after
circulation of the dilution refrigerator was started. That
means that all of our samples were cooled slowly; See C.lL
Lee's thesis (ref. 8) for details.

2) WWW
consisting of a commercial direct-current comparator
modified by D. Edmunds et al. (ref. 60) and a very sensitive
SQUID (Superconducting Quantum Interference Device) null-
detector, capable of detecting voltages less than 10’15V, or

current less than 10‘9A. The ratio of two currents (slave

57

current to master current) could be read with a precision of
0.1 ppm using a set of eight decade-dials. Nearly 0.01 ppm
could be achieved by interpolating beyond the last dial
using a computer averaging technique.

3) A_screened_rggm for screening out radio frequency
noise that affects the operation of the SQUID. l) and 2)
were enclosed in the screen room, and all the pumps were

kept outside (ref. 8).

3.1.2 Thermometers

Temperature measurements were very important in this
study. Three germanium resistance thermometers GRT2, GRT4,
and GRTS were used.

The calibration of these three thermometers was done by
C. W. Lee and M. L. Haerle (ref. 8,22). GRT4 and GRTS were
calibrated simply by carefully comparing them to GRT2. GRTZ
itself was calibrated in three steps. First, it was
carefully compared to a Cryocal CRSO thermometer, calibrated
by J. L. Imes (ref. 61) and G. L. Neiheisel (ref. 62) from
0.065K to 4.2K by using the He3 vapor pressure and a
susceptibility thermometer. A fit of the data with the

equation,
9 .
1n R = 2 Ai (1nT)1 (3-1)
i=0

was used for calculating the temperature from 1.3K to 4.2K.
Below 1.3K, this fit was not accurate enough for the
present study. Second, using a set of Superconducting Fixed

Point Devices SRM 767 and SRM 768 from the National Bureau

58
of Standards (ref. 63) to get 6 absolute temperature points
from 0.099K to 1.17K and using a susceptibility thermometer
to interpolate between these six points. The fit of this
set of data, from 0.059K ot 1.24K, with the above equation
(3-1) has been used for measurements below 0.5K. Third, for
(LSK to 1.3K calibration, a large powdered sample of cerous
magnesium nitrate and four of the fixed points between
0.519K to 3.414K were used to again fit Eqn. (3-1).

The accuracy of these calibrations was tested by using
the Wiedemann-Franz law and a Ag—0.1 at% Au Alloy. One end
of this alloy was attached to the mixing chamber at a fixed
temperature, and GRT2 was mounted to the other end for
measuring the temperature there. The calibration gave the
product TAT to within an average value of 0.47% with a
standard deviation of 1.2%. The maximum deviation from the
Wiedemann-Franz law was 2.6% (ref. 22).

To ensure that there were no problems with temperature
measurements while taking the data, the Wiedemann Franz law
was also used to double check the thermometry for every run
(see section 3.4).

The resistances of GRT2 and GRT4 were measured with two
SHE conductance bridges in the 4-terminal mode. The
bridges, using low excitation voltage (10 to 100 uV), were
accurate to better than 0.5%. They were self-balancing and
had a differential output, which was also used for regulat-

ing the mixing chamber temperature (section 3.5.1).

59

3.1.3 IhWWW

The alkali metals are all very reactive. They react
rapidly with air and water vapor. Secondly, in this study,
samples were measured under different circumstances: under
Ar, He, or vacuum, and either bare, or encased in, or
touched by, plastic. Therefore, some special equipment was
needed for sample preparation.

The main equipment included: a) two glove boxes sup-
plied by Vacuum Atmospheres Company; b) two different kinds
of stainless steel presses used for making wire samples and
potential leads; c) two sample cans, used to protect the
samples from reacting with air; d) a high vacuum system,
with all metal parts, capable of getting a 10'9 torr vacuum,
used to pump out the sample can and clean some plastic
tubings and pieces, glass tubings and molecular sieves, all
needed for sample preparation.

The details of the glove boxes, presses, sample cans,
sample preparation, measurement method, and uncertainty

analysis will be given in what follows.

3.2 IHE_§LQ¥E_BQKES_AND_RBESE§S

Two glove boxes supplied by Vacuum Atmospheres Company
were used for sample preparation. One had a built-in
Pedatrol pressure control, Dri-train Mo 40-1 inert gas
purifier with circulation rate of 50 cfm for He, and A0 316-
C Oxygen analyzer. It was used under He atmosphere with an

oxygen contamination level less than 0.5 ppm. Fresh

60

potassium exposed inside this glove box stayed shiny for at
least 2 hours. The other glove box was evacuable and had
only a built-in Pedatrol pressure control and locally built
inert gas purifier with circulation rate of about 1 cfm. It
was used under Ar atmosphere. Fresh potassium exposed
inside of it stayed shiny for at least 2 hours also. All
the samples were made inside these two glove boxes.

Two kinds of stainless steel presses were used. Figure
3-1 shows the structure of these presses. The big one,
which is threaded, was much more powerful than the other and
was used for making samples which could not easily be
extruded, such as Li or thin Na samples. When the big one
was used, it was stabilized in a big vice, and a long wrench
was used to screw the piston in.

There were two ways to put the material into the press.
For K, Rb, or Na, which were originally in ampoules, the
material was melted in the glove box with a hot-plate and
then poured into the press. For Li or Na, which were not
originally in ampoules, the surface was first cleaned with a
stainless steel knife, and the metal was then cut to
reasonable size and put into the press. After the material
was poured or put into the press, it was melted and allowed
to solidify into one piece.

Different sample thicknesses were produced by using

stainless steel dies with different hole diameters.

61

 

 

 

 

 

_

 

<—- Piston

 

 

/

 

GI ‘10. \Q-"\\’I-.§’(\’O-C \Il'l

’0 0|.

w>>>~If

.u""ilv O

...---..- ..

f (o) \It I
‘

 

 

 

 

 

 

 

 

\
.-
J. s.
\c....
o s
.. ..I
J-
\._
I
........ JL. '0”
w r
. .
. .
o .
. .
o 4
0 Q
xx

 

presses.

Two kinds of stainless steel

Fig. 3-1.

62
3.3 SAMELE.CAN

Two sample cans were used. Figures 3-2 and 3—3 show
the structure of sample can number 1, used for K, Rb, and Na
samples. Its sample holder has two separated sides, each
with two c0pper supports for current leads, and two more for
voltage leads. Sample can number 2, used for Li, is a
modified version of sample can number 1. The main differ-
ence is that sample can number 2 has 8 copper clamps instead
of 8 copper supports, because Li is not soft and thus not
easily cold welded to copper. All eight copper supports or
clamps were carefully cleaned with alcohol and filed just
before being brought into the glove box, and then filed
again inside the glove box.

Before the samples were made, a little material was
extruded out to clean the stainless steel die. Then more
material was squeezed out and smeared (cold welded) onto all
copper supports or clamps. After a sample was made, it
could be mounted onto either side of the sample holder in
three different ways (Figure 3-4). In method (a), the
sample was first cold welded onto the current supports.

Then two potential probes of the same material with the same
diameter were extruded and cold welded first to the sample,
typically 5 cm apart, and then onto the voltage supports.

In method (b), used for mounting thin samples, two thick
(d=l.0 or 1.5 mm) leads were made from the same material,
and each was cold welded onto a current support and the

neighboring voltage support. Then a thin sample (d30.25 mm),

63

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64

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66
typically 0.5-1.0 cm long, was cold welded onto those two
thick leads. As described later, for 0.25 mm K samples, the
data were same by methods (a) and (b). In method (c), used
for Li, the sample and potential leads were first clamped by
the copper clamps. The sample was then cold welded to the
potential leads by pinching them together with a stainless
steel tweezer.

Our samples were cooled in one of three different
atmospheres: Ar atm., He atm., and partial vacuum (~10qu
of He).

Under Ar atmosphere, the sample can was simply closed
with an indium "O” ring.

Under He atmosphere, some cleaned Linde Molecular Sieve
Type 13X pellets were left inside the can to adsorb the He
gas at low temperature, so as to prevent heat exchange
between the samples and the can. The molecular sieve was
cleaned by pumping it down to a vacuum of less than 10'6
torr in an Erlenmeyer flask with a stopcock, and then heated
to about 200°C under vacuum for two days. The flask was
then allowed to cool to room temperature, sealed by closing
the stopcock, moved into the glove box, and opened just
before closing the sample can. After the pellets were
poured into the sieve chamber, a piece of stainless steel
screen was pressed in to hold the pellets in place.

To evacuate the sample can down to 10 ”Hg, and then
seal it properly, a piece of indium-filled brass tube was

needed. In making these tubes, first the brass tube was

67
filled with molten indium. After the indium solidified, a
hole was drilled through it and the tube was soldered onto
the sample can as shown in Figure 3-3. [See M. L. HaerleT;
thesis for details (ref. 22).] An extension tube with a
‘valve was soldered to the indium tube before the can was
moved into the glove box. The sample can was sealed at room
temperature by closing the valve, and then was taken out of
the glove box and hooked onto a high vacuum system. In
order to prevent the diffusion of the air through the pump
to the can, the vacuum system was first completely
evacuated, then the input valve of the pump was closed and
the valve on the can was opened, this procedure was repeated
until the sample can was evacuated properly; typically,
about 10 u-Hg helium gas at room temperature was left inside
the can as exchange gas for the cooling process. The indium
tube was then pinched to seal the can. The indium cold-
welded to itself, providing a leak-tight seal at the lowest
temperature in this study. .Finally, the valve was removed
from the sample can, and the can was attached to the mixing
chamber of the dilution refrigerator through a copper well
and teflon ring (Fig. 3-2).

As noted above, temperature measurements were critical
to the results of this study. The two thermometers, GRT2
and GRT4, used for measuring the temperatures of the
samples, were clamped with copper clamps onto two c0pper
thermal feedthroughs outside the sample can. To ensure that

each of the thermometers was at the same temperature as the

68
sample it was measuring: 1) The thermal resistance between
the sample and the thermometer was minimized by using a well
annealed copper rod, 8 cm long and 3 mm in diameter, as the
thermal feedthrough; 2) Each feedthrough was sealed onto a
thin wall stainless steel tube using stycast 3850 GT epoxy,
and the stainless steel tube was then soldered onto the
sample can (Fig. 3-2). The samples were thus thermally
isolated from the sample can, because both epoxy and
stainless steel are very effective thermal insulators at
very low temperatures; 3) The sample can and dilution
refrigerator were both thermally isolated from liquid He by
enclosing them inside a vacuum can. To cut down thermal
radiation from the vacuum can to the thermometer sitting on
the sample can, the sample can was shielded by a thin copper

can, which was attached to the mixing chamber.

3.4 SAMELE.£BEEABAILQN

All of our high-purity potassium (99.95%), rubidium
(99.95%) and some of the high-purity sodium (99.95%) were
obtained from the Gallery Chemical Company, sealed in glass
ampoules under Ar gas. Some high-purity sodium was kindly
supplied by Prof. J. C. Garland of Ohio State University,
packed under paraffin oil in a plastic bottle. Our high-
purity lithium (99.99%) was obtained from Atomergic
Chemetals Corp. in a low sodium dry pack.

In order to keep the glove box clean, the oil bottle of

Na was first moved into a glove bag filled with Ar gas, the

69
Na pieces were taken out of the oil with a clean stainless
steel tweezer, soaked in a beaker of petroleum ether for a
few minutes to dissolve the oil, dried, sealed in a glass
container, and then transferred into the glove box.

The common way of preparing bare pure K, Rb, Li, and Na
samples was already described in section 3.1. In this
section, the details of each sample are given in Table 3-1
(for K), Table 3-2 (for Rb), Table 3-3 (for Li), and Table
3-4 (for Na). The details of making pure K samples encased
in plastic tubings, coated by oil, or touched by various
kinds of plastic pieces, and the details of attempting to
grow a single crystal K sample and of making some K-Rb alloy
samples will be given next.

In order to make a K sample inside a polyethylene,
teflon, or glass tube, an ampoule of K was opened inside a
glove box, heated with a hot plate until the potassium
melted, and then the molten potassium was sucked up into the
tube using a syringe (Figure 3-5). The tubes had been
freshly cleaned by placing them in a flask, pumping the
flask down to less than 10'”6 torr, and heating the tubes
with a heat lamp to above 100°C for two days. In order to
determine effects of different plastics, each ampoule of
potassium was allowed to come into contact with only one
kind of plastic. And, after a tool touched one kind of
plastic, it was cleaned by alcohol before being used for
dealing with another kind of plastic. The surface of a K

sample touching polyethylene was always shiny during all the

Table 3-1

70

Characteristices of bare free hanging pure K samples.

(RRR :

R(295K)

K(u.2K))

 

p(u.2x)

 

Ampoule Circumstance d [Pb] Mount
Samples No. at R.T. (mm) (nflcm) RRR method Other
K-7 # 1 Ar atm. 1.50 1.41 5100 (a)
[1.13]
K-8 4 1 Ar atm. 1.50 1.50 4790 (8)
[1.22]“
K-9 # 2 Ar atm. 1.50 1.72 4180 (a)
[1.441‘
K-lO # 2 Ar atm. 1.50 1.47 4890 (a)
[1.1913
x-11 #11 Ar atm. 2.00 1.26 5710 (a)
[1.01]
K-Hla # 3 He atm. 1.50 1.70 4230 (a)
(1.421“
K-Hlb # 3 He atm. 1.50 1.65 4360 (a) Second run
[1.371“ of K-Hla
K-HZ # A He atm. 1.50 1.27 5660 (a)
[0.991“
K-H3 # 5 He atm. 1.50 1.97 3650 (a)
[1.591“
K-Hu # 5 He atm. 1.50 1.22 5890 (a) Annealed
[0.9“]‘ at 50°C
for 30 min.
K-H5a # 8 10 qu He 1.50 1.27 5660 (a)
[1.02]
K-H5b # 8 10 )1Hg He 1.50 1.39 5170 (a) Second run
of K-HSa
K-H6a # 8 He atm. 1.50 1.23 5850 (a)

(0.951“

Table 3-1.

(Continued)

71

 

 

 

P(4.2K)
Ampoule Circumstance d [PB] Mount
Samples No. at R.‘1‘. (mm) (nSZcm) RRR method Other
K-H6b # 8 10 )1Hg He 1.50 1.32 5406 (a) Second run
[1.05:1'~ of K-H6a
K-H6C # 8 100 )ng He 1.50 1.35 5330 (a) Third run
[1.10] of K-H6a
K-0.9H6a # 5 He atm. 0.90 1.29 5570 (a) Done by M.
[1.01]“ Haerle
K-O.9H6b # 4 He atm. 0.90 1.37 5250 (a) Second run
[1.051“ of K-O.9H6a
K-0.5H # 5 He atm 0.50 1.79 4020 (a)
[1.511“
K-.25H1a # 6 He atm. 0.25- 2.37 3030 (a)
(2.051“
K-.25H2a # 6 He atm. 0.25 2.36 3050 (a)
[2.081‘
K-.25H1b # 6 He atm. 0.16“ 3.13 2300 (a) Second run
K-.25H2b # 6 He atm. 0.16' 3.27 2200 (a) Second run
of K-.25H2a
K-.25H3 # 6 He atm. 0.25 2.25 3200 (b)
[1.98]
K-.25H4 # 6 He atm. 0.25 2.23 3220 (b)
[1.96]
K-.1OHla # 9 He atm. 0.10 11.60 620 (b)
[11.30]
K-.10HZa # 9 He atm. 0.10 13.80 520 (b)
[13.40]
K-.10H1b # 9 He atm. 0.09' 11.60 620 (b) Second run
[11.30] of K-.10H1a
K-.10H2b # 9 He atm. 0.09. 13.00 550 (b) Second run

[12.70]

of K-.10H2a

72

 

 

 

Table 3-1. (Continued)
P(4.2K)
Ampoule Circumstance d [Po] Mount
Samples No. at R.T. (mm) (anom) RRR method Other
K-.25V1 # 7 ~10 )ng He 0.25 2.11 3410 (b)
. [1.86]
x-.25v2 # 7 ~10 pHg He 0.25 2.25 3200 (b)
[1.99]
' K-.10V1 # 7 -10 qu He 0.10 7.70 930 (b)
(7.421“
K-.1OV2 # 7 ~10 pHg He 0.10 7.16 1000 (b)
(6.881“
K-.1OV3 # 7 ~10 )1Hg He 0.10 8.83 810 (b)
[8.49]
K-.10V4 # 7 ~10 pHg He 0.10 7.83 920 (b)
[7.49]
K-.10A1 #10 Ar atm. 0.10 6.22 1160 (b) Without
[5.91] molecular
sieve
K-.10A2 #10 Ar atm. 0.10 8.31 870 (b) Without
[7.99] molecular
sieve
K-.10A3 #11 Ar atm. 0.10 3.67 1960 (b) With
[3.37] molecular
sieve
K-.10A4 #11 Ar atm. 0.10 5.00 1440 (b) with
[4.59] molecular
sieve

 

'Diameters of corroded versions were estimated by the change of room
temperature resistances.

“Values are determined by p
are determined by P0 z P(~

=p(4.2K) - 0.28 nil-cm. all other values

1K).

73

Table 3-2. Characteristics of bare free hanging pure Rb

samples (from one ampoule)

 

 

 

9(4.2K)
Ampoule Circumstance d [Pb] Mount
Samples No. at Rm. Temp. (mm) (nflcm) RRR method
Rb-Hl #1 He atm. 1.50 41.8 300 (a)
[35.61“
Rb-HZ #1 He atm. 1.50 44.8 280 (a)
[38.61“
Rb-V1 #1 ~10 pHg He 1.50 31.0 403 (a)
[24.7]
Rb-VZ #1 -10 (ng He 1.50 31.0 403 (a)
[24.9]

 

“Values determined by P =P(4.2K) - 6.252n em, all others
determined by PC as MAR).

Table 3-3. Characteristics of bare free hanging pure Li samples
(from same batch)

 

 

 

Circumstance d P(4.2K) Mount
Samples at Rm. Temp. (mm) (nflcm) RRR method Other
Li-Hla He atm. 3.0 9.0 1030 (c)
Li-HZa He atm. 3.0 10.8 860 (c)
Li-H1b He atm. 3.0 8.3 1120 (c) Second run
of Li-la
Li-H2b He atm. 3.0 10.4 900 (0) Second run

of Li-2a

 

74

Table 3-4. Characteristics of bare free hanging pure Na samples.

 

 

 

P(4.2K)
Circumstance d [Po] Mount
Samples Source at Rm. Temp. (mm) (nflcm) RRR method
Na-Hl Gallery He atm. 1.0 12.40 380 (a)
Chemical
Company
Na-HZ Callery He atm. 1.0 13.00 360 (a)
Chemical
Company
Na-H3 J. C. He atm. 1.0 1.02 4660 (a)
Garland [1.01]
Na-H4 J. C. He atm. 1.0 1.00 4720 (a)
Garland [1.00]
Na-.25A1 J. 0. Ar atm. 0.25 1.18 4010 (b)
Garland [1.18]
Na—.25A2 J. C. Ar atm. 0.25 1.14 4160 (b)
Garland [1.14]
Na—.25H1 J. C. He atm. 0.25 1.42 3350 (b)
Garland [1.42]
Garland [1.49]
Na-.10H1 J. c. He atm. 0.10 2.08 .2270 (0)
Garland [2.08]
Na-.10H2 J. C. He atm. 0.10 2.57 1850 (0)
Garland [2.57]

 

75

N!-OQ .1909 9! '0

 

 

 

 

 

Fig. 3-5. The way of sucking potassium into a tube: l.

Melting
potassium, 2. Tube, 3. Syringe.

76
experiments. The surface of K samples touching teflon
turned black right away, but the rest of the sample below
the very thin surface remained shiny during all the
experiments.

Two Na samples encased in polyethylene tubes were made
in the way described above.

One K sample was coated by cleaned paraffin oil and
several were measured while in contact with plastic.

To try to grow a single crystal K sample, a beaker of
0-122 White Heavy Paraffin Oil from Fisher Scientific
Company was carefully cleaned in the system shown in Fig.
3-6. The oil was heated to about 150°C while it was being
pumped by a mechanical pump for about 3 days until no more
bubbles came out. Then the input of the mechanical pump
was closed and liquid nitrogen was poured into the thermos
and the input valve of the cold trap was opened to better
evaluate the system. Some crumbs of clean potassium were
put into the oil to clean out left over dissolved air or
other impurities. This cleaned paraffin oil was used sepa-
rately also for coating a K sample, (sample K-0). A piece
of precision bore pyrex capillary, with internal diameter
d=l.5 mm from Wilmad Glass Co., Inc., was carefully cleaned
in the same manner as the plastic tubing described above.
The cleaned oil and pyrex capillary were moved into the Ar
filled glove box. First, a little cleaned paraffin oil was
sucked into and pushed out of the capillary with a syringe

to lubricate the internal wall of the capillary. Second,

77

.muo__oa xm. oa>h o>o_m cm_soo_oz otc_m cacao—o

     

 

 

 

 

 

 

ea_z to___c auto e_oo .e .toaoto_e e_ae_s .m .o>_o> .5 .esoa so. .m
.oum_auuo: .N .__o e.mmmtmm ._ "__o e.muatma mc_>m_tsa ecu Ecum>m one .mum .m_m
1.1. L
.l .. N
u. u. 4.. . .. .m

i m m ”m F a.

m .

Doooo‘. . .J

I'
l
J
i
l
l
l

 

 

 

 

  

 

 

 

 

aEsc .mo_cmzooe oh

78
molten potassium was sucked into the capillary and allowed
to solidify. Third, the cleaned paraffin oil was poured
into a clean hot-beaker, the part of the capillary with
potassium in it was submerged in the oil, and the oil was
heated to about 70°C, a few degrees above the melting point
of potassium. The capillary was then slowly pulled out of
the hot oil at the rate of 0.5 cm per half an hour (see Fig.
3-7). After growth, the sample was pushed out of the capil-
lary with a syringe. Its ends were carefully scraped with a
cleaned surgical knife to clean off the oil and were cold
welded to the current and potential leads as in method (b)
of Fig. 2-4. After the measurements were completed, a 1 cm
long piece of this sample was cut out and put into a thin
wall glass tube filled with cleaned paraffin oil to protect
the sample from reacting with air. Then it was tested on
GE. XRD-S x-ray machine with X= 0.7107 A and beam diameter
d = 0.4 mm. The result showed a sharp peak at 29 = 15.6°.
From Bragg equation nk = 2dsin0, d = 5.2 A was obtained for
n =- 2, thus, at least, within the spot of x-ray beam, a
single crystal part with d a 0.4 mm was found. Unfortunate-
ly, since only one spot of the sample was tested, the veri-
fication of the whole piece as a single crystal K sample was
not established, but Dr. P. A. Schroeder (ref. 64) and his
collaborators did grow some single crystal K samples in the
Netherlands using the same technique with even faster pull-
ing speeds. Their samples were tested with x-ray diffrac-

tion at three different spots along each sample.

79

 

:IEZEE

 

 

 

 

 

 

 

'_1C)

 

 

 

 

 

 

 

 

 

in.
1 ::
‘l. .. -Lz- --.
2__k
\
34""‘*'::::E::::::

 

Fig. 3-7. The system for pulling single crystal potassium
sample: 1. Precision bored PX capillary,
2. Cleaned paraffin oil, 3. Hot-beaker.
4. Manipulator with adjustment in two perpendicular
directions.

 

80

The K(0.077 at% Rb) alloy was obtained by first making
a.KK1.3 at% Rb) mixture and then adding some of this mixture
to pure potassium to form a.KK0.077 at% Rb) alloy. The
K19.40 at% Rb) alloy was obtained by adding Rb directly to
pure potassium. The percentages above were calculated from
the weights of each material, which were weighed on a preci-
sion scale inside the glove box (ref. 22).

The samples of the alloys were extruded through a die
and mounted in the same way as the pure K samples described
earlier. For the KK9.40 at% Rb) alloy, the resistivity was
high. Since a high resistance sample would cause big heat-
ing effects at low temperature when a current passes through
it, the samples and potential leads were thick (d a 3.0 mm)
and short (potential leads were ~l.8 cm apart, each sample
was ~4 cm longJ to keep the resistance low. The sample
holders were shortened for holding those short samples.

The characteristics of the possible single crystal K
sample, of the K samples coated by clean paraffin oil, and
those enclosed in polyethylene or teflon tubes, are given in
Table 3-5. The characteristics of the K-Rb alloy samples
are given in Table 3-6. The characteristics of the Na
samples encased in polyethylene tubes are given in Table

3-7.

81

Table 3-5. Characteristics of K samples in contact with oil or plastics

 

 

P(4.2K) Days

Ampoule Circumstance d Mount 9 Rm.

Samples No.

at Rm. Temp.

(mm)

[ P ]
(nflgm)

RRR

method

Temp.

Other

 

K-S

K-O

K-Phla

K-Phlb

K-Phlc

K-Phld

K-Phle

K-PhZa

#12

#15

#8

#8

#8

#8

#8

#15

Try to
be grown
as a
single-
crystal
sample
Ar atm.

coated by
cleaned
paraffin
oil

Ar atm.

in a

polyethy-
lene tube
10 pHg He

in a

polyethy-
lene tube
10 pHg He

in a
polyethy-
lene tube
He atm.

in a

polyethy-
lene tube
10 pHg He

in a

polyethy-
lene tube

100 qu He

in a
polyethy-
lene tube
He atm.

1.5

1.5

1.6

1.6

1.6

1.6

1.6

.9

1.17
[0.91]

1.32
[1.07]

2.97
[2.70]

1.68
[1.42]

1.18

1.20

1.20
[0.95]

3.30
[3.02]

5150

5450

2420

4280

6090

5990

5990

2180

(b)

(a)

(a)

(a)

(a)

(a)

(a)

(b)

13.0

0.5

11.0

24.5

26.0

27.0

0.5

Second
run of
K-Phla

Third
run of
K-Phla

Fourth
run of
K-Phla

Fifth
run of
K-Phla

Table 3-5. (Continued)

82

 

 

 

P(H.2K) Days
Ampoule Circumstance d [ Po] Mount 9 Rm.

Samples No. at Rm. Temp. (mm) (nflcm) RRR method Temp. Other

K-Ph2b #15 a. in a .9 1.35 5330 (b) 2.5 Second
polyethy- [3.02] run of
lene tube K-PHZa

b. He atm.

K-Ph2c #15 a. in a .9 1.32 5060 (b) 13.0 Third
polyethy- [1.17] run of
lene tube K-PHZa

b. He atm.

K-Ph2d #15 a. in a .9 1.28 5620 (b) 73.5 Fourth
polyethy- [1.03] run of
lene tube K-PHZa

b. He atm.

K-TH1a #16 a. in teflon 1.5 8.31 855 (b) 0.5

tube [8.10]
b. He atm.

K-TH1b #16 a. in teflon 1.5 5.17 1390 (b) 2.5 Second

tube [n.87] run of
b. He atm. K-TH1a

K-TH1c #16 a. in teflon 1.5 1.62 5370 (b) 13.0 Third

tube [1.33] run of
b. He atm. K-TH1a

K-TH1d #16 a. in teflon 1.5 1.20 5990 (b) 73.5 Fourth

tube [0.99] run of
b. He atm. K-TH1a

K-PA1a #1“ a. in a .9 5.u6 1320 (b) 0.5
polyethy- [3.02]
lene tube

b. Ar atm.

K-PA1b #1” a. in a .9 1.u5 H960 (b) 9.5 Second
polyethy- [1.20] run of
lene tube K-PA1a

b. Ar atm.

Table 3-5.

(Continued)

83

 

 

 

P(“.2K) Days '
Ampoule Circumstance d [ 90] Mount @ Rm.

Samples No. at Rm. Temp. (mm) (chm) RRR method Temp. Other

K-PA1c #1“ a. in a .9 1.39 5170 (b) 16.0 Third
polyethy- [1.1“] run of
lene tube K-PA1a

b. Ar atm.

K-PA1d #1“ a. in a .9 1.57 “580 (b) 19.0 Fourth
polyethy- run of
lene tube K-PA1a

b. Ar atm.

K-PA1e #1“ a. in a .9 1.36 5290 (b) 97.0 Fifth
polyethy- . [1.12] run of
lene tube K-PA1a

b. Ar atm.

K-PAZa #1“ a. in a 1.6 5.“8 1310 (a) 0.5
polyethy- [5.20]
lene tube

b. Ar atm.

K-PAZb #1“ a. in a 1.6 1.22 5890 (a) “.5 Second
polyethy- [0.97] run of
lene tube K-PAZa

b. Ar atm.

K-PAZc #1“ a. in a 1.6 1.38 5210 (a) 16.0 Third
polyetny- [1.12] run of
lene tube K-PA1a

b. Ar atm.

K-PAZd #1“ a. in a 1.6 1.23 5850 (a) 19.0 Fourth
polyethy- run of
lene tube K-PA1a

b. Ar atm.

K-PA2e #1“ a. in a 1.6 0.95 7570 (a) 97.0 Fifth
polyethy- [0.73] run of
lene tube K-PA1a

Ar atm.

84

Table 3-5. (Continued)

 

 

 

p(“.2K) Days
Ampoule Circumstance d [ Po] Mount @ Rm.
Samples No. at Rm. Temp. (mm) (nflcm) RRR method Temp. Other
K-KA #17 a. In con- 1.5 1.28 5620 (a)
tact with [1.03]
two pieces
of Kel-F
b. Ar atm.
K-PPA a. Bare sam- 1.5 1.22 5890 (a)
ple with [0.96]
potential
leads in
polyenty-
lene tube

b. Ar atm.

 

85

 

 

 

Table 3-6. Characteristics of bare free hanging K-Rb samples
. P(“.2K)
Ampoule Circumstance At % d [ no] RRR(“.2)Mount
Samples No. at Rm. Temp. Rb (mm) (nflcm) [RRRO] method
K—Rb1 K: #58 #6 He atm. 0.077% 0.25 -- --- (b)
: #2 [11.2] [6“0]
K-Rb2 K: #58 #6 He atm. 0.077% 0.25 --- --- (b)
Rb: #2 [11.6] [620]
K-Rb3 K: #13 Ar atm. 9.“OO% 3.20 1010 8.30 (a)
Rb: #3 [1010] [8.30]
K-Rb“ K: #13 Ar atm. 9.“005 3.20 10“0 8.10 (a)
Kb: #3 [10“0] [8.10]

 

86

 

 

 

Table 3-7. Characteristics of Na samples encased in polyethylene tubes
P(“.2K) Days
Circumstance d [ 90] Mount 6 Rm.
Samples Source at Rm. Temp. (mm) (nflcm) RRR method Temp. Other
Na-PA1a J. 0. Ar gas 1.6 2.27 (a) 12.0
Garland
Na-PAZa J. C. Ar gas 1.6 2.52 (a) 12.0
Garland
Na-PA1b J. 0. Ar gas 1.6 2.29 (a) 33.0 Second
Garland [2.29] run of
Na-PA1a
Na-PAZb J. C. Ar gas 1.6 2.“O (a) 33.0 Second
Garland [2.39] run of

Na-PAZa

 

87
3.5 MEASQBEMENI_MEIEQQ

3.5.1 W

More than fifty runs were made in this study. In each,
we measured PULZK), Po (usually), and both dP/dT and G from
0.07K to 4.2K. In each run, two twin samples or similar
samples were measured.

3.5.1.1 R§295K)

First, the room temperature resistance of sample was
measured using a constant current supply and a Keithley
digital nanovoltmeter. The room temperature T was also
measured. Then the room temperature resistance was
converted to 295K resistance R(29SK) by assuming a linear

temperature dependence, i.e.,

P(295K) x L(295K)

 

 

 

31.22252. . “295‘“ 0.2220. (3-2)
R(T) _ MT)
P(T) x L(T)
A(T)

where L and A are the length and cross-section area of
sample respectively, and L4295k)/A(295K) = LATO/A(T) to a

precision better than 0.1% at room temperature. Hence,

 

_ 9(295K)
R(295K) - R(T) -_3(TT_
= R(T) 1 + 21.80 (3_3)

l + a(T - 273.2)

assuming

88
P('I‘) = P(273.2K) [1 +oz(T - 273.2)] (3-4)

Actually, we happened to use the approximation R(295K) =
R(T) x 295/T for some measurements of potassium, this made
R(295K) not more than 0.5% too big in our room temperature
range.

The a and 9(295K) used for K, Li, Rb, and Na were those
as listed in Table 3-8 (from Landolt-Bornstein Numerical
Data and Functional Relationships in Science and technology,
Group III, Volume 15).

3.5.1.2 Circuit for Measuring P(4.2K), Po, and
dP/dT.

When the samples were cooled to 4.2K or lower, the
circuit in Fig. 3-8 was used. This circuit consisted of a
SQUID (Superconducting Quantum Interference Device) null
detector and three resistors wired in series. These three
resistors were the samples, R1 and R2, and a standard
resistor RSt =.LJ34 #9 made of a tin-indium alloy, chosen
to be superconducting at about 3.8K. For thin pure samples
and thick alloys with high resistances at low temperature,
an inductor (Lt z 50 pH) was inserted in series with the
three resistors to keep the relaxation time of the circuit
long enough to have the SQUID lock properly. The inductor
and all the wires in this circuit were made of Niomax CN, a
multifilament Nb-Ti superconducting (Tc > 4.2K) wire with
Cu-Ni cladding made by Imperial Metal Industries (ref. 22).

To keep the noise introduced by stray magnetic fields

as low as possible: 1) Rst and most of the wiring except

89

 

 

 

Table 3-8. The electrical resistivities of some alkali
metals
Metals (273.2K) a P(295K) (calculated)
K 6.45 .0050 7.19*
Li 8.50 .00445 9.32
Rb 11.25 .0051 12.50
Na 4.29 .00485 4.74

 

* This number should be 7.15 from the calculation, but we
used 0(295) = 7.19 for K to be consistent with van Kempen
et al. (ref. 14).

90

 

Mixing Chamber

 

 

 

 

 

 

 

 

 

 

Thermal Link
with Electrical
Isolation -—=)
EKJLHE)
U1 Controller U2
3::::;:3 ‘ ‘ [:;::::
% RL1 81.2%
Q 50010
:23 a:

 

 

 

 

 

 

 

 

 

 

 

 

 

1: G] Sample Can G2 I

_.__-__-__--..___...__...._.---J

Fig, 3-8. The low temperature circuit. The components inside
the broken line are inside the sample can.

91
that near the samples were shielded in superconducting lead
(Pb)('Tc = 7K) tubing. 2) All the wires leading to the
samples were carefully tied or varnished down to reduce
vibration and thus reduce magnetically induced currents.
3) The vacuum can, with the sample can inside, was shielded
by one Nb can, and the whole refrigerator was surrounded
with another u-metal can. Both the Nb can and the u-metal
can had openings only at the tops.

The wires were drawn out of the can through a special
electrical feedthrough. This feedthrough was made by
running several pieces of insulated Niomax wire through a
clean 1/8” diameter stainless steel tube, which was then
inverted in a small cup of liquid Stycast 1266 epoxy that
was then allowed to harden.

The ratio of the resistances for any two of the three
resistors in this low temperature circuit could be measured
by using the current comparator (mentioned in section 3.1).
Let Rm and RS represent the resistances of these two
resistors respectively; When the current comparator was
used to compare them, the master current Im was passed
through Rm while the slave current IS was passed through RS.
Here, IS = CIm, where C was the switch setting of the cur-
rent comparator. A standard current reversal technique was
used to eliminate thermal EMFs generated in this circuit as
follows. For currents going one way (the directions of I
currents through two resistors in this circuit should be

opposite to each other), the SQUID voltage is

92
v+ = I Rm - I R + v
m s s T

ImRm - cxmas + vT (3-5)

where VT represents the stray voltage due to any thermal
EMF. After the currents on the two sides are reversed, the

SQUID voltage is

V

-ImRm + ISRS + VT

—ImRm + CImRS + VT (3-6)

C was adjusted to make V+ = V‘. Then,

ImRm - CImRs = -ImRm + CImRs (3-7)

and hence, Rm/Rs = C independent of VT'

At 4.2K, the resistances of two samples, R1 and R2 in
Figure 2-8, were measured by using the current comparator
to compare each of them with Rst' Then the ratio of the
4.2K resistances to the 295K resistances were calculated.
neglecting the change of L/A between 4.2K and 295K (the

error caused by this neglecting will be discussed in section

3-5), one has
P(4.2K)/P(295K) = R(4.2K)/R(295K) = l/RRR.
For pure samples,

P(295K) Ppure (295K),1and

(”4.2K) P(295K)/RRR.

For K-Rb alloys, assuming Matthiessen's Rule (MR) as a first

approximation,

93

P(295K) P(4.2K) + PpureK(295K).

Therefore,
P(4.2K) a PpureK(295K)/(RRR-l).

The impurity-dependent deviation from MR might cause an
error as large as 1-3% when RRR is as small as 8 for a 9.4%
K-Rb alloy (ref.65).

The ppure(295K)s for K, Li, Rb, and Na are listed in
Table 3-8.

As described in section 3.1, there are two ways to
study the temperature dependence of resistivity at low
temperature. One is to measure P directly, the better one
is to measure dP/dT. We measured dP/dT in this study.

To obtain dP/dT of a sample, (e.g., R2), the current
comparator was used again to compare R2 (as Rm) to R1 (as
R5). The reference R1 was kept at a constant temperature T1
by regulating the mixing chamber temperature with a tempera-
ture controller monitored by GRT4. The temperature of the
sample R2 was changed from T2 to (T2 +70T) either by using
heaters U2 and L2 alternately with the same power input into
each or only L2 with two different powers. (These two
methods were tested several times in each run, to make sure
both of them gave consistent resultsd When the sample was
at temperature T2, the ratio of its resistance to the refer-
ence resistance R(Tl) was measured as C(Tz) = R2(T2)/R(T1).
When it was at T2 + AT, the ratio was measured as C(T + AT)

3 R2(T ‘1' AT)/R1(T1)o Then,

94

 

C(Tz)

= = —— = — 3’8
R2(T2) R2 92 ( )

 

since the change of L/A in AT, typically 0.1K, was less than

10-7%, which is negligible. Hence,

dP _ . AP _ AC
PdT " £15.30 PAT " CAT (3 9)

and,

(39 PAC

dT CAT

The current through the sample was usually 50 mA, or
occasionally, to test for current dependence, 20 mA. There
was never any measurable current dependence in dP/dT.

In order to get optimal precision, the noise caused by
the reference resistor must not be worse than that of the
sample. For this reason, the two samples were normally made
as similar as possible, and each of them was the reference
for the other. Even so, the Johnson noise (x/IFTREf) in
the reference would be larger than that in-the sample if the
temperature of the reference was higher than that of the
sample. To minimize this noise, the reference was cooled
down together with the sample to a given temperature, and
then kept at this temperature while the sample was heated to

higher temperatures for taking data. The Johnson noise in

95
the reference resistor was thus always smaller than that in
the sample.

At the ends of most runs, 903 of the samples were
measured by stopping the 1K pot pump and the circulation of
the dilution refrigerator when the temperature of the sample
was below 1K. When the standard resistor Rst' which was
attached to the 1K pot, became normal (TC = 3.8K), while the
temperature of the samples was still around 1K, the ratio of
resistance of each sample to the resistance of the standard
was measured using the current comparator. The resistivity
at about 1K was then calculated as PULzK) was done above.
According to van Kempen et al. (ref. 14), W.P. Pratt, Jr.
(ref. 59), and our new results, when the temperature is
below 1K, the electron-phonon scattering contribution to
the resistivity was negligible and the electron-electron
scattering contribution was at least 3 orders of magnitude
smaller than no in K, Li and Na. Thus, P(lK) was chosen as
80'

As already mentioned, temperature measurements were
critical in this study. Even though the thermometers were
already well calibrated (ref. 22), the Wiedemann-Franz (W-F)
Law was used for in-situ checking to be sure that no
systematic errors crept into the temperature measurements.
The procedure was as follows. In Fig. 3-8, RLl and RLZ,
both Ag-0.1 at% Au alloys, were used for weak thermal links
between the mixing chamber and the samples. For each link,

the U-heater was placed at one end (called the U end), which

96
was connected to the mixing chamber. Another identical
heater, the L-heater, was placed at the other end (called
the L end). The reference resistor and the mixing chamber
were kept at a constant temperature. The sample was first
heated to a temperature T by running a current through the U
heater. In thermal equilibrium there should not be any heat
flowing through RL (see Fig. 3-8). There should thus not be
any temperature difference between the U and L ends. Then
the sample was heated to T + AT by running the same current
through the L heater, this time there was a continuous heat
flow from the L heater through RL to the mixing chamber,
which caused a temperature difference between the U and L
ends. In these two cases, the temperature of the U end
should not change since the heating power in these two cases
were the same, so that the heat current flowing from the 0
end to the mixing chamber did not change. Thus, we can say
that the temperature difference between U and L ends in the
second case was equal tolAT. Applying the W-F Law, K/OT =

L0, to RL' one has:

RL = LOT AT/Q = LOT AT/RhIz (3-10)

where Rh was the resistance of the heater, I was the current
through the heater, and Tave = T + AT/Z was taken as T. The
resistance of the alloy RL was dominated by its residual
resistance R0 at low temperatures, so Tave AT/RhI2 should
not change for every datum we got. This was used for in-

situ checking the product T-AT and thus AC/CTnAT. The

97
uncertainty in T is estimated much less than 1% except near

0.1K where the uncertainty is closer to 1% (ref. 59).

3.5.2 Thsrmoelsstris_Ratic_§_and.lhermsscwsr_s.

The thermoelectric ratio G of each sample was also
standardly measured in each run, since it was easily
obtained. ‘The only additional requirement was a G heater
at the hot end of each sample (Fig. 3-8).

As mentioned in section 1.1,

c = -%— , i.e., = —¥— (3-11)

q 8&0 Q E=0

G was measured by sending a current IG to the G heater
to heat the sample from one end, and then sending another
current I through the sample to cancel out the resulting
thermal voltage. This voltage was detected by the SQUID.
The master current was normally'set to be 50 x 0.1 mA, the
cancelling current was supplied by the slave side of the
current comparator, (i.e., the cancelling current was C x
5 mA. The resistance of the G heater, a Dale resistor, was
measured at liquid helium temperature, and stayed constant
(change less that 0.1%) below 4.2K. The heating power was

calculated from

RGIé (3-12)

0.
ll

thus

0 = c x 5 mA/RGIé (3-13)

98
The thermopower S of only one K sample (K - 1/2 H) was
measured in this study. The G heater mentioned above was
still needed. In order to measure the temperature differ-
ence across the sample, two Germanium Resistance Thermom—
eters GRT4 and GRTS were placed on the copper supports of
potential leads (section 3-3).

As mentioned in section 1.1,

E V
S a — ' i.e., S = ‘—

S was measured by sending a current IG to the G heater
to heat the sample from one end, and then sending another
current I through the reference to cancel out the resulting
thermal voltage. This thermal voltage was detected by the
SQUID. 'The cancelling current was the same as in the G
measurements mentioned above, i.e., I = C x 5 mA. So the
cancelling IR drop was C x 5 mA x Rr' The different
temperatures of the two ends of sample were measured by GRT4
and GRTS. Then AT was obtained. Thus,

C x 5 mA x Rr

3 = AT (3-14)

 

To determine S, it is thus necessary to determine Rr
('1‘). Rr was measured only at 4.2K. Since [Rr (4.2K) -
Rr (OKil/R[(4.2K) = [p£(4.2K) - Pr(OK)]/Pr(4.2K), where
Pr(4.2K) - Pr(0K) = 0.28 nflcm for potassium (ref. 12,59).
Rr(0K) was then obtained as Rr(0K) = Rr(4.2K) [l —

0.28 nSZcm/Pr(4.2K)]. Furthermore, assuming R(T) = R(OK) +

99
Ce'ei‘r/T with 9* = 20K, where C could be obtained from
R(4.2K) = R(OK) + Ce'20/4'2, R(T) was calculated for
temperatures between 0K and 4.2K. The uncertainty of this

calculated R(T) is estimated as 3%.

3.6 W

3.6.1 W

In G and S measurements, our attention was focused on
the temperature dependence of G or S.

As mentioned in section BJLZ, G was calculated by G =
C x 5 mA/RGIé. The 0.1 x 50 mA current setting of the
current comparator had 0.16% uncertainty. The uncertainty
in the resistance of the heater RG was 0.1%. The error in
measuring IG was due mainly to the digital round-off of the
DVM used to measure it; this could cause 0.5% error at the
worst. Finally, the uncertainty of the resistance used to
measure this current was 0.05%. Taken together, these
uncertainties sum to, at worst 1%.

The major source of uncertainty in G was in the deter-
mination of C, due to thermal EMF noise and Johnson noise.
This uncertainty was less than 1% when the temperature was
above 1K, but the percentage error got worse as the magni-
tude of C and the heat input got smaller at lower tempera—
tures. At the lowest temperatures, the uncertainty in C
could be as large as 5%.

The major uncertainty in T measurement was random error

due to thermal fluctuations and thermal drifts in the

100

system. The worst case was when T was around 3K, since the
dilution refrigerator had not been completely started, the
cooling power was varying with time, thus temperature regu-
lation could not be used. This uncertainty in T is esti-
mated as only 0.5% in the worst case.

As mentioned in section 3JL2, S was calculated by S =
(C x 5 mA x Rr)/AT. The uncertainty in 0.1 x 50 mA current
setting was 0.16% (see above). The uncertainty in C was
less than 1% in whole temperature region for this S measure-
ment. The uncertainty in AT is estimated as 1% (section
3.5.1, ref. 59) in the worst case. The major uncertainty in
S was in determination of Rr' This uncertainty is estimated
as 3% (see section 3.5.2).

3.6.2 Uncertainties in P(4.2K),PQ, and dP/TdT

Measurements

P(4.2K) was determined from P(4.2K) = P(295K) x
R(4.2K)/R(295K), where R(295K) was measured by R(295K) = R(T)
(l + 26.8 xoz)/(1 + a(T - 273.2))or R(295K) = R(T) x 295/T
(see section 2.4.1). 90 was determined similarly. So, the
uncertainties in 1K4.2K) and 90 were dependent on three main
factors: a) The uncertainty of R(295K) was mainly due to
the uncertainty of the room temperature resistance R(T)
which was always measured with better than 1% precision.
in EH4.2K) or R0 were measured against Rst within 2%, due to
the temperature dependence of Rst near 4.2K (ref. 22).
RUL2K) was also affected, in lesser degree, by the small

temperature variation of the liquid He bath due to changes

101
in atmospheric pressure. c) The main error was the
systematic error due to ignoring the change of L/A when. the
sample underwent thermal contraction from room temperature
to 4.2K or 1K. The room temperature linear thermal
expansion coefficients of the samples (in units of 10"6 K'l)
were 83 for K, 71 for Na, 66 for Rb, and 45 for Li. These
coefficients would stay constant from room T to about their
Debye temperatures @Ds (330K for Li, 160K for Na, 114K for
K, and 65K for Rb), then start turning down, at about 0.161)
they would decrease as fast as T3. Calculations showed this
caused about 2% errors in FK4.2K) and Do. The 0.5% opposite
systematic error from using R(295K) = R(T) x 295/T as an
approximation is negligible. In summary, the error in
1N4.2K) and 90 measurements could be as high as 4%,
including a correctable systematic error of 2% due to c).
But, any error in 1M4.2K) and Po would not affect the form
of plots of the final quantity dP/dT versus T either as
p(4.2K) AC/C AT or PO AC/C AT. The only effect of this
error would be to multiply dP/dT by a constant close to 1.

In the calculation of the quantity' PAC/C AT, C was

ave
shown to have precision better than 0.1 ppm earlier, so the

main uncertainties came from 9. AC, and AT. a) P(1K) was
chosen to be P0 in this calculation. As already mentioned,
our attention was focused on the temperature region where
electron-electron scattering was dominant, in this tempera-
ture region the temperature dependent part of P was at least

3 orders of magnitude smaller than no or P(lK) (see section

102

3.5.1). So this caused less than 0.1% random error. Some-
times P(4.2K) was used instead of Po; and 9(4.2K) could be
20% bigger than no. This, however, represents only a dif-
ferent renormalization of the quantity of interest. The
systematic errors in P(4.2K) and PO measurements were al-
ready discussed above. b) In many cases of this study,

AC could be very small or even zero, so the percentage
uncertainty of AC could be very large. To reduce the
uncertainty in AC when AC was small, a micro-computer
averaging. technique was used to measure C to within 0.01
ppm. Typically, except at the lowest temperature, the un-
certainty in AC was less than 1%. c) The uncertainty in
AT, which was estimated using w-F law in-situ checking was

within 1-2% (ref. 59).

CHAPTER 4

Experimental Results and Analysis

W

As mentioned in section 1.2.1, C. W. Lee et al. (ref.
8) used the same technique as described above to measure
free hanging, bare, high-purity, thick (d = 0.9-3.0 mm) K
samples under Ar gas. Their dP/dT data varied closely as T,
(i.e., pm = A132, from 1.3K to 0.35K. "A" did not change
much from sample to sample, having a mean value of 0.24 i:
0.02 chm/Kz. This value is consistent with that expected
for electron-electron scattering, 0.17 pacm/K2 (ref. 7,21).
Below 0.35K, their data showed deviations from T2. Their G
data had the approximate form C = C0 + B*T2 + (C*/T) x
exp(-6*/T), with co = -0.03 i 0.03v‘1, 13* = -o.3o i
0.01v‘1K‘2, and 9* a 23 3: 2K.

Using a better glove box, an improved sample can (see
section 2.3), and a thin copper can attached to the mixing
chamber to shield the sample can and the thermometers on it
from heat radiation (see section 2.3), we measured 14 free
hanging bare K samples with d = 1.5-2.0 mm; 4 were still
measured under Ar atnh; 6 under He atm. (one among these was
sample K-H4 annealed at 50°C for 30 minutes); and 4 under
partial vacuum.

103

104

4.1.1 Ih£_B£SiSLiXi£¥.

The results of resistivity measurements on these thick
samples are shown in Fig. 4-1. Again, p(T) varied closely
as AT2 from 1.1K to 0.2K-+0.4K (for different samples), with
A = 0.24 1; 0.02 pflcm/K2 for Ar gas (the same as C. W. Lee
found), A = 0.25 i 0.02 chm/K2 for He gas without annealing
at 50°C, and A = 0.22 ¢_0.01 pflcm/K2 for partial vacuum.

The sample annealed at 50°C showed the smallest p(4.2x) =
1.22 x 10"9 Gem and the smallest A = 0.19 chm/Kz. Below
(L2K-+0.4K, all the samples showed an anomalous turn-up
similar to that found by Lee. The parameters of the samples
are given in Tables 3-1 and 4-1.

In the data of our 6 samples cooled in He gas, the
higher coefficient A corresponded to higher residual
resistivity no. But our samples cooled in Ar gas and
partial vacuum did not show any simple relation between A
and Po. We plot A versus 100 [no a p(4.2K) - 0.28 chm/KZ]
for 5 of our samples cooled in He (Fig. 4-2). These samples
are consistent with the constant value 0.25 :_0.02 chm/KZ.
However, they also fit A = A0 + cab (the solid line in Fig.
4-2) with A0 z 0.17s poem/K2 and c = 5.7 x 10‘5r’2. This
latter fit would imply a Po-dependent part in A of about
A1100) = A - A0 = 5.7 i 0.5 x 10‘590. This is somewhat
larger than that expected for electron-defect inelastic
scattering (section 2.2AD. With either fit, A has a dominant

component AO due to electron-electron scattering. The

linear fit in no would suggest an additional component due

105

 

 

 

 

110—- : K... x -~a
o - K-9 K-H4
0 K—10 K -H5a
$100 ' K-11 K -H5b
E 0 K- H1a K - H63
c K- H1b K- H6b
{SQOL . 0 K- H2 K - Hsc
r—5 E -- K- 4a
2 2° }01 C.W. Lee
\ :o —--- K- 53
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V v ' A A A A O
50— . ' . ' ' . .
4G .11., l l l i, .1 I
0 0.2 0.4 0.5 0.8 1.0 1.2
T [K]

Fig. “-1.. (P(“.2K)/PT)(AP/AT) versus T for free hanging bare high-

purity thick K samples.

106

Coefficient A and other parameters of thick K samples. To
determine A, we assume Po = P(“.2K) - 0.28 nSlom. t = time at
room temperature.

 

 

 

Diam t P(“.2K) A Circumstance

Sample (mm) (days) (nflcm) (chm/KZ) at room T.
K-8 1.5 0.5 1.50 0.25 Ar atm.
K-9 1.5 0.5 1.72 0.2“ Ar atm.
K-lO 1.5 0.5 1.“? 0.2“ Ar atm.
K-11 2.0 0.5 1.26 0.22 Ar atm.
K-H1a 1.5 0.5 1.70 0.26 He atm.
K-H1b 1.5 26.0 1.65 0.25 He atm.
K-HZ 1.5 0.5 1.27 0.23 He atm.
K-H3 1.5 0.5 1.97 0.27 He atm.
K-H“ 1.5 0.5 1.22 0.19 He atm.

annealed at

50°C for

30 min.
K-H‘Sa 1.5 0.5 1.27 0.22 10 flHg He
K-HSb 1.5 11.0 1.39 0.21 10 qu He
K-H6a 1.5 0.5 1.23 0.23 He atm.
K-H6b 1.5 2.0 1.33 0.22 10 {1H3 He
K-H6c 1.5 3.0 1.35 0.22 100 pHg He

+ molecular
seive

 

107

to electron—defect or electron-impurity inelastic
scattering. However, in view of the small number of data
points in Fig. 4—2 and the lack of any correlation of A with
pO in samples cooled in Ar or vacuum, we cannot rule out the
possibility that the observed variation with PC shown in
Fig. 4-2 is just apparent, in that a large number of data
points might end up randomly fluctuating the dashed line.

The anomalous turn-up below 0.35K might be associated
with defects in the samples. Comparing K-Hla (the first
run of sample K-Hl) and K-Hlb (the second run of sample K-Hl
after annealing at room temperature for 26 days), we see
that K-Hlb has a much smaller turn-up than K-Hla. Similar-
ly, sample K-HSb has a smaller turn-up than K-HSa. This
result is consistent with what M. L. Haerle (ref. 22) found;
deformation in pure K samples not only brought the coeffi-
cient A up, but also enhanced the anomalous turn-up at low
temperatures.

More discussions of the anomalous turn-up will be given
in section 4.6, together with the similar anomalous turn-up

in Na, Li, and Rb.

4.1.2 The Thermal-M '

G measurements were made for samples K-ll, K-HSa, K-
HSb, K-H6a, K-H6b, and K—H6c from about 4.2K down to about
70 mK, and for samples K-lO, K-Hla, and K-HZ from about 1K
down to about 70 mK. All the data are shown in Fig. 4-3.
They have the same form as C. W. Lee's data, except the data

of K-HSa and K-H6b, which have small bumps at temperatures

108

.o: c_ po_00u mo_dEmm x xo_;u m to“ JR m> <

 

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Trudi at
2 3 a... .
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law. V
m...
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llllllll O llllllllllllllllllllllll 1mm.
0

 

 

109

.mm_QEmm x xo_cu >u_tsa1£m_z scan mc_mcm£ worm sod p m> o

c: ._.

fie QM 9~

all .3...

 

m A

U I,
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made 5.--:
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110
around 1.5-2.0K. At T < lK, the data of all the samples
almost overlap. .Above 1K, the data diverge from sample to
sample, but can be divided into two groups. In the first
group, data of samples K-HSa, K-HSb, K-H6b almost overlap
and are close to sample K-l of Lee from 2.0-4.2K. These
data stay less negative than the others. These three
samples were under 10 qu He gas without molecular sieve
before being cooled down. The second group of samples, K-ll
(under Ar atm. at room temperature), K—H6a (under He atm.
with molecular sieve at room temperature), and K-H6c (under
100 qu He with molecular sieve at room temperature) almost
overlap each other over the whole temperature range, and are
more negative at T > 1K than the other data. Both groups

have the general form G = G0 + B*T2 + (C*/T) Exp (~68/T)

(see section 2;”. Assuming 9* 23K, for the first group

we got cg = -0.17 i 0.01v‘1, 3* -0.314 1 0.0004v"1z<"2 and

c* = 4100 i 100v‘1K for the second group 00 -0.075 1;

0.015v‘1, 3* = -0.484 33 0.005v‘1K‘2, and c* = 6200 i
140V'1K. These two fits are given in Fig. 4-3 by the dashed
and solid line, respectively.

The reason for the difference between these two groups
might be explained as follows. Without the molecular sieve,
the little amount of He gas (104qu at room temperature)
could be incompletely adsorbed, the heat current from the G
heater only partly passed through the sample, and part of it
through the left—over He gas. But in the calculation, we

still assumed that all the heat current passed through the

111
sample, (iJL, we divided the electric current I by too big
a heat current, so we got a smaller absolute value of G).
The small bumps could be explained as a kind of condensation
of left-over He gas occurred from 2K down to 1.5K, so the
data fell back to the position where they should be, and
overlapped with the second group.

The heat current passed through the left-over He gas
mentioned above could be roughly calculated as follows:
first we assume all the surface exposed to the 10 uHe gas
inside the sample can can only adsorb one layer of He atoms
at 4K. The total surface area is roughly calculated as: S
= 25 cm x 10 cm x 2 (the height of the sample can z 10 cm,
the perimeter of the sample can 2:25 cm, the areas of two
bottoms, sample holders, and samples are all included in the
factor 2). The atom spacing of liquid He is about 7.53.

So, the number of He atoms that could be adsorbed is:

 

The total number of He atoms inside the sample can is

23 -3
n = 6.023 x 10 x 10 x 10 mm x 500 cm3

22.4 x 103 cm3 760 mm

 

 

z 18 x 1015. (where v z 500 cm3)

So, the left-over He gas would be half as much as at room
temperature. To find the thermal conductivity (I(=

l/3 CVIV) of the left-over He gas, we need to calculate ,

112

 

 

CV, and V:
‘l = 8 cm (the diameter of the sample can)
Cv = n x l/2k
6.023 x 1023 10 x 10'3 mm
= .5 x x x .5 x
22.4 x 103 cm3 760 mm

1.38 x 10"16 erg/K
= 1.2 x 10‘9 J/Kcm3

Using the velocity formula of ideal gas,

 

v = 1.6 x/RT/fl = 1.60 :v%.3128 x 4 x 107 erg/4g
= 15 x 103 cm/s.
Thus,
x: 1/3 cvlv z 4.7 x 10‘5 watt/cm-K.

So the heat current passed through the assumed left-over He

gas would be approximately

Q = KSAT/AX ==4.7 x 10"5 watt/cm-K x 3 cm2 x 0.1 K/4 cm

s 3.5 x 10‘6 watt.
This is comparable to the heating power of the G heater:

96 = 123 = (40 pA)2 x 400052 = 6.4 x 10'6 watt.

4.2

we
(a
u
p
['1
Z
C
Ii
0
l
.

111G-‘RTY

4.2.1 W111i!

As mentioned in section 1.2.1, Rowlands et al., unlike
other investigators, found that below 1.3K, their p(T) data
of pure K samples were better fitted by p(T) at T3/2 than
by p(T) at T2. Subsequently, their data were interpreted as
evidence for electron—phason scattering (section 2.2.3). We
noticed that their samples were thinner than ours, with d =
0.79 mm. After reviewing the old data of our group, we
found that the plotting of dp/dT versus T for two pure K
samples (d = 0.9 mm) of 1.1. Haerle cooled in He gas also
showed deviations from a straight line, i.e. deviations from
T2 behavior. In hOpes of discovering the reason for the
difference between the data of Rowlands et a1. and our thick
sample data, we set out to measure a set of thin samples
with d = 0.5, 0.25, 0.16, 0.1, and 0.09 mm in three
different circumstances, Ar atm., He atm., and partial
vacuum.

We found that data for wires cooled in He gas displayed
a clear pattern of unusual behavior which is consistent with
that reported by Rowlands et al. in the region of overlap,
but more complex in form. The characteristics of all these
samples are shown in Table 3-1.

Fig. 4-4 shows a normalized dP/dT (ref. 59) plotted
versus T from 0.07 to 1.8 K for selected samples with

0.0938515 mm, prepared in He gas and cooled down with

114
molecular sieve. They were, in sequence from t0p to bottom

in Figure 4-4: samples K-Hla with d==145 mm 00, K—l/2H

with d 0.5 mm (I), K-1/4H3 with d = 0.25 mm (x), K-1/4H2a

with d 0.25 mm (c), K-1/4HZb with calculated d = 0.16 mm

(o), K-l/10H2a with d = 0.1 mm (9), K-l/lOHlb with
calculated d = 0.09 m (it), and K-1/10H2b with calculated
d = 0.09 mm (o). The characteristics of these samples are
given in Table 3-1. Two samples of each diameter were
measured concurrently. The data for such sample pairs
always agreed quite well with each other; one of the largest
disparities is indicated by the different symbols ({tand<o)
for the two thinnest (d = 0.09 mm) samples.

For comparison with data of K-Hla with d = 1.5 mm, the
dotted and dashed line indicate C. W. Lee's data for 3.0 mm
and 0.9 mm samples, respectively, in Ar gas (ref. 8).
Together with data of K-Hla, these data lines indicate the
range of variation for all "thick" samples of our group
(including old data (ref. 8) and new data in this study) in
both Ar (d = 0.9-3 mm) and He (d = 1.5 mm). Two solid lines
indicate data for two independent d = 0.9 mm samples in He,
measured by M. Haerle. Samples K-1/4H2a and K-l/4H3 both
had d = 0.25 mm but were mounted in different ways.
K-l/4H2a was mounted using method (a), while K-l/4H3 with
method (b) (section 3-2 and Fig. 3-4). Data for our other
d = 0.25 mm samples approximately spanned the range between

these two sets of data.

115

 

 

 

 

E
E
C} o
f; 0
*- , to...
g: .
216.710 Symb.4.2(pflm) o o*0.16m11o1°
Q“ + 17.0 ° ° °
-20- v 12.9 3’
. 13.7 ° m+0.10mm
‘30— I 17.9
x 22.5 o 1".
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0 31.3 Q. o
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o 130 ° .
’6'0 * 116 . .
J 1 I I l I 1 1 i...
0.2 0.4 0.5 0.8 1.0 1.2 14 1.6 1.8
T(K)
Fig. “-4, P(“.2K)(AlnPWAT) vs T for thin wires of K cooled

in a He atmOSphere.

 

116
Fig. 4-5 compares data of Rowlands et al. with the data
of Fig. 4-4. The samplesof Rowlands et al. had d = 0.8 mm

and comparable purity (residual resistivity ratio (RRR) =

I

[R(295K)/R(4.2K)] ~ 6000) to ours. Both the form and
magnitude of their data are consistent with what we expect
for samples of d = 0.8 mm in He gas, sitting right between
d = 0.9 mm data and d = 0.5 mm data. However, the more
complex form of our data rules out the simple T3/2 form for
p(T) that they originally proposed.

To calculate the mean-free-path of electron-impurity

scattering (lei)' we used the equation
lei = (rs/a)2 x 92 A/pu (4-1)

(ref. 4), where rs is the radius of a sphere whose volume
is equal to the volume per conduction electron, a = 0.529 x

10"8 cm, pp

thickest samples had RRR's of about 6000 and 19pl of about 1.2

x 10'3 at temperatures lower than 4.2K. From equation

is the resistivity in microhm centimeters. Our

(4-1) we find a corresponding lei of about 0.2 mm at
temperatures lower than 4.2K. Samples with d S. 0.25 mm,
comparable to or smaller than lei' displayed negative values
of dp/dT in the vicinity of IX, followed by a rapidly
increasing dP/dT above 1.3K due to electron-phonon
scattering. The changes in sign of dp/dT from negative to
positive shown in Fig. 4-4 indicate resistivity minima in
these very thin samples. At lower temperatures, dp/dT

increases more slowly with decreasing temperature. The

117

 

10.0"

m/K

c179“

(i

6.0-

AlnP/AT
s
I

9..
5}

 

 

 

 

 

 

 

 

l I l
02 0.4 0.6 0.8 1.0 1.2 1.4 1.6

T(K)

 

 

Fig. “-5. fK“.2K)(AlnPWAT) vs T for two K samples of Rowlands et al.
(rcf.l3) with d = 0.08 mm cooled in a He atmosphere. For
comparison, the solid lines represent data from Fig. “-4
for K samples having the diameters indicated.

118
other changes in sign of dP/dT at the low temperature end
indicate local resistivity maxima in these samples.

The variations with sample diameter shown in Fig. 4-4
cannot be due to plastic deformation during either
fabrication or cooling of these thin samples. According to
M. Haerle (ref. 22) and the discussion in section 4.1,
plastic deformation makes dP/dT more positive, exactly the
opposite of what is seen here.

To test for effects of surface corrosion, two different
thicknesses of samples were allowed to thin by means of such
corrosion, which occurred naturally inside the sample can.
In Fig. 4L4, the two samples with d = 0.16 mm (K-l/4H2b:o)
and dv= 0.09 mm (K-l/10H2b:o) were corroded versions of the
two samples with d = 0.25 mm (K-1/4H2a:c) and d = 0.10 mm
(K-l/lO H2a:o), respectively. The reduced diameters of
these corroded samples were inferred from their increased
room-temperature resistances. The changes seen with
decreasing thickness due to corrosion are similar to those
seen with decreasing thickness in freshly extruded samples.
The important quantity must thus be the size of the
uncorroded portion of the sample.

To test whether the electron mean free path, lei' is
also an important parameter, measurements of dP/dT were made
on d = 0.25 mm samples of K-0.08%-Rb alloys in He gas.

These alloys have lei =:0.04 mm, obtained from the equation
(4—1) used above. Fig. 4-6 shows the results of these

samples, the solid line indicates sample KRbhc of M. Haerle,

119

 

10.0—

 

 

l l

l

 

.——--

--.- *-——...—-._-

 

 

 

Fig. “-6.

0.2 0.4

P(“.2K)(AlnP/AT) vs T for two K-0.08at%Rb alloy samples
with d = 0.25 mm.

0.6

0:8
7 [10

120
which had the same composition but with d = 0.9 mm. No size
effect was seen; dp/dT increased linearly with T and had the
same magnitude as data of thicker samples with the same
composition. It can thus be concluded that the size effects
described above only occur in samples with diameters compar-
able to lei'

The complex form of the data in Fig. 4-4 and Fig. 4-5,
and especially the negative dP/dT for d g 0.25 mm samples,
are incompatible with most of the models pr0posed to explain
either previous data on K below 1K or "size effects" in
other metals. The standard model (section 2.1) cannot
explain either the general form of our data or the
appearance of a negative dp/dT. The anisotropic electron-
electron scattering model of Kaveh and Wiser (section 2.22M
requires a simple T2 variation, and has no way to explain
the negative dP/dT. It was mentioned in section 2.253 that
the electron phason scattering model, which was used to
explain the T3/2 behavior reported by Rowlands et al.,
cannot describe either the appearance of a negative dP/dT in
our thinnest samples, or the general form of our data. The
Knudsen model of Rowlands et al. cannot give an explanation
either. At first glance, the formula iKT) = BT2 - CT4,
suggested by Rowlands et al., seems able to provide a nega-
tive term for dP/dT. But, Rowlands et al. proposed this two
term formula only for large r/k ratio, where r = sample
radius and k= the mean-free-path of electron-electron scat-

tering (section 2JL4). They argued that for small r/k

121
ratio, the contribution of a term in (r/x)2 or high order
could be neglected. This led to FKT) = BT2 and thus to a
positive dP(T)/dT. Only when r/). is large should the term
in (r/K)2 to be taken into account to give P(T) = BT2 - CT4
(see section 2.2.4 for details). In our case, thick samples

had dP/dT > 0. Thin samples had smaller r (iJL, smaller

ylw

ratio) and thus less contribution from any (r/x)2 term.
According to the Rowlands et al. model, our thin samples
should have had a more positive dP/dT, exactly the opposite
of what is seen. Finally, none of the models of deviations
from Matthiessen's rule used to describe size effects in
other metals can generate a negative dP/dT (see section
2.2.5 for details).

The only model we know, which might explain what we
see, involves the combination of contributions from (a)
UEES, giving a positive term, 2AT, in dP/dT; and (b) NEES
plus surface scattering, giving a negative term in dP/dT,
[suggested by Gurzhi] (section 2JL6), which at low enough
temperatures increases in magnitude with increasing
temperature [as calculated by Black (ref. 38)].

The main difficulty in directly comparing our data with
this model is that the best estimates of 1126, (Normal
electron-electron scattering mean free path) for K yield lge
~lO-100 mm at 1K (ref. 39). The diameters of our samples
which show size effects were d = 0.09-0.9 mm. This means
that we are not in the Gurzhi limit l§e<<d, but rather in

. . . N g .
the OppOSlte limit 1 ee>>d. A Monte Carlo calculation by

122
Black (ref. 38) can be used to estimate behavior in this
limits.Table 4-2 shows some of the results from Black.

Since 1 (mean free path of electron—impurity scattering)

ei
is temperature-independent in first approximation, and d
changes only 2% from room T to lowest T, lei/(d/Z) is roughly
constant. Unlike lei' 129 is temperature-dependent, lge
T'Z. Thus, lge/(d/Z) decreases with increasing temperature.
This means that in Table 4-2, the direction of decreasing
lee/(d/Z) (to the right in Table 4-2) is the direction of
increasing temperature. Since Pal/leff, then dP/dT a:
-(l/leff)2 dleff/dT, where 1eff is the effective mean-free-
path of electrons. This means that an increase of leff
with increasing temperature gives a decrease in p and thus,
a negative dP/dT. In Table 4-2, when lei“d' which is the
situation for our thinnest samples, negative values of dp
/dT persist to at least lg‘e/(d/z) 210. When lei/(d/Z) a 5,
Table 4-2 shows a reduction in the total resistivity P of
about 1% (the limit of accuracy of Black's calculations)
from 1ge/(d/2) sec to lge/(d/z) = 10. In the case of our
thinnest samples, lei/(d/Z) z 4.4, (d = 0.09 mm and lei =
0.2 mm calculated above), life/mm)- 200-2000 at 1K, and the
reduction of p from T a 0K to T = 1K (i.e., from 126/(d/2) =
aoto lg‘e/(d/z) z 200-2000) is only 0.003%. So, the
calculation of Black could be consistent with our data.

The fundamental theoretical questions are whether for
lei ~ d the Gurzhi effect dominates the Knudsen—flow effect

at such large values of lge/(d/Z) and, if so, whether the

123

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124
Gurzhi effect is large enough to be seen. A much more
accurate Monte Carlo calculation will be needed to answer
these questions. If the Gurzhi effect is not the source of
the behavior we see, then there currently exists no
satisfactory explanation for what we see.

As mentioned above, since we are not in the Gurzhi
limit lge<<d, there is no guidance for finding a closed form
to fit our data.

Assuming electron-phonon scattering is negligible for
Tgl.lK (ref. 6,16), and since there are no anomalous low
temperature deviations (described in section 4Ju1) for
Tz0.4K, we tried to fit our. data from 0.4K to 1.1K. First,
we tried a fit which has an electron—electron scattering

term plus an additional term, as in the form

”4.2 if
p dT

 

= 2A'T - CT“. (4‘2)

with 2A'==(L6l pncm/K2 deduced from the data of thick
sample K-Hla, and C, n as two adjustable parameters. The
computer results showed n z 4/3 and C ~ l/x/d for all our
samples.

Considering that A varies a little from sample to
sample, even for our thick samples, we then tried

p4 2 an 55

' -— = 2A'T - ———-- T4/3 (4-2*)
P dT ‘Vd/l mm

 

125
and left only A! as an adjustable parameter. The computer
results showed that Eqn. (4-2*) is a good fit for all our
data from 0.4K to 1.1K (see Fig. 4-7) and that A' varies not
much from sample to sample (see Table 4-3).

Table 4-3. Coefficient A! deduced from the fitting of
Equation (4-2*)

 

 

Samples K-1/2H K-l/4H2a K-l/4H2b K-l/10H2a K-l/10H2b

 

2A'
(pflcm/Kz) 0.61 0.59 0.64 0.70 0.70

 

So far, data discussed in this section are all for K
samples cooled in He gas. To test whether these observed
size-effects are unique for samples cooled in He gas,
measurements of dP/dT were also made on thin samples cooled
in Ar gas or in partial vacuum.

Four thin K samples with d = 0.1 mm cooled in Ar were
measured in this study. They were samples K-l/lOAl, K-l/lO
A2, K-1/10A3, and K-l/10A4. Six other thin K samples cooled
in ~10 qu He were measured. Two of them with d = 0.25 mm
were samples K-l/4Vl and K-l/4V2. Four of them with d =
0.1 mm were samples K-l/lOVl, K-l/10V2, K-l/10V3 and K-
1/10V4. The characteristics of these samples are given in
Table 3-1.

Fig. 4-8 compares data of 8 of these samples with the
data of K wires cooled in the He from Fig. 4—4. The solid

lines in this figure denote the data of K wires cooled in

126

 

(iQm/K)

 

‘o

AlnP/AT

4.2

 

 

 

 

 

1 l

1 1 l l 1 | l 1_
0.2 0.4 6.6 0.8 1.0 1.2 ] .4 ].6 1.8
T (K)

 

Fig. “-7. P(“.2K)(AlnP%AT) vs T for samples indicated in
Fig. “-4 with Fitting curves of Eqn. “.2.

127

 

(film/K)

pmAln P/AT

 

 

Symb d(M) Gas P099111)

 

 

 

.10 Ar 59.1
‘7 ~10 Ar 79.9
A .10 Ar 33.7
A .10 Ar 46.9
' .10 Vac. 74.9

0 .10 Vac. 84.9
+ .25 Voc. 18.6
X .25 Vac. 19.9

I 1 l I l I I 1

 

 

 

Fig. “-8.

0.2 0.4 0.6 0.8 1,0 1,2 1,4 1,6 1.8
T (K)

P(“.2K)(AlnFWAT) vs T [or thin wires of K cooled in a Ar
atmosphere or a partial vacuum. For comparison, the solid

lines represent data from Fig. “-4 for K samples having the
diameters indicated.

128
He. (v)s indicate sample K-l/lOAl, (Vls indicate sample K-
1/10A2. These two samples, both with d = 0.1 mm, were
measured in a same run. Each of them was used as the
reference for the other. These two sets of data are quite
different. The data of sample K-l/lOAl are close to those
of typical d = 0.9 mm samples cooled in He; while the data
of sample K-l/10A2 are close to those of d = 0.16 mm samples
cooled in He. Data of another pair of samples with d a
0.1 mm cooled together in Ar are indicated by (Ala and (A)s.
The data of K-l/10A3 (A) are close to data of d = 0.5 mm
samples cooled in He; the data of K-l/10A4 (A) are close to
data of d = 0.25 mm samples cooled in He. The plusses and
crosses individually indicate data of a-pair of samples, K-
1/4V1 and K-l/4V2, with d = 0.25 mm cooled together in
partial vacuum. Both of them are close to data of the same
diameter samples cooled in He. But data of samples K-l/10V3
and K-1/10V4 with d = 0.1 mm cooled in partial vacuum,
indicated by filled and open circles individually, are close
to data of d = 0.5 mm samples cooled in He.

It can be seen in Fig. 4-8 that 1) The form of all
these data is similar to that of data for thin K wires
cooled in He. 2) The data of some of these samples: .
samples K-l/10A2, K-l/10A4 cooled in Ar and sample K-l/4V2
cooled in partial vacuum, show negative dp/dT in the
vicinity of 1K like those seen for samples cooled in He.

3) Data of 6 samples with d = 0.1 mm cooled in Ar gas or

129
partial vacuum tend to behave like data for samples in He of
larger diameter and to show greater variability for fixed d.

The reasons for point 3) above are not yet clear.
Possible contributions to an explanation include:

1) Since the contribution of electron-surface
scattering to electrical resistivity is only from diffuse
surface scattering, less diffuse scattering would make dP/dT
behave like that in a sample of larger diameter, and make PO
smaller. He gas might somehow make the ratio of diffuse
surface scattering to specular surface scattering large and
stable, while Ar and partial vacuum might make this ratio
smaller and less stable. One possibility'is that small He
atoms could diffuse into the surface layer of K samples and
then provide a good condition for diffuse scattering;
without He atoms, the surface would not provide a good and
stable condition for diffuse scattering. This model could
be used to explain a) that the 6 samples with d = 0.1 mm
cooled in Ar or partial vacuum had smaller Po and behaved
like thicker samples in dP/dT than what would be expected
when they were cooled in He gas. b) Sample K-l/lOAl had
smaller po than sample K-1/10A2, and behaved like a thicker
sample than k-1/10A2 in dP/dT.

2) The influence of impurity concentration could also
not be ruled out, since more impurity concentration tends to
quench these size effects, as shown in K-Rb dilute alloy
samples with d = 0.25 mm described above. Impurity input

from the sample surface could occur during extrusion or

130
during surface corrosion. This will cause sample-dependence
of impurity concentration. This source might contribute to
why samples K-l/10V3 and K-l/10V4 had larger 90 than samples
K-1/4Vl and K-1/4V2, and behave like thicker samples than K-
l/4Vl and K-l/4V2 in dP/dT.

In order to see a size effect in the residual
resistivity po, 90 is plotted versus l/d for selected
samples in Fig. 4-9. Data of other samples with particular
values of d spanned the range bounded by the plotted data
for samples with the same d. In Fig. 4-9, (els indicate
data of samples cooled in He gas; (x)s indicate data of
samples cooled in Ar; and (A)s indicate data of samples
cooled in partial vacuum.

The residual resistivity is expected to have two
components: one due to electron-impurity and electron-
defect scattering, called the bulk residual resistivity'pw,
and the other due to electron-surface scattering, called

80(9); In the first approximation, pb(d) « l/d: SO

In a plot of 100 versus l/d, a straight line with SIOpe
c and intercept an is expected. Woods (ref. 66) suggested
c z 5 x 10‘11 52cm2, which is indicated by the dashed line in
Fig. 4-9. For data of our samples with d z 0.16 mm, a
smaller c: c 2:3.4 x 10'11 52cm2 is obtained (see the solid
line in Fig. 4-9); this is comparable to c = 2.9 x

10"11 9cm2 (ref. 67) found in another study. For reasons

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132
not yet clear, the 90's of our thinnest samples (d g 0.1 mm)
showed a lot of scatter, with most of the pb's much larger
than expected. Possible explanations include: (1) Uncer-
tainty in determining the diameters of these thinnest
samples. The diameters were determined by the hole size of
the dies. The smallest die (d = 0.1 mm) could easily get
partially blocked or slightly damaged, which would reduce
its hole size and make the extruded samples thinner. The
733 of these samples would thus be higher. (2) Corrosion of
sample surfaces. This would thin the samples. For the
thinnest samples, the thinning due to corrosion would be
significant. [(1) and (2) together could make the real d
half as small as the hole size of 0.1 mm dieJ (3) Impurity
input from the sample surface during extrusion. This would
give the thinnest samples the largest fractional concentra-
tion of impurities, which would increase their figfls.

4 . 2 . 2 The Thermeilww '

G measurements were made for each thin K sample from
about 4.2K down to about 70 mK. Fig. 4-10 shows G data for
the same samples shown in Fig. 4-4, which were cooled in He
gas. For K-Hla (d = 1.5 mm), G measurements were made only
for T 1 1K, so data of another thick sample, K-H6a (d =
1.5 mm), are added to Fig. 4-10. All these data have

similar form, and can be well fitted by

G = 60 + B*T2 + (C*/T) Exp(-9*/T) .

133

 

 

 

 

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134

But, the thicker samples show larger magnitudes of G at high
temperatures and smaller magnitudes at lower temperatures
than do the thinner samples. Table 4-4 shows the
coefficients of the fits Go' 8*, C* for each sample with 9*
fixed as 23K, lines in Fig. 4-10 show the fits for three
samples: K-1/10H2a, K-l/4H2a, and K-H6a. The fit for
sample K-H6a was already shown in Fig. 4-3 (section 4.1.2).

Table 4-4 shows that as the diameter of a sample
decreases: 1) the magnitude of Go gets larger, and 2) the
magnitudes of B* and C* get smaller. "1" can be explained
by the Gorter-Nordheim rule (Eqn. 2-45) 8 = (pisi + Pdel/P
(see section ZdLl), where S is only the diffusion term of

the thermoelectric power, S-

1 = thermopower due to impurity

scattering, pi = electric resistivity due to impurity scat-
tering, Sd = thermOpower due to another temperature-
independent scattering (here considering only surface scat-
tering) = pd + pi = total residual resistivity; When d
decreases, pd increases. If Sd does not change in magnitude
and always has the same sign as Si' then S would increase in
magnitude. But, since 8 = GOT, this means that Go would
also increase in magnitude. "2" can be explained as fol—
lows. When a sample is thin, electrons and phonons are
scattered from the surface more often, phonon drag (both
normal and Umklapp) gets weaker, and the magnitudes of B*
and C* get smaller.

Fig. 4-11 shows G data of four K samples with d =

0.1 mm cooled in Ar gas, (ladata for these samples were

135

 

 

 

Table “-“. The coefficients from fits to the 0 data of thin K samples
cooled in He gas

Sample d(mm) 00(V'1) 3*(V‘1K’2) c'(V"K)
K-H6a 1.5 -0.075 1 0.015 -0.“8“ 3; 0.005 6200 _-1_-_1“0
K-1/2H 0.5 -0.166 _-1_-_ 0.010 -0.“65 : 0.00“ 6090 _-1_-_ 120
K-1/“H3 0.25 ~0.301 ;1-_ 0.013 -0.“18 :t. 0.006 5360 i 160
K-1/“H23 0.25 -0.302 x 0.008 -0.382 1'. 0.00“ “8“0 ; 110
K-1/“H2b 0.16 -0.322 -_1-_ 0.010 -0.329 1 0.00“ “020 1,100
K-1/1OH2a 0.10 -0.“88 1 0.01“ -0.257 1; 0.00“ 3280 g 100
K-1/10H2b 0.09 -0.“83 i 0.023 -0.251 3; 0.011 3190 1 220
K-1/10H1b 0.09 -O.“87 1 0.022 -O.255 ; 0.011 3260 1 210

 

136

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137
shown in Fig. 4-8). These data show similar form to the

data shown in Fig 4-10, and can also be well fitted by
G = 60 + B*T2 + (C*/T) exp(-e*/T).

The fits of samples K-l/lOAl and K-l/10A4 are indicated by
two solid lines in Fig. 4-11. Table 4-5 shows the
coefficients of the fits of each samples with 6* = 23K fixed
again.

Table 4-5 shows that for four thin samples with d =
0.1 mm cooled in Ar gas, (1) unlike dP/dT, Go is almost
sample-independent, and is close to the Go of samples with
the same size cooled in He gas; (2) B* and C* show sample-
dependence, and tend to behave like data for samples in He
of slightly larger diameter. The reasons for (l) and (2)
are not yet clear. A trial explanation is as follows. As
described in section 4.241, for dP/dT, more impurity
scattering and less diffuse surface scattering of electrons
would make the samples behave like one with thicker size.
For 60' things are different: from the Gorter-Nordheim rule
(Eqn. 2-45)

1
S = '73:)" (9181 'i' Pde).

More impurity scattering will bring 01 and no up, and less

diffuse surface scattering will bring pd and 0 down. Both

0

together could keep S unchanged, and thus Go unchanged. For

phonon drag, more impurity scattering of both electrons and

138

 

 

 

Table “-5. The coefficients from fits to the C data of thin K samples
cooled in Ar gas
Sample d(mm) 00(771) B*(V'1K‘2) C'(V'1K)
K-1/10A1 0.10 -0.““5 ,1; 0.027 -0.259 1:. 0.010 3500 3, 2“0
K-1/10A2 0.10 -0.“66 : 0.019 -O.261 .t 0.007 3280 i 150
K-1/10A3 0.10 -0.“19 _~1_-_ 0.03“ -0.3“1 3 0.011 ““90 1 220
K-1/10A“ 0.10 -0.“65 3 0.0111 -0.312 1., 0.011 11070 i 250

 

139

phonons will make phonon drag weaker, and less diffuse
surface scattering of electrons will enhance phonon drag,
these two factors coming together could explain why B* and
C* were unstable, but less unstable than dp/dT.

Fig. 4-12 shows G data of four K samples with d = 0.10,
0.25 mm cooled in partial vacuum (10 “Hg He), (Ivdata of
these samples were shown in Fig. 4-8). Each set of these
data shows a "bump" in the vicinity of 1K. A possible
explanation for these bumps was already given in section
4JL2. For thinner samples the heat conductance is smaller,
so a smaller part of the heat generated by the G heater
passes through the samples; thus the error in the G
calculation (discussed in section 4Ju2) is bigger and
brings the data further down in magnitude. Except for the
bumps, the shape of these data is similar to that of the

data in Fig. 4-10 and Fig. 4-11.

4.2.3 W

The thermoelectric power 8 of one K sample (K-l/2H) was
measured in this study to find the form of L(T) and the low
temperature limit of L(T) = Lo (Eqn. 1.8) in K. The reason
for choosing a sample with d = 0.5 mm is as follows. Since
S is calculated from S = V/ATI 1:0, the sample must be thin
enough so that its heat conductance is small enough to give
an accurately measurable AT across the sample with a small

enough heat input to keep the sample cold. But if the

.E::oc> _m_utmd m :_ pm_ooo x we mmt_3 :_£u to; h m> u .~_1: .m_m

ed on a: b oA.N 3
_

 

140

 

 

A 7
code 8;; 583616 N>1m1x .
sources": 85363 tfiwux l-
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++ + U n .m. X o 9
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+
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O O O O O O ”0.
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do . o o o o o o #1. o 00 0 o
0 +x
$ 0 o o i .1
c w .o + N+~

 

 

141
sample is too thin, it would show a strong size-effect. d =
0.5 mm was thin enough, but not too thin.

Fig. 4-13 shows a plot of S versus T. The dashed line
in Fig. 4-13 indicates a fit of s = A'T + 3'T3 + c'e'9*/T
(section 2.3.1) with 6* fixed as 23K, as used for G data
analysis above; the coefficients obtained were: A! = -0J78
i 0.15 x 10‘8 v‘l, 3' = -0.681 5; 0.034 x 10'8 v‘lx‘z, and
c' = 11530 i 770 x 10"8 v‘lx. A better fit was found when
6? was left as a variable parameter; the coefficients were
then: 9* = 17.4 i 0.4K, A' = -0.433 1 0.081 x 10‘BV'1,

3' = -0.957 3: 0.045 x 10‘8 vx‘z, and c' = 4070 i 250 x 10"8
V’lK. This fit is indicated by the solid curve in Fig. 4-13.
Fig. 4-14 compares measured G with G* calculated by

G* = S/LOT. A divergence occurs when T > 1K, this implies
the temperature-dependence of L (see section 1.1) for K at
T>1K. In order to see the form of L(T) more clearly, we
plot L(T)/Lo versus T in Fig. 4-15. The values of L(T)/Lo

were obtained from
G*/G = (S/LOT)/(S/L(T)T) = L(T)/Lo.

In Fig. 4-15, one can see L(T) z LO only for T < 1K. From
about.lK up to about 4K, L(T) decreases with increasing
temperature. Each value of L(T)/Lo in Fig. 4-15 was checked
by the Wiedemann-Franz Law as follows.

Since,

0 = xAAT/l, R = pl/A, and xp/T = L(T) (section 1.1);

142

.:~\_ux m_aEmm LOm h m> m .m_n: .m_m

 

 

. . v: .
oe on on Q_ o
_ _ _ u
1O.m—
S
5.2 1
m.
0.
A8
M.
on [
r/l/J/IIJ

 

3-5

 

 

143

.:~\_nx m_aEmm tom k m> Apob\m uv¥o wen u

.57: .2“.

 

 

o6 o_.m c: e ow. o_._ o
_
. m .e 13..
. e .
. 9
o o o - l._o.N.l
. N,
. m. -3-

 

 

144

.:~\_-x ..eEm. tot e m) ol\AeV5

o.—

 

 

d

.93 .3“.

 

 

145

SC,
QR/TATLO = Kp/TLO = L(T)/Lo,

where Q can be calculated by Q = R6175; as mentioned above.
Table 4-6 compares two sets of values of L(TL/Lo, calculated
from G*/G and from the Wiedemann-Franz Law. One can see

these two sets of L(T)/LO are very close to each other.

4.3 NQMlflAL.31flfiLE:QBX§IAL.K_§AflELE

4.3.1 J2h3__BgS.i..3.‘s’..i.’:L.'L.t’..lL.w

As mentioned above, in an attempt to get a longer mean-
free-path of electron-impurity scattering, so as to see
whether the size-effect could be observed in a thicker
sample, we tried to make and measure a "nominal" single-
crystal K sample with d = 1.5 mm: sample K-S. The word
"nominal" is used here because, since only one X-ray spot on
the sample was tested, the verification of the whole sample
as a single crystal was not established (section 3am.
Sample K-S had one of the lowest Po's and one of the largest
RRR's for K in this study (p0 = 0.92 chm, RRR = 6150) (the
details of its characteristics are given in Table 3-5).
However, the mean-free-path of K-S was not much longer than
the previous ones; lei = 0.5 mm was obtained from Eqn.
(4-1), still much smaller than d (d = 1.5 mm). As expected,
no clear size-effects were seen in this sample.

Fig. 4-16 shows a plot of dP/dT versus T. The data at

around T = 0.7 K look a little low, but not low enough to

146

Table 4-6. L(T)/Lo values calculated from G*/G and from
Wiedemann-Franz Law for sample K-l/ZH

 

 

 

T(K) L(T)/Lo from G*/G L(T)/LO from W-F Law
0.482 1.00 1.00
0.714 0.98 0.98
0.948 0.96 0.96
1.182 0.95 0.94
1.368 0.91 0.91
1.483 0.91 0.89
1.741 0.81 0.81
1.941 0.76 0.78
2.201 0.76 0.75
2.490 0.67 0.66
2.808 0.62 0.61
2.977 0.58 0.56
3.874 0.42 0.40

 

 

147

 

 

 

 

6,0— /
_ /
r /

eso- ,/
C} /
c: /
/

’l: 4.0" / Z
< /
2 A
:4 3.0— / ,
Q. ,A
g’ / /

2.0- / a

/ /
1.0— / / .
///A
()t’ A I l l l .J i
O 0.2 0.4 0.6 0.8 1.0 1.2
T [K]

Fig. 4-l6. (IKh.2K)/P)(AP%AT) vs T for the nominal single crystal
K sample. '

148

show clear evidence of size-effects; while the data at T <
0.2 K show a clear turn down. In order to see the anomalous
turn down, dP/TdT is plotted versus T in Fig. 4-17. It is
clearly seen in Fig. 4-17 that the anomalous turn down
occurs when T < 0.3 K. This anomalous turn down is
completely different from the size-effects described above.
In section.4.4iwe will see similar turn-downs for K samples
encased in polyethylene tubes. Further discussion of this

turn-down will be given in section 4.4.

4.3.2 W

G measurements were made for sample K-S from about 4.2K
down to about 70 mK. Fig. 4-18 shows a plot of G versus T
for samples K-S (A) and, for comparison, K-H6a (x) which was
plotted in Fig.14én. These two sets of data almost overlap
each other for T > 1.5 K. For T < 1.5 K, the data of sample
K-S stay slightly less negative than K-H6a, a result
consistent with the Gorter-Nordheim rule (Eqn. 2-41) (see

section 4.2.2).

4.4.1 W57.

As mentioned in section 1.241, van Kempen et al. (ref.
14) and Greenfield et al. (ref. 18) measured F>of high-
purity K samples inside polyethylene tubes down to about 1K.
The details of the results from these two groups are given

in section 1.2Ju Tb investigate further the variations in

149

 

 

 

1 l 1

 

Fig. 4-17.

I
0.4 0.8 1.2
T [K]

(P(L+.2K)/PT)(AP/AT) vs T for the nominal single crystal
K sample,

150

.m_asmm x .mum>tu e.mc_m .mc_eoc ecu to» k

U: ._.

2, o .23 .2“.

 

 

(I:

 

 

151
magnitude of the T2 coefficient reported by both of them, we

measured four pure K samples clad in polyethylene tubes down
to 0.07K. To test for annealing effects, each sample was
measured several times, with intervening anneals for varying
periods of time at room temperature under Ar gas, He gas, or
partial vacuum. Sample K-PHl was in a 1.6 mm dia polyethy-
lene tube and was measured 5 times under He gas and partial
vacuum. After the second measuring run it was taken out of
the sample can. It was remounted just before the third run
with a new bare K sample as the reference. Sample K-PHZ was
in a 0.9 mm dia polyethylene tube and measured 4 times under
He gas with a reference K sample inside a 1.5 mm dia teflon
tube. After the third run, one of its connections broke and
the sample can was opened and the sample repaired. It was
remounted just before the fourth run using the same refer-
ence sample. Samples K-PAl and K-PAZ were in 0.9 mm dia and
1.6 mm dia polyethylene tubes respectively, and were meas-
ured concurrently 5 times under Ar gas. To test for the
rolling effect reported by Greenfield et al. (ref. 18), both
of these samples were taken out of the sample can after the
third run, rolled, left at room temperature for about 20
hours, and then remounted just before the fourth run. After
the fourth run, both samples were again taken out of the
sample can, and later remounted for the fifth run. The
details of the preparation of these samples are given in
section 3.4. The details of the characteristics of each

sample in each measurement are given in Table 3-5.

152

Fig. 4-l9 and Fig. 4-20, in different scales, show a
normalized dP/TdT (ref. 59) plotted versus T for five runs
of sample K-PHl. Similarly, Fig. 4-21 and Fig. 4-22 for
four runs of sample K-PH2; Fig. 4-23 and Fig. 4-24 for five
runs of sample K-PAl; and Fig. 4-25 and Fig 4-26 for five
runs of sample K-PA2. For comparison, the data of sample K-
S plotted in Fig. 4-17 were added in Fig. 4-19; and the data
of samples 2a, 2b, 2c from van Kempen et al., which were
also measured under He gas, are added in Fig. 4-21. At T >
1.1K our data in Fig. 4-21 are consistent with the data for
samples 2a and 2b of van Kempen et al., which had Po's
comparable to ours. Sample 2c of van Kempen et al. had a
lower F5, and its data stay well below our data (see Fig. 4-
21). The dashed curve in each of these figures shows, for
comparison, typical behavior of a free-hanging, bare, high-
purity, thick K sample (section 4.1.1). At T < 1K, the data
presented in these figures, including the data of sample K—
S, which was melted in clean paraffin oil during its growth,
exhibit similar deviations from the dashed curve. For
samples encased in polyethylene tubes, sample annealing at
room temperature significantly brings both the normalized
value of dP/TdT and Po down until dP/TdT becomes negative
and large. To fully developlthe negative deviation requires
about 10 days of annealing. The temperature at which dP/dT
crosses zero corresponds to a resistivity minimum, which in
the fully deve10ped samples occurs between 0.3 and 0.5K.

When first cooled and measured, the samples appeared to have

 

 

 

 

 

l
l A
\ A
\\ A A A ‘ 0
8°0— \\ o I
H ‘ \ 0 A
N ‘\~~_‘__-__—-____-_.--_--_-----
' at
g * ‘k * . O . 3 A
E 00 I I ‘
e . _- .
1.74.0— *°
Q - ‘
Q. A _ .
£5 * o Nocfldays 1%
g at roomT (ngcm)
E A ‘ A K-PH'Ic 0.5 2.70
63.2.0L' . K-PHlb 11.0 1.42
0 . ° K-PHic 24.5 0.90
*A - K—PHld 26.0 0.95
' K-PHle 27.0 0,95
0 e
*r K-3
0
2.0—
. u l 1
0 0.4 0.8 12
T [K]

Fig. h-l9. (P(h.2K)/T)(AlnPh&T) vs T for sample K-S and five runs of
sample K-PHl.

154

 

 

\“ ‘A-——---O~ “’A ——————————
O.— —-AQ25—5-‘—°LA—OA— _‘ _____________
AA 00‘ A
.- ‘
‘1'" 8 '
X g' ‘
E ' .
g 40.0-
““ A
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éaoo— 4
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SI.
‘ A K-PHla
A K-PHlb
120.0-
1 0 K-PH‘I:
‘ . K-PHld
I K-PHle
‘ 1 l 1

 

160.00

Fig.

h-ZO.

0.4 0.8

[K]

1.2
T

A version of Fig. 4-19 to larger scale.

 

 

155

 

 

 

 

 

 

F No of days
10.0" at room T p" ("9““) °
0 K—PHZa 0.5 3.02 "
i
I K-PH2b 2,5 LIO A .
A K-PHZC 13.0 |.l7
8.0- ‘ 0 K-PH2d 73.5 I.04 w
‘7; x *
\ 0
£5 \
c ‘.‘ I O
u— ‘\ o 0
" EMO“ "
A ““~ ______ __,Q_._——-(-)———-I-—
l2] 0 7° .
\\ A *
QE 0 ' A t 0
a ' . - '
I 4.0— A.
< o . O
Q; I
V o A O
253'“
O
G . Annealing P( )
o time(daysl ° chm
A Data 11' 2a 2.39
from
van it 2b 2 |.|3
O Kempen! 0 2c 80 l 0.88
_ l
2'0 0.5 1.0 1.5
T [K]

Fig. 4-21. (P(L+.2K)/T)(AlnP/AT) vs T for four runs of sample K-PHZ.

For comparison, the data of samples 2a, 2b, and 2c of van
Kempen et al. are indicated.

 

156

 

10.0 \
\\\\“~-___O _______ O---S——.—A——--
o o . g I'
O I
0 00 72.5;
‘L o “z
E .-
c —— A
t- .-
A IA
% -2o.o——
o. '-
5 A
4 e
a: __ I o K—PH2a
\\Q . I K-PH2b
_40.0__ A K-PHZC
' o K-PH2d
1 l J i 1 J i
0 0.4 0.8

Fig. 0-22.

 

 

 

T (K)

A version of Fig. h-Zl to larger scale.

 

157

 

 

 

 

 

__ No. of days R,
10-0 at room T (chm) o
O K-PAla 0.5 5.|9
- K-PAlb 4,5 1,20
A K‘PAlc l6 Lid
0 K-PAld I9 L32 A
8.0- o K-PAle 97 |,|2
‘ O
l
3‘ O
7 \ '
=1 6.0— X o -'
E “~__---._- 0 o
c O -—-——-—---——---—--—-—-— A
t o I O
C
4.0— ° 0°
e ° ' .
E C
< I
.. . 3
0- o 0.
\~ 20' .
s“ ’ 1 -
g ‘2
l
0i i
l
I A
1
l
l
-2.0 ' ' l
0 0.5 1.0 1,5

T (K)

Fig. L1-23. (P(L+.2K)/T)(AlnP/AT) vs T for five runs of sample K-PAl.

158

 

 

 

 

201
o 00 E oIA
o 5
__ 8
q.“ o
5: _ .2
E5 ,
C} _ q.
“3 A
1— - __ .°
Q 50
Q.
E ‘— as
Si __
.2
‘\:: _.
SE? __ 8 K-PAla
K-PA1b
4001- : 4......
._ K-PAld
14 1 1 l 1 1 l
0 0.4 0.8
T (K)
Fig. A-ZA.

A version of Fig. A-23 to a larger scale.

 

159

 

 

 

 

10.
No. 0! days R,
at room T (chm)
0 K-PAZa 0.5 5.20
o
I K-PAZb 4.5 0.97
8.0— o K-PA2c I6 Ll? II
I K-PAZd l9 0.98
0‘ K-PA2e 97 0.73 o
1 o
\‘ O .
A \ O I
N \ O A
I _ \
x 6.0 \\‘ Q I A O
E ““““““““““““ .‘ """"" o
C} . °
0
u— 0
V o I A0
o
O
p 4.0- . A
<1
o. 0 ° -
E.
S) ° -
1: ' °
\2I0_
0‘. O
V
V C
I A
0
o
o
A
_2. 1 1 1
G0 0.5 1.0 1.5
T (K)

Fig. A-ZS.

(P(A.2K)/T)(AlnP/AT) vs T for five runs of sample K-PAZ.

 

(meK'2)

( ”/T) (AInP/AT)

160

 

 

 

 

20
d) t"---1L---‘ .VK“‘§“‘
c 8 ° .3
0 I
O A
_ O I~A
I
1— IA
I
A
'40; 'A
I
I
o K-PA2a
— I
A I K-PA2b
A A K-PAZC
‘80P- <>K-PA2d
1 1 L 1 1 l
0 0.4 0.8

T

Fig. 4-26, A version of Fig. 4-25 to a larger scale.

 

161

a rather large concentration of impurities or defects, as

indicated by the relatively large value of P .About 3

o'
days of room temperature annealing seemed adequate to bring
Podown to its normal value for free-hanging, bare, thick K
samples. For reasons not yet clear, sample disturbance
during sample remounting at room temperature seemed to bring
dP/TdT up, particularly for data at low temperature end, but
did not affect 13 significantly. The effect of sample
rolling was not simple. For K-PAl, after rolling, no in-
creased, but dP/TdT did not change significantly; while for
K—PAZ, 90 did not change significantly, but dP/TdT
increased.

For a possible Kondo effect, the electrical resistivity
at low enough temperature is expected to have the form p=
p0 + AT2 - BlnT (section 2.2.7). Thus, dP/TdT = 2A - BT72.
In this case, a plot of dp/TdT versus T"2 should be a
straight line of negative slope -B and with an intercept of
2A at T'2 = 0. Figures 4-27, 4-28, and 4-29 show plots of
dP/TdT versus T"2 for samples K-PHl and K-PHZ, K-PAl and K-
PA2 respectively; All the data in these figures are well
fitted by dP/TdT = 2A-8T‘2, as indicated by the broken
lines. It is also clearly seen in these figures that sample
annealing at room temperature enhances the effect (brings
the magnitude of coefficient B up) while sample disturbance
appears to weaken the effect (brings the magnitude of B

down). This can also be seen in Table 4-7.

'162

 

0—27.

Fig.

163

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h —————— a ————— -—
O’RE
Q
&\
\
\\\
%£>
\
-20.- °\\\s
\ \
\\ \
\\ \
,. Wx 1
N 9 \ l
1" \\\\\\ 1
—-4o__ \ K \ l
E \\\ \5 l
C: Q \\ l
1: \\ \ ‘
\ \ i
‘ \ \ .
\ \ \ 1
1 \ \ \
’60?- \ \ \ \
:- ° \ \ ‘0
\ \ \ l
<\] \ \ \ i
% \ \ \
— \ \ o
$1 ‘1 \ \ 1
,._£3 L_. IQ'PAJ \ \
a o O \A .\ \
.' I \ \
A C \ \
0 e \\ \\\ \
\ .-
‘4C“D" \\ A\ i
\ \ \
\ \
\ \
\ \
\ \ .
10:1
_ l l l -
O 20 40 60 80
T"2 (K-zl
Fig. h-28. (P(h.2K)/T)(AlnPA5T) vs T.2 for sample K-PAl.

164

 

 

 

 

 

 

  

 

 

 

 

—o-——_—-1> ——————
0* _____ —0————_.a._
0 ‘1 \ _____
\‘ \\o.
\ \
\R \\ \O\
\ 'K \\\
\ °\\
\ \\ \\
a— X \
a 0x
| -20 \\ \K \\
X \ \\ \
E \\ ‘1\
C} A \ l
u— \ ‘\
\e. \
9 40 \\ \K
<" i \
Q- \\o \ I
s \ \
g \\ \\
; K‘PA2 \\ \\
\“6 o O \\
q \
Q: I b \A\
A C \
o e \\
b.
-80
fi 1 l 1
0 20 240 2 60
T (K )
Fig, 4-29. (P(L+.2K)/T)(AlnP/AT) vs T-2 for sample K-PAZ.

 

 

 

165

Table 4—7. The coefficient B from fits to the 9 data of K samples
encased in polyethylene tubes

 

 

 

No. of

d days at After sample Po B
Sample Gas (um) room T disturbance (nflcml (farm)
K-PHla 10 1.1119 He 1.6 0.5 Yes 2.97 0.14
K-PHlb 10 “Hg He 1.6 11.0 No 1.68 0.90
K-Pch He atm. 1.6 24.5 Yes 1.18 0.32
K—PHld 10 pHg He 1.6 26.0 No 1.20 0.47
K-PHle 100 pHg He 1.6 27.0 No 1.20 0.47
K-PHZa He atm. 0.9 0.5 Yes 3.02 0.16
K—PHZb He atm. 0.9 2.5 No 1.10 0.72
K—PH2c He atm. 0.9 13.0 No 1.17 0.87
K—PH2d He atm. 0.9 73.5 Yes 1.04 0.55
K-PAla Ar atm. 0.9 0.5 Yes 5.19 0.06
K-PAlb Ar atm. 0.9 4.5 No 1.20 1.39
K-PAlc Ar atm. 0.9 16.0 No 1.14 1.50
K-PAld Ar atm. 0.9 19.0 Yes 1.32 -
K-PAle Ar atm. 0.9 97.0 Yes 1.12 1.23
K-PA2a Ar atm. 1.6 0.5 Yes 5.20 0.05
K-PA2b Ar atm. 1.6 4.5 No 0.97 0.83
K-PA2c Ar atm. 1.6 16.0 No 1.12 1.08
K-PA2d Ar atm. 1.6 19.0 Yes 0.98 -—
K-PAZe Ar atm. 1.6 97.0 Yes 0.73 0.27

 

166

To test whether the effect described above only occurs
for samples in contact with polyethylene and samples melted
in clean paraffin oil, we measured samples encased in a
teflon tube (K-THl) under He gas, a sample in contact with
Kel-F pieces (K-KA) under Ar gas, and a sample coated by
cold clean paraffin oil (K-O) under Ar gas. No similar
effect was found in any of these three cases. To test
whether there is some kind of impurity continuously diffus-
ing from the polyethylene tube into the K sample and causing
the ”turn down" effect, we measured a bare sample (K-PPA)
with d = 1.5 mm and with potential leads encased in poly-
ethylene tubes which were left in room temperature for 1.5
days before the sample was cooled down under Ar gas. No
effect was found here either.

Figure 4-30 shows a plot of dP/TdT versus T for these
last four samples. Sample K-THl was measured 4 times con-
currently with sample K-PH2 described above. The details of
the characteristics of each sample in each run are given in
Table 3-5. The data of samples K-KA, K-PPA, and K-O are
similar to those of bare, free-hanging, thick samples. The
data of K-THla (the first run of sample K-THl with d =
1.5 mm and large p0) stay more positive than those of bare
thick samples. ‘When the annealing time at room temperature
increased, the residual resistivity Do dropped, and the
values of dD/TdT shifted downward with maximum shift at
temperature around 1K, so that the data no longer formed a

horizontal straight line. Since the smallest Db of sample

167

 

 

 

 

10.0-
on 0 A
O
c?“ __ o
x89 .
E1 0
I ‘ ‘
g '— ..O ‘ a o
o c 8 0 0 .
x . o 0 if:
X 5*: ‘ X ‘ x X ' ,0 X if
6. {3* 9* o o f: 0 oft * A
|:- b t t a fit a
A O
Q _ ‘ .0 . .
CL _ o o
.5 s K-KA
<1 01_
: 4. at K-PPA
'- x K-O
\N' _ No. of days Po
.7 at room T (ngcm)
0 K-Tch 0.5 8.!0
2.04- o K-THlb 2.5 4.87
‘ K‘TH‘C I3 L34
- o K-THld 73.5 0.94
01 l l l
0 0.5 1.0 1.5
T (K)
Fig. h-30. (P(h.2K)/T)(AlnPAdT) vs T for K samples in contact with

other plastics.

 

168
K-THl was 0.94 nacm, which yields an electron mean free path
0.26 mm (Eqn. 4-1) much smaller than the diameter of 1.5 mm,
it seems unlikely that the deviation from horizontal line
behavior is due to a size effect similar to that seen in the
bare thin K samples. The reasons for the deviation from the
horizontal straight line are not yet clear.

Further experiments recently done in our lab (ref. 68)
showed that application of an average magnetic field of 0.1T
longitudinally'to polyethylene encapsulated samples shifts
the resistance minimum to much lower temperatures. Thus,
these samples seem to be exhibiting a Kondo effect due to a
magnetic impurity which has an effective magnetic moment
equal to or greater than one Bohr magneton. Identification
of this magnetic impurity is still in progress. It seems
to be a product of the reaction between potassium and

hydrocarbons.

4.4.2 W

G measurements were made for each sample above, in each
run, from about 4.2 K down to about 70 mk. Fig. 4-31 shows a
plot of G versus T for five runs of sample K-PHl; Fig. 4-32
for four runs of sample K-PHZ; Fig. 4-33 for four runs of
samples K-PAl and K-PAZ. For comparison, the data of a free
hanging bare thick sample (K-H6a) shown in Fig. 4-3 are also
shown as a broken line in each figure above.

For the samples which have been annealed for only 0.5
days, the plots have the same form as the broken line; the

deviations from the broken line, more negative at low

169

 

._:dlx o_dEmm com A m> o ._m-: .m_u

00

4.0 u .

IN A.

x O o

, ~ Imlv. 9

I )
I / A
z -
a (l
I I.
/o
o z
I An
I o z
. , 3
’0 0 II 0
I /4
I I
I I pp 0 I
/¢ .
a / o .o
13 I .
I/. .4.» I WK
/
l I
./ v 80’
I

 

 

 

170

.delx o—dEmm to» h m> o .lea .m_m

ON our 0

 

 

 

\ I x
\ 4 Q I/
x .. I/
\. l I
/ '1‘ 8
x
. ,, .. on
\n ,
x I
\I dI/ Dd
\ Ix.
/ n-
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x N0
0
/
szlx

 

10.09

(1.111

 

171

 

 

o x
8 3 0 ‘
O O
OI“) .0 \
A O ‘
o I o .
‘ O \
o“
. \
A O.
I 0 ‘
‘ A
‘ \
\
O
I
00
O “
0 ° \‘0 \.
\ t
o ‘\ x
I. ~ I ‘\ \O
. .‘ . . \ \
00 o O \
I‘ o o 0 ‘
\
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\
I— J .A‘
. 4;) ° V
g \
\
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\
\
<5 ‘ °
\
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A— '
'ZQ 0‘

‘—l(-PA2
00
-b
AC
ed

00

 

0 ‘L0

Fig. “-33. G vs

2!)

T for

K-PAl -—I

00
-b
AC
od
09
O
O
a
I
I
I
. I
I
I
.A
I 0 ’
I
I
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samples K-PAl and K-PAZ.

CD

 

172

temperature end and less negative at high temperature end,
[i.e., with smaller magnitudes of Go' B*, and C* (see sec-
tion 4.2.2)], can be ascribed to a high level of impurities
(see section 4JL2 for details), as suggested by the large
value of Po.

Above 1K, further annealing causes the data in Figs. 4-
31, 4-32, and 4-33 to follow more closely the broken line,
but below 1K they exhibit significant negative deviations
which increase when the values of B obtained from Figs. 4-
27, 4-28, and 4-29, increase. A Kondo effect can produce
such negative anomalies in.CL Below 1K, our data can be
well fitted by G = G0 + B*T2 + D*/T, where the D*/T term is
what is generally expected for a Kondo effect. Fig. 4-34
shows the data below 1K for two selected runs of each
sample, the solid curves in this figure indicate the fits.
The values of parameters Go, 8*, and D* obtained from com-
puter fittings are given in Table 4-8.

The small bumps at T z 1.5K in the plots of samples K-
PHla and K-PHld in Fig. 4-31 are similar to the ones for
their reference samples: K-HSa and K—H6b, discussed in
detail in section 4.1.2.

The plot of G for samples K-KA, KAPPA, K-O, and four
runs of sample K-THl are very similar to those of free-
hanging, bare, thick samples. They are not provided in this

thesis.

173

 

.x. uaonm zo_on ocaumcanou um mo>cau m:_uu_m
eo_z ~<a-x oeo ._<¢-x .mza-x ._:a-x $.95m rot e o> o .sm-s .m_c

 

 

a: e
or md o QF md o
_ _ _ 1 1_._- A _ A A _ A A _ A _ 4 4 _ _
04+. A mv- -
l - lV.—.l
.. . -3- - . -
- \ - - 1 -

- T as; \. - 0.7 - \ -
no-.. Ill-K .. - of
- .// -. l \\ -

- //// \. lmo-. n/ 4182+: \\. -

h.AJ

 

 

 

 

 

 

174

Table 4-8. The coefficients from fits to the G data of K samples encased
in polyethylene tubes

 

 

 

sample G(V‘1) B*(V'1K'2) D*(V’lK)
K-PHlb -0.228 g; 0.055 -0.l66 i 0.058 -0.065 a; 0.014
K-PHle -0.116 1; 0.007 -0.354 r. 0.010 -0.055 1; 0.002
x—mzc -0.115 3,. 0.006 -0.379 1 0.006 —0.052 3: 0.001
K—PHZd -0.060 1 0.013 -o.393 3,. 0.011 -0.061 at 0.003
K-PAlb -0.177 i 0.009 -0.333 3; 0.010 -0.126 1 0.003
K-PAlc —0.157 i 0.019 -0.358 i 0.017 -0.149 1 0.006
K-PA2b -0.125 g 0.010 —0.380 1 0.011 -0.075 1 0.003
K-PAZC -0.192 1; 0.011 -0.379 1 0.013 —0.089 1. 0.003

 

175
4.5 K:BD_ALLQX_§AMEL§§

In 1980 C. W. Lee et al. from our lab (ref. 8,23) found
that P = PO + AT2 + AiPoT2 for K-Rb below lK (see section
1.2.1 for details). Assuming G = Go + G3 + G3, they also
found that phonon drag terms (Gg and G3) are quenched in K-
Rb more and more as the impurity concentration increases and
that Go is positive and roughly obeys the Gorter-Nordheim
rule G0 = Gi + FP/Po (Gp-Gi) (see section 1.2.2 for details)

In this study we have improved the absolute accuracy of
our temperature scale, and also our measurement precision by
nearly an order of magnitude, so that we can now detect
changes in p(T) of ~2 parts in 108. These improved
conditions allow more detailed examination of the form of
iflT) and G(T) in K-Rb alloys. Moreover, using improved
glovebox facilities, including an accurate, built-in
weighing scale, the atomic percentage of Rb impurity can be
determined more precisely.

Four K-Rb samples were measured in this study. Samples
K-Rbl and K-Rb2 were 0.077 at.% Rb alloys with d = 0.25 mm;
samples K-Rb3 and K-Rb4 were 9.4 at.% Rb alloys with d =
3.2 mm. The details of the characteristics of these samples

are given in Table 3-6.

4.5.1 Wis-tints:

Fig. 4-35 shows a plot of residual resistivity (Po)
versus the Rb impurity concentration (c) for samples K-Rbl,
K-Rb4, and two samples of M. L. Haerle (with 0.38 and 1.3

at.%Rb respectively). The stars indicate the two data

176

 

i—‘I-T-i

 

0.2L

   

 

 

C (at%)

Fig. 4-35. PO vs c for K-Rb samples.

177
points from M. L. Haerle; the solid straight line indicates
the best literature value of no per atomic percent Rb
impurity (= 0.13 uQcm/at%) (ref. 67); the dashed line
indicates the value of no per atomic percent Rb impurity
obtained by C. W. Lee in 1980 (ref. 8,23). It is clear in
Fig. 4-35 that: (a) the value of no per atomic percent Rb
impurity for our new dilute K-Rb alloy samples (including
the two samples of M. L. Haerle) is now in closer agreement
with the best literature value than what C. W. Lee
previously obtained. (b) For our 9.4 at% samples, the value
of no per atomic percent Rb impurity departs from the solid
straight line. This departure might partly be due to the
finite concentration effects which yield no = a(l-c)c in
the simplest model. Only for dilute alloy the impurity
concentration c is very small, so (1-c) z 1, and thus no =
ac, yielding a straight line with a slope of a in a plot of
no vs c; for c = 9.4 at.% as in our case, (l-c) = 0.91, ifa
= 0.13 Mom/at% as given above, then a' = a(l-c) = 0.12 1.1.9
cm/at% should be the effective value of no per percent Rb
impurity. Actually, the value deduced from Fig. 4-35, would
be 0.11 mum/at%.

Fig. 4-36 shows a plot of normalized dn/dT (ref. 59)
versus T for sample K-Rbl (with 0.77 at.% Rb) and two
samples of M. L. Haerle (described above). The broken
line indicates typical data for a free-hanging, bare, high-

purity, thick K sample. The small turn up on the broken

line occurs for T < 0.3K, which we tentatively associated

178

 

 

 

40 ‘
x
x x
1.3%Rb x x x X x
X x
30— -—
A X
Nx X
A
Q
T." .-
<
\ 20— —'
i3:
% 0.38%Rb D D
E U DD U D C] D U D
_._. u D D
E!
10— --
\ 0.077%Rb A
‘\ A ‘ A AA A A A
K
o l l
O 0.5 T(K) 1.0
Fig. 0-36. (P(A.2K)/T)(AlnPAAT) vs T for dilute K-Rb alloy samples.

 

 

 

 

179

with residual dislocations (see section 44LJJ. In
contrast, in the K-Rb alloys there is a low temperature
turn-down which becomes progressively more apparent as the
Rb concentration increases. This turn-down is very obvious
for the 9.4 at.% samples, samples K-Rb3 and K-Rb4 shown in
Fig. 4-37.

‘ The coefficient (A) of the T2 term was deduced from the
data in the region close to a horizontal straight line in
Figs. 4-36 and 4-37. We tried to fit A by A = A0 +
Aipo (see section 2.2.8). A0 = 0.24 i: 0.04 chm/K2 is
obtained, consistent with the one for thick pure K samples
(see section 4.1.1), and A1 z 11 1; 2 x 10"6K‘2 is obtained
for all of our four K-Rb samples. This value of Ai is
slightly larger than Ai 2:8.5 x 10"6K"2 obtained by C. W.
Lee, and clo‘ser to the theoretical values of 13.7 x 10‘6K“2
from P. L. Taylor (ref. 24) and 12.5 x 10'6 from Kus and D.
9L Taylor (ref.25).

In order to intercompare the Po dependence of dP/dT for
all of the dilute alloys, we have normalized the data for
each sample shown in Fig. 4-36 to the data for sample K-Rb3
by multiplying by pK_Rb3(4.2K)/p(4.2K) and then displacing
each curve vertically to bring it into near coincidence with
sample K-Rb3. Although the scatter in the data for the more
dilute alloys is large, it is clear that the curves now
roughly match. This means that the turn down is approxi-
mately proportional to 90. Such a dependence is quite

3 5/2

different from the no and no expected from either

250

180

 

 

A.
I C
‘ 'b [3 C] I x
u C —
x
D
A.
.A “ X X
X
.A
is ' .A

' K-Rb
t I 9.4 % p=l.O4p.Q-cm
. 94 7° p =].01pQ-cm

 

 

1001— old {X 1.3 % -
data 00.38%
A0077%
0
so ‘ I l
0 0.5 T(K) 1.0

Fig. h-37,

NormalizedrflPAdT vs T for K-Rb alloy samples.

 

 

 

181
localization or electron-electron interaction effects (see
section 2.2.9).

We tried to fit the data by p = Po + A'T2 + f(T) with
various different functions fKT). The best fit, shown as
solid curves in Fig. 4-38, contains f(T) = -CT. Such a term
would be expected from localization (see section 2.2.9).
This fit is better than one with f(T) = -DTl/2, which would
be expected for electron interactions (see seciton 2.2dn,
or f(T) = -BlnT, which would be expected for a Kondo effect
(see section 2.2.7), but the scatter in the data is such
that neither of these two alternatives can be completely
ruled out. Fig. 4-38 shows the data of samples K-Rb3 and K-
Rb4 in plots of normalized dP/TdT versus T'l, T‘3/2, and
T‘z, with these three alternative fits in the forms as:
dP/TdT = 2A - C/T, dP/TdT = 2A - D/2T3/2, and dP/TdT = 2A -
B/Tz. The coefficients deduced from Fig. 4-38 are C a 2.2 i
0.4 pocmx‘l, D = 2.0 i 0.5 pnemx‘l/Z, a = 0.3 i 0.1 poem for
sample K-Rb3, and C = 2.3 i 0.8 chmK’l, D = 2.3 1;

1.2 chmK'l/z, B a 0.5 1; 0.3 pflcm for sample K-Rb4.

Comparing the magnitudes of our anomalies with predic-
tions for localization or interaction effects, we found that
the measured anomalies for 9.4 at% samples are about one
order of magnitude larger than predicted for the interaction
effects and two to three orders of magnitude larger than
predicted for localization effects. The calculations are as

follows:

 

 

 

 

 

 

 

 

-2 .2
T (K )
Fig. 4-38. Three different trial fittings for (P(L+.2K)/T)(A|nP/AT)
of two K-9.4at.%Rb alloy samples.

 

183
According to Kaveh and Mott (ref. 69), the conductivity
contribution of electron-electron correlation effects in

disordered metals is:

e 1

"2h I"int(T)

 

«mm = f(x)

where Lintu‘) = (hD*/kT)1/2 with D* = (1/3)VF1e' and f(x) is
typically of the order of 0.5 in high-resistivity metallic
glasses. Using VF = 0.9 x 108 cm/s for K, and 1e z 2 x
10"5 cm calculated from Eqn. 4-1 for our samples with poz
1 Mom, we got D* s: 6 x 102 cmz/s, then G(T) = (0.2 {Idem-1
K'1/2)T1/2. Since the resistivity contribution Ap= —pga-(T)
= -DT1/2, so D = pg (0.2 n‘lem'lx‘l/Z) a 0.2 pocmrc'l/2 for
our samples with 90 = 1 pacm. This calculated value of D is
one order of magnitude smaller than our measured value of
about 2 pacm/Kl/z.

According to Howson (ref. 56), the conductivity
contribution of localization for high-resistivity metallic

glasses has the form

1
h Li(T)

m

 

G(T) =37 with Lfim = (l/2)1eli
7"

In our case 1e 2 x 10'5 cm as above and 1i = mVF/nezAilf’oT2
with A1 z 10'5 K'2 (see section 2.2.8), thus om z 0.005
(fzch)'l T. Since the resistivity contribution AP=

— pg em = -CT, (2 = pg (0.005(szch)’1) z 0.005 pncmx'l for

our samples with p0 = lyncm. This calculated value of C is

184
two to three orders smaller than our measured value of about
2 pach’l.

Considering the higher than linear power dependence on
90 predicted for electron correlation or localization
effects, the discrepancy will be even larger for the more
dilute alloys.

A Kondo effect could be proportional to no, but we did
not see any marked anomalies (e.g., see section 4.4.2) in
the G's of our samples, such as normally occur in Kondo
systems. Also, experiments more recently done in our lab
(ref. 68) showed that the turn-down of dp/dT at low
temperature end for K-Rb dirty alloy was not affected by a
magnetic field of 0.2T; thus it cannot be a Kondo effect.

In 1983 Cochrane and Strom-Olsen (ref. 70) used recent
scaling theories of the metal-insulator transition, (which
take account of both localization and electron-electron
interactions), to analyze the low temperature resistivity of
their Y-Al metallic glass samples and several other
representative alloy systems. They argued that at finite
temperature the scaling theories lead to a conductivity:
o'(T) = 0(0) (1 + Tl/z/A). They also claimed that they found
the correlation gap A~p"2, leading to a contribution to the
conductivity: Ao(T) 2 6T]'/2 (Qcm)'1.

Since the resistivity contribution Ap= 43(2) AO'(T), thus
AP z -69(23T1/2. In our case, po 2: l pflcm, this gives us a

coefficient D z 6 chmK‘l/z, which is of the same order as

185
our measured value of D 2 2 pach'l/z. Again, however, the

no dependence of our data is not as predicted.

4.5.2 W

G measurements were made for samples K-Rbl, K-Rb2, K-
Rb3, and K-Rb4 from about 4.2K down to about 0.1K. Fig. 4-
39 shows a plot of G versus T for these samples. For
comparison, data of two samples from M. L. Haerle et al.,
(with 0.38 at.% Rb' and 1.3 at.% Rb), are plotted in the same
figure.

The data of each pair of our samples with the same
concentration of Rb impurity almost overlap each other over
the whole temperature range. The data of samples K-Rbl and
K-Rb2 with 0.077 at.% Rb are well fitted by a solid curve of
c = so + 3*1'2 + (C*/T) exp(-23/T) with Go>° in Fig. 4-39.
The coefficients obtained from a computer fitting are: G0 =
0.353 1, 0.051 v‘l, A* = -o.205 i 0.014 v‘lx‘z, 3* = 2570 i
270 v‘lx.

Some other results for Go and phonon drag terms can be
seen in Fig. 4-39. For Go: (1) The value is positive for
each sample. (2) The magnitude increases with increasing
p0, and seems to have a limit of about 0.5 V‘l. For phonon
drag terms (both normal and Umklapp): (1) They are quenched
more and more as the impurity concentration increases, i.e.,
the magnitudes of these two terms decreases and thus the G

data below 1 K are more positive and the data above 3K turn

up more slowly with increasing temperature. (2) the G data

186

 

 

 

 

 

0.6—
7m.
OV.
(331‘?\\ “23
__ \\.-:.
‘6
PF 02— \. .2, a 9.4%
:4 _ \c -' a 'v :7 5 v.
0'— ‘3
o ’_ I
- . 1.3%
—0'2_ ' I I
-o.4-— .
_ 00.38% 0
‘Qér—
—o.8.-— .
_ \omm .
‘10 l l \l A l
' 1.0 2,0 3,0“ 40
T [K]
Fig. h-39. G vs T for K-Rb samples.

 

187

of K-Rb3 and K-Rb4 with 9.4 at.% Rb can no longer be well
fitted by G = Go + Al‘T2 + (B*/T) exp(-6*/T).

4.6 W
W

According to the standard theory of rflT) of high-purity
metals (see section 2.1), when the temperature is low
enough, electron-electron scattering should dominate pun,
yielding p(T) at T2. Approximate T2 behavior of p(T) has
been reported for K (ref. 14,18,19), Na (ref. 18), and Li
(ref. 20) in the vicinity of 1K. This variation has been
attributed to electron-electron scattering.

If electron-electron scattering were the only source of
iKT) at very low temperatures, then the T2 variation of pCT)
should persist to the lowest temperatures studied. For our
thick high-purity K samples, INT) varied closely as T2 only
from 1.2K down to 0.3K; below 0.3K significant deviations
from T2 behavior occurred (see section 4.1.1).

In order to see whether there are, as in K, anomalous
deviations in Na, Li, and Rb, from T2 behavior of p at low
temperatures, we measured four freely hanging, bare, thick,
high-purity samples of each of these metals from 4.2K down
to about 0.1K. G measurements were made for each sample,
concurrently with fix The details of the characteristics of

these samples, are given in Tables 3-2, 3-3, and 3-4.

188

4.6.1 Ihfi_Bfiai££ixiL¥.

Fig. 4-40 shows a plot of (Po/pT) (dP/dT) versus T from
3.6K to 0.1K for four thick (d - 1.00 mm) Na samples from
two different sources (see Table 3-4). The two dotted lines
indicate, for comparison, the T2 coefficients reported by
Levy et al. (ref. 18). The lowest temperature they reached
was 1.1K. Each set of their data followed a horizontal
straight line from 1.1K to 2.1K, above which the data turned
upward due to electron-phonon scattering. The open circles
indicate data for our sample Na-H3 with RRR a R(295K)/R(OK)
4700. The data follow approximately a horizontal straight
line from 1.21: to 1.9K with a nominal T2 coefficient
(A z 0.175 p(zcm/Kz) slightly smaller than those of Levy et
al. Above 1.9K the data turn up, as did those of Levy et
al., as electron-phonon scattering becomes more important.
Below 1.2K, the data also turn up, as in the case of K. A
few solid circles indicate data of sample Na-H4, the twin of
Na-HB. The open and filled triangles indicate data of
samples Na-Hl and Na-HZ with RRR 2:400. These samples
behave qualitatively like samples Na-H3 and Na-H4, but with
larger apparent T2 coefficients (A.z 0.215 pncm/KZ), and
smaller turn-ups at the lowest temperatures.

The observed variations in the nominal T2 coefficients
of the different Na samples shown in Fig. 4-40, as well as
the lack of a large temperature range over which a simple T2
dependence is observed, mean that considerable uncertainty

remains concerning the magnitude of the "intrinsic"

189

14&)

 

12.0—3‘ o N a

0.0— “‘3

(film/K”) _.
.m

o

l l

[e/IllAInp/Arl
.0»
‘?

P
(D
l
o

l
o

 

 

 

Fig. “-40. (PO/T)(AlnPAdT) vs T for free hanging,
bare, thick, high-purity Na samples.

190
electron-electron scattering contribution to the resistivity
in Na.

Fig. 4-41 shows a plot of (94.2K/p) (dp/TdT) versus T
from.2.8K down to 0.07K for several Li and Rb samples and,
for comparison, for typical K samples of diameter 1.5 and
3 mm.

For Li, two samples were initially measured (Li-Hla:A
and Li-HZa:v) and then remeasured after nine days at room
temperature (Li-Hlb:A and Li—H2b:v). For all four sets of
measurements, the data are consistent with a horizontal
straight line from 4.2K (the highest temperature measured)
down to about.l.6K, below which the data rise anomalously
above the extension of the horizontal line. Our data are
the same to within experimental uncertainty for all four
sets of measurements and are also in good agreement with the
data of Sinvani et al. (ref. 27) in the region of overlap
above 1.2K. The horizontal dashed line in Fig. 4-41 indi-
cates the T2 coefficient (A 2:3.0 pncm/Kz) reported by
Sinvani et al. The high Debye temperature of Li (9D(Li)z
330K) (ref. 69) ensures that electron-phonon scattering is
negligible below'4.2K, the highest temperature measured in
the present experiments.

In contrast, for Rb, which has a very low Debye
temperature (6D(Rb) z 65K) (ref. 70), electron-phonon
scattering remains important down to about 0.5K, below which
we see anomalous behavior. All three samples measured (Rb-

H2:-, Rb-Vl:o,.and Rb-V2:o) show qualitatively similar

191

 

 

 

 

 

130—-._ , °
110—,_ - °
: I
90— _ °
% '- '
A 70_ o . o Rb
< '- .
E o 8
g SO—v ‘3 0°
" =5 5; ~170
r—z—I .
E 8. ,
5. ‘. L I —l30
51 as. _
'3 ‘~ *
t3?) .v H “‘90
' V.
i " “ 21+“ 4 “‘Wrfi'?‘ 3‘4—1‘+“F+FF‘+14
9.0—é To
l"
7.04:0. K .
b'rofi 0. °.,°. 3; . '
50 l 1
° 1.0 2.0

 

 

T(K)

Fig. h-hl. (P(h.2K)/T)(AlnPAdT) vs T for Rb, Li, and K samples.

 

192
behavior, with no temperature range over which a T2
variation is dominant.

For K, included for comparison, the data are horizontal
from about 1.2K down to about 0.3K, as described in section
4JLJ, below which they rise anomalously above the extension
of the horizontal line. No measurements were made on sample
K-4A (I) of C. W. Lee (with d = 3.0 mm) and our sample K-Hla
(o) (with d a 1.5 mm) between 1.1K and 2.8K, but measure-
ments on our sample K—H6a (x) and some other samples (see
Fig. 4-1) have established that at about 1.2K the data begin
to rise rapidly above the horizontal line with increasing
temperature as electron-phonon scattering becomes important.
The Debye temperature of K (6D 2: 114K) (ref. 71) is in
between those for Na and Rb, which is consistent with
electron-phonon scattering becoming important in K at about
1.2K.

The standard theory of ART) provides no explanation for
the very low temperature anomalies in Fig. 4-40 and Fig.
4~41. One possible explanation involves low energy
excitations associated with residual defects, such as
dislocations, which can scatter electrons inelastically (see
section 2.2.2).

As noted in section 1,2;1, in their studies of deformed
, K, M. L. Haerle et a1. (ref. 22) reported that the low
temperature resistivity of the deformed samples was found to
be fit reasonably well by adding to a T2 variation a model

of inelastic electron scattering due to dislocation

193

vibration (see section 2.2.2). In this model, a single-
frequency local-phonon mode of frequency Pb is associated
with each dislocation and yields a resistivity contribution
of P8 (C/4T) sinh (hwo/ZkT) (section 2.2.2). Another
mechanism of inelastic electron scattering can be due to
additional bound states for electrons with energy slightly
larger than the Fermi energy existing near the cores of
dislocations. This model yields a resistivity contribution
of P: o:(l +5exp(e/kT))'1 (section 2.2.2). These two
models yield maxima in both (l/T) (dP/dT) and dP/dT, with
the maximum occurring at higher temperatures in the latter
case. A maximum in dP/dT was found in deformed K at about

0.2K (ref. 22).

Fig. 4-42 shows a plot of (p4.2K/P) (dp/dT) vs T from
0.1K to 1.4K for the Na, Li, Rb, and for comparison for K
samples shown in Figs. 4-40 and 4—41.

The K data shown in Fig. 4-42 do not display a maximum
even in dP/dT. It is not yet clear whether this anomaly in
undeformed K is due to residual effects of incidental
plastic deformation during cooling, in part because this
anomaly does not extend over a sufficiently large tempera-
ture range to allow discrimination between different models
fits. Also, we do not yet have a plausible mechanism for
producing enough dislocations to cause such an anomaly in
the undeformed K.

The undeformed Rb data shown in Fig. 4-42 shows a hint

of local maxima in dP/dT near 0.2K, but both the uncertainty

194

 

 

 

 

 

 

 

 

 

120“
80—— o
Rb . I, if
40?— .;,°° ' ' 7:80
0— 'v ' ' //’I “—60
E .‘."‘ /// —40
E .‘ ' Li
l- " ///I 7 —20
Q /’
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a ‘ ‘ ........... o [’3'” “—4.0
.. . 0 I ‘A . o O .L. ...... 6. I”’
.*° .° --/-°',":”’ ——2.0
f .I..;.”’
w”l l l 1 1 1.1 o
0 0 2 04 0.6 0.8 l 0 1.2
T(K)
Fig. h-AZ. P(4.2K)(AlnPAAT) vs T for Rb, Li, K,

and Na samples. The data are the same

as in Fig. h-QO and Fig.24-hl.
dashed lines indicated T

The

resistivity

variations inferred from all of the

available data.

195

in the data and the large contribution of electron-phonon
scattering (the very-low-temperature form of which is not
known for Rb) preclude any meaningful analysis until meas-
urements on higher purity samples are extended to still
lower temperatures.

The undeformed Na data shown in Fig. 4-42 show maxima
which occur at very different temperatures for Na from our
two different sources.

A plausible generation mechanism for the residual
defects needed to produce the observed anomalies in Na and
Rb could be the martensitic transformation that Na undergoes
at about 35K and Rb might undergo at about 4.2K (ref. 72).

In undeformed Li, the anomaly sets in at a high enough
temperature (21.6K) that a meaningful fit can be attempted,
and the simple straightforward T2 form of INT) above about
1.6K makes the fitting procedure highly selective between
different alternatives. In addition, as recently suggested
(ref. 20), the martensitic transformation that Li undergoes
at about 75K provides a plausible generation mechanism for
the dislocations needed to produce such an anomaly.

We tried to fit our Li data in both dP/dT and dP/TdT
forms with several alternatives. Fig.«4—43 shows 4 differ-
ent fits for the data in dP/dT form, while Fig. 4-44 shows
these fits in dP/TdT form for our data. First we tried to
fit our Li data with a T2 term plus a single-frequency
local-phonon mode term, i.e., P= po+ AT2 + (C/4T)

sinh'2(ht./2kT) (see section 2.2.2), thus,

196

 

 

 

 

 

 

 

2
./ a
120- single local er/T _, .
[phonon mode ”l/ ’ ,,/
0”
( ) ’/" so"
30‘ A ./C / ,.// —.120
I y/ / ',./
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I n ‘f/ / ""‘/
I I, ‘ I / / /v./"/
40". *g’ ' ' / __80
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:5 T X j:
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phonon modes r/{/,.,z ./.
v / ’
80.“ (c ) ,/'// / (.4/ #120
/ / ,,(/
v" / /‘//"/
/'// / y/
40‘“ "‘3‘" / / / "‘80
f. / / 2’ /
,/ vf‘”// /’
04“ ’ ~40
i / ——-—->
3‘ f/[tfilo bound electron states]
/
L5 1 l l ‘ l 1 1 1 1 I 1 1 1 0
o 1.0 2.0
T [K]

Fig. 4-h3.
samples.

Trial fittings for P(L+.2K)Alnp/AT of Li

197

 

   

 

 

 

 

 

     

 

 

 

 

 

 

if: —l70
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Fig. h-hh.

Trial fittings for (P(l+.2K)/T)(AlnP/AT)

of Li samples.

198

dP/dT ZAT + (C/4T2)sinh'2(hw/2kT) x

((hw/kT) ctgh(haM2kT) - 1) (A)

and,

dP/TdT = 2A + (C/4T3)sinh-2(hw/2kT) x
((hw/kT) ctgh(hw/2k'1‘) - l) (1)

This model described our data reasonably well from 2.8K down
to about 0.2K, see the solid curves in part (A) of Fig. 4-43
and part (1) of Fig. 4-44, below 0.2K the theoretical curves
dropped off more rapidly than the data. As an alternative,
we also tried a fit with a T2 term plus a term associated

with bound electron states at dislocations, i.e.,

p: p°+ AT2 + a(1 +5exp(E/kT))"l,
thus,

dP/dT = 2m: + (aflE/k)exp(E/k‘1‘) (1 +flexp(E/KT))"2 (B)*
and,

dP/TdT = 2A + (afiE/kT)exp(E/kT) (1 +fiexp(E/kT))'2 (21*

with ,8(the spin degeneracy of the additional level) fixed

as 1, we have
dP/dT = 2AT + (aE/k)exp(E/kT) (1 + exp(E/kT))"2 (B)
and

dP/TdT = 2A + (aE/kT)exp(E/kT) (1 + exp(E/kT))"2 (2)

199

These fits shown as the solid curves in part (B) of Fig. 4-
43 and part (2) of Fig. 4-44, are even worse at the lowest
temperatures. The problem with both models is that at
temperatures well below their maxima in dRVdT, their single
exponential decay for dP/dT is too rapid for our data.

These two fits can be improved by using two local-phonon
modes, or two bound electron levels, with adjustable
energies and coefficients, but still fail at the lowest
temperatures, as shown in parts (C) and (D) of Fig. 4-43 and
parts (3) and (4) of Fig. 4-44. This improvement could mean
either that we have a range of dislocation lengths in our
samples, or just that we have introduced enough parameters
to describe almost any smooth, peaked behavior in dP/dT.

The coefficients deduced from each fit above are given in
Tables 4-9 and 4-10.

The differences between the residual resistivities
(measured at about 1K in this study) and 4.2K resistivities
are very different for different alkali metals. As we
mentioned above, this difference is about 0.28 nncm for
potassium. For Rb, this difference is as big as about
6.2 nncm, since Rb has a much lower Debye temperature than K
has, so the resistivity contribution of electron-phonon
scattering in Rb samples at temperatures lower than 4.2K is
much larger than for K samples in the same temperature
range; and 1K might not be a low enough temperature for
measuring the residual resistivity of Rb. On the other

hand, since the Debye temperature of Na is significantly

 

200

 

 

 

 

 

 

 

 

 

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201

higher than that of K, the resistivity contribution of
electron-phonon scattering at 4.2K for Na is expected to be
smaller: our results (Table 3-4) showed this contribution is
less than 1% of residual resistivity. For Li, we did not
measure the 1K resistivity, but the Debye temperature of Li
is even higher than that of Na, so FK4.2K) should be usable

as the residual resistivity.

4.5.2 Ih£_Ih£LmQ£l£QLLis_BaLiQ_9

G measurements were made concurrently with p from 4.2K
down to about 0.1K for all samples described in section
4.6.1.

The G data for the four Na samples are shown in Fig.
4-45. The data for samples Na-Hl and Na-HZ are similar to
each other in both form and magnitude, and can be well
fitted by G a G0 + B*T2, (since the Debye temperature of Na
is as high as 160K, it is plausible that the Umklapp phonon
drag term might be negligible below 4.2K). The lower solid
curve in Fig. 4-45 indicates such a fit to sample Na-HZ, and
the deduced coefficients are co = -0.296 3; 0.003 (V4) and
3* a -0.0176 1 0.0004 (V'lK'z). The data of samples Na-H3
and Na-H4, from another source, are similar in form to each
other, and for reasons not yet clear, show a small turn down
at the lowest temperatures. The upper solid curve in Fig.
4-45 indicates a fit of G = G0 + B*T2 for sample Na-H3. The
deduced coefficients are G0 = 0.063 i 0.021 (v-1) and B* =
-0.065 i.0-0027 (V'lK’z). In contrast to K samples, the Na

samples with smaller 1%3 show more positive Gos.

202

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203

The G data for all four Li samples are quite similar to
each other, not only in the form but also in magnitude, as
shown in Fig. 4-46. Since Li has an even higher Debye
temperature than Na, the G data of Li samples are also
expected to be fitted by G = G0 + B*T2. The solid curve in
Fig. 4-46 indicates that the data of sample Li-HZb can be
very well fitted by such an equation, and the deduced

+ 0.0005 (v‘1) and 13* =

coefficients are G0 = 3.7367
-0.00707 1 0.00006 (V’IK'Z). The slight turn-up in the c;
data of samples Li-Hla, Li-H2a, and Li-Hlb at the very
lowest temperatures is not yet understood.

In the Rb samples, on the other hand, the G data show
different forms from those for Na or Li. Since the Debye
temperature of Rb is as low as 65K, the G contribution of
Umklapp phonon drag will not be negligible even in the

vicinity of 1K. The data are then expected to be fitted by
G = G0 + B*T2 + C*/T exp(-6*/T).

The G data for all four Rb samples are shown in Fig. 4-47.
The solid curve in this figure indicates that the data of
Rb-HZ are almost perfectly fitted by the equation given just
above. The deduced coefficients from the fit are G0 =
1.0566 i 0.0025 (v‘1), 13* = -o.211 i 0.006 (v‘lx’Z), c* =
561 1 32 (V'lK), and 6* = 9.71 i 0.12 (K).

204

 

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A
o Li-Hla
3.6— . li-H2a
A li-Hlb
A i. i-H2b
l l 1 1
0 1.0 2.0 3.0 4.0
T (K)
Fig. h-h6. G vs T for Li samples.

 

205

 

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206

4.7 i.. .:t k =::. . \ . .-- - . _. - E

In section 4.2 we described the size effects in f>and G
for free hanging, bare, thin, high-purity K samples cooled
in Ar or Be gas. In order to test whether similar size-
effects can also be seen in pure Na, we made 9 and G
measurements for six thin Na samples cooled in Ar and He
gas. Samples Na-l/4Al and Na-1/4A2 (d = 0.25 mm) were
cooled in Ar gas, samples Na-l/lOHl and Na-l/lOHZ (d a
0.1 mm) were cooled in He gas. The details of the

characteristics of all these samples are shown in Table 3-4.

4.7.1 The_BeaistixiL¥
Fig. 4-48 shows a plot of normalized (dP/TdT) versus T

for these six thin Na samples from 3.6K down to about 0.1K.
For comparison, the data of one thick Na sample made of Na
from the same source (Na-H3 with d = 1.0 mm) are also shown
in Fig. 4-48 by (x) with a solid curve through them.

The data of samples Na-l/4Al (a) and Na-l/4A2 (o) are
quite close to the solid curve, no obvious size-effects are
shown. 0n the other hand, data of samples Na-l/4Hl Cfil and
Na-l/4HZ (*) are lower than the solid curve for T 2.1K,
showing a hint of the same size-effects seen in K samples.
But, for reasons which are not yet understood, the data of
still thinner samples, Na-l/lOHl (A) and Na-l/lOHZ (A), are
a little higher than those of Na-l/4Hl and Na-l/4HZ,
contrary to what is expected for the size effects.

To see the possible size-effects more clearly, in Fig.

4-49 we plot normalized dP/dT versus T from about 3.3K to

 

207

 

 

 

 

A o Na-H3
" O . Na-l/4AI
12.0-E O Na-I/4A2
L Tr Na-I/4Hl
r—\
«x — 1 * Na-I/4H2
A
\E g . Na-I/IOHI
c100"" A A Na-l/IOH2
:: _ ..°
0
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'21 i
— to
a u.
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d 60". f
1: _ i 0 °
\ 11... 0
OS *2‘ 0 .
h—J 4.0L_ :0 O O
4- o - it
- M" " AA
W4. . it *
2.0—
l l L J l l l l l l L l l l
0 10 2.0 - 30
T (K)
Fig. ’-+-’-18. (PO/T)(AlnP/AT) vs T for thin Na samples.

 

 

208

 

 

 

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8.0-
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7.0- /.
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/
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()/ 1 l 1 l 1 l
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T [K]
Fig. 4-49.

(PO/P)(AFVAT) vs T for selected thin Na samples.

 

 

209

about 0.1K for three selected samples from above. Again,
for comparison, the data of Na-H3 are indicated by Open
squares, and the coefficient of the nominal T2 term is
indicated by the slope of the broken line. The data of all
three thin samples are lower than the broken line for
temperatures between about 1K and 3.3K. According to the
theory of inelastic scattering of electrons by defects and
impurities, the sample with larger 1% is supposed to have a
larger coefficient of T2 term, but here we see contrary
results. These results can be explained by electron-surface
scattering effects, (i.e., size effects) (section 4.2.1).
On the other hand, all six thin Na samples showed no nega-
tive dP/dT. This means that any size effects in thin Na
samples are not as strong as these in thin K samples. A
partial reason could be that the electron mean free-path in
bulk Na samples is smaller, about 0.14 mm calculated from
lei = (1/10,‘,)(rs/a0)2 x 92 A (Eqn. (4-1); ref. 4). But this
could not be the main reason, because this is only a factor
of 0.65 smaller than 0.2 mm, the electron mean-free-path in

bulk K samples.

4.7.2 Ih£_Ih2LmQ§l££LLiQ_BaLiQ_§

G measurements were made concurrently with P for all
six thin Na samples described in section 4JLJ.from about
4.2K down to about 0.1K.

Fig. 4-50 shows a plot of G versus T for these six

samples. For comparison, the G data of sample Na-H3 (made

 

210

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211

of Na from the same source), originally shown in Fig. 4-45
are shown again in Fig. 4-50 by means of the best-fit curve.
As in K samples (section 4.2.2), in general, the
thinner Na samples show less negative G at high temperatures
and more negative G at low temperatures than do the thicker
samples. This means that the thinner samples have more
negative Go and less negative 8*. The possible explanations

of this are given in section 4.2.2.

4.8 Na_EAM2LES_ENQASED_IN_£QLXEIEXLENE_IHBEE

In section 4.4 we described likely Kondo effects in P
and G for K samples encased in polyethylene tubes. In order
to test whether Kondo effects can also be found in Na in
contact with polyethylene, we measured tiand G for two Na
samples encased in 1.6 mm dia polyethylene tubes under Ar
gas (Na-PAl and Na-PAZ). To test for annealing effects,
each sample was measured twice, allowing it to anneal for
two different periods. The details of the characteristics

of each sample in each measurement are given in Table 3-7.

4.8-1 W

Fig. 4-51 shows a plot of normalized dP/TdT versus T
for Na-PAla, Na-PAZa, Na-PAlb, and Na-PAZb from 3.6K down to
about 0.1K. For comparison, data of sample Na-H3 are indi-
cated by open circles in Fig. 4-51. All data of these four
samples almost overlap each other and overlapsthe ones of
Na-H3 for temperature lower than 2.2K, no Kondo-effect-like

turn down (see section 4.4) have been seen in Fig. 4-51. In

212

 

 

 

 

14!) o
— A No -PAlo
12.0- A No -PAia
O Na
_ v No -PA2b
v Na-PAQb
10.0— C;
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2 s
C'. 8 '
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2.0-
o N0-H3
1 1 v L,l 1 1 1 1 l 1 1 1 1 I 1 1
0 1 3.0

T(1<)2'o

Fig. li-Sl. (Po/T)(AlnP/AT) vs T for Na samples encased
in polyethylene tubes.

212

 

 

 

 

14.0
0
F A No -PAlo
12.0“ A No -PAlc
. Na
__ V No -PA2b
v No-RAZb
1CHD—' '1
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1 1 1 l 1 1 1 1 l 1 1 1 1 l 1 1
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T (K)
Fig. h-Sl. (Pb/T)(AlnP%dT) vs T for Na samples encased

in polyethylene tubes.

 

213

order to see the data at the low temperature end more clear-
ly, we plot dP/dT versus T in Fig. 4-52 for samples Na-PAla
and Na-PAlb. For comparison, data of samples Na-Hl and Na-
H3, originally shown in Fig. 4-42, are also shown n Fig. 4-
52. Data of sample Na-PAla in Fig. 4-52 show a clear turn-
down at temperatures lower than 0.3K, giving a hint of a
Kondo-like effect. But for reasons not yet clear, data of
the same sample annealed for a longer period show, unex-

pectedly, a low temperature turn-up.

4.8.2 Ih2_Ih£LmQ£1££LLiQ_BaLiQ_§

The plots of G versus T for samples Na-PAla, Na-PAZa,
NaePAlb, and Na-PA2b are quite similar to the ones of free—
hanging, bare, thick, pure Na samples and no negative
anomalies at low temperatures were seen. These plots are

not given in this dissertation.

214

 

 

 

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CHAPTER 5

Summary and Conclusions

The general outline of this present thesis work is
described in section 1.3. The conclusions of this study are
as follows:

1) For free hanging, bare, thick, high—purity K
samples cooled in Ar, He, or partial vacuum, with improved
techniques we found that a) the temperature-dependent
electrical resistivity p(T) varied closely as AT2 from 1.2K
to 0.2-0.4K (for different samples), with A varying from
0.19 to 0.24 pacm/K2 for samples under different
circumstances; b) below 0.2-0.4K, all the samples showed an
anomalous turn-up in a plot of dP/TdT versus T; c) the
thermoelectric ratio G of these samples could be well fitted
by the theoretical formula G = Go + B"’T2 + (C*/T)Exp(-9*/T).
These conclusions agree well with what C. W. Lee et al.
found for thick pure K samples cooled in Ar gas. With an
improved sample can and other techniques, we believe that
our G data have higher accuracy than C. W. Lee's data. We
have co = -0.075 1; 0.015 v’l, 3* = -0.484 3; 0.005 v'lx'z,
and c* = 6200 i 140 v‘lx, with 9* fixed as 23K.

2) For free hanging, bare, thin, high-purity K samples

cooled in He gas we found that the data of normalized dP/dT

215

 

216
displayed a clear pattern of unusual behavior which is
consistent with that reported by Rowlands et al. in the
temperature and diameter regions of overlap, but more com-
plex in form. dP/dT even became negative in the vicinity of
1K for the thinnest samples whose diameters were comparable
to or smaller than electron mean free path (~0.2 mm). The
more complex behavior of our data rules out both the simple
T3/2 form for p(T) that Rowlands et al. originally proposed,
and all previously published explanations for the data of
Rowlands et al. The only model we know, which might explain
the negative dP/dT we observed, is an interaction between
surface scattering and normal electron-electron scattering
as first prOposed by Gurzhi and then calculated by a Monte
Carlo method by Black. However, the data are not in a
regime where a closed form expression for P has been de-
rived. We therefore tried various trial fits to our data.
The best fit was obtained as: (P(4.2K)/P) (dP/dT) =- 2A'T -
55T4/3/x/d7'I—Tn'n7, where A' ranged from 0.59 to 0.70 pSzcm/K2
for samples with different diameters. For reasons not yet
clear, similar samples cooled in Ar gas or partial vacuum
tended to behave like data for samples in He of larger
diameter and to show greater variability for fixed sample
diameter. We also found expected size effects in the resi—
dual resistivity po. #5 = .QD+-c/d was obtained for samples
with d20.l6 mm, where c a (3.4 i 0.5) x 1011 Qcmz was com-
parable to the best literatue value (c = 2.9 x 10"ll Qcmz).

For reasons not yet clear, the £%'s of our thinnest samples

217
(d$0.l mm) showed a lot of scatter, with most of the 90's
much larger than expected.

All G data for these samples could still be well fitted
by c a Go + 3*T2 + (C*/T)Exp(-9*/T) with 9 fixed as 23K, but
the thinner samples show larger magnitudes of negative Go'
smaller magnitudes of negative 8*, and smaller magnitudes of
C*. This behavior could be explained by the Gorter-Nordheim
rule plus a surface scattering effect on phonon drag.
Further investigations on size-effects, including making
thin samples under vacuum and measuring thin film samples
etc. are in progress in our lab.

The only S data taken on a 0.5 mm dia K sample, indicate
that L(T) :1: Lo, i.e., S = LOGT, is approximately valid only
at T 5 1K for potassium.

3) For pure potassium samples prepared and cooled in
contact with hydrocarbons, such as encased in polyethylene
tubes or melted in clean paraffin oil, a Kondo-type effect
was seen in both Pand G measurements. The data of P can be
well fitted by p a po + AT2 - BlnT where the -BlnT term is
the Kondo effect term. The data for G show a negative
anomaly below 1K. The data below 1K can be well fitted by G
= G0 + B*T2 + D*/T, where D*/T term is attributed to the
Kondo effect. A stronger Kondo effect was seen: (a) in K
samples encased in polyethylene tubes compared to one melted
in paraffin oil, (b) in samples encased in thinner polyethye
lene tubes compared to ones encased in thicker tubes, and

(c) in samples encased in polyethylene tubes and held at

218
room temperature for a long period compared to those held
for only a short period. K samples in contact with non-
hydrocarbon plastics such as teflon or Kel-F showed no Kondo
effect in either l’or G. An average magnetic field of 0.1T
applied longitudinally'to polyethylene encapsulated samples
shifted the resistance minimum to much lower temperatures.
Thus, these samples seem to be exhibiting a Kondo effect due
to magnetic impurities. The magnetic impurities seem to be
a product of the reaction between potassium and hydro-
carbons. Further investigation attempting to identify the
magnetic impurities is in progress in our lab. In addition
to increasing the size of the Kondo-type effect, sample
annealing at room temperature also brought both p5 and A
down, as also observed by van Kempen et a1. Po changed by a
factor of about 3, consistent with the results of van Kempen
et al. However, the change in A was only about 30%, much
less than the factor of 3.6 reported by van Kempen et al.

4) For K-Rb alloy samples, a plot of Po vs c shows
that the value of Po per atomic percent Rb impurity is in
close agreement with the best literature value of 0.13
imam/at% for our dilute alloy samples. Plots of normalized
dP/TdT vs T showed that (a) each set of data follow a
horizontal straight line roughly from about 0.5K to about
1.2K, yielding p= p0 + A'TZ. Here A' can be well fitted by
A' = A0 + Aipo, with A0 found to be 0.24 i 0.04 p$2cm/K2 for
an electron-electron scattering contribution, and A1

found to be 11 x 10"6 K"2 close to the theoretical values of

219

13.7 x 10’6 K"2 from P. L. Taylor and 12.5 x 10"6 K"2 from
Kus and D. W. Taylor for inelastic electron-impurity scat-
tering contribution; (b) a low temperature turn-down became
progressively more apparent as the Rb concentration in-
creased. The reasons for this turn-down are not yet clear,
but our data can be fairly well fitted by adding to P 8 Po +
A'T2 an additional term having any one of the alternative
forms - CT, -DT1/2, or -BlnT. The -CT term would be ex-
pected from the localization effect; the -DT1/2 term from
the electron-electron interaction effects; the -BlnT term
from the Kondo effect. However, our turn-down is approxi-
‘mately prOportional to p0, not proportional to pg or PCS/2
as expected for localization or electron-electron interac-
tion effects. And, the G measurements and further meas-
urements under a magnetic field of 0.2T did not show any
behavior such as normally occurs in Kondo systems. More-
over, for the 9.4 at.% samples, the calculated value of C
from localization effects is two to three orders of magni-
tude smaller than the deduced value of C from our data, and
the calculated value of D from electron-electron interaction
effects is one order of magnitude smaller than the deduced
value from our data. Considering the higher than linear
power dependence on Po predicted for electron-electron
interaction or localization effects, the discrepancy will be
even larger for the more dilute alloys.

We also found that the G data of dilute K-Rb alloy

samples could still be well fitted by G = G0 + B*T2 +

220
(C*/T)Exp(-9*/T), where Go is positive and increases with

increasing po to a limit of about 0.5 V"1

, and the magnitude
of both 8* and C* decrease with increasing pb, (iJL, the two
phonon drag terms are quenched more and more as the impurity
concentration increases). These variations in Go' 8*, and
C* can be explained by the Gorter-Nordheim rule plus an
impurity scattering effect on phonon drag. Measurements on
still more concentrated K—Rb alloys and a search for a
similar anomaly in K-CS alloy samples are planned.

5) For free-hanging, bare, thick, high-purity Na, Li,
and Rb samples we made l7and G measurements from 4.2K down
to 0.07K. A T2 dependence was found for Li from 1.6K to at
least 4.2K (the highest temperature in the measurements)
and, perhaps, for Na from 1.2K to 1.9K. No T2 dependence of
p was found for Rb. We found anomalous turn-ups at low
temperatures in a plot of dP/TdT vs T, as seen in K samples,
for all Li, Na, and Rb samples. The turn-up is largest in
Li and smallest in Na. Above 0.2K, the data of p for Li can
be fairly well fitted by adding to a T2 variation a model of
inelastic electron scattering due to local phonon mode of
dislocation vibration or due to bound electron states
associated with dislocations. Below 0.2K the theoretical
curves dropped off more rapidly than the data. The fits can
be improved by using two local phonon modes, or two bound
electron levels, with adjustable energies and coefficients,

but still failed at the lowest temperatures.

221

We also found the G data of Na, Li, and Rb to be well
fitted by the theoretical form G = G0 + 3*T2 + (c*/T) x
Bxp(-6*/T). For Na and Li, which have relatively high Debye
temperatures, the third term due to Umklapp phonon drag was
found to be negligible. Because different metals have dif-
ferent Debye temperatures, and thus electron-phonon scatter-
ing would die off at different temperatures, the differences
between 1K4.2K) and p5 (measured at about 1K) were found to
be very different for Rb, K, Na, and Li. We found this
difference to be about 6.2 chm for Rb; about 0.28 nacm for
K, (consistent with what van Kempen et al. found): and less
than 0.01 nflcm for Na. For Na and Li, 4.2K is low enough
for measuring the residual resistivity p0. On the other
hand, for Rb, 1K might not be quite low enough for measuring
Po accurately.

6) For thin Na samples we found possible size effects
in both P and G, which, if they really exist, are much less
strong than the ones in K samples. No negative dPVdT was
found in thin Na samples.

For Na samples encased in polyethylene tubes, no

obvious Kondo effect was found in p or in G.

10.

11.

12.

13.

14.

15.

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