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“E5“ I Laminar I
Michigan grate
‘ ._.~ University l

112 °
J

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£ .
i

 

—I-O

This is to certify that the
dissertation entitled

THE EFFECTS OF PLASTIC DEFORMATION ON THE
ELECTRICAL RESISTIVITY AND 'I‘I-IERMOEIEC'I‘RIC
RATIO OF POTASSIUM FROM 0.07-K TO 4.2K

presented by

Mark Lyndon Haerle

has been accepted towards fulfillment
of the requirements for

Ph.D. Physics

degree in

 

 

Major professor

 

Date March 027, F733

 

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

 

 

RETURNING MATERIALS:

MSU Place in book drop to”

LIBRARIES remove this cher' ~- 'vn

= w" "'1' -' rec

 

 

I? m 7 I " I —-—--e.-__ m. -

THE EFFECTS OF PLASTIC DEFORMATION ON THE ELECTRICAL

RESISTIVITY AND THERMOELECTRIC RATIO OF POTASSIUM

FROM 0.07K TO 4.2K

 

BY

Mark Lyndon Haerle

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1983

/37-937&

ABSTRACT

THE EFFECTS OF PLASTIC DEFORMATION ON THE ELECTRICAL
RESISTIVITY AND THERMOELECTRIC RATIO OF POTASSIUM
FROM 0.07K TO 4.2K
BY

Mark Lyndon Haerle

Samples of pure polycrystalline potassium were
plastically deformed at 4.2K and at 60K. One sample of
polycrystalline potassium - rubidium alloy was deformed at
60K. Such deformations had a significant effect on both the
electrical resistivity and the thermoelectric ratio.

In the resisitivity, deformation enhanced the electron
— phonon term from 4.2K to below 1K causing a DT5 term to
appear in [3, possibly due to the quenching of phonon drag.

Furthermore, a new term was seen ( fig ), which dominates

-dis
the temperature dependent resistivity in deformed samples
below 1K. This new term was interpreted as being caused by
dislocations, and could be fit resonably well by a model in
which the electrons scatter inelastically off localized
phonon modes associated with the dislocations.

The effects of dislocations on the electron - electron
term in the resistivity were obscured by F%-dis' Until
fiDe-dis is more fully understood, it will remain difficult

to identify any variation in the electron - electron term

caused by dislocations.

For the thermoelectric ratio, both the normal and
umklapp phonon drag terms were suppressed by the
introduction of dislocations. In addition, the magnitude of
the diffusion term was increased by both point defects and
dislocations, with the systematic addition of dislocations
giving agreement with the Gorter - Nordheim relation.

Finally, a resistance minimum near 0.2K was seen for
potassium in contact with various plastics. The added term
to the resistivity followed a -ln(T) behavior, which is seen
in Kondo - effect systems. The thermoelectric ratio for

these samples was variable and followed no clear pattern.

 

ACKNOWLEDGMENTS

I wish to thank both Dr. Pratt and Dr. Schroeder for their
Supervision and invaluable aid in all aspects of this
dissertation. I am particularly grateful for the many late
nights both worked while taking data.

I extend a thanks to Dr. Bass for many helpful
discussions, to John Zwart who helped take data on several
occasions, and to Vern Heinen who helped calibrate GRTZ.

Finally, I acknowledge the financial support of the

National Science Foundation and Michigan State University.

TABLE OF CONTENTS

Page
LI ST OF TABLES O O O O O O O O O O O O 0 O ..... O O O O O O O O O O O 0 O O 0 O O 0 O O O O 0 v
LI ST OF FIGURES O O O O I 0 O O O O O O O O O O O O O O 00000 O 0 O O 00000 O O O O O 0 vi

Chapter

IntrOducti-On OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...l

1 Previous work on potassium ................. ..... 3
1.1 Resistivity ........... ..... ..... ................ 4
1.1.1 The residual resistivity ............... 7

1.1.2 Electron-phonon scattering ............. 9

1.1.3 Electron-electron scattering .......... 17

1.1.4 Electron-phason scattering ............ 24

1.1.5 Electron-dislocation scattering ....... 25

1.2 Thermoelectric power ............ ...... ......... 28
1.2.1 Diffusion thermopower ........ ......... 28

1.2.2 Phonon drag thermopower ............... 31

1.2.3 Thermoelectric ratio .................. 32

2 Experimental techniques ........................ 36
2.1 Sample can ..................... ........ ........ 37
2.2 Plastic deformation of the sample ........ ...... 41

2.3 Removing He gas from the sample can ............ 44
2.4 Sample preparation ............................. 48
2.5 Measurements ................................... 56
2.6 Heat flow in the sample can .. ..... ............. 65
2.7 Thermometry .......... ..... ..................... 70

iii

 

2.8 Uncertainties .................................. 77

3 Experimental results ........................... 80
3.1 Resistivity .................................... 81
3.1.1 The residual resistivity .............. 84

3.1.2 The electron - phonon resistivity ..... 86

3.1.3 The low temperature data .............. 90

3.1.4 The intermediate temperature range ... 110

3.2 Thermoelectric ratio results .................. 111
3.3 Resistance minimum ............................ 120

4 Conclusions ................................... 126

LIST OF REFERENCES ..... 128

LIST OF TABLES

Table Page
2.1 Sample can parts ................................ 38
2.2 Materials in direct contact with the samples. ... 49
2.3 Time table of making samples .......... ..... ..... 51
2.4 Chemical analyis of potassium ................... 51
2.5 Sample parameters ............................... 57
2.6 Alternate heat flow paths ....................... 69
2.7 Current comparator error in absolute current .... 78
3.1 Recovery of annealed samples .................... 85
3.2 Effective A for the samples ..................... 96
3.3 Parameters from fits to the resistivity data ... 105
3.4 The coefficients from fits to the G data ....... 116

 

LIST OF FIGURES

Page

11
15

23

39
40
43
46
52

55

58
61

67

71

75

76
82
83

Figures

1.1 An edge dislocation ................................ 6

1.2 Normal and umklapp scattering .....................

1.3 Comparison with theory for the electron-phonon ....
term

1.3 Comparison with theory for effects of dislocations
on A

2.1 Sample can ........................................

2.2 Sample area .......................................

2.3 The He pressurization system for the bellows ......

2.4 Indium filled brass tube ..........................

2.5 Stainless steel press ............... ........ ......

2.6 He gas handling system for pumping out the sample .
can

2.7 The low temperature circuit .......................

2.8 Temperature dependence of the copper reference ....
re51stor

2.9 A) Weak thermal link assembly .....................
B) thermal connection tab

2.10 The voltage of the silicon diode vs temperature ...

2.11 W-F law check of TZXT calculated using the ........
GRT2 calibration

2.12 wa vs T ..........................................

3.1 High temperature comparison with previous groups ..

3.2 Low temperature comparison with previous results ..

3.3 The residual resistivity vs the pressure of ......

deformation

vi

87

The effects of deformation on the electron - 89
phonon term

(1/T)(dfl)/dT) vs T for Kh8 .................. 91
(l/T)(dfl)/dT) vs T for Kh9 ......... 92
(1/T)(dP/dT)vsT for KRbh 93
ZXA' vs the change in the residual resistivity .. 97
cap/am vs T for Kh8 ......................... 100
dp/dT vs T for Kh9 .. ..................... 101
dp/d'r vs T for KRbh 102
Comparison of the best fits of Equations ........ 106
1.26 and 1.23 to the data of Kth

Comparison of A with Kaveh and Wiser‘s theory ... 108
The thermoelectric ratio for Kh8 112
The thermoelectric ratio for Kh9 113
Gvs T2 .. ......... ......... 114
Modified Gorter - Nordheim plot ..... .. 117
dG/dT vs T .................... ........... 119
(1/T)(dp/d'r) vs T for Kh2e, Kh3, Kh4, and Kh7 121
(1/T)(dp/dT) vs T-Z, a Kondo system? 123
G vs T for the resistance minimum samples ....... 124

vii

 

Introduction

This dissertation is an experimental study of the
effects of plastic deformation on the electrical transport
properties of polycrystalline potassium, from 70mK to 4.2K.
The two parameters measured as a function of temperature
were the electrical resistivity and the thermoelectric
ratio, with the main emphasis on resistivity.

Potassium has long been a material of considerable
theoretical and experimental interest. The reasons for this
are clear. Potassium is believed to have a nearly spherical
Fermi surface which does not touch the Brillouin zone
boundary. Thus the nearly free electron model is a good
approximation for many calculations, making transport theory
comparatively easy to carry out. Furthermore, in contrast
to potassium; lithium, sodium, and possibly even rubidium(1)
undergo martensitic phase transformations at low
temperatures which complicates the interpretation of
measurements. Also potassium is less reactive than rubidium
or cesium and thus is easier to work with experimentally.
Finally, Overhauser(2) believes potassium is a good testing
ground for the existence of charge density waves (CDW),
which result in broken translational symmetry of the ground

state. If found to exist, CDW's would radically change our

1

2
basic understanding of transport in metals.
The main motivation for this study was a theory by
Kaveh and Wiser(3) which predicted for electron - electron

scattering an AT2

term in the resistivity, where T is the
absolute temperature. At low temperatures when scattering
off phonons can be ignored, the resistivity was predicted to
be,
_ 2
p-po+AT
where the coefficient A was dependent on the relative level
of impurities and —dislocations. It was believed that a
systematic variation in the number of dislocations would
provide a good test of this theory. Plastic deformation at
low temperatures is a good way to produce dislocations, so

it was proposed that we plastically deform potassium and

measure the change in the resistivity.

 

CHAPTER 1

Previous work on Potassium

This chapter is an overview of the previous work on

potassium. As in all studies of electrical transport, the

fundamental basis of the work are the transport equations,

J Lee E + LetVT (1.1)

o Lte E + LttVT (1.2)

where J is the electric current density, Q is the heat flow
current density, and E is the electric field. The L
coefficients are in general tensors, but since potassium has
cubic symmetry, they can be replaced with scalars. The two
quantities of importance to this study are [3 the electrical
resistivity and S the thermopower. We define the electrical
resistivity when V7T=0 by the equation,

a = 0‘ E= p‘1 E (1.3)
so, (T: p'1 = L (1.4)

ee

Similarly, we can define the thermopower generated by a

4

temperature gradient when J = 0.

_ -1__ -1
s - 6(K7T) - [ Lee] L (1.5)

et

1.1 Resistivity

Using Matthiessen's rule, the resistivity of potassium

is generally interpreted as a sum of terms,

p= ,00 + ,Oep(T) + peeVr) + nether (1.6)
where LEN the residual resistivity, is a constant
independent of temperature. It is due to scattering off

various static imperfections in the crystal such as
impurities, other point defects, and dislocations.

The second term ޣp(T) is due to electrons scattering
off phonons, and at low enough temperatures (below 1.3K in
undeformed potassium) this term becomes negligible. In a
pure sample it accounts for 20% - 30% of the total change in
the resistivity between 1K and 4.2K.

[3ee(T) is the resistivity due to electrons scattering
off other electrons in the sample. At 1K this term is

4

~10 times smaller than the residual resistivity in pure

potassium. A careful study of this term requires
temperatures below 1.2K and at least one part in
107 precision.

The last term, D

Other could easily represent a sum of

terms itself. The first of these is the

'Oe-phason'

 

5

resistivity due to electrons scattering off phasons, the
elementary excitation of charge density waves. Although
recent evidence(4) suggests there is no need for such a
term, its existence in potassium is still under debate.
Another possible temperature dependent resistivity

would be pram,

due to interactions between dislocations
and electrons. A dislocation is an extended line defect in
a crystal. On deforming potassium, one expects the
dislocations added to be mostly edge dislocations.
Figure 1.1 shows an edge dislocation at the surface of a
crystal, the line of the dislocation is perpendicular to the
page. Electrons can interact with dislocations through
several possible mechanisms. For example, the dislocations
might vibrate about their equilibrium positions and the
electrons could scatter inelastically off the dislocations
by changing the energy of the vibrations. Resonant
scattering of phonons by dislocations has been observed by
A.C. Anderson(5). Moreover, Motowidlo et a1.(6) have
observed the effects of electrons damping the motion of
dislocations. Thus the interactions between electrons,
phonons, and dislocations are well established. Several
models for such interactions exist. Another mechanism
proposed by Brown(7) would be the scattering of electrons
off localized electron states associated with the
dislocation cores. Dislocations may also modify the other

terms, such as pep and ,Oee.

 

A. :11“ -. 1 .
.l’l SOFT: 0151313951711

 

 

1.1.1 The residual resistivity

The first term I will discuss is [Do a temperature
independent resistivity due to scattering off static
imperfections in a crystal. It is quite often considered a
good measure of the purity of a sample, for the smaller the
concentration of impurities the lower will be the residual
resistivity.

A theory relating the change in [30 due to adding
dislocations was proposed by Basinski et al.(8). In this
theory one calculates the mean displacement from equilibrium
due to the strain field around a dislocation. Then one asks
what temperature is needed to give the same mean
displacement due to random thermal motion. One then assumes
that the increase in fi% due to the dislocations is the same
as the resistivity due to the electron - phonon interaction

at this temperature. One gets,
pd=Wde (1.7)

where fig is the increase in F% due to Nd dislocations.
This theory predicted for potassium that W =4 x 10—19 (Icm3.
The above equation was tested indirectly, and the resulting
experimental value of W was in good agreement with theory.
In other metals, where this theory has been tested directly,
the agreement with experiment is quite good. Unfortunately,

no direct measurements of dislocation densities in potassium

have been made.

Brown(9) arrived at a similar result using a very
different argument. He calculated the resistivity due to
resonance S-wave scattering of the electrons. The resonance
being created by virtual bound states associated with the
dislocation cores. His theory predicted, for potassium, a
coefficient of 8 x 10.19 {Icm3, off by a factor of two from
the experiment. But Brown argued against a correction
factor used by Basinski to obtain the experimental results,
which if omitted would bring Brown's theory in closer
agreement with experiment.

An early study of the effects on [30 of annealing a
wire of deformed potassium was made by Gurney and Gugan.(10)
In this study the residual resistivity of the sample was
measured, then the sample was plastically deformed at 4.2K,
and again fi% was measured. Then the sample was warmed
slightly, and cooled to 4.2K where the resistivity was
measured. This process, of warming to increasingly higher
temperatures and then cooling to 4.2K, was continued until
the residual resistivity had recovered to its initial value.
Gurney and Gugan claimed to have found five annealing stages
as they rewarmed the sample.

The first stage occurred below about 7K and was
attributed to annealing out of interstitials, for this stage
only about 5% of the total recovery of L% occurred. The
next stage, from 7K to 20K, was attributed to long range

migration of vacancies and included 40% of the recovery.

 

9

The third and fourth stages, 20K to 40K and 40K to 80K,
covered a 30% recovery and were attributed to the
interaction between impurities and vacancies, with possibly
some dislocation relaxation. The final stage, 80K to 150K,
was believed due to the annealing out of dislocations and
included about 25% of the total recovery of fi%.
The temperature ranges given are rough guesses based on

the breaking points in the complicated annealing curves of
[)0 vs the annealing temperature. There were variations
among samples with one not entirely recovering until 200K.
Furthermore, many mechanisms used to explain the various
recovery stages were largely unsubstantiated. Nonetheless,
the study does provide a basis on which the effects of
dislocations on the temperature dependent resistivity may be

examined.

1.1.2 Electron-phonon scattering

The second term I will discuss is due to electron -
phonon scattering. These interactions are usually divided
into two groups, normal and umklapp. To better understand
these interactions let us look at a collision in momentum
space. Let K be the momentum of an incoming electron, K'
the momentum of the scattered electron, q the momentum of a
phonon which interacts with the electron, and G a
reciprocal lattice vector. Then the change in momentum of

the electron will be,

10

K'-K=q+G (1.8)

Potassium is believed to have a nearly spherical Fermi
surface which does not touch the Brillouin zone boundary.
This is illustrated in Figure 1.2, which shows phonon
annihilations for the two types of electron - phonon
processes. For a normal process, G is zero, and the
scattering is within a single Brillouin zone. For an

umklapp process, G is not zero and we have,
K'-G=K+q (1.9)

The scattering in this case extends outside one Brillouin
zone. Clearly for an umklapp process to take place a
minimum value of phonon momentum (qmin) is required.

A theory was worked out for the electrical resistivity
by Bloch(1l) who assumed an equilibrium phonon distribution,
the relaxation time approximation of the Boltzmann transport
equation, no umklapp scattering, and the Debye theory for

phonons. He predicted that,
_ v 5 .
,OeP(T)— D (T/ed) 35(6/Td) (1.10)

where D' ia a constant, and ed is the Debye temperature, and

ed/T
jx(y) = z [(exp(z)-l) (1-exp<z)>1‘1az (1.11)
0
is the Bloch-Gruneisen function. Since the Debye

temperature for potassium is ~100K, and we will be

Umkhpp

 

 

 

 

 

 

Siniplified
BriHouin

Zones
Normal

 

 

 

 

Figure 1.2 Normal and umklapp scattering

 

12
concentrating on results below 4.2K, we will be in the low
temperature regime. Then j5(ed/T) approaches a constant and

the resistivity becomes,

5

F£p(T) = D T (1.12)

Measurements by Gugan(12) (T>1.2K), Ekin and Maxfield(l3)
(T>l.5K), and van Kempen et al.(14) (T>l.1K) on potassium
did not showed this T5 term. Instead their results could be

fit to an equation of the form:

n

fig-p = B T eXP(‘e*/T) (1.13)

with n~1 and e*~2ox

The main reason for the failure of this early model was
attributed to the presence of phonon drag(12). Basically
phonon drag arises when the phonon distribution is disturbed
from its of equilibrium value. This happens when the
phonons gain momentum from collisions with the electrons and
have no scattering mechanism through which equilibrium might
be restored. Then the electrons and phonons drift along
together, and so the resistivity due to electrons scattering
off phonons is reduced. At high temperatures this effect is
quenched, because phonon - phonon umklapp scattering
equilibrates the phonons with the crystal. At low
temperatures in potassium, phonon drag becomes important,

5 term in the electrical

and effectively suppresses the T
resistivity.

The exponential behaviour seen experimentally can be

13
understood in terms of electron - phonon umklapp scattering,
which is scattering between two Brillouin zones. This
scattering requires phonons with at least the minimum

momentum (q ) to jump the gap between the two cells.

min
Phonons obey Bose-Einstein statistics, and at low
temperature the density of phonons which can participate in

an umklapp process goes as,

exp(*fi v q /kT) (1.14)

min
where v is the velocity of the phonons. This factor
dominates the electron - phonon resistivity seen
experimentally.

Detailed calculations have been made including the
effects of phonon drag and both umklapp and normal
scattering. Orlov(15), K. Frobose(16), and Kaveh and
Wiser(l7) have all derived slightly different equations to
describe the resistivity for electron - phonon scattering
below 4.2K. All of these theories were brought together in
a paper by Kaveh, Leavens, and Wiser(18), who stated an

empirical expression,
ޣp(m) = B Tnexp[-e*/T] (1.15)

where the value of n was related to that of 8*. They noted
that increasing n by one, would reduce 6* by about 2.8K. On
adjusting the value of n and 6*, the earlier predictions
(15-17) for electron - phonon scattering were reproduced.

There are mechanisms other than phonon - phonon umklapp

14

scattering which can pull the phonons into equilibrium. For
example a high concentration of impurities or dislocations
(R. Taylor 1978) could interact with the phonons providing a
way for the phonons to lose the momentum given to them by
the electrons, resulting in an equilibrium distribution.

Detailed theories using this mechanism have been worked
out for potassium by R. Taylor et al.(19) and Danino et
al.(20) predicting the effects of dislocations on the
electron - phonon scattering term. These theories give the
same dislocation densities for which at least 90% of phonon
drag is quenched, Nd ~ 4 x 1011 /cm2. These theories are
compared with experimental results in Figure 1.3. 2p is
the electron - phonon term of an annealed sample, where the
phonon drag would be maximal. The solid lines are from the
theory of Danino et al. The dot-dash curve is from Taylor
et al. (The curve representing the theory of Taylor et al.
was obtained by taking the ratio of two plots from their
paper. The difficulty of obtaining accurate values from the
plots gives an uncertainty to the line of 1 0.1.) In the
high Nd limit, both theories give about the same result.
For a sample with no dislocations, both theories would
predict a horizontal straight line of value 1.0. The data
of Gugan(21) for sample K5d has the same general form as the
theories, but the fit is not very good over the whole
temperature range. Moreover, the data plotted include the
effects of electron - electron scattering, whereas the

tzheories do not. Correcting the data by subtracting the

 

15

 

Eco. coccgc I :cEcch c5. (EL .fcocp Liz :cmiccEcc ... A Pita:
2 c
0.? O.m O.N
I E I I _ I I _ _( O.—

 

 

ll'll Ill

8Hez

'0'
I I II,

_o .o

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05:00

2:50

    

0,.
I 0

we. u

0.3. o

p . w ufiozom. In

v.0 uiozozcxx I ll

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In
«.599 x n I z

     

 

 

ON

0.
d)

c
Q

 

0.

Q0

01m

l6
electron - electron term would increase the ratio
significantly only below 2.5K on this graph, and at 2K it
would add ~0.5 to the ratio.

Both theories require dislocation densities of
~5 x log/cm2 to ~5 x lolo/cm2 in order to fit the data.
Such a high level of dislocations is debatable. These
theories require at least an order of magnitude more
dislocations than indicated by the expression
pd: Nd x 4 x 10.19 Qcm3 obtained by Basinski(22), which
would predict 0.6 x log/cm2 for sample KSd and
3 x log/cm2 for sample KSb. However, both theories used
Klemens formula to approximate phonon - dislocation
scattering, and Taylor has pointed out that this formula
does not include core scattering, so the actual scattering
strength will be much higher, reducing the dislocation
densities required to fit the data near 4K.

Another mechanism by which dislocations may modify the
electron - phonon scattering term was suggested by
Hans-Lennard Enquist(23) and later by Danino, Kaveh, and
Wiser(24). In these papers it is argued that the normal
electron - phonon scattering term (DTS) is enhanced by the
anisotropy of electron - dislocation scattering. Enquist
predicted the result in two limits. The first is where the

phonon drag is quenched by dislocations.
p~po+[1+6Q]2DT5 (1.16)

where Q is a measure of the anisotropy of the scattering and

 

17
D is the coefficient from Bloch's low temperature formula
for normal electron - phonon scattering (see Equation 1.12).

The second is where one has the full effects of phonon drag,

5

p ~po + [15/4192 D T (1.17)

with the same values for Q and D. These two equations show
that dislocations enhance the coefficient of the T5 term.
The dashed line shown in Figure 1.3 is a fit of the theory
by Danino, Kaveh, and Wiser to the data of Gugan. This fit
includes the contribution of electron - electron scattering
in that an AT2 term was added to both pep and ’06:). It
gives better agreement with the data than the first

mechanism discussed

1.1.3 Electron-electron scattering

The third term is due to electron - electron
scattering, which becomes important below about 1.5K as
/)ep becomes small. Both the form and size of this term has
been controversial. A theory was worked out for potassium
by Lawrence and Wilkins(25) for a screened coulomb
interaction. The theory was later refined by MacDonald
et al.(26), who included both screened coulomb scattering
and phonon - exchange scattering. MacDonald et al. found
that the screened coulomb interaction gave a much smaller
contribution than the earlier work of Lawrence and Wilkins

had suggested, and instead that the dominant term was due to

18

phonon - exchange scattering. Both theories predicted an
AT2 term with A=0.17p)§2cm/K2.
The experimental results of van Kempen et al.(27) and

Levy et al.(28) supported the existence of a T2

term, but
the coefficient A was seen to be sample dependent. Also
their analysis was complicated by a significant contribution
due to (Dep since the lowest temperature they could obtain
was 1.1K. The work by Rowlands(29) showed, for temperatures
as low as 0.5K, a term of the form AT“, where the best fits
were obtained for n~l.5 with a variable A. The most
definitive work to date on undeformed potassium by Lee
et al.(30), from 4.2K to 70mK, showed a term AT2 where A was
relatively constant among samples.

In order to explain the variation in the observed
values of A, several suggestions were made: size
effects(31), the presence of dislocations modifying the
electron - electron scattering(3), and electron - phason
scattering(4l).

The study of Lee et a1. suggested that size effects
were not important for samples larger than 1mm in diameter.
Very recent unpublished work by the same group(32) has since
shown that a notable size dependence can be seen for samples
less than ~lmm in diameter. Thus part of the variation in A
can be attributed to a size effect since the samples of
previous groups were less than 1mm in diameter.

However, a sample of van Kempen et al.(33), upon rapid

cooling to liquid nitrogen temperature, showed a sizable

 

19

increase in both [)0 and A. After the sample was annealed
at room temperature and then slowly cooled, both decreased
to the lower values previously obtained. Annealing at room
temperature should remove dislocations from the sample.
Danino, Kaveh, and Wiser(34) attributed the observed
increase in A as being due to dislocations caused by strain
induced in the sample during the rapid cooling. They
contended that the data of this sample supported their
theory; that is, dislocations modify electron - electron
scattering. But Gugan(35) has claimed that the high level
of dislocations needed by Kaveh and Wiser was not
necessarily present in the sample. At slow cooling rates
impurities may freeze out in clumps, and have little effect
on the resistivity, so the large increase in L% might be
due to there being a larger fraction of impurities in
solution rather than a large number of dislocations

Before discussing the theory of Kaveh and Wiser, we
first need to define the concept of normal and umklapp
scattering for electron - electron interactions. Following
an argument similar to that used for the electron - phonon
case, we let K1 and K2 be the initial states of the
electrons, and K3 and K4 the final states. Then for an

umklapp process,
X + K = K + K + G (1018)

where G is a reciprocal lattice vector. For the normal

process the electrons must remain in a single Brillouin

20
zone, so G is zero.
From Ziman(36) we know that the approximate solution to
Boltzmann's equation for normal electron - electron

scattering contains an integral with a factor of the form,

. 2
{[(t(xl)xl+t(xz)x2- t(K3)K3-t(K4)K4] U }

x 5(x1+x2-x3-x4) (1.19)

where K1, K2 are the initial states and K3, K4 are the
final states. t(K) is the relaxation time, and U is a
unit vector in the direction of the electric field. Clearly
when t(K) is a constant the integral will vanish, so to
first order a normal process with an isotropic relaxation
time does not contribute to the resistivity.

In the second order approximation, umklapp scattering

can take place, and the delta function is replaced with,
5 (K1+K2-K3-K4+ G ) (1.20)

Then umklapp scattering gives a finite, but small value for
A, even if t(K) is isotropic. But if t(K) were anisotropic,
the above integral for the normal process would not, in
general, vanish. Since the resulting term is of first
order, one would expect it to be quite large. Kaveh and
Wiser(37) estimated the coefficient A to be 8 times larger
in the anisotropic t(K) (normal) limit than it is in the
isotropic t(K) (umklapp) limit. Thus for even small levels
of anisotropy, the normal process may be significant

compared to the umklapp process.

21

Kaveh and Wiser(38) in their theory argue that the
magnitude of the electron - electron scattering term depends
on the relative contribution to F% of isotropic and
anisotropic scatterers. They define poi as the component
of the residual resistivity due to isotropic scattering and

[308 as that due to anisotropic scattering, where

p0 = poi + poa (1°21)

By isotropic Kaveh and Wiser mean the scattering time {t(K)}
of the electrons is constant over the fermi surface. They
claim A is a function of poi/ poa’

To understand how this dependence of A on the dominant
scattering mechanism occurs, let us take a look at the
significant scattering mechanisms present near 1K.
Electrons are scattered predominantly by defects since
electron - phonon and electron - electron scattering make
negligible contibutions to the residual resistivity. The
important defects are impurities and dislocations.

The residual resistivity added by dislocations is
assumed to be predominately anisotropic, whereas impurity
scattering is almost totally isotropic. The amount of
anisotropy in t(K) depends on the dominant scattering
mechanism. For impurity scattering where one only has
isotropic scattering, clearly t(K) will be a constant and a
particular value A=Ao will result. Similarly, when the
scattering is dominated by anisotropic scattering (due to

dislocations, for example), another limiting value

 

22
A=AO+An will result. The relationship Kaveh and Wiser(38)

predict is,
_ -2 '
A — A0 + Al( 1 + 1001/ poa) (1.22)

where A0 is the value predicted for dislocation free
potassium, and Al is an adjustable constant. Figure 1.4
shows the result of the Kaveh and Wiser theory with the data
of some research groups.

The major difficulty with this theory was that it
required at least two orders of magnitude more dislocations
in strain free samples than were believed could easily
exist. Furthermore, there was no way to definitively
separate Chi from (Doa' so each data set plotted had one
adjustable parameter, making it easier to fit the theory.

It was clear that a study of samples that were
systematically deformed and annealed was needed. At about
the same time the group at MSU started working on this
project, a group in the Netherlands, headed by van
Kempen(39) also began work. They finished their study
first, but the limited temperature range, down to only ~O.9K
made their analysis of the data questionable, since the
electron - phonon term appears to be quite large in these
deformed samples in this temperature range. Furthermore, we
were able to obtain at least 50 times higher precision.
Comparison of their results with ours will be made in the

experimental results chapter.

 

 

23

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"OhioQ

o.—

:.H sesame

 

_~_L~_ _

_o .0 :anox :0)
_U .0 «mu—.3308
.0 3 xi:

.0 .o no.

 

_____ _ .—

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0

   

 

 

24

1.1.4 Electron - phason scattering

An alternate theory first derived by Boriack and
Overhauser(40) and later corrected by Bishop and
Overhauser(4l) was used to explain the AT1'5 term in the
resistivity of potassium, observed by Rowlands et al.(29).
This theory contends that there are periodic oscillations
(charge density waves -~ CDW) in the electronic density,
which are incommensurate with the crystal lattice. If these
CDW's were to exist in a metal, then the properties of the
metal would be greatly changed, for instance the fermi
surface would no longer be spherical, but instead a complex
interconnected one.

The existence of CDW's is a matter of considerable
controversy. CDW theory can explain many observations on
potassium which conventional theory has not explained(2).
But other measurements on potassium are at variance with CDW
theory(2). Since the scatter in the data of Rowlands et a1.
does not preclude a simple AT2 behavior, it is not
absolutely necessary to invoke CDW theory to explain the
observed resistivity data for undeformed potassium. In
addition, the recently observed size effect by our group(32)
can explain why the 0.8-mm-diameter samples of Rowlands

1'5 behavior. So far, the CDW

et al. appear to have a T
theory has not addressed the observed size effect.

Bishop and Overhauser(41) have claimed that CDW's

 

25
should enhance electron - electron scattering. This
enhancement depends on the relative orientation of the 'Q'
domains (regions over which long range correlation of the
CDW's exist) and the electic field. Since the orientation
of the 'Q' domains is sample dependent, the electron -
electron scattering should also be sample dependent. The

details of this theory have not been worked out.

1.1.5 Electron - dislocation scattering

This section is a discussion of the other ways
electrons and dislocations can interact. For some of these
mechanisms no analytic theory has been worked out.

The first theory to be discussed was proposed by
Brown(42), who suggested dislocations could have virtual
bound states for electrons with energies slightly above the
fermi surface. He estimated these energy levels for
potassium to be at about 10-4 eV. These energy levels would
be localized near the cores of the dislocations. Gantmakher

and Kulesko(43) derived an equation for elastic scattering

off such states,

1

ޣ_dis = a [ 1 + b exp(E/kT)]- (1.23)

where a, b, and E are constants, and b is related to the
spin degeneracy of the localized energy levels at energy E
above the fermi energy.

Other calculations have been performed for interactions

 

26
with localized energy levels in the crystal. Fulde and
Peschel(44) calculated the resistivity due to electrons
scattering inelastically off localized energy levels
produced by a crystalline electric field. This theory may
be applicable to scattering off local electron states caused
by dislocations such as the virtual bound states predicted
by Brown. Theory predicts that,
0(T) = (To [1— _c;_2__gfl_] (1.24)
sinh(D/T)
and if one applies Matthiessen's rule in the derivation one

gets,

CT(T) = CTo 1 + G2 '1

l+(2/3)[sinh2(D/2T)]

(1.25)

where G and D are constants, and (To is the residual
conductivity. D is the energy level splitting for the
assumed doublet.

As mentioned earlier, another mechanism is inelastic
scattering off local phonon modes associated with the
dislocation. A. Anderson and Ohara(45)(46) have made
measurements of thermal conductivity on superconducting
metals, which showed resonant phonon absorption at certain
energies. This work supports the existence of dynamic
dislocations at low temperatures. These local modes can be
pictured by considering a dislocation which is pinned at
both ends. There is a restoring force pulling the

dislocation back to a straight line. The oscillations of

27

the dislocation around this equilibrium line form the local
modes. The lowest energy state is one that has the longest
wavelength. The pinning sources can be other dislocations
or impurities and other point defects. Clearly the
frequencies of the local modes will depend on the distance
between these pinning sources. Hence adding more
dislocations and impurities should increase the fundamental
frequencies of the system. Also there will be a
distribution of lengths, due to the random placement of the
dislocations and impurities.

The inelastic interaction of these local modes in the
phonon spectrum with the electrons was calculated by
Gantmakher and Kulesko(47) who approximated the local modes

by a single frequency oscillator. They found,
pe_dis = (C/4T) sinh—Zfiiw/kT) (1.26)

where C is a constant and w is the ground state frequency of
the oscillator.

The above are the only cases where the resulting
electrical resistivity for a dislocation scattering
mechanism has been calculated. A few of the other possible
scattering mechanisms include: the tunneling of
dislocations, and the scattering of electrons off localized
electron energy levels produced by the interaction between

dislocations.

 

28

1.2 Thermoelectric power

In this section previous work on the thermopower of
potassium will be discussed. When there is a temperature
drop across a sample, there will be a thermoelectric
voltage. The thermopower is this voltage divided by the
temperature drop. One can view the thermopower as being the
sum of two terms. The first term is diffusion thermopower
and the second, phonon drag thermopower. Phonon drag
thermopower arises as phonons (heat) flow from the hot to
the cold end of the sample. As phonons flow they interact
with and transfer momentum to the electrons. This causes
the electrons to flow along with the phonons until enough
electrons have moved to form an electric field which stops

any additional net motion of the electrons.

1.2.1 Diffusion thermopower

Diffusion thermopower is generally written using the

Mott(48) expression as,

s . = Jrrzkzrr 61;. (2(3) (1.27)
diff _——7T‘_— _
e 6E E—E:f
(where Ef = the fermi energy, CT: conductivity) Expanding

the above equation, Guénault and MacDonald(49) obtained,

29

sdiff = -7T2k2Tfain[n(E) v2(Ell +alnt(E) (1.28)

3 e Ef\ lnE alnE E=Ef
where t(E) is the relaxation time, n(E) is the density of
states, and v(E) is the average electron velocity at energy
B. When impurity scattering dominates one expects for
potassium that the quantity in the parenthesis will be
independent of temperature. Thus one expects the diffusion
thermopower in potassium to linearly increase with
temperature.

The measurements carried out on potassium by Guénault
and MacDonald(49)(50) have shown this linear term to exist,
but the coefficient varied from sample to sample. This
variation in the coefficient has been attributed to a small
shift in the concentration of impurities which can easily
change the dominant scattering mechanism. Guénault and
MacDonald argued that slight variations in the preparation
of the sample could easily change the impurity content.
More recently, Gugan(Sl) has pointed out that the rate of
cooling of a sample may affect the number of impurities
which come out of solution (with the majority of impurities
clumping together for slow cooling and thus giving a small
contribution to the residual resistivity or the
thermopower).

The effect of having more than one scattering process
in a sample can be viewed as follows: If Matthiessen's rule
applies, one can express the resistivity as a sum with each

term due to a separate process,

30

p= (01* ,0d (1.29)

where we will take Ffl as the initial impurity scattering
and ‘LQ as that from adding a new type of impurity or
dislocation. Using Mott's formula,

5 = 71‘2 kZT Oinpm) (1.30)
——w 3 e 69E E

one gets the Gorter - Nordheim relation.

5 = 51’ pi ( 51' sd) (1.31)

,0

For fixed Si and Sd' a plot of S vs l/fi) should yield a
straight line. Such a plot is called a Gorter - Nordheim
plot. This relation was observed by Guénault and
MacDonald(52) for rubidium impurities in potassium.

For diffusion thermopower one might expect the
temperature independent dislocation scattering to act in
much the same way as impurity scattering, so one would
expect to obtain a straight line on a Gorter - Nordheim plot
when I) is increased by the addition of dislocations.

An additional term may be present besides the usual
diffusion term. Fulde and Peschel(53) calculated the
thermopower due to inelastic scattering off crystal field
split states, and their equation may apply for the virtual

bound states predicted by Brown. They get,
(1.32)

where,

31

51 = -b27(‘T/(3 e sf) (1.33)

and,
$2 = - a D tanh{(D/2T)[1+(D/27TT) Im(y(in/27FT)]} )
1.34

where Y(z) is the trigamma function, D is the level

splitting energy, and both a and b are constants.

1.2.2 Phonon drag thermopower

Phonon drag thermopower becomes important at slightly

higher temperatures in potassium. It is usually divided
into two parts, one due to normal electron - phonon
processes and the other due to umklapp processes. The

theory in both cases tends to be rather complicated;
fortunately we will be working in the low temperature limit.

The normal processes involve scattering of electrons
within a single Brillouin zone. In the low temperature
limit this contribution to S goes roughly as the lattice

specific heat, which has a T3 temperature dependence.

(1.35)

The umklapp process is more complicated. To scatter
from one Brillouin zone to another, a minimum phonon wave
vector (qmin) is needed. The number of these phonons

available goes as,

32

N(q) = exp(-thmin /kT) ~ exp( -fivq

l-exp(-th /kT) min /kT) (l 36)

 

min
This exponential decay is clearly the predominant

factor in umklapp scattering. An approximation was worked

out by Guénault and MacDonald(54), who obtained for low T,

 

u
sph = l/te (.200)[_k_] :9: 5+ 3 + 6_'I_‘__ exp(-e*/T)
*
l/te+l/ti e ed T e
(1.37)
Where te= phonon - electron relaxation- time, ti: phonon -

impurity relaxation time, ed = Debye temperature, and 6*: a
constant.

More detailed theories have been worked out by
Ziman(55) and Bailyn(56), but so far, fitting to the data
has been done using only the simplest forms for the umklapp
scattering process. Guénault and MacDonald fit their data

to a simple equation of the form,

S 3

SOT + BT +C exp(-8*/T) (1.38)

which gave fairly good agreement with their data where
*
e ~21K. For pure potassium, they found, as expected, that

SO and B were negative while C was positive.
1.2.3 Thermoelectric ratio

The actual measurements taken for this thesis were of

the thermoelectric ratio. Very little theory has been

33

worked out for this quantity,

but fortunately it can be

related to the thermopower using a simple expression to be

derived in this section.

As we saw earlier,

J = Lee E + Let§7r

Q = Lte E + Ltt§7T
G, the thermoelectric ratio, is
electric field, so G=Let/Ltt’ S is

Relating the two,

G: Let/Ltt=LetLee/(LttLee)

= '5 Lee/Ltt = S w /fi)

w (=l/Ltt) is the thermal resistivity.

defined as,

L= p/(wT)
so,
G= S/(LT)
Thus the theory for thermopower

thermoelectric ratio.

defined

defined as

may be

(1.39)
(1.40)

as J/Q at zero

S=-Let/Lee’

(1.41)
(1.42)

The Lorenz ratio is

(1.43)

(1.44)

related to the

Ideally L would be a constant, and at

low temperatures when impurity scattering dominates this is

indeed the case. Using our

potassium,

earlier

equation for pure

 

 

34
,0: p0 + AT2 + B T exp(-6*/T), (1.45)

and the expression from Newrock and Maxfield(57) for the

thermal resistivity in potassium,

 

w = E /T + F T2, (1.46)
one can calculate,
'k
L = p0 + AT2 + B T exp(-e /T) (1.47)
E + F T3

Clearly L approaches a constant value L at low

temperatures.
LO= (00/ E (1.48)

This limiting value for L has been calculated assuming

elastic scattering(36).

LO=7T2 k22 ~ 2.445 x 10‘8 v2/x2 (1.49)
3 e
We have measured L in pure undeformed potassium. At 1K,
L/Lo ~ 0.97 with the ratio becoming closer to 1 at lower
temperatures.
Therefore, at low temperatures where the umklapp

scattering is frozen out, G will have the following form.

6 = GO + b T2 (1.50)

2 is the normal phonon

where Go is the diffusion term and bT
drag term. A departure from this form would indicate the

presence of some other scattering mechanism which has not

35

been taken into account in the above theory.

 

CHAPTER 2

Experimental techniques

This chapter will cover the techniques and equipment
used to study potassium. It will also discuss the
uncertainties in those measurements.

The main purpose of this study was to examine the
effects of plastic deformation on electron - electron
scattering. This placed a high demand on the system used
for taking data. First it was essential that one go below
1K to eliminate the effects of electron - phonon scattering.
Also, very high precision was needed, since

fiLe/ 1fi% ~ 10-4, near 1K.

All of these requirements were met in this experiment.
The measurements were taken in a dilution refrigerator which
was locally built by W.P. Pratt,Jr.. This refrigerator was
useable from 4.2K to as low as 60mK. The cryostat was
equipped with a commercial Superconducting Quantum
Interference Device (SQUID model RMPC, from S.H.E.

Corporation(58)), which was used in measuring both the

36

37
resistivity and the thermoelectric ratio. This SQUID was

9

capable of detecting currents less than 10_ A and thus

provided the high sensitivity needed.

2.1 Sample can

Since potassium reacts with air, and dislocations
anneal out at temperatures above 160K, a self-contained
sample can was needed, which would permit both measuring the
electrical characteristics of the samples, as well as
permitting plastic deformation of the sample. Figure 2.1
shows a drawing of the sample can which was designed to
permit deformation of the potassium sample while mounted on
the dilution refrigerator. The sample can and dilution
refrigerator were both mounted inside a main vacuum can,
which was surrounded by liquid He. Some features of this
sample can were as follows. First note that the sample was
mounted horizontally in the sample area (K) between two
plates, the lower of which (I) was movable. An expanded
view of the sample area is shown inverted in Figure 2.2. The
sample was a 0.9mm diameter potassium wire which was
connected to permit 4-probe resistance measurements, and
also to permit measurement of the thermoelectric ratio. The
electrical connections to the samples were made by cold
Welding the potassium to one silver (V) and three copper
probes. The G heater was a 4kII wire-wound resistor mounted

on a copper current probe.

38

Table 2.1 Sample can parts
Label description

stainless steel bellows

brass cap

He gas line

stainless steel push rod

movable copper squeeze plate

copper squeeze plate

brass flange

copper press plate guide

movable copper plate

stainless steel support rod
potassium sample area

brass top flange

brass can

copper rod for attaching to mixing chamber
connection to mixing chamber
stainless steel support for thermal feed through
copper thermal conductor

electrical feed through

motion detection spring

copper switch plate

copper clamp to hold spring in place
silver wire

teflon isolator

indium 'O' ring seals

NE‘CCHmwO'UOZZF‘xLH-‘IOWJPJUOWJ’

 

39

 

I?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

40

Voltage Probes

 

 

 

l‘G Heoier

 

Potassium Sample

Vol’roqe Probes

3 l. ...
I-—-> <‘~I
{ \ \{Conloci
/ N Moteriol
Silver Wire

Copper
Teflon

 

 

 

 

 

 

 

41

Temperature measurements were critical to the outcome
of this study. The thermometer (GRT2) used to measure the
temperature of the samples was mounted outside the sample
can, by clamping it to the copper thermal feedthrough (Q).
It was important that GRT2 be at the same temperature as the
sample, so the thermal path between the sample and GRT2 was
designed to be of low thermal resistance and the sample was
to be well thermally isolated from the sample can. Thermal
isolation of the sample was provided by the stainless steel
support (J) holding the upper squeeze plate (F), and also by
the stainless steel support (P) holding the thermal

feedthrough (Q).

2.2 Plastic deformation of the sample

When the sample was to be deformed, the bellows (A) at
the bottom of the can was contracted by admitting He gas,
thus forcing upward the bottom squeeze plate (E). This
squeezing could be done at any temperature from 4.2K to room
temperature. The bellows was purchased from the Metal
Bellows Corporation(59). It could handle a maximum pressure
of 45 psi and had an effective area of 2.1 square inches, so
the maximum force the bellows could generate was 95 pounds.
Gas was introduced between the bellows (A) and the brass cap
(B) through a capillary (C) connected to an external
pressurization system. When the bellows moved up, it turned

on a switch by grounding the copper switch plate (T) through

 

42
the spring (5) attached to the plunger (D). The copper
switch plate was connected to a wire which ran out of the
sample can through an electrical feed through (R). Using
this switch, the initial motion of the bellows could be
detected. Thus one could be sure the pressurizing line was
not plugged.

The plumbing for the He pressurization system connected
to the bellows is shown in Figure 2.3. Normally when
operating the bellows, valve #1 was closed and He was slowly
added using valves #2 and #3. This two - line System was
built for two reasons. If a solid air plug should form when
pressurizing at 4.2K, the bellows could still be evacuated
using the second line since any such plug would take place
in the first line above the liquid He level, which would be
above the place where the two lines joined to form one line.
Secondly, the pressure was measured using gauge #1, which
was connected to a line which had no gas flowing down it,
and thus it would give an accurate measure of the pressure
down to where the two lines were joined at the top of the
main vacuum can. The ballast tanks were 14 cubic inches in
volume and were added just before run Kh9. Their purpose
was to damp out Taconis oscillations which had occurred in
run Khl. These oscillations made accurate measurement of

the bellows pressure virtually impossible.

 

43

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Liquid He
+— *5
on C)
=9 a
C
CD CD E; c:
0 a
E a»
O.
‘3 \ E
> \8
l/

 

 

Q PreSSUl'e gouge

Pump He 905 ® Volve

 

Figure 2.3 The He pressurization system for the bellows

44

2.3 Removing He gas from the sample can

The glove box in which the potassium samples were made
was filled with He gas, and thus when the sample can was
sealed in the glove box it contained He. The heat
convection and conduction through this gas was large enough
to make it impossible to accurately measure the sample
temperature. This He gas had to be removed. Several
methods were used to remove the He. In the first two, the
He was adsorbed either on loose 13X molecular sieve
pellets(60) or cylinders made by compacting zinc shavings
and sieve pellets. The major difficulties with these two
methods were outgassing from the adsorbing materials which
corroded the sample, and the heat load these materials
placed on the sample can at the lowest temperatures, since
these materials cooled slowly. The final and most
successful method used was to pump the majority of the He
gas out at room temperature, and use a sintered silver disk,
made from 700A silver powder, to adsorb any residual gas at
low temperatures.

When sieve was used it was necessary to outgas it.
This was done by placing the sieve in a sealable glass
flask, which was heated to 200°C and pumped with a diffusion
pump for ~2 days. This was important since any water or
oxygen released from the sieve would react with the

potassium sample. The sieve, while still under a vaccuum,

 

 

45
was then placed in the glove box without exposing it to air.
Next it was exposed to the high purity He atmosphere for at
least one day prior to making a sample to allow for any
additional outgassing. Prior to mounting the sample, the
sieve was placed in a stainless steel cage which was screwed
to the top flange of the sample can.

When cooling the refrigerator and the sample can to
4.2K, it was found that the sieve cooled very slowly. It
was hoped that if one made zinc-sieve pellets, that the zinc
would help in cooling the sieve. These zinc-sieve pellets
were made by grinding the molecular sieve to a coarse grit,
and mixing it in a ratio of 2 parts ground sieve to 3 parts
#50 mesh zinc. This mixture was then placed in a press and
compressed with several tons of force to 1/2" diameter by
~3/4" long cylinders. These cylinders were then outgassed
in the same way as the plain molecular sieve, with the only
exception being that at the end of the bake out, hydrogen
gas was added to sinter the zinc. The cylinders were then
mounted in a copper holder which provided good thermal
contact with the sample can.

Even the pellets with their improved thermal contact,
cooled slowly and caused difficulty in attaining the lowest
temperatures. Consequently, neither the molecular sieve nor
the zinc-sieve pellets were used after sample Kh2.

After sample Kh2, the He gas was always pumped out of
the sample can before transfering liquid nitrogen. To

facilitate this, special indium filled brass tubes were

 

46

 

l ll l/l/
l/é Iona __ 8 DiameTer
brass Tube

Indium

 

 

OUTside coaTed
wiTh lead-

iin solder

 

 

 

Figure 2.4 Indium filled brass tube

47

prepared (see Figure 2.4). The purpose of these tubes was
to permit pumping out the He gas in the sample can, and then
to seal the sample can by crimping the tubes shut. When
crimped, the indium inside the tubes would cold weld to
itself providing a leak tight seal even at liquid He
temperatures. In making these tubes, first the outsides of
brass tubes were coated with 50/50 lead/tin solder, to allow
easy soldering later. The tubes were heated to ~200°C and
filled with indium. "Stay Clean" Flux(61) was found to give
the best bond between the brass and the indium. A wire was
run down the tubes to ensure that no air bubbles formed.
After the tubes were filled with indium, they were allowed
to cool, washed under running water, and placed in a beaker
filled with bicarbonate of soda for several hours to
completely neutralize any trace of the highly corrosive
flux. Finally the inside of the tubes were drilled, first
~3/8" deep at each end with a #30 drill and then completely
through with a #53 drill bit.

The indium filled tubes once finished were soldered in
place. The first one, used to remove He from the sample
can, was soldered with Rose's alloy between a copper tube
coming out of the sample can and a valve assembly. This low
temperature solder was used because it ensured that the
indium would not melt and plug the brass tube. These solder
joints and the valve assembly were then thoroughly tested by
checking first that gas would flow through the system,

second that the joints were leak tight, and third that the

 

 

48
valve after sealing was He leak tight. The second tube was
soldered on the input line (C) to the bellows(A), just
outside the sample can. At the end of the run when the
sample was to be removed from the refrigerator, this tube
was crimped shut. Thus the bellows remained under a vacuum
and the sample can could be disconnected from the
pressurizing system without contracting the bellows and

further deforming the sample.

2.4 Sample preparation

Before placing the sample can in the glove box, sheets
of sample contact material were prepared. These were the
materials in direct contact with the potassium. Table 2.2
lists what was used for each sample. The sheets were
cleaned with high purity ethyl alcohol. Next the copper
electrical contacts were cleaned by carefully filing them
until they were shiny. The upper layer of contact material
was then placed between the teflon isolator plate and the
copper contacts. Figure 2.2 shows the sample area which was
normally inverted, as shown, when the sample was being
prepared. Then the copper contacts were screwed on the
teflon disk with 000-120 screws thus holding the contact
material in place. The thick teflon disk in turn was
screwed onto the copper base plate. Another piece of this
contact material was then taped to the guide collar holding

the movable copper squeeze plate (I) in place. The sample

 

 

49

 

Table 2.2 Materials in direct contact with samples
Sample material

Khl teflon

Kh2 "Handiwrap" (a clear plastic)

Kh3 "Handiwrap"

Kh4 parafilm

Kh6 nothing (this sample was free hanging)
Kh7 polyethylene

Kh8 teflon

Kh9 teflon

KRbh teflon

 

50
area and tools which were to come in contact with potassium
were then cleaned with ethyl alcohol to remove any grease or
other contaminants. The sample can and tools were then
placed in the glove box.

The glove box was made by Vacuum Atmospheres
Company(62). It contained a high purity He atmosphere which
had a nominal oxygen contamination of less than 0.4 ppm.
The water vapor content was not directly measured but
exposed potassium remained shiny for hours. Thus in the
30-60 minutes it took to make a sample, no noticable surface
deterioration could be seen. To allow the sample can to
outgas and to allow the purification system time to remove
any residual contamination brought in with the can, all
needed materials were placed in the glove box well in
advance of making the sample (see Table 2.3 ).

The pure potassium samples were made of 99.95%
potassium obtained from Gallery Chemical Company, a division
of Mine Safety Appliances Company(63). Table 2.4 shows the
chemical composition of a similar batch of potassium. The
potassium came in glass ampoules sealed under argon gas.
The ampoules were opened in the glove box and the potassium
melted and transfered to stainless steel presses (see
Figure 2.5). Two separate ampoules of potassium were used,
both were from the same manufacturers batch. One was used
for samples Khl to Kh6a+b. The second one was used in
samples Kh6b+d to Kh9. The potassium from the two ampoules

was directly compared in run Kh6, where a,b represents the

51

 

 

Table 2.3 Time table of making samples

Sample time sample can was time from making sample
in glove box before to transfer of liquid
making samples nitrogen

Khl over night 2 days

Kh2 few hours 3 days

Kh3 few hours 7 hours

Kh4 3 days 12 hours

Kh6 2 days 5 hours

Kh7 10 hours 14 hours

Kh8 13 hours 6 hours

Kh9 60 hours 9 hours

KRbh 17 hours 7 hours

Table 2.4 Chemical analysis of potassium

Element PPM Element PPM Element PPM

Fe <5 Cr <5 Sr <1

B <10 Si 25 Ba <3

Co <5 Ti <5 Ca 8

Mn 1 Ni <5 Na 15

A1 <2 Mo <3 Pb <5

Mg 2 V <1 Zr <10

Sn <5 Be <1

Cu <1 Ag <1

This information is from the Callery Chemical Company.
< means less than this level of impurities could not be

detected.

 

Piston

 

 

body

 

 

 

die

Figure 2.5 Stainless steel pre

U)
U)

53
first ampoule and b,d the second.

The KRbh sample was obtained by first making a 1.3at%Rb
mixture using Rb from Callery Chemical Company and then
adding some of this mixture to pure potassium, to form a
K(0.077at%Rb) alloy. The first potassium ampoule and a
third potassium ampoule were used. This 0.077at% mixture
was used in sample KRbh.

A 0.9-mm-diameter die was then screwed into the press
and a sample extruded. The samples were ~2 inches long and
were cold welded between the copper and silver current
contacts, see Figure 2.2. Next, two potential probes of
potassium were extruded and cold welded first to the
sample ~2cm apart and then to the copper voltage contacts.
Additional potassium was added to the copper and silver
contacts to make sure the cold weld surfaces were entirely
coated with potassium. These cold welds were very reliable,
and it proved nearly impossible to completely remove the
potassium from the contacts short of disolving the potassium
in alcohol or oxidizing the potassium and filing the
contacts clean.

A sheet of the contact material was placed over the
sample and then the potassium potential probes were
flattened to ~.1mm. This allowed the total force generated
by the bellows to be applied directly to the sample area.
Finally, the guide collar (H) was mounted on the copper
squeeze plate (F) with screws and the sample can sealed

using an indium 'O' ring.

54

The sample can was then removed from the glove box and
mounted in the dilution refrigerator. The room temperature
resistance of the sample was measured by passing a known
current through the sample and measuring the resulting
voltage across the sample with a Keithley 180 nanovoltmeter.
The bellows pressurization line was connected and leak
tested. Then the bellows movement was tested by
pressurizing the bellows until the switch showed that the
bellows had moved. Finally, the bellows was evacuated, and
left in this state until the first deformation of the
sample.

In the case where the He gas in the sample can was to
be pumped out, a cryopump ( Figure 2.6 ) was placed in a
liquid He storage dewar and connected to the sample can
using the gas handling system. This system was then pumped
out and leak tested. It was then flushed with He gas and
pumped out using the cryopump for ~40 minuties. Then the
valve on the sample can was opened and the sample can pumped
to about 300 microns pressure. At that point the indium
tube was squeezed shut, providing a leak tight seal between
the sample area and the valve. The cryopump system was then
removed, and the valve was unsoldered from the indium tube
and removed from the sample can.

Next the refrigerator was closed and leak tested in the
usual manner, with liquid nitrogen being transfered that
day, thus limiting the time the sample stayed at room

temperature (see Table 2.3). This reduced the amount of

 

 

32 sec...

 

Sec oHaEnm asp

 

CCU

oEEom

p30 uzflaea Lora :5me 2129?: m ”a. m:

 

 

 

 

 

 

 

 

 

 

 

 

.ooccOU xoSU

0.x mmsaflh

 

9.2-0 Lone: HH—

o>_o> g

o>o_m

bO-DUQ-OE xm— 1

QE:Q

sonata

 

Lo
32:053.:

0:
.636:
a.
L
N.
.
s I
X's
—.
(I)

 

 

 

 

 

mom GI

 

 

 

 

 

 

 

 

 

 

56

oxide or hydroxide which could form due to outgassing.

2.5 Measurements

This section is intended to explain how the various
parameters of the samples were measured.

The first parameter is A/L, the cross sectional area of
the sample divided by its length. This was obtained by
measuring the room temperature resistance of the sample
using a constant current source and a Keithley digital

nanovolt meter.
A/L = ,0(300K)/ R(300K) (2.1)

[)(300K) was taken to be 7.19 x 10_6§2cm, the value
measured by Gugan(64). To obtain the resistivity of the
sample when T was not 300K, a linear temperature dependence
was assumed for [3, which for the few degree correction
needed was a good approximation. The A/L values are listed
in Table 2.5. The room temperature resistance was also
measured after the sample had been deformed. When a sample
was only deformed once during a run, the final value for A/L
was used for all but the first undeformed measurement. In
Kh2 the A/L was unchanged. A/L should remain constant
during deformation, since the deformation will not alter A
and it is not expected to change L. The handling of Kh8
will be discussed in the next chapter.

Once the sample was cooled to 4.2K or lower, the

57

 

Table 2.5 Sample parameters

sample A/L loo -’O4.2 Pressure of Temp. of
deformation anneal or

deforma-
tion
(10 3 cm) (n(lcm) (nilcm) (inches Hg)

Khla 1.98 4.19 4.56

Khlb 6.30 6.75 95+ 4.2K

Kklc 4.76 5.17 60K

Khld 1.91 4.12 4.48 155K

Kh2a 1.61 1.87 2.19

Kth 2.04 2.38 23.6 60K

Kh2c 2.28 2.62 54.1 60K

Kth 2.55 2.89 94.8 60K

Kh2e 1.62 1.95 2.27 160K

Kh3 1.48 1.04 1.30

Kh4 1.96 1.05 1.30

Kh6a 1.14 1.30

Kh6b 1.60 1.38

Kh6c 1.12 1.44

Kh6d 1.54 1.36

Kb? 1.38 1.24 1.51

Kh8a 1.54 1.32 1.62

Kth 1.38 1.68 7.1 60K

Kh8c 1.46 1.75 14.1 60K

Kth 1.53 1.82 21.1 60K

Kh8e 1.81 2.11 50.1 60K

Kth 1.40 2.05 2.36 84.3 60K

Kh9a 1.99 1.09 1.35

Kh9b 2.21 2.53 95 4.2K

Kh9c 2.20 2.51 7.5K

Kh9d 1.86 2.16 20K

Kh9e 1.49 1.77 60K

Kh9f 1.39 1.66 80K

Kh9g 1.16 1.42 120K

Kh9h 1.90 1.09 1.35 165K

KRbha 3.06 10.75 11.07

KRbhb 11.52 11.86 95 60K

KRbhc 10.78 11.10 160K

+ Due to the Taconis oscillations during Khl the pressure is
probably much larger than this.

 

Mixing Chamber

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Thermal Short
‘th
E’.’ .-. . 9 WW
e l l
C r. 0 Controller fl HU2
lsolahon
HUl m— g RWL
‘—-m HL2
Q SQUID
H“ m—
?— '—- GRT2
GRT4 -—' RSn/ln j: '3 !— —-§ilic;n -—l
>
’ l
l >< *——‘- Speer l
l
> _) f < I
I1 Rref I RK
, l
l 1
I l
H62

E: T hermometer
ESZSZ Heater

Figure 9 7 The low temperature circuit

|—.

L Sample Can

 

59
circuit shown in Figure 2.7 was used. This low temperature
circuit consisted of the SQUID (Superconducting QUantum
Interference Device) null detector and 3 resistors wired in
series. These three resistors were 1) Rk which was the

potassium sample to be measured, 2) R which was a

ref
copper-silver alloy resistor designed to have a resistance

with low temperature and current dependence, and 3) R a

Sn/In
resistor made of a tin-indium alloy chosen to go
superconducting near 3.8K. The wires connecting these
resistors were made of Niomax CN, a multi-filament Nb-Ti
superconducting wire with Cu-Ni cladding made by Imperial
Metal Industries(65). To keep the noise introduced by stray
magnetic fields low, the tin-indium resistor and most of the
wiring except that near the sample were shielded in
superconducting lead tubing. The wires leading to the
sample were all carefully tied or varnished down to reduced
magnetically induced currents and heating due to vibration.

The wires exited the can through the electrical
feedthrough (R, see Figure 2.1). This feedthrough was made
by running the Niomax wire through a clean 1/8" diameter
stainless steel tube which was then inverted in a cup of
Stycast 1266 epoxy(66).

R was made of an oxygen annealed Cu 0.02 at% Ag

ref
alloy. This alloy was drawn down to .034" diameter wire,
which was made into a 4-probe resistor by welding to it
0.030" - diameter oxygen - annealed high purity copper wires

about 1.3cm apart. By slowly stretching and twisting the

 

60
resistor, a 4.2K resistance of 1.5 x 10’6() was obtained.
The temperature dependence of the final resistor is shown in
Figure 2.8. The resistance minimum was probably due to iron
that diffused into the resistor while spot welding on the
voltage probes.

The ratio of the resistances for any two of the
resistors in the above circuit could be measured. This was
done using a commercial current comparator modified by
D. Edmunds et al.(67). This device generated two currents
whose ratio was stable to better than 0.1ppm precision.
When comparing resistances, one current (Im) passed through
what was called the master side resistor (Rm), and the
second current (15) passed through the slave side resistor
(R5). The resistances Rm and Rs represent here any pair
wise combination of Rk' Rref’ RSn/In‘ The master side
current Im could be ramped slowly to a predetermined value.
The current Is was C x Im where C was the switch setting
of the current comparator. C was adjusted until the SQUID
output signal indicated a null condition at its input.
Thermal EMFs generated in the low temperature circuit were
eliminated by the following procedure. Let Vs be a stray
voltage in the circuit due to a thermal EMF. Then for the

currents going in one direction the SQUID measured,
V=IR-IR+V (2.2)

and for the reversed currents, one had

 

 

MA— __ 7 _._

 

 

 

 

 

10
l 1* l + +1
0 +
Ex + + + + _
vc + + + + +
r.- -10 l— + —-(
Q +
-20~ __
U)- +
‘7‘
‘3
(LB
~40- -
'50 + _.
‘60 l l J
O 1 2 3 4
HM
Figure 2 3 Temperature dependence of the capper reference

A '15-‘-
PCSIDUOT

 

62

V = ‘I R + I R + V (2.3)
C was adjusted until V+ = V- was obtained. Then

IR-IR=-IR+IR (2.4)

21R=2IR (2.5)

(2.6)

independent of Vs.

On the initial cooling to 4.2K the resistance of the
sample and the copper-silver reference was measured using
the tin-indium resistor. This double checked the copper
silver reference and allowed. Lu 2, the resistivity of the
potassium sample at 4.2K, to be calculated using the A/L

obtained earlier.

{94.2 = $24.2 x p(300k) / R(300K) (2.7)

On subsequent deformations or annealings of the sample, just
the ratio between Rref and the sample was measured, with
this information being used to calculate 41.2. To cool the
system below 4.2K, the 1K pot was pumped. At this point,
the tin-indium resistor, which was thermally connected to
the 1K pot, was cooled below its superconducting transition
temperature.

po of the sample was measured directly by first

stopping the circulation of the dilution refrigerator, after

 

 

63
the temperature of the sample can was less than 1K. Then
the 1K pot was warmed up to 4.2K, ‘at which point the
tin-indium resistor became normal. Usually the sample
warmed little during this process and stayed below 1K. Then
the ratio between the tin-indium standard and the potassium
or the copper resistor could be measured.

As the reader has already seen, one could measure the
ratio between any two resistances in the low temperature
circuit (Figure 2.7). This section explains how this ratio
was used to obtain a temperature dependent resistivity for
the potassium sample. Normally other research groups keep
one resistor at 4.2K and just measure the ratio between it
and the sample, while varying the temperature of the sample.
The disadvantage of that method is that the Johnson noise
generated by a resistor at 4.2K can be greater than the
noise in the sample at 70mK. To minimize this, one would
like to cool the reference to the same temperature as the
sample, but then one has the problem of separating its
temperature dependence from that of the sample. To avoid
this problem, we regulated the temperature of the reference
(T') keeping it slightly cooler than the sample. Then some
heat was added to H 2 and the ratio C measured at

U

temperature T. Next the heat was switched to H so that a

L2
temperature difference [XT was produced across Rwl' and
C+Z§C was measured at temperature T+Z§T. The current
through the sample was usually 50mA, but checks were made to

be sure there was no current dependence by using 25mA

 

 

64

occasionally. We then calculated the quantity:

AC =[ 2 ] [lc+Ac—Ql (2.8)

CaveAT (c + c + At) (T +AT - T)J

but c= Rk / R (T')

ref

 

 

= [ 2 arefm') ] [(Rk(T+AT)-RR(T))]

c
CaveAT (Rk(T+AT) + Rk(T)) (AT Rref(T'))

= (Akal = 1 dpm = ding (2.9)
Rk(T+AT/2)AT p(T) dT T

This quantity was normalized by multipling through by loo in

 

 

the plots of the data.

This technique permits measuring the logarthmic
derivatives of resistance to high precision by reducing the
thermal noise of the reference. Another advantage is it
permits double checking the thermometry, as will be
discussed in section 2.7.

Another quantity measured was G, the thermoelectric
ratio. This was measured since it was easily obtained, and
the only additional equipment required was a heater at the
end of the sample as shown in Figure 2.2. G is the ratio of
electric current to heat flow at zero voltage drop across
the sample. It was measured by sending heat (Q) through a
sample and then a current (I9) to counteract the resulting

thermal voltage, as indicated by the nulling condition at

the SQUID input.

 

65
I I

G = = g (2.10)
‘63 E=0 Ih x Ih th

where Ih is the heater current and Rh is the heater

 

resistance.

2.6 Heat Flow in the Sample Can

The primary low temperature thermometer used in these
experiments was a GR-200A-30 germanium resistance
thermometer from Lake Shore Cryotronics(68). We called this
thermometer GRT2. It was without a doubt the most important
thermometer, since it was used to measure the temperature of
the sample. It was mounted in a copper holder, and this
holder was placed in a copper block which clamped it to the
copper thermal feedthrough rod (Q, see Figure 2.1) This
provided a strong mechanical support and a good thermal
connection between the thermometer and the sample.

It was important that GRT2 be at the same temperature
as the sample. This meant that the heat flow along the path
from the sample to GRT2 had to be very small or a
temperature gradient would have developed. Thus the heat
flow in the sample can needed careful consideration. The
flow of heat from the sample to the mixing chamber went as
follows. The sample was cold welded to the silver wire (V)
inside the can, providing good electrical and thermal
contact. Two internal thermometers were connected to and

supported by this silver wire. This wire was then spot

 

66

welded to an 1/8 inch diameter copper bar (Q) which took the
heat outside the sample can. This copper bar was welded to
another silver wire which took the heat to a silver tab
which was pressed hard against the silver weak thermal link
assembly (see Figure 2.9) with a brass screw. This assembly
contained the U2 and L2 heaters and a thermal resistor
(56.4511fl). The other side of this assembly was screwed to
two silver tabs welded to the copper mixing chamber of the
refrigerator. The largest thermal resistance and thus the
largest temperature drop was due to the 56l1§l resistor.

Let me now discuss the various parts of this thermal
path. The silver wire was made of 99.9999% silver which was
rolled to ~1 mm square and then oxygen annealed for 3 days
at 900°C to give a very high thermal conductivity. This
wire was bent as little as possible while shaping and
welding in place. Thus few dislocations should have been
added, keeping the thermal conductivity high.

The copper rod was made of OFHC copper which was
machined to 1/8 inch diameter. This rod was oxygen annealed
for 3 days at 950°C to remove dislocations and oxidize
magnetic impurities. The electrical resistance of a similar
rod was measured at 4.2K to be ~0.07fi1§), a very small
resistance compared to the weak link. This copper rod was
part of the thermal feedthrough which conducted the heat
outside the sample can. It was supported by a teflon
Spacer, for electrical isolation, in a copper block and

sealed in place using Stycast 2850 GT epoxy(69). The Stycast

 

O\
\l

 

 

 

 

 

 

1““ .
A _1 (7 Location of
6 upper and lower
0 l hea‘l’ers
Weld Points
for Ag.16t%Au J
Wll'e

 

 

Q _ O
F:—-———'Vespel

 

 

 

 

B _._

©

Figure 2.9 A) Weak thermal link assembly
B) thermal connection tab

68
provided a helium leak tight seal between the two copper
parts. The copper block had been hard soldered to a 1/4
inch diameter thin wall stainless steel tube (P) 1.25 inches
long, which was soldered to the top flange (L) of the sample
can. This stainless steel tube provided thermal isolation
for the feedthrough, as well as mechanical support.

The last part to be discussed is the weak thermal link
assembly and the silver tabs (Figure 2.9). The electrical
resistance between the tabs and the silver blocks was
measured to be ~0.341§?. The tabs and the two silver blocks
were made of 99.9999% pure silver. The silver blocks were
separated by a square Vespel rod(70), which had a very high
thermal resistance. Between the two silver blocks was spot
welded a thermal resistor made of Ag 0.1at%Au wire, with a
resistance of ~56fllfl. The two blocks each had a 4k§2Dale
resistor(71) as a heater.

On looking at the design, one can see that there are
two other paths for heat from the sample to take. One is
through the stainless steel tube (P, see Figure 2.1)
supporting the copper thermal feedthrough (Q) and the other
is through the stainless steel tube (J) supporting the
copper squeeze plate (F). Table 2.6 shows the relative
sizes of these heat leaks compared to the heat flowing
through the weak thermal link at various temperatures used
during the experiment. It should be noted that these ratios
are upper limits since no account has been taken of the

Various contact thermal resistances in the alternate heat

 

69

Table 2.6 Alternate heat flow paths

Temperature 4K 1.0K 0.4K 0.1K
thermal feedthrough r = 2.6% 2.8% 1.9% 1.6%
stainless steel support r = 2.2% 2.4% 1.6% 1.5%

r is the ratio of heat flow through the indicated path to
heat flowing through the weak thermal link.

 

70

flow paths. Since the thermal resistance of the path
between the potassium sample and the weak thermal link was
small, and very little heat flowed down this path, there was
practically no temperature difference between the sample and
GRT2. The estimated temperature error cased by this was
less than 0.2mK for the worst case.

Since GRT2 was mounted outside the sample can, thermal
radiation and conduction of heat through the residual

He3

gas from the surrounding walls at 4.2K could cause
significant heating of this thermometer. To eliminate this
problem, a copper shield was placed around GRT2 and the
sample can, and this shield was thermally connected to the
mixing chamber. After transfer of liquid helium, the

He3

exchange gas in the vacuum can of the dilution
refrigerator was extracted until a He leak detector showed

less than 10—7 cc/sec of He3 coming out of the system.

2.7 Thermometry

There were three thermometers outside the sample can
and two thermometers inside the sample can. The first
thermometer inside the sample can was a silicon diode from
Lake Shore Cryotronics, Inc.. This thermometer, used
between 4.2K and room temperature, was thermally connected
to the cold end of the sample. In use, the voltage across

5

the diode was measured with 10- A reversed biased current.

Its temperature characteristics are shown in Figure 2.10.

 

71

 

mazpmamcEmp m> opofle cooflfiwm one a: vampac> one CH.H madman

00 e

 

com com 00— o
a _ _ _ o
r. _l 0; An
\A/
0
l 1 Hw
/S\
F 1 ca
_ _ _ _

 

 

 

72
Below ~30K it had a sensitivity of ~50mV/K, and above 30K
about 2.8mV/K. It was calibrated from 4.2K to 80K against a
germanium resistance thermometer, and from 80K to 300K
against a platinum resistance thermometer. This diode
thermometer was used solely for measuring the temperature at
which the sample was deformed or annealed.

The other thermometer inside the sample can was a
Speer(72) lOOi? carbon resistor, which was calibrated
against a germanium resistance thermometer (GRT2). The
Speer's calibration was good to ~5mK at 90 mK and to ~20mK
at 1.0K. No long term drifts due to thermal cycling were
observed on comparing it to GRT2. This thermometer was also
thermally connected to the potassium sample by a silver wire
and was used to make sure that there were no large
temperature differences between the sample inside the can
and GRT2 outside.

With the exception of the silicon diode, the
resistances of the thermometers were measured using two
(digital conductance bridges from S.H.E. Corporation. An

uricalibrated Speer carbon resistor was mounted on the mixing
chanmer, and it was used when regulation of the mixing
Chamber temperature was required. The calibration of GRT4
(see Figure 2.7) was made by carefully comparing it to GRT2.
1Wfi.s calibration was not important for most work since GRT4
was only used to maintain the reference at a constant
tenmoerature. The only exception is that samples Kh6b&d were

measured using GRT4 .

73

Next the calibration of GRT2 will be discussed. The
first step, taken by J.L. Imes(73) and G.L. Neiheisel(74),
was to calibrate a Cryocal CR50 thermometer using the
He3 vapor pressure scale and a susceptibility thermometer.

GRT2 was directly compared to the CR50 thermometer from
0.065K to 4.2K. The data from this comparison was fit with

the equation,

9

In R = EE:A1 (lnT)i (2.11)

i=0

This first fit has been used for all measurements from 4.2K
to 1.3K. However, there were indications that the
calibration below 1.3K was not accurate enough for this
study. Thus GRT2 was recalibrated below 1.3K.

The first step in the recalibration of GRT2 involved
the use of a set of Superconducting Fixed Point Devices, SRM
767 and SRM 768 from the National Bureau of Standards<75).
These devices gave an absolute value of temperature at 6
points, from 0.099K to 1.17K. Interpolation between the six
points was performed using a susceptibility thermometer made
of an irregularly shaped single crystal of CMN where 90% of
the Ce was replaced by La. This second set of data, from
0.059K to 1.24K, was then fit with the above equation, and
the resulting coefficients used for calculating the
temperature below 0.5K.

The second step in the recalibration of GRT2 employed a

 

74
large powdered sample of cerrous magnesium nitrate and four
of the fixed points between 0.519K and 3.414K. This last
set of measurements was used to provide the 0.5K to 1.3K
recalibration of GRT2.

The accuracy of these calibrations was then tested
using the Wiedemann-Franz (W-F) law. This was done by
mounting GRT2 at the end of a resistor made from a
Ag 0.1at.%Au Alloy (which had previously been measured and
shown to follow the W-F law by Edmunds et al.(76)) and
attaching the other end to the mixing chamber, which was
kept at a fixed temperature. Then heat was alternately
switched from one side of the resistor to the other. The
average temperature (Tave) and the change in temperature
( [110 were calculated using GRT2 and the above fits. Then
the Lorenz ratio was calculated from the equation:

L: R o
TaveAT (2.12)

R~79.8[LII. Figure 2.11 shows the results in terms of
percent deviation from the W-F law. Thus the calibrations
gave the product TZXT 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%.

Finally, to ensure that there were no obvious problems
while taking the data, the W-F law was used to double check
the thermometry for every run. Figure 2.12 shows the

results where the value of R the weak thermal link

wf’

75

 

 

 

 

 

3%» I l I
o A
2%>b _
. A
1%.r- . o.) _
o ..
310,070 g. .. T
I I
-p% _ _
I
A
-296- I _
-3% L l L
04) 1.0 21)
T (K)
Calibrations
o 0.0 — 0.5
I O~5--l.3
A 1.3 — 4.2
Figure 2.11 W — F law check of T T calculated using the

m r ° :
GR12 calibration

76

 

 

 

 

59 1 1 1 1
I-+(h9|1
58 :
,—.. F'. . ¢.-Kh§>0 -
C. u .
i I I
~.— 0
g 57L 4
' : : '
56u45*-- -- -- :'- -—- -- ‘-- --
. I
56 h- I ‘“
I C I
C
55 l l l J
O l 2
T (K)

Figure 2.12 Rw vs T

77
resistance (see Figure 2.9), has been calculated using the

W-F law,

_ L
wa — o (2.13)

0 °_ 2
Q TaveZXT Q-RLZ I

 

This is compared with the measured value of the resistor of
56.45filg2. One should note that there is an uncertainty of
~1% in the values of wa plotted, since the resistance of
heater L2 is only known to 1%. The values for wa have a
range of 3%-higher to 2% lower than the measured value of
this resistor. The main systematic problem in measuring the

temperature of the sample is from the calibration of GRT2.

2.8 Uncertainties

The major uncertainty in G was random error due to
thermal fluctuations and thermal drifts in the system. I
estimate this to produce 2% uncertainty in G, based on
temperature measurements made while taking data. The next
largest error was in measuring the power to the heater,
which for the very lowest measurements (~80mK and lower) was
as high as 2%, and was due to the digital round off of the
DVM used. Also at the lowest temperatures, Johnson noise
limited our null detector resolution to ~l% in G. The other
sources of error were mostly systematic and very small in
comparison with the random uncertainties. The value of the
heater resistance was measured to within 0.1%. The

uncertainty of the resistance used to measure the current

78
through the heater was 0.05%. The current Ig was produced
by the current comparator (C. C.) system and was measured
to be within 0.16% of the value set. A current setting of
0.1 x50mA was used while measuring G, see Table 2.7. The
drift in the zero current of the current comparator was much

less than the absolute uncertainty in the current.

Table 2.7 Current comparator error in absolute current

 

C.C. Setting Current output - 1
C. C. Setting

0.1 x SOmA +0.16%

0.2 x SOmA +0.18%

0.5 x SOmA +0.22%

1.0 x SOmA +0.20%

The random errors in measuring /30 and ޣ.2 were as
follows. First R4.2 and R0 were measured to ~2% against
RSn/In’ the limiting factor being the temperature dependence
of RSn/In near 4.2K. To a lesser degree the measurement of
R4.2 was also affected by the slight changes in the
temperature of the He bath due to changes in atmospheric
pressure. Additional systematic error arose from the
measurement of A/L. The room temperature resistance of the
samples was always measured to better than 1%, but on
deforming the samples the A/L sometimes shifted. This was a
correctable problem for all but Kh8, where we had to
interpolate to obtain the values for A/L. This problem is
discussed in the next chapter.

The precision of C has been shown to be better than 0.1

ppm(76). To reduce the uncertainty in ZXC, an averaging

79
technique was used, which permitted interpolation to less
than 0.01 ppm in C. In some cases, however, the averaging
technique was not used for small [XC and a much larger

uncertainty resulted. In the calculation of the quantity

1 Ac

Cave [31‘ (2.14)

the largest uncertainty (outside of the uncertainty in £5C)
was usually due to the systematic error in [§T, which was
estimated using the W-F law in the last section, to be less
than 2.6%. So the uncertainty in this quantity is usually
less than ~3%.

Finally, the uncertainty in measuring the heat produced
by the upper and lower heaters used for the in-situ W-F law
checks is 1%, the uncertainty in the resistance of the 4k§2

Dale resistors.

CHAPTER 3

Experimental results

In this chapter the experimental results will be
presented and interpreted in light of the many theories
discussed in chapter 1. Two pure samples of potassium Khl
and Kh9 were deformed at 4.2K and then annealed in a series
of increasingly higher temperatures. The samples, Kh2 and
Kh8, were subject to a series of plastic deformations at
60K. One potassium - rubidium alloy was deformed at 60K and
then annealed at 160K. The remainder of the samples were
not deformed and were used as a study of the effects on the
resistivity of contact with various materials.

For the samples deformed at 4.2K, it was hoped to see
the relative effects of both point defects and dislocations.
By annealing these samples at 60K, where most point defects
are unstable, one could then see the effects of just
dislocations. For samples deformed at 60K, one expects
mostly dislocations to be introduced.

For samples Kh2 and Kh8 the dependence of dislocation

80

81

scattering on the concentration of dislocations was sought.

3.1 Resistivity

It is of some interest to compare our results for
undeformed samples with the earlier experimental work on
potassium. This comparison will be made in two temperature
regions; first above 1.1K to show we get the same electron -
phonon term and second, below 1.2K to compare the electron -
electron term.

Figure 3.1 shows this comparison of samples Kh8a and
Kh9a at high temperatures with the previous work of Lee
et al.(77), van Kempen et al.(78), and Rowlands et al.(79).
Below about 1.5K the temperature dependence seen is due
primarily to the electron - electron term in [3. Above 1.8K
the electron - phonon umklapp process predominates, and the
data follow the expected fi)~exp(+e*/T) dependence. Our
data above 1.8K agree very well with the previous work, so
the samples in this study have the same electron - phonon
term as measured by other groups.

Figure 3.2 shows a comparison below 1.5K with the 0.9mm
samples of Lee et al.(80). In this figure an AT2 behavior
for #3 would produce a horizontal straight line if A is
independent of T. Note that samples Kh8a and Kh9a have, on
average, lower values of A than the samples of Lee et al..
Also the departure from the A = constant expectation is more

marked for Kh8a and Kh9a. Recent work at MSU has shown that

82

 

 

 

 

*1 T I ll 1 47
.- oKéb Lee et al —
0K”: Rowlands e9 II
on
O
o n K2b van Kempen ct ol
0
1000—- ‘Khao "
08
E 00
C:
Et' 00
mo—- D -
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< a _
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é 0 0
Q Q.
0
‘, o
0 o
q o
u— 0 DD -‘
D ' "
O
O
O
1 J J J J J .J
.2 .3 .4 .5 .6 .7 3 9
T" (K">
Figure 3.1 High temperature comparison with previous groups.

33
QU

mpfizmmfl @6393 :33 com. TELEoe o.H:p,mLocEep Bod

O: ._.

 

 

 

m_ 0_ mo 00
_ a q _ a _ q a a O
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4:4.
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1 a3. 0 a 1
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r. . m mLSwfik

84
this lowering of A and the temperature variation of A become
more pronounced for samples which are thinner than
0.9mm(32). Since the diameter of the extruded samples are
determined by the size of the exit hole on the press and
since we do not have a direct way of measuring the diameter
of the samples, it is possible that Kh8a and Kh9a had
slightly smaller diameters than the 0.9mm sample of Lee

et a1.

3.1.1 The residual resistivity

It is of some interest to observe the effects of
deformation and annealing on the residual resistivity and to
compare these results with the earlier work of Gugan(81).
Table 3.1 lists the recovery of the residual resistivity for
samples Khl and Kh9.

Both samples showed complete recovery by 165K, in
agreement with Gugan, who found the last stage of annealing
to take place below 150K for most samples. For annealing
above 80K, Gugan found ~25% recovery in good agreement with
our results. However, we find ~30% recovery from 7.5K to
20K, and 42% from 20K to 80K, whereas Gugan found 40% and
30% respectivly. Thus for our sample slightly more
annealing occurred at higher temperatures. Also less
annealing is seen in the 4.2K to 7.5K stage, which may be
due to a slight heating of our sample during the

deformation, near 4.2K. Our samples showed the same general

 

85

Table 3.1 Recovery of annealed samples
Sample recovery of fik> Temperature
Khla ---- ----
Khlb 0% 4.2K
Khlc 78% 60K
Khld 100% 155K
Kh9a ---- ----
Kh9b 0% 4.2K
Kh9c 1% 7.5K
Kh9d 31% 20K
Kh9e 64% 60K
Kh9f 73% 80K
Kh9g 94% 120K
Kh9h 100% 165K

of anneal

(deformation)

(deformation)

 

86
features for recovery of the residual resistivity as Gugan.

Now we will look at the dependence of the residual
resistivity on the force used to deform the samples at 60K.
Figure 3.3 shows a plot of the residual resistivity vs the
pressure applied to the bellows. In this plot corrections
have been made to the pressure to remove the force needed to
move the bellows.

As mentioned before, the A/L for sample Kh8 shifted
during the experiment, most likly due to an unexpected
change in L during deformation. No Such shift took place in
Kh2. Since Kh2 showed a roughly linear increase of ‘fi% with
pressure, it was decided to correct the A/L of Kh8 by
requiring L% to be proportional to the pressure. The
straight line drawn through the Kh8 data represents this
requirement. We know that this line is correct for both end
points, since A/L was measured directly for them. The
difficulty comes for the points in between, with the maximum
correction to fl% being 12% for Kh8e if the initial A/L were
correct. It is likely that the estimates of A/L are always

more accurate than this worst case.

3.1.2 The electron - phonon resistivity

As the reader has already seen, we reproduced the high
temperature data seen by other research groups for
undeformed samples. The next question is how well our

deformed sample data compare to the theories on ޣ_p in the

 

 

 

(nSZ cm)

 

 

 

 

 

of 1.0
A kh2
t o kh8 initial A/L _
I kh8 final A/L
J J J l J I J J
O 50 100
Pressure (inches Hg)
Figure 3 3 T“ I"?

88

limit of a high concentration of dislocations for T>l.7K.

Figure 3.4 is a plot of our data compared to the theory

of Danino, Kaveh and Wiser(82). The solid curves represent
their theory for the indicated dislocation densities, Nd'
For this plot ‘08:) was calculated by fitting our undeformed
pure samples (Kh3, Kh9a, and Kh9h) to an equation of the

form,

a.
paw) = ATZ + BT exp(-6*/T) = AT2 + 'Oep (3.1)

where 2<T<4.2K, and the superscript "a" refers to the
undeformed samples. The parameters obtained were:

A= 1.30 1 .007 x 10'13 Qcm/x2

B: .773 1 .038 x 10'9 Qcm/x

6 = 20.3 i .2 K

The resistivity [ng was calculated by subtracting the

AT2

term from the experimental 13(T), using the coefficient
listed above. The assumption that A does not change
appreciably under deformation will be justified later.
Sample KRbh is not shown, since it was measured against a
different reference resistor, the resistance of which has
not been measured as a function of temperature.

One can see from the graph, that the experimental ratio
of fi£_p/ figip increases at low temperatures, and is in
rough agreement with theory near 2.0K for Nd~ 5 x lolo/cmz.
However, near 4K the data and the theory are not in

agreement for this value of Nd' In addition, using

Basinski's result(83), one would predict from the change in

 

 

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90
£k>for sample Kh9e that Nd~ 8 x 108/cm2 which is almost a
factor of 100 smaller than the above number. Thus the
theory does correctly predict that the ratio will increase
as the temperature is lowered, but the theory does not give
the correct overall temperature dependence or a reasonable
value for Nd.

The more recent theory by Enquist(84) and also by
Danino, Kaveh, and Wiser(85) would appear to give a more
correct form for our data. This theory was discussed in
section 1.1.2. The dash curve shown in Figure 3.4 is the
result of fitting this theory (DTS) to sample Kh9e. For
this fit only the parameter D was varied. This theory
follows the form of the data reasonably well over the entire
temperature range. The coefficient obtained (D=30 f§2cm/K5)
was less than the maximum theoretical value predicted by
Ekin and Maxfield(88) (D=35.4 f§)cm/K5) for the case of full
quenching of phonon drag. Moreover, the value obtained for
D was larger than that obtained from the low temperature
fits discussed in the next section. This is not suprising,
since only a crude estimate of A was used for the data of

Figure 3.4.

3.1.3 The low temperature data

Figure 3.5, Figure 3.6, and Figure 3.7 show the
results for Kh8, Kh9, and KRbh respectively. The main

feature of the graphs is the very large low temperature

2)

(lo-”9cm/K

91

 

 

 

 

 

 

If)
U)
°q Sample p0 (chm) Remarks
0 D Kh8a 1.32 no deformation
‘3. o 1(th 1.38 P=7.l" Hg

0 4 Kh8c 1.46 P=14.1"

o -+- Kth 1.53 P=21.1"
5:: x Kh8e 1.81 P=50.1"
0‘ O Kth 2.05 P=84.3"

8
3 7"
o"

X

X
a 0
0'1
0 X
"3‘ o
0
I3 X
._‘ o
O
o A" x
“J. +
D + .
m o + "
"?_4 o
O o b

+ X
0
A O
2 o + x O O
c“ 0 ‘ + x x. o o o ; x
0 A + x x x +
A * * ‘ A

"’ ° . ‘ . : r : .
D 0 ° 0 o o 9 °

5 DO U D a U I D
8
“b. 0 20 0 40 0160 (7.30 {.00 R20 1 4o {.60

T (K)

Figure 3.5 (l/T)(d,O/dT) vs T for K118

.80

2)

-H

(I0

dP
d

Dem/K

T

T

 

 

 

 

 

8
6
32
=‘ p
: Sample 0 (n()cm) Remarks
:3 ‘A
:« D Kh9a 1.09 no deformation
‘ O Kh9b 2.21 deformed at 4.2K, P=95” Hg
0 A Kh9c 2.20 annealed at 7.5K
g -+ Kh9d 1.86 annealed at 20K
01 ‘p x Kh9e 1.49 annealed at 60K
(0 Kh9f 1.39 annealed at 80K
o Q Kh9g 1.16 annealed at 120K
3. 1 X Kh9h 1.09 annealed at 165K
9
U)
m +
0"
X
a 2
61
0
U)
N X
5‘ +
I
D
N O
0'.
3
u: 0 1
'7. I
D ‘ 2
° 9
o ’ + 0 i
3‘ ’ 2 + X
9 o o o ’
° * 3 I o o i
0 x x X X x 4 X
8 xx 4 ’ o o o 0 ° m
c‘ ‘ o , ’ o . . ¢ ’ B
hula B B I I 3 a I a a
S
c0.00 0120 6140 0160 0130 1100 {120 1.40 x 60
T (K)
Figure 3.6 (l/T)(d/D/dT) vs T for Kh9

 

.80

2)

qulacm/K

dP
TdT

 

93

 

”.lw'

 

 

 

 

 

U3
LO
a"
Sample ‘0 (n(Tcm) Remarks
D
2‘ D KRbha 10.75 no deformation
OKRbhb 11.52 deformed at 60K, P=95" Hg
A KRbhc 10.78 annealed at 160K
U)
a.
3
5+
5
g); 0
8 o
D"
8
D"
O
O
“.‘q o
O
0
O
33 o
:3" o o o
0 l
o o o 0 o
o I! O
5‘ 0
A Q
‘ 6 a 0 e a I I I i '
g B
o.“ A
D
3
‘b .00 0120 0.40 0160 0100 1100 1120 T40 1160 1.00
T (K)
Figure 3'7 (l/T)(d/D/dT) vs T for KRbh

 

94

divergence after deformation. Since these divergences are
not seen for these samples in the undeformed state, we may
associate them with some new scatterer introduced through
plastic deformation. Another point of interest, seen in Kh9
and KRbh is that after annealing the samples near 160K, one
recovers almost completely the temperature dependence prior
to deformation. This recovery coincides with the obseved
recovery of (Do (discussed in Section 3.1.1). Hence, it is
concluded that dislocations are the major contribution to
the divergence.

Looking at Kh8, one can see that as one increases the
number of dislocations, the size of the divergence
increases. Although it is somewhat harder to see, the
slopes also increase for T> 1.2K, which is expected since,
as discussed in section 3.1.2, (Dep increases under
deformation.

If one were to have electron - electron, electron -
phonon, and electron - dislocation scattering, the

resistivity would be of the form,

_ 2
,O _ '00 + A T + e-p + pe-dis (3.2)

d ,0 /dT = 2 A T + [O'e-p + p'E‘dis (3.3)

In the limit where F’e-p and. fi)e-dis were negligible the
data would follow a horizontal straight line, on these
graphs. All these figures show such a region, but the
actual size of the electron - electron term can not be

easily deduced from the height of this plateau, since the

 

95

low temperature tail of ID'e-p and high temperature tail of
[D'e-dis are present in this region. However, the height of
the plateau does give a maximum value for the size of the
electron - electron term. Let A' be this maximum value of
A, as listed in Table 3.2. Sample Kh9 shows that annealing
at temperatures upto 20K, has little effect on the height of
the plateau. Thus the annealing out of point defects below
20K does not influence A' appreciably.

Next, our results will be compared to those of the
Netherlands group. When van Vucht et al.(86) analysed their
data, they were not aware of the low temperature
£De-dis term, since they could not measure below 0.9K.
Because of the plateau, it is understandable why this group
chose to interpret their data as having AT2 behavior near
1K. Thus the values they quoted for A should be compared to
A'.

Figure 3.8 is a comparison of A' of Kh8, Kh9, and KRbh
with the data of van Vucht et al. This graph shows the
change in A' ([3A') vs the change in the residual
resistivity (Ape). Plotting the data this way removes
differences between the samples due to different initial
impurity levels and different initial sample sizes in their
undeformed states.

Sample K7 (measurements 7 to 9) was deformed at 4.2K
with several levels of deformation. One sees a somewhat
non - linear increase of AA' with APO. Comparing this to

sample Kh8, which was deformed with progressivily higher

 

Table 3.2

Effective A for the samples

 

Sample fl% A
(n Qcm) (10 13 Q cm/KZ)

Khla 4.19 3.00
Khlb 6.30 6.96
Khlc 4.76 5.75
Khld 4.12 3.06
KhZa 1.87 2.25
Kth 2.04 3.23
Kh2c 2.28 4.09
Kth 2.55 4.81
Kh2e 1.95 2.24
Kh8a 1.32 1.52
Kh8b 1.38 2.07
Kh8c 1.46 2.60
Kh8d 1.53 3.01
KhBe 1.81 3.92
Kh8f 2.05 4.42
Kh9a 1.09 1.45
Kh9b 2.21 3.79
Kh9c 2.20 3.76
Kh9d 1.86 3.85
Kh9e 1.49 3.19
Kh9f 1.39 2.89
Kh9g 1.56 1.93
Kh9h 1.09 1.51
KRbha 10.75 3.65
KRbhb 11.52 6.38

KRbhc 10.78 3.72

 

(me/K2)

I

11A

 

D3

09

 

Id
lb

O.

K7 3'5

von Vucht et ol.

K7 7"? _-

Kh8

Kh9 -‘

KRbh

 

 

Figure 3.8

[XIV vs

the change

LO

(nflcm)

in the residual resistivity.

 

98

pressures at 60K, one sees that ‘ZXAfi also increases with
[3,Cg, in a non - linear fashion. The relatively small
differences between the two curves are most likely due to
the different temperatures at which the samples were
deformed. The smaller overall slope of these van Vucht
et a1. data is consistent with the fact noted earlier that
the defects which anneal out below 20K have little effect on
A'. Since these defects would still contribute to [311“
one would expect a smaller dependence of AA' upon APO.

Next, sample K7 (measurements 3 to 5) will be compared
with sample Kh9. Both samples were deformed at 4.2K and
annealed at several higher temperatures. K7,3 and Kh9b are
the unannealed samples. The samples K7,4 and Kh9d were
annealed at 18K and 20K respectively, and samples K7,5 and
Kh9f at 78K and 80K respectively. The data of van Vucht
et al. showed a gradual recovery of A' with increased
annealing temperature, whereas, Kh9 showed little recovery
until it was annealed above 20K. After 20K, Kh9 does show
the same gradual recovery as K7. One may conclude that the
behavior of the deformed (0.9-mm-diameter) samples in this
study is rather similar to those of van Vucht et al., even
though their 3-mm-diameter samples were twisted instead of
squashed. Finally, one should note, that KRbhb had about
the same increase in the residual resistivity as Kh8f, and
showed about the same increase in A'.

Next let us compare A' with the theory of Kaveh and

Wiser. Samples Kh8 and KRbh were deformed at 60K. In both

 

 

 

99

samples, there was a similar increase in the residual
resistivity and A'. These increases in A' are much larger
than would be predicted by Kaveh and Wiser (see Figure 1.4,
page 23). Moreover, Kaveh and Wiser claimed that A was a
function of fibi/ fibd’ Sample KRbh should be in the
impurity dominated limit, since the undeformed sample had
l30~10.75n§7cm which was many times larger than the
residual resistivity of a pure sample (1.32nf2cm for the
pure undeformed Kh8a). Thus, while adding 0.73nf7cm of
dislocations to a pure sample like Kh8 should have a large
effect on A, one would expect to see little change for
sample KRbh. Since the change in A' is the same for both
samples, one must conclude that some mechanism other that
that proposed by Kaveh and Wiser causes most of the change

in A'.
One must remember that A' is most likely an over
estimate of the correct value of A. To make a realistic
comparison with the theory of Kaveh and Wiser, the

p

e-dis and the pe_ph terms must be examined.

Figure 3.9, Figure 3.10, and Figure 3.11 show plots of
the resistivity data for Kh8, Kh9, and KRbh, which emphasize
the behavior of i£-dis° All three graphs show similar
behavior: a peak in the data for the deformed samples near
0.2K. On these graphs the electron - electron term ’would
appear as a straight line (2AT) going through the origin.

Again one sees in samples Kh9 and KRbh that annealing near

160K brings back the initial temperature dependence of the

 

-)T

(10 9cm/K)

100

 

 

 

‘33
o'
G
(D
3. Sample po (chm) Remarks 4
D Kh8a 1.32 no deformation
:5 O Kh8b 1.38 P=7.l" Hg
6* A Kh8c 1.46 P=l4.1" 4
+ Kh8d 1.53 P=21.1"
x Kh8e 1.81 P=50.1" °
3‘. O Kth 2.05 P=84.3" 1
o.

.10

 

 

 

 

 

I I T I I
.00 0.20 0.40 0.Tso (K 0.00 1.00 1.20 1.40
)

droo

Figure 3.9 d/D/dT vs T for Kh8

(I0 “cm/K)

101

 

  
   

   

 

 

 

v
‘5 0
Sample p0 (chm) Remarks
N
g. D Kh9a 1.09 no deformation
0 Kh9b 2.21 deformed at 4.2K, P=95" Hg
A Kh9c 2.20 annealed at 7.5K
+ Kh9d 1.86 annealed at 20K
9 x Kh9e 1.49 annealed at 60K
0‘ O Kh9f 1.39 annealed at 80K
9 Kh9g 1.16 annealed at 120K
X Kh9h 1.09 annealed at 165K
0
O
51
ID
D.
0‘
V
D.
o‘
N
c
0'1
(3
D.
‘33.00 0.20 0140 (Re 0.00 1.00 1.20 1 40
T (K)

Figure 3.10

dflD/dT vs T for Kh9

102

 

 

 

1‘.

O

N Sample ‘8 (n(2cm) Remarks a

N

w

o CJKRbha 10.75 no deformation 0
CDKRbhb 11.52 deformed at 60K, P=95” Hg

0 AKRbhc 10.78 annealed at 160K

N

o.

 

-H

()0 0cm/K)

Q
.0

dT

 

 

 

 

T 1 I 1 T
0.20 0.40 0.50 0.00 1.00 1.20 1.40
T (K)

cp.00
1':
D

Figure 3.11 dID/dT vs T for KRbh

 

103
undeformed samples.
Sample Kh8 shows a gradual increase of dfl3/dT, as one

increases fikfi The form of fig changed markedly when

-dis
sample Kh9 was annealed at 20K and 60K, but for later
annealings the main effect was to reduce the size of dflD/dT,
not its shape. Thus it seems unlikely that this peak is
caused by point defects such as vacancies or interstitials
but rather it is mostly due to dislocations, with the form
being affected somewhat by the presence of point defects.

To further understand the form of (De-dis' an equation

of the following form was fitted to the data for T<1.3K.

p: p0 + AT2 + ’Oe-dis + DTS (3.4)

where DT5 represents the effects of quenching of phonon
drag by the defects and. fik~dis represents the different
equations listed in Chapter 1, which might decribe the
resistivity due to scattering off dislocations. The umklapp
electron - phonon term has been neglected in this equation
since below 1.3K it is not expected to make a major
contribution.

The Equation 1.26 from Gantmakher and Kulesko for
electrons scattering off localized phonon modes gave the
best fit to the low temperature data, and these fits are
shown as solid curves in Figure 3.9, Figure 3.10, and
Figure 3.11. The discrepancy between the curves and the
data at ~1.5K is due to the onset of the umklapp electron -

phonon term, which was not included in the fits. These fits

 

104

showed the presence of the T5

term in deformed samples.
Furthermore, for these fits, the value of A varied little
when compared to A‘ ( Table 3.3 lists the coefficients
obtained).

Figure 3.12 compares the best fits for the two
equations of Gantmakher and Kulesko (Equation 1.26 and 1.23)
to the data of sample Kh8f. No set of parameters for the
localized electron model (Equation 1.23) gave a better fit
to the data. The normal electron - phonon term (DTS) was
forced to zero in this fit, which is inconsistent with the
T>2K analysis. To compansate for D=0, the program then
chose a value of A greater than A'. It is possible that
some combination of different energy levels might produce a
better fit.

The agreement of the vibrating dislocation model with
the data is nonetheless surprising. Gantmakher and Kulesko
assumed a single energy of vibration, but one would expect a
distribution of dislocation lengths and thus a distribution
of energies. The closeness of these fits may be fortuitous.

The energies (E=hw) obtained from the fits follow an
interesting pattern. Sample Kh8 shows a steady increase in
E as one adds more dislocations. This is consistent with
having shorter dislocation lengths due to the dislocations
pinning each other. One would expect point defects also to
pin the dislocations and thus increase E. But, when sample
Kh9 was annealed at 20K (Kh9d), presumably removing point

defects, E increased. Perhaps the defects that anneal out

105

 

 

Table 3.3 Parameters from fits to the resistivity data
Sample 6% A E=fiw C/4 D

(n 0 cm) (p Q cm/KZ) (K) (10-130 cm) (10-130 cm/KS)
Kh8a 1.32
Kh8b 1.38 .107 i .007 .250 i .105 .085 i .066 .118 i .016
Kh8c 1.46 .087 i .006 .278 i .049 .187 i .062 .195 i .017
Kth 1.53 .093 i .012 .353 i .129 .378 i .250 .208 a .017
Kh8e 1.81 .105 1 .003 .395 i .008 .677 t .028 .248 i .010
Kth 2.05 .134 z .007 .340 i .006 .571 i .024 .233 i .023
Kh9a 1.09
Kh9b 2.21 .150 i .005 .268.: .011 .259 i .021 .186 i .013
Kh9c 2.20 .154 i .004 .275 i .006 265 i .012 .185 a .010
Kh9d 1.86 .192 i..014 .353 i .014 442 i .045 .095 i .030
Kh9e 1.49 .060 i .004 .307 i..005 .389 t .014 .206 1 .009
Kh9f 1.39 .070 i .004 .274 i .008 .249 i .015 .186 i .009
Kh9g 1.16 .114 t .003 .256 i .025 076 i .014 .077 i .007
KRbha 10.7 .366 i .002 ---------------------- .082 i .096
KRbhb 11.52 .393 i .010 .486 i 014 .874 i .059 .221 i .023
KRbhc 10.78 .370 i .001 ---------------------- .082 i .096

‘8 was not obtained from this fit, and is listed for
comparison purposes on this table. The constant C is
the coefficient of Equation 1.26.

106

 

 

   

 

 

 

7 l
Sazple Kth equation 1.23
-13
a = 0.5538 x 10 I? or.
E__ b= 05 ._
3. E = 0.388 K _lr, 2
A(e-e) = 0.461 x 10 'Qcm/K
‘2
\
E
U
“_d
'70
v If‘
9- -—
Q1—
'01)
:3 J ‘
0.5K 1.0K
T (K)
Figure 3.12 Comparison of the best fits of equations

1.26 and 1.23 to the data of Kh8?

107
at 20K form small dislocation loops which might increase E.

The values for A obtained from the above fits (listed
in Table 3.3) can be compared to the theory of Kaveh and
Wiser. Figure 3.13 shows a comparison with samples Kh8 and
KRbh. One must remember that (phi is unknown, and its
common value for each sample was obtained by finding the
best fit to the curve. If one ignores Kh8a (undeformed
sample) and Kh8b (slightly deformed sample) the data of Kh8
would give a reasonable fit to the theory. It is difficult
to estimate accurately the value of A in an undeformed
sample, since it does not follow exactly a T2 behavior due
to the previously mentioned size effect(32). In the well
deformed samples, one would expect the mean free path of the
electrons to to be sufficiently decreased that the size
effect would be no longer significant.

The theory of Kaveh and Wiser does have difficulties
with sample KRbh. The A values shown in Figure 3.13, are
the values from the above fits, with a contribution to A due
to vibrating impurities (0.091pf7cm/K2) subtracted out(87).
If one chooses to assign only 2.04n Q cm of the residual
resistivity as being due to the impurities, a reasonable fit
to the curve in obtained. However, it is very unlikely that
the impurity contribution to ,06 will be this low, since the
effects of adding rubidium to potassium has been observed in
many laboratories to produce a much higher contribution to
the residual resistivity. A more reasonable estimate of the

impurity contribution to the residual resistivity, as shown

 

n 1111111

 

 

 

 

 

 

 

 

 

 

O'JT]TITTI t IIITTHI
+ kh8 ,00i : 1.00 nncm
0.4._. o kah poi: 2.04 nncm _.
I kah 0,: 9.75 nncm
0.3—
N
X
\
E
U
C:
\‘b 0.2--
<
0.1»—
O.O1)JJHJ J 11111111 1 JlJJlJ
0.03 0.1 1.0 10.0
poi
Po.
7 4 wit. Kaveh and ‘iser's trecr

 

109

in the figure, causes the data to deviate significantly from
the theory. If Kaveh and Wiser were to choose a higher
value for A0 of 0.27chm/K2 (see Equation 1.22), then the
theoretical curve would be shifted up; and the data of KRbh
would agree better with this theory. This choice of a
larger value of AO seems reasonable since we have observed
in thick, pure samples A=0.24 i 0.02 pf7cm/K2(4). Since the
size effect makes uncertain the proper value of A in the
pure samples, it is not clear whether a conflict exists
between the theory and the data of Kh8.

A direct comparison of the results of Kh9 cannot be

made with the theory of Kaveh and Wiser, since both

p

oi and fiha vary as one anneals out the various defects.

Nonetheless, a qualitative comparison is possible. Starting
with Kh9b, one sees that annealing out point defects at 20K
increases A, as listed in Table 3.3. This is what Kaveh and
Wiser would predict, because the ratio figi/' figa is
decreasing, so A should increase (see Equation 1.22). On
annealing at 60K, A drops substantially. One should still
be annealing out point defects, so Kaveh and Wiser would
predict that A would increase. Furthermore, on annealing at
120K, the value of A increases. Since figa is decreasing
at these temperatures, Kaveh and Wiser would predict A
should decrease, in contrast to the result seen.

All of the above comparisons assume that the Gantmakher
and Kulesko formula, for electrons scattering off the local

phonon modes of dislocations, correctly describes the low

110
temperature resistivity data. This may be an incorrect

assumption.

3.1.4 The intermediate temperature range

Finally, one should note that the fits do show an
increase of the DT5 term, as one increases the number of
dislocations. This is consistent with the quenching of
phonon drag. The dash line in Figure 3.9 is the fit to Kh8f

with the T5 5

term subtracted out. The DT term gives a large
contribution to d fi)/dT near 1.5K. On examining the
coefficients from Table 3.3, one sees that D increases for
Kh8 as one adds more dislocations. The maximum value
obtained of 0.233 x 10-13Qcm/K5 is much less than the
theoretical value one would expect when phonon drag is
completely quenched. Ekin and Maxfield(88) predicted for
queched drag that D=0.354 x 10-1311 cm/KS, so the numbers
obtained for D are consistent with only partial quenching of
phonon drag. Sample KRbha shows a normal electron - phonon
term even before deformation. One would expect this, since
the added point defects would provide a way for the phonons
to lose their momentum. For sample Kh9, little reduction of

5

the DT term is seen (aside from Kh9d, which has a large

uncertainty for D) until one anneals at 120K (Kh9g). Thus

the largest effect on the DT5

term is due to dislocations.
This is reasonable, since the normal electron - phonon term

is predicted to be enhanced both by the anisotropy of the

 

lll
dislocations and by their shortening of the phonon mean free

path (see page 16).

3.2 Thermoelectric ratio results

From Chapter 1, the thermoelectric ratio can be written

as a sum of terms:

2

c = [ Go + B T + [C/T] exp(-e*/T) + 0x ](LO/L)

(3.5)
where Gx would be an additional term possibly due to
dislocations or the size effect. Figure 3.14 and
Figure 3.15 show the results of measurements of G on samples
Kh8 and Kh9. The turn up in these plots at ~3.5K is due to
the exponential term. For T<1.0K, the exponential term is
negligible, and L=Lo to within 3%, so a plot of G vs T2 as
shown in Figure 3.16 should give a straight line assuming
Gx=0' There is a slight deviation from T2 behavior at low
temperatures, which is most noticable in Kh9e.

Except for the low temperature deviation from T2, the
data can be fit by the above equation. For these fits,
L/LO was calculated using Equation 1.47. The value for F,

the coefficient for the T3

term in the thermal resistivity,
was obtained from a fit to the data of Newrock and

Maxfield(89).

—3 -6
F= (2.4154 x 10 ) — (.12966 x 10 )( [3300/ 704.2) cm/W
(3.6)

The resulting ratio L/LO was compared to the the

 

112

 

 

 

 

D
c?
O 1 fi 1 1 I W 1 7
a...“
o
a? fi‘of’frmi 0 Sample Q) (chm) Remarks
?" 60+ 6 0 + ‘
4, ‘9 D Kh8a 1.32 no deformation
:1 u ‘0 O 1(th 1.38 P=7.l" Hg
D W" " x x 49,. A 1(th 1.46 P-14.i"
1113 .0 ° 00 x . -+- 1(th 1.53 P-21.1"
q). 3* x Kh8e 1.81 P=50.1"
1 o 1(th 2.05 2434.3"
X
D
‘1‘
74
C)
"3
>1”
\
D
c?
N.
OI
D
Y
N.
l
D
‘3
N.
l
O
“3
m.
l
O
(.0
fl; 1 J J 1 l 1 1L J
'0 1 2 3 4
T (K)

Figure

3.14

The thermoelectric ratio for Kh8.

 

 

(l/V)

G

113

 

 

       

 

 

 

C
D
O. I T T T ' ‘7 T f
...-...“- .. Sample po (chm) Remarks
2 '0 O 0 Q . 1.:
=34 "- D Kh9a 1.09 no deformation .
‘ , ,.. ... " , o Kh9b 2.21 deformed at 4.2x, P=95" Hg
’;0 . . A Kh9c 2.20 annealed at 7.5x
o ‘”“ ‘x . +-Kh9d 1.86 annealed at 20x
“3 u. o x Kh9e 1.149 annealed at 60K
C.” .4» . y 3;. o Kh9f 1.39 annealed at 80K -
’1‘ * g f Kh9g 1.16 annealed at 120K
- " .. 4 X Kh9a 1.09 annealed at 165}:
c “ a .
“3
T“ “ .
O
‘9
T‘ a
O
c?
N.
I
O
".
N4 '1
I
D
‘3
N4 4
I
D
“3
ma 4
l
D
‘9
T1 .
C)
c?
7 1 1 l l 1 l L J
O T 2 3 4

Figure 3.15 The thermoelectric ratio for Kh9.

|.J

 

°-° 1 7 I

 

 

 

 

0 )1th

.... ._

‘po 0 O Q '—
”1.0 _._ kh9b

41—. —1

J l J
0.0 0.5 1.0
T2

Figure 3.16 G vs T:

 

115

experimental measurements performed at MSU, and this ratio
agreed with the data to within the experimental scatter of
the points, never deviating more than 3% from the data.

Equation 3.5 was fit to the G data of samples Kh8 and
Kh9. Some of the curves obtained are shown in Figure 3.14
and Figure 3.15. The resulting coefficients are listed in
Table 3.4. For these fits, no account was taken of the low
temperature deviation from T2(Gx=0), so it is not surprising
that the fits do poorly near 0.2K for the deformed samples.

First, let us examine the effects of dislocations on

Go' Figure 3.17 shows a modified Gorter - Nordheim plot for

 

sample Kh8. Adding more dislocations does increase the
magnitude of Go in proportion to l/ ’Oo’ per the Gorter -
Nordheim relation (see Equation 1.31). From sample Kh9
(Figure 3.15), one sees that the point defects, which
annealed out at 20K and 60K, make a significant contribution
to the diffusion term (Go). Thus the diffusion
thermoelectric ratio behaves much as expected.

Next the phonon drag terms will be examined. On
looking at the coefficients B and C listed in Table 3.4, one
sees that the dislocations added in sample Kh8 suppressed
both the normal and umklapp terms. Furthermore, the results
for Kh9 show little recovery of either B or C, outside
experimental error, until one starts annealing near 80K
(Kh9f). Thus one may conclude that dislocations effectively

suppress the phonon drag terms, and point defects have a

much smaller effect on them.

116

The coefficients from fits to the G data.

Table 3.4
sample Go
(l/v)
Kh8a .0641 i .012
Kh8b .182 i 010
Kh8c .287 i .010
Kth .336 i .009
Kh8e .539 i .007
Kh8f .656 i .008
Kh9a .123 i .006
Kh9h .877 i .006
Kh9c .879 i .007
Kh9d .795 i .008
Kh9e .513 i .009
Kh9f .435 i .008
Kh9g .238 i .010
Kh9h .153 i .009

B c
2 3

(l/V/K ) (10 l/V/K)
.419 1 .007 1.21 a .02
.257 a .005 1.05 a .03
.195 1 .004 .889 1 .031
.168 e .003 .801 a .023
.111 1 .003 .606 1 .030
.0848: .004 .512 1 .052
.409 i .003 1.18 1 .09
.127 1 .003 .656 1 .027
.126 a .003 .715 a .039
.125 i .004 .637 a .036
.149 1 .005 .726 1 .044
.167 1 .004 .776 i .037
.257 1 .004 .980 1 .024
.362 1 .005 1.15 1 .02

(K)

15.8
17.6
18.1

18.9
19.2

15.8
18.1
18.4
18.0
18.2
18.1
17.2
16.2

H-H-H-H-H-H-

H'H-H-MH-HH-H-

[_J
}_1
‘4

 

0 1 1 r
Kh8
A.
l> P.
v
0 —

 

 

 

 

1

I

Figure 3.1

\

Modified Sorter — Nordheim plot

118

One may ask how the suppression of the phonon drag
terms in the thermoelectric ratio relates to the enhanced
normal electron - phonon term in the resistivity. In the
resistivity case the phonon distibution is pulled into
thermal equilibrium through interactions with the
dislocations. Thus the rate of momentum loss for the
phonons is higher when dislocations are present. Since the
phonons are losing momentum more rapidly to the
dislocations, the electrons will receive less, so the
thermopower due to such interactions will be reduced.

A more detailed examination of the low temperature

deviation from T2 behavior is needed. At low temperature,

_ 2
G-GO+BT +6)( (3.7)
g§_= 2 B T + de
dT dT

Figure 3.18 is a plot of dG/dT vs T. This plot shows
roughly a linear behavior, with a non zero intercept. This
suggests that Gx may be linear in T. Since the intercept is
positive, the coefficient would be positive. The sample
Kh9a is undeformed and has an intercept on this graph only a
factor of two smaller than the intercept for the heavily
deformed samples. The reason the assumed linear term is
more noticable for high levels of dislocations is that the
T2 term is depressed, and the linear term is larger,

permitting easier observation of the linear term. No

theories have been discovered which predict a linear term in

 

119

 

   

 

 

 

0.4
I 1 1 Tf
« .1
:2 _
'2,
0.0— _-
LWF- _
‘O‘O .—
i— I —
-os__ —4
A
" . Kh8 f ‘
— ‘ Kh 9 O A _‘
-o.9 1 1 1 l I 1 1 1 1
0.0 0.5 1.0

T (K)

Figure 3.18 G/dT vs T

120
G. It is quite possible that this is only the low

temperature limit of a more complicated function.

3.3 The resistance minimum

The resistivity of samples Kh2e, Kh3, Kh4, and Kh7
shown in Figure 3.19, deviate from any previous published
results for potassium. All of these samples show a
resistance minimum near 0.2K. Many possible sources for
this minimum were examined.

The final conclusion was that a contaminant was
entering the potassium from the contact material. The
resistance minimum was seen when we used Handiwrap,
Parafilm, and polyethylene. All of these are hydrocarbons,
whereas, teflon, which does not cause a resistance minimum,
is made of a fluorocarbon. It was thought that possibly a
hydrogen compound or other impurity from the hydrocarbons
was diffusing into the sample. These materials all had been
carefully cleaned using high purity alcohol.

The reason for using a material other than teflon was
that teflon reacted with the potassium. The teflon reacts
with the surface of the potassium sample leaving a layer of
carbon. Since none of the hydrocarbons showed signs of
reacting with the Sample at room temperature, it was felt
they might be better to use.

In the end, we simply used teflon as our contact

material, even though a black layer was left on the sample,

 

121

 

'5 T 1 1 l I T 1 1 I
+, + + +
+ o o
_ +
A ‘ —
+ o ‘ . . . '

 

 

 

 

 

 

 

T
+ Kh 2e Hondiwrop
‘5 '_ Q . Kh3 Hondiwrop _-
0 Kb 4 Porofilm
‘ Kh 7 polyethylene
-.75 1 l 1 l i l l I l
O 0.5 1.0

T (K)

Figure 3.19 (1/T)(dp/dT) vs T for Kh2e, K113, 10114, and Kh7.

122
since the low temperature data agreed better with the
results of Lee et al.(4) when nothing was in contact with
the potassium. An interesting aside, is that many of the
previous groups had encapsulated their samples in
polyethylene, but they had not gone low enough in
temperature to see this effect.

There are several possible theoretical explanations of
the results. The most obvious is the Kondo effect(90) for
scattering off magnetic impurities. The Kondo effect should
produce roughly a -lnT term in [3. Figure 3.20 is a plot
which should give a straight line for such a theory. Within
the scatter in the data, a straight line is seen. Thus it
seems likely a magnetic impurity is entering the sample from
the plastics. To date, no attempt has been made to identify
the impurity.

It is of some interest to see the effect on G of the
mechanism which caused the resistance minimum. Figure 3.21
shows this data. The solid curves on this graph are drawn
to guide the reader. No measurement of G was made on sample
Kh7. The non-reproducibility of the curves is surprising.
If this were a Kondo system one would expect an anomolous
value for G, but not this large a variation among samples.
In both KhZe and Kh3, the phonon drag terms appear to be
supressed, but not for Kh4. Only sample Kh3 shows a local
minimum in G near 1.2K.

Additional measurements are needed to identify the

source of the resistance minimum, and the cause of the

.50

 

O

(pilcm/ K2)

:50

 

 

'7
.5o

 

 

12M

 

-O
I

D Kh2e Handiwrap
O Kh3 Handiwrap
A Khé Parafilm

~9-80

-l.20

.60

 

- 1‘

I
u

0 (UV)
-§.20 —g.eo v2.00

3.60

13.00

 

 

 

~‘JO

C
h

T (K)

Figure 3.21 G vs T, for the resistance minimum samples.

variation in G.

125

CHAPTER 4

Conclusions

This study has been an investigation of the effects of
plastic deformation on the electrical resistivity and
thermoelectric ratio of potassium from 70mK to 4.2K.
Plastic deformation does have a significant effect on both.
Plastic deformation increased both the residual
resistivity and the temperature dependence of resisitivity.
The residual resistivity on annealing showed the same
general recovery as in the earlier work of Gugan(lO).
Deformation enhanced the electron - phonon term from

4.2K to below 1K by causing a DT5

term to appear in [3 near
1K, possibly due to the quenching of phonon drag. The
coefficients obtained for D, were always less than its
theoretical maximum value computed by Ekin and Maxfield, for
completely quenched phonon drag.

Furthermore, a new term was seen ( L) ), which

e-dis
dominated the temperature dependent resistivity in deformed
samples below 1K. This new term was interpreted as being

126

127

caused by dislocations, in that it disappears when
dislocations are annealed from the sample. This new term
can be fit reasonably well with a model of Gantmakher and
Kulesko(44) for scattering of electrons off localized phonon
modes associated with the dislocation. Although the fits to
the data are quite good, it is difficult to understand why
the dislocations are dominated by a single vibration
frequency, as assumed in the fit.

The effects of dislocations on the electron - electron
term in the resistivity predicted by Kaveh and Wiser cannot
be easily tested, since fié-dis obscures the AT2 term.
Until fiQ-dis is more fully understood, it will remain
difficult to identify any variation in the electron -
electron term caused by dislocations.

Plastic deformation of potassium also has a large
effect on the thermoelectric ratio. Both the normal and
umklapp phonon drag terms are suppressed by dislocations and
are effected relatively little by point defects. In
addition, the magnitude of the diffusion term is increased
by both point defects and dislocations with the systematic
addition of dislocations giving agreement with the Gorter -
Nordheim relation.

Finally, a resistance minimum near 0.2K was seen for
potassium in contact with various plastics. The added term
to the resistivity followed a -ln(T) behavior, consistent
with a Kondo system. The thermoelectric ratio for these

samples was variable and followed no clear pattern.

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