’ ' “‘7‘wa ,3“ - 7;"
7 kltf'fl‘fig 5151777777“? ,7
"-71 .‘

7"”7'7557 ‘7" 7777-7777753;
'v‘.; . 2:1,

I |. . . I an!
77'I'7‘Iv7r7 ‘ ,I ‘ ¥ ' ..II.
.I ' 7 .4177“ I" f; ‘77}:‘I'7L77777:7417‘577777‘7“
'I‘J‘ 7 ' I J".."A. '

   
  

      
  

   

 
  
   

‘

'.'.I V . .- 7 0 I.‘.
.I_77 .',’7. "‘ 7'I‘LH" 7.77 577:7" " ‘ VLVV777777777:777 77:: 77.77
I "It'.

   
 

   

   
    
  

  

~ _
IIII

Y » l
-h ; 7777.777 -7

   

   
 
      
     
 
 
       
    
 
 
       

 

. 7'17'17777Qh77777'777 ' . V

'7" 7' .7": ' II '7-I77~::I7' '177'777 77 I ' 777.77 I-77 " .I.
j 3;}:- 5775.775»: ' .II V".'rVIV:* , 747:.- " 7..‘. ..'.. 77"'7-77777777'1775I"77V I
”'5 I} 77'7'I‘1I-II '73 1‘77. "7777*” Mr. .7""7 7 77‘777‘7.7_?"-".

 

  

IH' .I
I77 I,‘, l' )J 3 t .V V V V
@511: In,’m‘:‘II-""I“r. 1' 75I' 7‘7III "-::"I. ' 'I.‘."II'I'I"V
'I'I ' .'..-.-"-'..II ‘-"I “II -"|777' '7I77I‘fv; '7' «VI-95g JJ'I~5I'I
T"V"V‘V\VV. I,I- IH.'I.‘.77"V t7"77777 I

I
"I .. IVV
I" ,,"VV VIV‘VVIV A.
I V.

II. ,
J 7 I '7' 7771;! O7 7“ 71’7“ '1: III
. II‘JI. 7l"77" v.3“ I
7} .7I77.V7.‘\'V.V.‘.,.' ;, 7‘," '75 "If’wn 7'3“” 51‘."le
'II7 ;I ‘ IV,“"I .7: ‘I: 773-77" I,' I7 ‘7".55'77755'7.‘ $777:
I”, " 7777“. 50" III ’1‘ If I' '
I VVVVV 7". . I 'VVIVV ".'V. 7'7JI7I7 AIV H
. '77-‘77177777IV7III7717"
77:75'7 VrV,II'.VfVV

. V. .5”, V.‘:' I (II. “77'”- ' ”SW"
577(17.'777 77E7‘ VV‘IV'7.:\::'I.‘I777-5I',,I71.?77'7m7777,‘7I,77‘|‘o771'77\I-7IV777717:):I77r777';
I
l
:'

 
   

    

".' 1’1," '40:.) V
777777.75.. 7‘7" 7'74?“ 7'77II7V‘7I
-"I "5.137 I 7.7.
7.7-7'7’---- ' "'II'I"iII V3.17"
@7777 ' :1 I7.

   
 
 
 

7775',” ‘577"'| ,VfVVTJ

9.75.
'77; A7.
I... VIIIWI

7777:7777777 V'B'J
. VAV'

 
   
   
 
 
  
    
   
  

 

t I

  
  
 

a

   
 
     

VVVVV N'I'Ih’fl'l 7"! -777|V777"V7’V V .- I. V
.VVVVVVVV-VVV .' 7VV,'I'VVVVVV\VVI‘V\ 7;:‘1; . VV V , I577",'\, .. f 7'»
77 77%277757'71‘777777 . *hl'fi. .17! 7 '57 I;-','7I77 b .77:7"77 Irpfl‘fl " 7I7:; I'lgn'c77' '0V177 #:‘j-‘a' '7:)II7I\'777‘V l’», I7-.-:
I7V75V7777 ”7 n l7. 7‘777'7I'Y7777 7'I_V777| 7“; VIVJVVE :‘7V7VV'7J7VVV-I7 7\'V7 VV.VVV 77%?V: I7 1'. .V I} VVV7VV7'9' VV '77'77‘7I I link?! , 'I. 777.7'77775'1'Nl77 7 5" Witfi'f“ 777771;,7'UVV' TGI'VVV

I' *if. Ir” 7777747777“ n. I0 NHL '7I.7.7.7 37,7 7.7-.',I_7:' iVVI'V'” 777.77 . ,I. .3377! ’VVI'J'VUVIV 7.77;?“ VVVJVVVIVV 7. I

"=13 -'I‘ l"'I’, ’ "In ’4“ ""- Irvt'

  
 
   
  

   

.(IVI
ll’ '\'

777'7VVlfif-II1fl-Vth-I7'I I _0
' I‘.'7 .ngfI 7

  

'. 77:17.7 V':7V 71-07;;

l7.7 .'7|
I

    
    
  
 
    
    
  
 
  

   

     

 

_ 777:7“: VV 7 “7";7771'70
I:‘- 't‘. I. WII’IIII'IIIIII III-III I3

'I-I‘77'7'5‘IVL‘ I..'Jfl_ IV'VLJ

    

.' '\

q. '-
VVVV l7 7‘77777757 I777777V77H 751777 _V
'- 1|" . . ' . \~ J7 'r.-,' . . A.-
77‘II7 17! I7 . ‘|,-,t,'|I \" II“; M. n l7u7ht$717lI1VD~h 'VV’KV-VIV. V7; 7“".17fl7 “if
:7l-77:/'77'77 ‘I V ‘7‘:Ih7717"‘V VVVII IIVI. 1' 3.. I77 I'77 V 7'7.’ .'.."7'7 :")VI7 '7'. l. IHIVV,~VV\fIVVI .7.
7?:9 fi7§|,"'..£.h .nI ‘.. I I 7"]!i'77 . .77 .. . V V V V7Y’V7 I 77: \'

   

I .. 4 I‘ ..

, I 44.]. a. I . .. .I. ,..-,~ . ... :.I.',"' I V_,j"-,I'I II‘ V.. 0,
II I 'fi‘ 4|" V . V . H. V .I,-. II I ‘ ‘7‘7f VII, ‘.. '7""'II"7';' I 7. V 577,7 7]:'I‘ 7:7V'I'II'I'I. (1,7. V‘V'VV'IVI'7 -7
'7. INN" I'" II - “ V§7'7.. [- I 7 ‘ ( ’I' ‘ ' .. a" I'I 'I I 'll 777777, v.7.7“ “y'-
-" I‘M «I. II“. II J .' mm» ‘l".';'~f'.'.‘. 'II-:-- ':'-;':I UIIIIII - ,5". I F v:

'5‘: " 7f77 ,7‘. .I 77'57 V VIVVI“: I7l7'7777777V ' ,_,' y, -I‘75'7 ‘ 7II I .‘1. 77>7 7V\"7I ,7 "'7” ' 777I'V'7V7' . ..

' I' l t' I' V‘ ' .4 .--I ' II' . l I

"'ch ' “7"I"'7 'I7' 7 7 '77 7 7 ”I"? 7])7 "7'7 7.7777717 ' 7I'V'7'V777l777 V7777'I~7 . '7'7I7 7'75“: 30:2: I kI' ’7'.”

.‘7I .w'"
VII ”'17,: Rh! h IH,”V.’ ”“7 H 7I77|:'7\ [I7 ‘I"7n V .I, 7_IVI
V |I|.|| , l" .n’ ”V [7' \‘V7V77‘. 7777.7:th Vic". I" 777“.” “77': 777:7'7777 VV It I‘
I-"IIIIJ “NIH ' . .. {527755 ,I’W WI: I-r-II‘III.III .

        
 
    
 
 
    

  

V0{l77 I. 7' ' 7V7" ‘. ,:V t)7' 7,7 7|. V, V . V777” ';' Vu. - V
.. I. ulV I v V .V1 7' I.7I‘ ,VI V V l-ID
77I7 77777 77 "'.‘I. I‘m" ""~‘II7' II‘VO '7'7',7"7' ~,I, 7'7'!""“""" ' 7777;7I'I7-III7I7I7 17.577 .H'I'i 3 7777775' . l JV-r..:7I77 . 'II’IzI'
7 57775777777717 5'7 |I7 77: 7|"77 77III'7I 77:17”; 7757 :*7' "' 77 7 |7V7J7T7777M 7'5““! "vhf: 77l7 .‘I 7" 7I'I'7 ‘77”‘77‘ I 'V .‘ 7' [ 7
77 £777'"777\7 ”7777 I 7'7‘ “hp. 7 77I.'7I'H'I 7' 775'7 ". ‘715”I.I .“I7"1'| 7, ,V I77: ’I I7 .
, , . II.I'.I.7V . m; .-'f{I‘.*'-..I‘ WI '2. 7737.7 II =:‘ :i'i'. I I: III'I'*':I '- II' : ... .
77 7‘“ 77 ‘.'""" "V': -:1 ~ “I ,.I' 'I . I. ‘I‘. III :II',.;:‘.'I‘:" I.
.- 7| .‘.- *1in I n. 7'7‘;{7777V I t' VV7'I7VV'}:VJ'.I7 77IIV7,7‘i7 Vy'VHJV'V .'.' . . I . 7
'prll' ".IIIV ‘7 .771” ‘VVV .7I7 I' '7III‘I'7V . ‘ 7.'.7.7 ' ' I
I II'

vet-0‘

 
 
 
     

  

LIBRAR Y
Michigan Sm:
Univcnicy

This is to certify that the

thesis entitled

The Electrical Resistivity and the Thermoelectric
Ratio of Potassium and Dilute Potassium-Rubidium
Alloys Below lK

presented by

Chi-Wai Lee

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics

has

Major professor

 

 

Date August 1, 1980

 

0-7639

OVERDUE FINES:
\

25¢ per day per item
fu‘é‘ékaL A

RETURNING LIBRARY MATERIALS:
Place in book returnt to remove
charge from circulation records

 

 

 

THE ELECTRICAL RESISTIVITY AND THE THERMOELECTRIC
RATIO OF POTASSIUM AND DILUTE POTASSIUM-RUBIDIUM

ALLOYS BELOW 1 K

By

Chi-Wai Lee

A DISSERTATION

Submitted to

Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1980

(.27 // (L762) “U

ABSTRACT

THE ELECTRICAL RESISTIVITY AND THE THERMOELECTRIC
RATIO OF POTASSIUM AND DILUTE POTASSIUM-RUBIDIUM

ALLOYS BELOW 1 K
By
Chi-Wai Lee

We have measured the electrical resistivity of K and
dilute K-Rb Alloys with a new high precision technique
(0.1 ppm) and the thermoelectric ratio of the same
specimens over the temperature range from “.2 K to 0.07 K.
Measurements were carried out on free hanging, polycrystal-
line specimens with emphasis on data below 1 K, especially
for the resistivity. Our resistivity data above 1 K are
comparable to previously published results by others.
Below 1 K, the resistivity due to electron-phonon scat-
tering becomes at least three orders of magnitude smaller
than the total temperature dependent resistivity, therefore
our results are not affected by the phonon effect. For
the pure K, we have measured one 1.5 mm diameter sample

and three 3 mm diameter samples. For the K-Rb alloys, we

Chi-W31 Lee

have measured five 3 mm diameter samples of nominal atomic
concentrations 2.2A%, 0.83%, 0.32%, 0.13% and 0.05%.

For the pure K, our resistivity data below 1 K for the

2

3 mm samples show a possible T dependence from 1.0 K to

0.4 K, but there is complicated behavior below 0.“ K.

In the temperature range from 0.08 to 0.42 K, best fits

1.910.03

with T were observed, and below 0.2 K, possible

T2 fits were again found. For the 1.5 mm sample, the

temperature dependence was found to be lower than T2

and higher than T from 1 K down to at least 0.2 K. Below

2 dependence. The

2

0.2 K, the data are consistent with a T
coefficients determined from the possible T terms for the
pure samples range from (2.57:0.5) x 10-13 t0 (3.2:0-3)

x 10"13 Q-cm/K2. These T2 terms are consistent with the
theory of simple isotropic e-e scattering, i.e., they are
of the right magnitude and roughly sample independent.
However, there are presently no satisfactory explanations
for the apparent Tl'9 dependence for the 3 mm samples and

the different behaviors of the 1.5 mm sample and the 3 mm

samples above 0.2 K.

2

For the K-Rb alloys, a term proportional to T and

to the residual resistivity is positively identified
below 1.3 K. This is consistent with the resistivity due
to inelastic electron—impurity scattering as predicted

by Koshino-Taylor. We obtained, below 1.3 K, the re-
sistivity p = p + (8.510.26) x 10-6 p0 T2 + (2.15:0.32)

o
x 10-13 T2 Q-cm. The term (8.5:0.26) x 10'6 00 T2 is

Chi-Wei Lee

best described by the theory of inelastic electron-im—
purity scattering. The coefficient is close to the theo-
retical value. We believe that this is the first time
the Koshino-Taylor term has unambiguously been observed.
The other T2 term (2.15:0.32) x 10"13 T2 Q-cm can be com-
pared with the results of the pure samples for which a
term (2.7:0.27) x 10—13 T2 Q-cm was found. This T2 term,
which is independent Of p0, could be associated with e—e
scattering.

The measurements of the thermoelectric ratio of K and
K-Rb alloys were made with higher precision than that
which has been attained before. we were able to unam-
biguously separate the diffusion component, the normal
phonon drag component and the umklapp phonon drag component.
For the pure samples, the thermoelectric ratio was found
to be of the form G = G0 + AT2 + % e'e*/T. From the data
up to 4.2 K, we obtained for the umklapp phonon drag
6* = 23:2 K. From the data below 1.0 K, we obtained the

1 and the normal phonon

drag component A = -0.30:0.01 V'lK’Z. These results are

diffusion term G0 = —0.03:0.03 V-

consistent with the results of MacDonald 33 al. For the K-
Rb alloys, we found the positive diffusion thermoelectric

ratio characteristic of Rb impurities in K to be G0 =

Rb in K
+ 0.A8 i 0.01 V”1 from a Gorter-Nordheim plot. The normal

phonon drag is quenched slightly and is no longer ac-

2

curately a T dependence. The normal and umklapp phonon

Chi-wai Lee

drag terms are quenched more and more as the impurity

concentration increases, similar to the data of Guénault

and MacDonald.

TO MY PARENTS

ii

ACKNOWLEDGMENTS

It is a great pleasure to acknowledge my thesis ad-
visor, Professor Jack Bass, whose guidance, support and
criticism at all stages of this research were invaluable.
I am indebted to Professor W. P. Pratt for his advice and
help in the construction of the dilution refrigerator
system, especially for his construction of the dilution
unit and the direct current comparator and the recalibra-
tion of the thermometer. I would also like to thank
Professor Peter Schroeder for his advice and aid in carry-
ing out this research. I wish also to thank Dr. John
Rowlands for his helpful advice and ideas in the design
and handling of potassium. Thanks are also due to Pro-
fessor Robin Fletcher for suggestions involving the glove
box. The time donated by Mark Haerle in helping to make
measurements is gratefully acknowledged.

Specific thanks go to Dr. Gopal Kote, Vernon Heinen,
Brent Blumenstock and Henry Elzinga for their help in
various stages of the construction of the dilution re-
frigerator system. Thanks are also due to Dan Edmunds
for his help in constructing various items of electronic
equipment, especially the current comparator, and to Boyd

Schumaker for preparing the Ag-Au alloy and Ag thermal

iii

links, as well as his help in various other ways. I
would like to thank all the guys of the machine shop for
their help in constructing the various pieces of appara-
tus.

I would also like to acknowledge the financial support

of the National Science Foundation.

iv

Chapter

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . .

LIST OF FIGURES. . . . . . . . . . . .

I.

II.

INTRODUCTION . . . . . . . . . . .

1.

l.

2

2.

1.

Basic Electronic Transport
Properties of Metals . . . . . . .

1.1.1. Electrical Resistivity

'0

1.1.2. Thermal Conductivity K
1.1.3. Thermoelectric Power S

Previous Work on K and K-Rb. . . .

1.2.1. Resistivity. . . . . . .
1.2.2. Thermopower. . . . . . .
Present Thesis . . . . . . . . .

1.3.1. Electrical Resistivity .

1.3.2. Thermoelectric Ratio G

EXPERIMENTAL TECHNIQUES . . . . . . .

.l.

2.

Dilution Refrigerator. . . . . .

High Precision (0.1 ppm) Resistance

Bridge . . . . . . . . . . . . .

2.2.1. SQUID (Superconducting
Quantum Interference
Device . . . . . . . . . .

2.2.2. Current Comparator
Resistance Bridge. .

Page

.Viii

\O\OO\U'1

13
15

17

3A

Chapter Page

2.3. Measurement Methods. . . . . . . . . . . . 36
2.3.1. Resistivity. . . . . . . . . . . . 36
2.3.2. Thermoelectric Ratio G . . . . . . A2
2.3.3. Thermometry. . . . . . . . . . . . uu

2.A. Sample Preparation . . . . . . . . . . . . A5

2.5. Error Analysis . . . . . . . . . . . . . . 58
2.5.1. Temperature Determination. . . . . 58
2.5.2. Heat Loss. . . . . . . . . . . . . 61

2.5.3. Thermal e.m.f. and Johnson
NOise. O O O O O O O O O O O O O O 6“

2.5.“. Uncertainty in p0 and Sample
Contraction. . . . . . . . . . . 65

III. THEORY . . . . . . . . . . . . . . . . . . . . 69
3.1. Basic Transport Theory . . . . . . . . . . 69
3.2. Resistivity p. . . . . . . . . . . . . . . 76

3.2.1. Resistivity due to Electron-
Phonon Scattering. . . . . . . . . 79

3.2.2. Resistivity due to Electron-
Electron Scattering. . . . . . . . 80

3.2.3. Resistivity due to Electron-
Impurity Scattering. . . . . . . . 82

3.3. Thermopower S. . . . . . . . . . . . . . . 83
3.3.1. Diffusion Thermopower. . . . . . . 83

3.3.2. Phonon Drag. . . . . . . . . . . . 85
3.3.3. Impurity . . . . . . . . . . . . . 88
3.“. Thermoelectric Ratio G . . . . . . . . . . 90

3.5. The Resistivity of K . . . . . . . . . . . 93

vi

Chapter Page
3.5.1. The Resistivity of K Above
105 K. o o o o o o I o o o o o o 9!"

3.5.2. The Resistivity of K Below
105 K. o o o o o o o o o o o o o 97

3.6. The Resistivity of K-Rb. . . . . . . . . . 101
3.7. The Thermopower of Pure K. . . . . . . . . 105
3.8. The Thermopower of K-Rb. . . . . . . . . . 109
IV. THE EXPERIMENTS AND RESULTS . . . . . . . . . . 114
4.1. Resistivity for Pure K . . . . . . . . . . 114
4.2. Resistivity for K-Rb . . . . . . . . . . . 133
4.3. Thermoelectric Ratio for Pure K. . . . . . 145
4.4. Thermoelectric Ratio for K-Rb. . . . . . . 151
4.5. Discussion and Summary . . . . . . . . . . 158
4.5.1. Resistivity for Pure K . . . . . . 158

a. Isotropic e-e Scattering . . . . . 159

b. Anisotropic e-e Scattering . . . . 160

c. Electron-Phason Scattering . . . . 162

d. Conclusion . . . . . . . . . . . . 165

4.5.2. Resistivity for K-Rb Alloys. . . . 166

4.5.3. Thermoelectric Ratio for
K and K-Rb . . . . . . . . . . . . 166

REFERENCES . . . . . . . . . . . . . . . . . . . . . 169

APPENDIX A . . . . . . . . . . . . . . . . . . . . . 173

vii

Table

2-3

2-4

LIST OF TABLES

Page

Characteristics of the samples.

(a) Pure potassium samples;

(b) K-Rb alloys;

(c) Samples of Guenault and

MacDonald(23) (for comparison). . . . . . . 54
Parameters of the alkali metals

(after Ziman(l)). . . . . . . . . . . 57
Transition temperatures of the

superconducting fixed point

devices used to recalibrate

the thermometer and the cor-

responding temperature indicated

by the thermometer. . . . . . . . . . . . . 59
The thermal conductance of the Ag-Au

wire, the Nylon support and the

stainless steel tube at 4 K and

1 K and the ratios to the thermal

conductance of the Ag-Au wire . . . . . . . .63

The values of AV/IR at tempera-

tures 0.1 K, 0.5 K, 1.0 and 3.0 K

viii

 

Table Page

2-5 and the values of Vn/IR at 1 K

for K-Rb 2.24%, K-Rb 0.83% and

Pure sample K 1 . . . . . . . . . . . . . . 66
4-1 The ratios of AT from the Wiede-

mann-Franz law to AT' from direct

measurements for the pure sample

K1, K2 and K4 . . . . . . . . . . . . . . . 116
4-2 The ratios of AT from the Wiede-

mann-Franz law to AT' from direct

measurements for the K—Rb 0.13%,

0.32%, 0.83% and 2.24% alloys . . . . . . . 135
4-3 The parameters from the best fits

to the term BTe-e*/T for the K—Rb

0.32%, 0.83%, 2.24% samples . . . . . . . . 138
4-4 The coefficients A of the T2 terms

for the K-Rb alloys and pure K sample

K4. . . . . . . . . . . . . . . . . . . . . 139
4-5 Comparison between results of our

data and the results of MacDonald,

_t _l.(22) for the pure K and of

Guenault and MacDonald<23> for the

K-Rb alloys, assuming

8' = GLOT

2 C -e*/T
(GO+AT +Te )LOT

A'T+B'T3+C'e'9*/T. . . . . . . . . . 150

ix

Table Page

4-6 The diffusion components GO
of the pure samples K1 and K2

and the K—Rb alloys 0 o o o o o o o o o o 0152

 

Figure

LIST OF FIGURES

Page

Conventional low temperature measure-

ment methods for resistivity and thermo-
power. (a) Resistivity measurement;

(b) Thermopower measurement . . . . . . . 3

“He solutions.

Phase diagram of 3He-
(After Radebaugh, NBS Technical

Note 362, 1967). . . . . . . . . . . . . . 20
The principal parts of a conven-

tional dilution refrigerator: 1.

Condenser, 2. Main impedance, 3. Still,

4. Still heat-exchanger, 5. Discrete

heat exchangers, 6. Phase boundary,

7. Mixing chamber, 8. 1.3K “He pot, 9.
Orifice, 10. Dilute phase, 11. Contin-

uous heat exchanger, 12. Concentrated

phase. . . . . . . . . . . . . . . . . . . 22
Bellow assembly for vibration isola—

tion . . . . . . . . . . . . . . . . . . . 27
Schematic diagram of the dilution

refrigerator gas handling system . . . . . 28

xi

Figure Page

2-5 The rms value of the minimum

detectable voltage as a function

of Rx for four different values of

Tx’ assuming that Af = 1 Hz, T6 =

4.2K, and Re = 12 mg. The limiting

behavior, <Ui(min)>l/2 = [4kAf Te Ri/Rell/2

is shown by the dashed line. (After
Lounasmaa<30)) . . . . . . . . . . . . . . 33
2-6 Simplified block diagram of resistance

bridge. Inside the region with a

bold outline, the components are at

4K or below and all the wiring is

superconducting. (After Edmunds,

gt a1.(38).) . . . . . . . . . . . . . . . 35
2-7 Schematic diagram of the set up of

specimens for resistivity measure-

ment using the temperature modulation

technique.G heaters are included for G

measurements . . . . . . . . . . . . . . . 38
2-8 Calibration curve for thermometer

Th2 (GR-200A-30) . . . . . . . . . . . . . 46
2-9 Potassium sample holder: 1. Copper

well for attachment to mixing chamber,
2. G heater electrical leads, 3. Ag

thermal link, 4. Current and potential

xii

Figure

2-9

2-10
2-11
3-1

Page

leads, 5. Brass flange, 6. 1/8"
stainless steel tube, 7. Nylon sup-
port, 8. OFHC copper pieces, 9. Po-

tassium samples, 10. G heater, 11. P0-

tassium cold welded to copper, 12. Super-

conducting leads . . . . . . . . . . . . 48
Dry box gas handling system. . . . . . . 50
Sample press . . . . . . . . . . . . . . 52

Schematic diagrams of the electron-

phonon normal and umklapp processes for
spherical Fermi surfaces that do not

touch the lst Brillouin zone boundary. . 81
Plot of resistivity vs T for the

data of sample K2C of Rowlands §£.e1-(lu)

The fitting curve with J4 is shown.

In the inset, extrapolations of

Tl°5, J2, J“, J5 fits to lower tem-

peratures are shown. (After Bishop

and Overhauser(15).) . . . . . . . . . . 102

121.0. .1.
Plots of log (p dT) vs T for sample

Kl. Similar results extracted from

data of van Kempen gt al.(l3) and

of Ekin and Maxfie1d(11) are also

shown for comparison . . . . . . . . . . 117

xiii

Figure

4-3

4-4

4-5

4-6

4-7

4-8

Page

Plots of % %% vs T below 1.6K for

the two runs on K1 (Kla and Klb) . . . . 119
Plots of l g% vs T below 1K for
0
sample Kl, showing both data using
measured AT and using AT from the

Wiedemann-Franz law. Open squares are

data obtained from differentiating

C(T) below 0.25K . . . . . . . . . . . . 120

Plots ofll g9 vs T below 1K for
p dT
sample K2, showing both data using

measured AT and using AT from the
Wiedemann-Franz law. Open squares

are data obtained from differentiating

C(T) below 0.25K . . . . . . . . . . . . 121
Plots of C vs T and T2 below 0.25K
for sample K1 (1.5 mm) . . . . . . . . . 123
Plots of C vs T and T2 below 0.25K

for sample K2 (3mm). . . . . . . . . . . 124

2

Plots of C vs T and T below 0.45K

for sample K3 (3 mm) showing both
data using old temperature calibra-
tion (.) and new temperature cali-

bration (o). . . . . . . . . . . . . . . 125

2

Plots of C vs T for sample K3 (3 mm)

showing three sets of measurements . . . 126

xiv

Figure

u-9

4-10

4-11

4-12

4-13

4-14

4-15

Page

Plots of C vs T and T2 below 0.25K

for sample K3 (3 mm), showing two

sets of measurements . . . . . . . . . . . 128

Plots of C vs T2, Tl'g, Tl'8 below

0.45K for sample K3. . . . . . . . . . . . 129

Plots of C vs T2, Tl'9, T below

0.45K for sample K4. . . . . . . . . . . . 130

1 do
Plots of E'dT vs T below 1K for

samples K2, K3 and K4. Open

symbols are data obtained from

differentiating C(T) . . . . . . . . . . . 131
1.99 .1. _

Plots of log (p dT) vs T for K Rb

2.24% and 0.32% samples. .’. . . . . . . . 136

The fitting curve for % %% data of

K-Rb 0.32%, assuming p = pO + AT2

- i
e 6 /T. Shown in the three

+ BT
temperature ranges: (a) 0 to 4.2K,

(b) o to 2.5K, (c) o to 1K.

6* = 19.6K . . . . . . . . . . . . . . . . 140
Plots of %'%% vs T for T < 1.3K for

K-Rb alloys 2.24%, 0.83%, 0.32%, 0.13%

and 0.05%. Typical error bars are

shown for the 0.13% sample . . . . . . . . 141

XV

Figure Page

4—16 The coefficient A of the T2 term
plotted against po for the K-Rb

alloys 0 O 0 O 0 O 0 O O O O O O O O O O O 1182

4-17 The slope M ( = EA) plotted against

1 0

po 0 O O O O I O O I C O O O O O O O O O O 1“”
4-18 G data for samples K1 and K2 and

the K—Rb alloys below 4.2K . . . . . . . . 146
4-19 The fitting curve for the G data

of sample K1, assuming G = G0 + AT2 +

_ i
g e 9 /T with 6* = 23i2K . . . . . . . . . 1u7
4-20 Plots of G vs T and T2 for samples

K1 and K2 below 1.2K . . . . . . . . . . . 148
4-21 G data below 4.2K for the K-Rb

alloys . . . . . . . . . . . . . . . . . . 153
4—22 Plots of G vs T below 1.2K for the

K-Rb alloys. . . . . . . . . . . . . . . . 154
4-23 Plots of G vs T and T2 below 1K

for the K-Rb 2.24%, 0.83%, 0.32%

and 0.13% alloys and pure sample K1. . . . 155

4-24 Gorter-Nordheim plot: G0 vs 5;
1 0
G0 = +Oou8i0001 (V) o o o o o o o o o 156
Rb in K
4-25 The fitting curve for % g%~data of K1
below 1K, assuming p = p0 + AT2 J2 (e¢/T),
With 6 = uoBSKo o o o o o o o o o o o o o 163

¢

xvi

Figure

4-26

A-3

Best fit to % %9 data of K1 below
1K, assuming p = p0 + AT2 J2 (6¢/T),
With e¢ = 605K 0 o o o o o o o o

The principal types of Josephson
Junction. a) Tunnel Junction. b)
Normal or semiconductor barrier Junc—
tion. c) Dayem bridge. d) Adjustable
point-contact. e) Solder drop 'SLUG'.
f) Crossed wire or hair pin Junction
(After Giffard et al., Prog. Quan.
Electr., 4, 301 (1976)).

a) The Josephson Junction in a
superconducting loop. b) Single
Junction, rf biased symmetric SQUID
of Zimmerman, Thiene and Harding<36)
¢/¢O as a function of ¢ext/¢o in

three cases: 211LJc = ¢ 3¢O and

o’
5¢O. Vertical lines with arrows
correspond to discontinuous changes

of one flux quantum in the fluxoid ¢' a
and to somewhat smaller changes in the
flux ¢. (After Lounasmaa<30)) . . . .
The staircase pattern. The

voltage Urf across the tuned

L C -circuit is shown as a

rf rf

xvii

Page

164

175

176

179

Figure

Page

function of the rf level. At

the plateaus the quantity plotted

is Urf(maX), the voltage Just before

a dissipative transition occurs.

(After Lounasmaa<3o)). . . . . . . . . . . 180
The "triangle" pattern. The
maximum voltage Urf(maX) vs

¢dc for different values of the
rf level. (After Lounasmaa<30)) . . . . . 182

Typical rf SQUID electronics . . . . . . . 183

xviii

I. INTRODUCTION

In this introduction, we first define the basic elec-
tronic transport properties of metals and then review pre-
vious work on K and K-Rb. We will indicate why we were
interested in measurements on K and K-Rb at very low tem-
peratures. Then the work in the present thesis is briefly
described. In Chapter 2, the experimental techniques
relevant to this work are described. In Chapter 3, the
basic transport theory applicable to potassium and K-Rb
alloys is given. The experimental results are presented

and interpreted in Chapter 4.

1.1. Basic Electronic Transport Properties of Metals

 

The electronic transport properties of a cubic metal
are completely determined from measurements of the electrical
resistivity, o, thermal conductivity K and thermoelectric
power S. Study of the temperature dependence of these
properties down to very low temperatures provides informa-
tion about the dominant scattering mechanisms of the
electrons in metals.

For a cubic metal, the macroscopic electronic transport

equations are given by

+ +
E = pJ + SVT (l-la)
6 = %§.E - K$T., (l-lb)

where E is the electric field, 0 the heat current density
and RT the temperature gradient applied to the metal. In
the following, we will define p, K, S as well as a new
quantity G, the thermoelectric ratio, based on these equa-
tions. We will only discuss these properties at low temp—

erature (less than 6D/30, where 6 is the Debye temperature).

D

1.1.1. Electrical Resistivity p

 

The electrical resistivity p is defined as

p =,% (1-2)

where E and J are in the same direction. Low temperature
electrical resistivity is best measured with a potentiom-
etric method. A conventional method is shown in Figure 1-la.
The specimen at temperature T is measured in a 4-termina1
configuration. A known current Ix is passed through the
specimen (through the current leads), and is nulled using

a very sensitive null-detector, by the voltage V across a

 
 
 

 

1 thermal link to cold sink at
(-———— controlled temperature

Fir—'4— Ir

 

reference
resistor at 4 K

 

 

 

 

l..__’_
I'
X
‘9
n

 

superconductinc leads

 

 

 

 

 

/ < .,
Tc ——
reference
AT Rf resistor
ll"—‘
é #
heater

b. Thermopower = - 33L
AT

Figure 1-1. Conventional low temperature measurement
methods for resistivity and thermopower.
(a) Resistivity measurement; (b) Thermo-
power measurement.

reference resistor produced by current Ir' The reference
resistor is normally held at constant temperature (e.g.,

4.2 K). p(T) is then given by

Ir

=-I—)-{-RT

pn>
pm>

m_1. _
om-I , (13)

X

where A is the cross-sectional area of the sample and A is
the separation between the potential leads.
At temperatures much below the Debye temperature,the

resistivity, according to basic transport theory, is ex-

(1)

pected to have the form

_ N U ,
C(T) - D0 + pe-e + pe-ph + pe-ph ' (1-4)

p0 is the residual resistivity caused by elastic scattering
of electrons by residual impurities or lattice defects in
the metal. The resistivity is said to be in the "dirty"
limit(2) when 90 is much greater than the other terms.

pe-e is attributed to electron-electron scattering and is
2

expected to have a temperature dependence of the form AT ,

where the coefficient A is essentially independent of p0(3).

N

pe-ph is due to electron-phonon scattering normal processes

(see Section 3.2 for a discussion of normal and umklapp
processes). From Bloch-Grfineisen theory, 02-ph = BT5.

In general, we have 82-ph m Tn, with n = 5. The last term
pg_ph is attributed to electron-phonon umklapp processes,
which in a metal such as K, with a spherical Fermi surface
completely within the first Brillouin zone and not touch-
(1)

ing the boundary, is expected to have the form

U -8*/T
'pe-ph « e .

where 6* is a constant. This term diminishes much faster
than the p§_ph indicated above and thus would normally be

expected to be negligible at sufficiently low temperatures.

1.1.2. Thermal Conductivity K

 

The thermal conductivity K due to electronic conduc-

tion is defined as

6'9
K = .. . (l-S)
VT E=0

(l)

K is related to p by the Wiedemann-Franz law

59 = L . (1-6)

When elastic scattering of electrons is dominant,

2 2
L = L = Il_3_ k = 2.145 X 10-8 VZK-2 0
L0 is known as the ideal Lorentz number.

1.1.3. Thermoelectric Power S

Thermoelectric power or thermopower S is associated
with the Seebeck effect, which is one of the three basic
thermoelectric effects: Seebeck, Peltier, and Thomson
effects. These three effects are related through the Kelvin-
(l)

Onsager relations and thus can all be determined when
one is known.

The thermopower S is defined as

.12

S = VT (1‘7)

+
J=O

When a temperature gradient AT/AX is established across a
metal, a thermoelectric e.m.f. AV appears between the hot
and the cold ends of the metal. For low temperature measure-
ments, the specimen is best measured in a 4—terminal con-

figuration as shown in Figure 1-1b. The thermal e.m.f. AV

across the potential leads is nulled by a nulling voltage,
such as that across a reference resistor, using a null
detector. The leads to the potential probes are super-
conducting wires so that there is no thermal e.m.f. due to
the leads(u). The temperatures at the hot and cold ends
are measured with thermometers,which allow the determina-

tion of AT. S(T) is then given by

__E_ -_-_9_‘l _
S‘VT AT ° (18)

(5)

At low temperatures, S(T) is expressed as

_ N U
where Sdiff is due to electron diffusion, s: is the normal

U
8

thermopower. Sdiff is expected to have the form

is the Umklapp phonon drag
(5)

phonon drag thermopower and S

diff = AT , (1-10)

and

s = BT3 , (1-11)

whereas for K
(1-12)

Another quantity associated with S, the thermoelectric
ratio G, which was first introduced by Garland and Van-
Harlingen(6), is defined as

G = 4 . <1-13)
Q

where J is the current density through the sample needed

to null the thermal e.m.f. caused by the heat current den-

sity Q. G is easier to measure than S in that there is no
need to determine AT and J and Q can be measured accurately.

In general,

S = GLT and

QIU)
l

- Kp. (1-14)

When elastic scattering is dominant, L = L0 and

U)
ll

so that determination of G implies determination of S.
At low temperatures, when elastic scattering predom—
inates, G is expected to have the form
n
-6 /T

2 Be (1-15)

2 term is

where G0 is due to the electron diffusion, the T
the normal phonon drag term and the third term is the um-
klapp phonon drag term appropriate to K.

In the next section, a review is given of previous work

on p and S for K, which shows that p deviates from the simple

temperature dependence indicated above, whereas S does not.

1.2. Previous Work on K and K-Rb

1.2.1. Resistivity

 

In the past decade, there has been a lot of interest
in the study of the resistivity of pure potassium. K is
a simple monovalent metal with cubic symmetry and a nearly
spherical Fermi surface (to within 0.1% according to deA
(7)).

measurements It is studied in preference to other
alkali metals because it has several advantages. Li and Na
undergo martenistic transitions (from b.c.c. to h.c.p.
structure) when cooled down from room temperature to liquid
helium temperatures. Rb and Cs are much more reactive than
K and therefore more difficult to handle.

Earlier resistivity measurements on K by MacDonald et
a1. (1956)(8) in the range 2 to 8°K, by Natale and Rudnick
(1968)(9) below 7K and by Garland and Bowers (1969)(10)
from 2 to 4K showed an approximate T5 dependence. However,
in 1971, more precise resistivity measurements were re-

(11)

ported by Ekin and Maxfield from 1 to 25K and by

10

(12)

Gugan from 1.2 to 4.2K. Both groups found clear devia-
tions from the T5 law, which were consistent with an exponen-
tial behavior below about 4.2K. Gugan analyzed his data

in the form

o(T) = N<T>T5 + U(T) e‘l/T = o” + p” .
Equally good fits could be obtained with and without the

T5 term. Assuming U(T) = U Tn, with n=0 the fit with and

0
without the T5 term yielded ¢ = 23.6 °K and ¢ = 23.1K,
respectively. Fairly good fits could be obtained up to

Tn is to reduce ¢ by
(12)

n=3 in U(T). The effect of U(T) = UO

about 2.8K for each additional power of T
(13)

Later,
van Kempen et a1. measured pure K with very high pre-
cision (ml ppm) from 1.1 to 4.2K and fitted the electron-
phonon resistivity (corrected for DO and a T2 term determined
from data below 1.75K, see below) to the form expected for
electron-phonon umklapp scattering

n -e*/T

pe—ph = BT e ’

with n = l and 6* = 19.9 i 0.2K. No normal electron-
phonon T5 term was required and the term BTn e'Ew/T was
consistent with the data of Ekin and Maxfield and of Gugan.
The absence of the T5 term is now understood from

the nonequilibrium nature of the phonons (phonon drag

ll

effect) and the theory is described in more detail in
Section 3.5.

Lawrence and Wilkins in 1972(3) had predicted for K at
temperatures below about 2K a dominant electron—electron
scattering term proportional to AT2 with A z 0.17 x 10-12
0cm/K2. The coefficient A was expected to be independent
of po and of other scattering mechanisms. Van Kempen

et al(13) fitted their resistivity data on K below 1.75K

to the formula

S -6*/T

p = 00 + AT + BTn e (1-17)
with n and 9* as given above. They pointed out that their
data could be equally well described with S between 1 and 2.
Using S = 2, as expected from the theory of electron-electron
scattering, the coefficient A they obtained showed varia-
tions from sample to sample by as much as a factor of 3.6,
in contradiction to theory. In 1978, Rowlands et a1.(lu)
extended measurements of the resistivity of K down to 0.5K,
also with very high precision (m0.l ppm). Their results
showed a possible deviation from T2 behavior, with a best
fit of the form ATl'S from 0.5K to 1.3K with A = (86 i 10)
-6 -1.5 (15)

x 10 p0 K . Bishop and Overhauser

such a form could be explained by the scattering of elec-

argued that

trons by phasons - the collective excitations of an incom-

mensurate charge density wave. Later, Levy et a1 (1979)(l6)

12

reported additional measurements on pure K and claimed to

2.0:0.l

observe from 1.1 to 1.4K a term AT attributed to

electron-electron scattering. They also found that the

coefficient A was sample dependent, with Ao=(od/po)2

3
where pd is the contribution to p0 arising from electron-
dislocation scattering, as predicted by a theory of Kaveh

(17). If the temperature dependence of the

and Wiser
resistivity data of van Kempen, et a1., and of Rowlands,
et a1., were assumed to be T2 (below 1.5K), Kaveh and Wiser
were able to explain the sample dependence of their results
also. 8

In order to resolve the apparent discrepancies discussed
above for data below 1.5K, one needs very high precision
measurements down to below 0.1K, so that other higher power
temperature dependent terms such as e-6 /T, etc., become
negligible, and a sufficient temperature range so that the
power of T can be definitely established.

No high precision resistivity measurements on any
alloys of K with known amountscfi‘known impurities at low
temperatures have been previously reported. According to

(18)

Koshino , also Taylor (19), an impurity concentration

dependent T2 term in the resistivity could be due to the in-

elastic scattering of electrons by impurities. Kus and

(20)

Taylor performed a more detailed calculation of this T

term for K—Rb alloys and obtained a coefficient A of the T

term which was linear with be. For K—l% Rb, A = 2.2 x

12

10' Q-cm/Kz.

13

1.2.2. Thermopower

 

About twenty years ago, MacDonald et al.(21’22’23)

at the National Research Council of Canada performed ex-
cellent work on the thermo-electricity of the alkali-metals
at very low temperatures. They measured the thermal e.m.f.
E of encapsulated specimens down to about 0.1K using a de-
magnetized salt as the cooling agent. For pure K, their
deduced thermopower S = dE/dT could be fitted, below 3K,

to the form

it

S = AT + BT3 + Ce"
The exponential term was attributed to electron-phonon
umklapp processes with 6* = 21K. The detailed form of the
coefficient 0 could be temperature dependent(22), but that
was unimportant because the exponential was the dominant
factor. The term BT3 was attributed to normal phonon drag
thermopower and the term AT to diffusion thermopower. Be-

cause of the difficulties involved in separating the various

terms, they were only able to obtain the coefficients A
8 V

to be (+ .5 to -1.0) x 10‘ —2- and B to be (-.15 to -.30)
x 10.8 3% . Ziman (1959)(24§ and Bailyn (1960)(25) were
K

both able to explain qualitatively the data of MacDonald
et al.(2l) between 2 and 20K by considering the contribu-
tions from electron-phonon normal and umklapp processes.

However Ziman required a significant Fermi surface

14

distortion in his theory and the normal and umklapp phonon

drag effects were not separable, i.e., simply additive.

The Fermi surface distortion requirement was not supported

by subsequent experiments(7). On the other hand, Bailyn

did not consider any Fermi surface distortion, but the normal

and umklapp processes were assumed to be additive. Bailyn's

treatment was the basis of the analysis using Equation (1-18).
Guénault and MacDonald (196l)(23) later measured the

thermopower of alkali-alkali alloys below 3K. The quenched

phonon drag Sph was fitted to

(2) _ l (1) 2 (l)
where
6/T -
S(l) = B(Ԥ')3 f X e 2 f(X)dX,
0 (l-e'x)
= E%; 6 is the Debye temperature of the lattice; f(x)

is the fraction of phonon-electron scattering among all

phonon scattering:

l5

¢L,T’ 8L,T denote the parameters for the longitudinal and
transverse polarizations of phonons. The diffusion thermo-
power was anomolously positive<23). Thornton et a1 (1968)(26)
explained the thermopowers of alkali-alkali alloys using a
phase shift analysis and found fair agreement with these
data, including the correct sign for the diffusion thermo-
power.

No extensive measurements of S for the alkali—metals and

alkali-alkali alloys have been made since.

1.3. Present Thesis

 

1.3.1. Electrical Resistivity

 

The electrical resistivity in metals attributed to
electron-electron scattering is predicted to have a T2
temperature dependence. Such a term has been observed at

low temperatures in transition metals, e.g., Pt<27). Law-

(3)

rence and Wilkins predicted such an e—e scattering T2
term to be observable below 2K for potassium. However,
experimental results below 1.5K by various groups showed
discrepancies with this theory, as described above in Sec-
tion 1.2.1. The difficulties in previous experiments

in determining the power of T in the temperature dependent
part of the resistivity were primarily due to their limited
temperature range (that is, they were not able to make

measurements to low enough temperatures to definitely

establish the temperature dependence); as well as to

l6

insufficient precision at very low temperatures. In order
to resolve these apparent discrepancies we began to ex-
tend resistivity measurements of K to lower temperatures
(about 70 mK) using a dilution refrigerator and a high
precision resistance bridge with 0.1 ppm precision and

15

sensitivity better than 10- volts (see Section 2.1 and
2.2). This high precision is essential for measurements
below 1K because the temperature dependent part of the

4 of the residual resistivity

resistivity is only 10‘5 to 10'
of K (in the dirty limit). A precision of 10‘7 of the resis-
tivity enables one to measure changes in resistivity below
1 K to about 1%.

Two samples with different diameters were measured as
a first order check to see if our data were size dependent.

(From the Blatt-Satz theory(28)

, size effects could produce
a resistivity approximately proportional to T2.)

In order to determine whether the various behaviors of
the resistivity below 1.5K described in Section 1.2.1 could
be caused by the residual impurities in the metal, we put
Rb as a controlled impurity into the potassium. Rb was chosen
because its atomic size is close to that of K, and because,
since Rb has the same valence as K, its presence does not
deform the Fermi surface significantly. The change in the
lattice parameter of K-Rb with increasing solute concentra-

tion is estimated to be 0.09%/at %.(29) Also, if the Koshino-

Taylor T2 term becomes dominant over other temperature

l7

dependent terms in sufficiently concentrated alloys, we

should be able to determine it directly with our precision.

1.3.2. Thermoelectric Ratio G

No extensive low temperature thermopower measurements
below 1K have been made on K and K-Rb alloys since the NRC
group. Since with our dilution refrigerator, we could reach
temperatures down to about 70 mK, and since G, which gives
essentially the same information as S in the elastic scatter—
ing regime, could be measured accurately with the available
SOUID null detector, we made a careful study of G on free
hanging K and dilute K-Rb alloys in an attempt to determine
the diffusion, normal phonon drag and umklapp phonon drag
terms. The G measurements were made during the same run
as the resistivity measurements. This allowed us to analyze

the two properties on the same specimen.

II. EXPERIMENTAL TECHNIQUES

In this chapter, a brief review of the basic principle
of the dilution refrigerator is first given, and the opera-
tion of the refrigerator with our locally built gas handling
system is described. Then the high precision resistance
bridge with commercial dc current comparator and SQUID
null-detector is described. The basic principle of the
SQUID is briefly reviewed. A description of the measure-
ment methods for the resistivity and thermoelectric ratio,
as well as the thermometry calibration are then given.
Finally, the sample preparation techniques are discussed

and error analysis outlined.

2.1. Dilution Refrigerator

To obtain temperatures significantly below 0.3 K
continuously for low temperature research work, the only
practical device is the 3He/uHe dilution refrigerator.(30)
The principle of dilution refrigeration was first suggested
by London in 1951(31) and by London, Clarke and Mendoza
(1962).(32) The first prototype of this type of dilution
refrigerator was built by Das, De Bruyn Ouboter and

Taconis in 1965.(33) It reached 0.22 K. Rapid development

18

19

followed shortly after that. The most modern, well de-
signed dilution refrigerator is capable of maintaining a

(34)

temperature of 2.0 mK continuously. An elementary one
today can easily reach 70 mK. A dilution refrigerator
system with commercial cryostat insert was built locally
for our experiments.

3He/uHe dilution refrigeration is based on the finding
of a finite solubility of 3He in liquid “He even at abso-
lute zero temperature. The phase diagram of 3He/uHe
mixture (Figure 2-1) shows that below about 0.86 K, phase
separation of the two liquids takes place. The phase which
is rich in 3He (the concentrated phase) will float on top
of the other which is rich in “He (the dilute phase).
Below 0.86 K, the left branch of the phase diagram curve
represents the dilute phase, which shows a solubility of
3He in ”He of 0.064 at T = 0°K. The right branch of the
curve represents the concentrated phase. Below 0.1 K,
X0 = 1 (0.99997 actually), $43,, the concentrated phase is
made of practically pure 3He, while the dilute phase con-
tains at least 6.4% of 3He in “He. The finite solubility
of 3He in liquid ”He at T = 0°K can be understood by the

3He quasiparticle concept. Liquid 4

He, having zero nuclear
spin, superfluid properties, and obeying Bose-Einstein
statistics, is effectively in its quantum mechanical ground
state below T = 0.5 K. Whereas the lighter 3He with nuclear

spin of 1/2, obeys Fermi—Dirac statistics. The concentrated

20

 

  
   
 
     

 

 

 

I l l I r l I I I l I f
O - de Bruyn Oubo/er er a/
D - Edwards a! a/ 5
L5 _ A ' Brewer and Keyston -
x'0”‘ —
o
F b—
O 5 —- ._.
PHASE SEPARA T/ON
bl REG/0N
O I o , u 1 I I I
02 09 X 0.6 08 - l.O

Figure 2-1. Phase Diagram of 3He-uHe solutions. (After
Radebaugh, NBS Technical Note 362, 1967).

2O

 

Ill . I N l I I T l I l T T I fl

 

 

 

._ O
O - de Bruyn Oubo/er er a/
. D - Edwards :9! a/ I
I6 — A ‘ Brewer and Keyston -*
ll Line
I— .4
O
-— -1
O
o O
r5 _ . ° .
e . T
i . _
O
>— . O —4
O 5 ’— e ._
PHASE SEPARA 770M
’ . REG/0N . ~
_ . _
... I
I
O i 1 ' l ' ' ' l ' ' ' l 1 ' l l . ' .

O 2 O 4 X 0.6 O 8 - IO

Figure 2-1. Phase Diagram of 3He-uHe solutions. (After
Radebaugh, NBS Technical Note 362, 1967).

21

phase, which at low temperatures is practically pure 3He,
can be thought of as Fermi liquid. 3He atoms, which are
bound more strongly to liquid “He than to liquid 3He,
dissolve across the phase boundary into superfluid “He.

4 4

3He atoms interact with He atoms to form a 3He- He quasi-

particle gas with an effective mass m*. The superfluid “He,
in its inert ground state, provides a background 'vacuum'.
We can thus think of the dilution process as analogous to
an ordinary evaporation process: 3He atoms evaporate from
the Fermi liquid and form a quasiparticle gas supported by
the inert “He background, which is at a pressure equal to

“He. The thermal properties

the osmotic pressure of 3He in
of the quasiparticle gas can be calculated from the proper-
ties of an ideal Fermi-Dirac gas. The cooling of the dilu-
tion refrigerator comes from continuously removing 3He
atoms from the dilute phase; thus 3He atoms continuously
'evaporate' across the phase boundary to keep the concen-
tration of 3He in the dilute phase in equilibrium. In a
dilution refrigerator, the removed 3He atoms are recircu-
lated through a mechanical pump at room temperature, re-
condensed and returned through heat exchangers back to
the concentrated side to keep the cooling process going
continuously

The principal parts of a dilution refrigerator are

schematically shown in Figure 2-2. The phase separation

occurs in the mixing chamber. 3He atoms in the dilute

 

Figure 2-2.

 

ft)
h)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

— - o o a a no u,
f... ..‘.ee~ 0.0... :‘e' .“

  

 

 

C I
' g' 0" '0'... O
fi‘- -- -- - ---

 

 

 

12
IO

 

 

 

The principal parts of a conventional dilution
refrigerator: 1. Condenser, 2. Main im-
pedance, 3. Still, 4. Still heat exchanger,
5. Discrete heat exchangers, 6. Phase”
boundary, 7. Mixing Chamber, 8. 1.3 K
pot, 9. Orifice, 10. Dilute phase, 11.
11. Continuous heat exchanger, 12. Concen-
trated phase.

He

23

solution, driven by the osmotic pressure, go through the
heat exchangers, reach the still and are pumped away by a
mechanical pump. The 3He gas is recirculated through the
pump. Incoming 3He is first precooled to 4.2 K and liqui-
-fied in the condenser which is attached to a pumped “He
(1.3 K) pot. Condensation is possible because of the flow
impedance below the condenser. The liquid then enters the
still exchanger at 0.7 K and passes into the heat ex-
changers, cooled by the dilute solution, and reaches the
mixing chamber again. “He in the circulating gas is sup-
pressed by the specially designed still in which superfluid
“He film flow is reduced. The circulation rate is con—
trolled by heating the still. The efficiency and per—
formance of adilution refrigerator depend critically on
the design of the heat exchangers.

The cooling power of the dilution process can be
understood from the enthalpy considerations in the mixing
(35) For the concentrated phase, the entropy and

chamber.

enthalpy are

T C3 J
s = f —)dT = 25 T — mol , (2-1)
c T 2
0 K
. T 2 J
H = f c dT = 12.5 T — mol , (2—2)

= 25 T (3— mol below 140 mK).

with C
3 K2

24

In the dilute phase, CD = 107 T J/(K2 mol), and the entropy

SD = A? DC/T dT = 107 T JK-2 mol-l. The chemical potential
“c (concentrated phase) and “D (dilute phase) of 3He in

both phases in the mixing chamber under equilibrium are

equal. Since the chemical potential per mole u = H-TS,

we have

Thus

HD = He + T(SD-Sc) = 94.5 T2 JK-2m01-l (2-3)

The cooling rate at the mixing chamber is
n3EHD(Tm) - Hc<Tm>3 <2-u>

where Tm is the mixing chamber temperature, and n3 is the

3He circulation rate in moles/sec. External heating is
from the heat leak Q and the incoming 3He which is at

temperature Tn' At equilibrium,
6 + n3tHc<Tn)-HC<Tm>l = n3EHD(Tm)-HC(Tm)]- <2-5)

Putting in values of HC and Hd, the cooling power 0 is given

by

25

[94.5 Tm2-12.5 Tn2] JK‘2 mol’l . (2-6)

,0.
ll

n3

H
Hi
38°
II
o
v
a
II

0.36 Tn' For a perfect exchanger, T = T

Tm = Q/(82 113).

This is the lowest temperature the mixing chamber can reach
for a given heat load and circulation rate n3 with an ideal
heat exchanger.

The most efficient type of heat exchanger is the con-
tinuous counter—flow heat exchanger. For such a heat ex-
changer, one can show that the mixing chamber temperature

is given by(3“)

Tm2 = 6.4 ka 53/6 + 6/(82 n3) , (2-7)

where ka is the Kapitza thermal boundary resistivity defined
as ka = AT/Q 0T3, in m2K“/watt, n3 in mol/sec, o is the

2

exchanger surface area in m and Q in watts. One sees

that the larger the exchanger surface, the lower the Tm.

The optimal circulation rate n t can be determined from

30p
the condition dTm/dn3 = 0.

In our system, the dilution refrigeration unit was
built locally by Dr. Pratt, using a continuous counter flow

tubular heat exchanger and a commercially built cryostat

26

insert unit (SHE 40/4000). This dilution refrigerator is
capable of reaching 60 mK. The gas handling system was
built locally. The cryostat is mounted on a vibration
isolation air mount (NeWport Research), the top of which
is filled with sand for additional weight and damping.

The pumping lines are all vibration isolated through flex-
ible metal bellows. For the large diameter pumping lines
(the still line and the l K pot line), special bellow as-
semblies are used. The assembly consists of two (1-1/2")
1/2" flexible stainless steel bellows and a 2-1/2" diameter
stainless steel bellow. They are arranged as shown in
Figure 2-3. The two 1-1/2" bellows, counter act each
other's contraction forces and provide isolation to vibra-
tions perpendicular to the bellows. The third bellow
(stiffer) provides isolation to vibrations along the two
smaller bellows. Thus vibrations along all directions are
damped. A diagram of the gas handling system is given in
Figure 2-4. The sealed pump for 3He circulation is an
Edwards ED660 rotary pump. It provides sufficient pumping
speed for the present system to reach 60 mK. However, a
booster pump is installed in case greater pumping speed
should be needed in the future. The gas handling system is
built on a rack anchored to the floor. The top of the rack
is sand filled to damp out vibrations from the pumps. The
whole system is enclosed in a screened room with the pumps

outside. The pumping lines from the pumps to the gas

27

.COfiumHomH soapmpnfi> pom mHnEommm zoaamm .mlm opswfim

\

 

\

 

 

 

 

_

«25:2. \ .

e338:
N

:—

 

 

 

 

 

a

26:2.

m
in

 

 

28

 

 

 

 

men, _.
e: .:::a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T...

 

 

.Eopmzm mafiapcmc mam nonmpowflpmop COszHHp map mo Empwmfip oHmeocom

 

 

.233:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.zlm ogzwfim

29

handling system are connected through bellows and enter the
screened room with non-conducting portions through appropri-
ately long metal pipes. The screened room is important in
screening out radio frequency noise that affects the opera—
tion of the SQUID.

In a normal experimental run, the sample can is first
mounted and all electrical connections are checked. The
outer vacuum can is then put on (indium O-ring is used for
the seal) and leak tested. After closing the cryostat,
the refrigerator is flushed with the 3He/“He mixture. The
continuity of the main flow impedance is then checked by
letting in mixture to the condenser side and noting the rate
of pressure rise in the still, indicated by the thermocouple
gauge, after the still is pumped out with the sealed pump.
The 1 K pot is flushed out with “He and the “He inlet
capillary continuity is similarly checked. The 1 K pot is

“He afterwards. The

left with a slight overpressure of
system is then precooled to liquid nitrogen temperature
with appropriate exchange gases in each part. The same
continuity test procedure is repeated at LN2 temperature
to get ready for the liquid helium transfer. After the
helium transfer, all the thermometers and heaters are
checked and the SQUID electronics set up. To operate the
refrigerator, the mixture is first precleaned through a

liquid nitrogen cooled molecular-sieve cold trap, and the

l K pot is pumped to 1.3 K. All the mixture in the storage

30

tank is then evacuated into the refrigerator. Circulation
is started by first using the circulation helium pump

main valve throttle to keep the condensing pressure close
to about -2" Hg. A safety valve connected to the storage
tank is switched on to ensure that the pump output pressure
is less than 1 atm so as to protect the sealed pump, which
is not designed to work above 1 atm, otherwise precious

3He gas might leak out of the system. The mixing chamber
cools as the condensing process proceeds. After all the
mixture is condensed and the mixing chamber temperature is
around 0.8 K, the still body and still orifice heaters

are turned on. The mixing chamber then cools down rapidly
to its lowest temperature. Measurements can be made during
the cool down by regulating the M.C. temperature at the

desired value.

2.2. High Precision (0.1 ppm) Resistance Bridge

The resistance bridge used in our measurement system-
consists of a commercial direct current comparator and a
very sensitive SQUID null-detector. The basic principle
of the operation of the SQUID and its limitations are
reviewed and then the complete resistance bridge is des-

cribed.

31

2.2.1. SQUID (Superconducting Quantum Interference

Device)

The SQUID is an important device in our high precision
measurement. It is used as a null-detector in our system.
It provides the sensitivity high enough to measure voltages
below 10’15 volts, limited only by the thermal Johnson noise
in the source resistance.

The basic working principle of the SQUID is based on
the Josephson effects. A description of how the SQUID
works is given in Appendix A. In our system, a symmetric
point contact r.f. biased SQUID, first developed by Zimmer-
man 33 a1. (1970),(36) is used. The SQUID probe used is a
SHE Model RMPC with SHE Model 330 electronics. The sym-
metric SQUID Magnetometer is used as a null detector by con-
necting a current source to the signal coil. The SQUID out-
put voltage is then proportional to the current through the
signal coil.

The mean square current noise can be expressed as

2 4kTeAf
(Jne> = T ’ (2'83)
e

where T8 is the SQUID temperature (usually 4.2 K),
Af the bandwidth and Re the equivalent noise resistance.
For a typical SQUID magnetometer, Re a 12 m0. The thermal

noise (Johnson-Nyquist noise) of the source resistance RX

32

is

4kTXAf

<3ix> = “‘l?“' , (2-8b)
X

where Tx is the temperature of the source resistance. So

the total mean square current noise

 

 

 

2 _ 2 2
<Jn> — <Jnx> + <Jne>
R T + R T
= 4kAf e x X 9 (2-80)
ReRx
or the mean square noise voltage
R R R R (R T +R T )
<Ui> = <Jr21> < e x )2 = 4kAf 6‘ X e x 2" 9 (2-9)
Re+Rx (Re+Rx)
The minimum detectable voltage VU:(min) = <Ur21>1/2 A

 

plot of"VU§(min) vs. R is shown in Figure (2-5). One

x
sees that for TX < l K, Rx << Re, voltage sensitivity of
10-15 V is possible and for most cases limited only by
the Johnson noise of the source resistance. The SQUID
null detector is much superior in sensitivity to other

devices such as superconducting chopper amplifiers, etc., and

together with the current comparator, providesggxxienough

33

 

(Unz- (manl) (V)

    

 

 

l
3 10‘

 

3 10'

Figure 2—5. The rms value of the minimum detectable
voltage as a function of Rx for four dif-
ferent values of Tx’ assuming that Af = 1 Hz,
Te = 4.2 K, and R6 = 12 m0. The limiting

2 1/2 2 1/2
behavior, <Ux(min)> = [“kAfTeRx/Re]
is shown by the dashed line. (After Lounas-
)(30)

1118.3.

34

precision for our measurements which would be impossible to

do with lesser techniques.

2.2.2. Current Comparator Resistance Bridge

A commercial dc current comparator which provides cur-
rent ratio resolution of 0.1 ppm is used with the SQUID
null detector in a 4-terminal resistance bridge. The cur-
rent comparator is based on the design of MacMartin and
Kuster (1966).(37) The main components of this comparator
are two ratio windings which carry the currents to be com-
pared, and a double toroidal core magnetic modulator which
is completely shielded by a magnetic shield. The ratio
windingseuwacoupled to the modulator. The magnetic modu-
lator senses the flux imbalance in the magnetic core caused
by the imbalance of current ratio in the ratio windings.
If the output of the modulator is fed back to one of the
current windings (slave current) to keep the flux zero,
then the ratio of the two currents would be given by the
turns ratio of the two windings. With a variable turns
ratio, we can get currents of various ratios. The cur-
rent comparator used in our system is a commercial model
from Guildline Instruments Ltd. It is modified for semi-
automatic operation with operating electronics built by
Edmunds e: al.(38) A block diagram of the resistance
bridge is shown in Figure 2-6. It provides precision of

0.1 ppm with currents from 10 to 100 mA when the current

35

7va 4m Mm .nossEom pooh:
ma mafinfiz on» Ham can zoaon no m: up one mucoOQEoo map .mCHHpso paon a
.owpfihn mocmpmfimmp mo Emnwwfip xooan pofiMfiHQEHm

cpfiz seamen map mcfimaH

 

 

.wCHpospcoopoQSm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    
        

 

 

 

 

F—_—'l

 

Aw

 

 

 

 

._
_

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

55: 9:2... 5.5a
. >32. . m . umoo :33
«0.25323
hzwceau
o;
5:33 , muaaaoo
acufido «65.23 348:6
.x #9»:
.
.
_
2350 3.39... g _ . 338 :35".
~33...“ use Enema _ .quecau 5.5;. 9:.
or 526.. u>3m zowwwwuo . $542 2. 526a
_
.
. a . . N
. "
copuupuo
3.: use onmwum.» I. 93
o... 532. to o.
H.338 23 5438
5&8 33:8 :m n. 43:36 5&3.
526a 526..
“23..” 5.552
_ fl _ 7
5qu
3.5.53 Haywoou cause .65on 44:05 3.5::
.. c - _ ._
» o> n. uxe or 530.. » 3 c:

 

 

 

 

 

 

 

 

 

.mlm omswfim

36

ratio is close to one. The precision decreases with larger

or smaller ratios. The detailscfi‘this resistance bridge

are published elsewhere. (38)
The resistance bridge and the dilution refrigerator system

are all enclosed in a commercial screened room (Erik A.

Lingren and Associates, Inc.). The screened room shields

out rf interference that would unlock the SQUID. The

walls are made of two sheets of metal, an outer galvanized

steel sheet and an inner copper sheet. The galvanized steel

sheet reduces lev frequency' electromagnetic fields and

the copper sheet reduces high frequency fields. The power

lines into the screened room are fed through r,f,I, filters

to eliminate interference that could be carried by the power

lines. Batteries are used in the power supplies to the

master and slave circuits as well as for the logic cir-

cuits. This eliminates the annoying problem of ground

loops and A.C. line interference. With the screened room,

we can operate the SQUID with optimal low noise. The

reduction of rf interference also reduces the problem of rf

induced self heating in our resistance thermometers, which

would otherwise require complicated filters.

2.3. Measurement Methods

2.3.1. Resistivity

 

In the conventional u-terminal resistance bridge, a

reference resistor kept at constant temperature, usually

37

h.2 K, is used. In order to get optimal precision, the
noise caused by this reference resistor must not be worse
than that of the specimen. The factors that affect the
accuracy and the precision of a measurement due to the
reference resistor are: Johnson noise, thermoelectric
voltage noise due to temperature fluctuations, temperature
dependence and current dependence of the resistor, as well
as long term stability. A good reference resistor with

low thermopower, low temperature dependence, low current
dependence and stable resistance upon thermal cycling is
difficult to make.(39) Also, in order to make full use of
the precision of the current comparator, the resistance of
this reference resistor must be close to that of the speci-
men; thus different reference resistors are required for dif-
ferent specimen resistances. The Johnson noise (/FkTfiKf7
of such a reference resistor kept at “.2 K would be much
larger than that of the specimen (of similar resistance)

at low temperature (<l K), thus the precision is degraded.
Therefore, a different method, that eliminates the above
difficulties involving the reference resistor, is necessary
in order to have 0.1 ppm precision for measurements below

1 K. A temperature modulation technique, suggested by
Edmunds gt al.(38) which makes use of a reference resistor
that is made nearly identical to the specimen, serves this
purpose. The set up of this method is shown in Figure 2-7.

Each sample is thermally linked through a Ag-Au alloy of

306:

SQUID

Figure 2-7.

 

38

 

 

 

 

 

 

 

MIXING
CHAMBER
* Illnermgl Illink
icn::'|:::dy
heaters /
U2 U I
:ZZ—J] 5
L2 R, (Ag-A1,; H
::_'J____' E

EIIEJ— .. e—‘L'IEL

 

 

 

 

 

 

 

 

thermal
‘— liflk‘ '9 sample can
”‘1“.;.;;.:.:""‘j"/'
/ \ ‘ r
J
shorted

 

—---d - - -‘O— -----—--‘_

 

 

V

_--—-—-—-_-—--—

 

 

 

------- —---—4

Schematic diagram of the set up of specimens
for resistivity measurement using the tem-
perature modulation technique.G heaters are
included for G measurements.

39

residual resistance R' to the mixing chamber. To make a
resistance ratio measurement, both are cooled down to the
desired temperature. The temperature of the ref.resistor

TS is then kept constant by regulating the M.C. tempera-
ture with a temperature controller monitoring thermometer

Th 1. Heat Q is first applied to heater U2. The temperature
of the specimen TX is measured with a calibrated germanium
resistance thermometer Th 2. The resistance ratio of the
two samples is then obtained by adjusting the current

ratio, as determined by the current comparator ratio,

to obtain a balance as indicated by the SQUID null-detector.
Standard current reversal technique is used to eliminate the
effects of spurious emfs. The ratio C(Tx) = Rx(Tx)/Rs(Ts)°
Then the heat is switched from U2 to L2. This heat current
through the Ag-Au alloy produces a temperature gradient AT.
The temperature Tx + AT is then measured with Th 2. The

new ratio C(Tx + AT)==RX(Tx + AT)/RS(TS) is again measured

with the resistance bridge. So

Rx(Tx+AT) Rx(Tx)
AP. _ C(TX+AT)-C(Tx) - RS(TS) RSKTSS
c ’ GIT I ‘
x Rx(Tx)
R (T35

S

 

 

 

HO

RX(TX+AT)-RX(TX) = ARX(TX) _ Apx

AC
77

 

p
The quantity

AC
EK_ (2—10)

can then be determined. AT can be determined from the
thermometer readings or it can be determinedusingtflma
Wiedemann-Franz law under the condition that all the heat
(Q) passes through the Ag-Au alloy. The Ag - 0.1 at %

Au alloy used has a residual resistance R' = 88.42 pH and
the Wiedemann-Franz law should be accurately valid for this
alloy (elastic scattering predominates). Using the Wiede-

mann—Franz law,

NIQ‘
I
.O'
33

AT

L‘"
*3

where

T = T + and L - 2 H5 x 10'8 K3
x 77 o ' '

K

M

We used the measured temperature difference at T < l.“ K,
when our thermometer is sensitive enough to check against.

this latter method to determine possible heat loss.

41

Since at low temperatures,
<< 1%

where pxo is the residual resistivity, we have

1 dpx - l dpx

SZ'YFF'? 3;; ?fi7

Then % %% can be analyzed as the scaled slope of the ox

XE T curve. This eliminates the problem of estimating

pxo and for a resistivity of the form
0 = 00 + AT

yields

(2-11)

one

SK?
ll

IE
*3

.2

H

One can determine the power law (N) more accurately this
way, even though the absolute accuracy of the coefficient A
is limited by the accuracy of pxo' For K and K—Rb,

% %% is very small and of the order 10‘“ to 10‘5 below 1 K.

With a AT 5 0.1 K, %g = 53-3 10—5, so only with a precision

0
high enough to resolve C to 0.1 ppm can we determine

%'%% to about 1%. The measuring current used (20 mA to

50 mA) is chosen so as to minimize spurious thermoelectric

42

effects due to self heating, Peltier effect, etc. No cur-
rent dependence was observed to 0.1 ppm on the resistance
ratio using 20 mA and 50 mA.

For temperatures below 0.2 K, the value of % %% is
very small,of the order of 10-5. So with a 10% tempera-
ture gradient AT m 0.02 K, %§-m 2 x 10-7. The uncertainty
in AC becomes unacceptable. A different method is thus
needed to obtain data down to about 70 mK. We kept the
reference resistor at constant temperature around 90 mK or
150 mK and varied the temperature of the specimen. The
resistance ratio C was determined at various temperatures
between 70 mK and 460 mK, similar to the conventional method.
Keeping the reference resistor at such a low temperature
reduces the Johnson noise significantly as compared with

that where it is kept at “.2 K; thus optimal precision

could be obtained for this limited temperature range. 'The

1 AC

result is then differentiated to obtain 5 AT'

2.3.2. Thermoelectric Ratio G

The current comparator is used as a precision current
source in G measurements. The set up is shown in Figure
2—7. The master current is shunted internally. The slave
current going to the specimen is then Just the product of
the dial ratio and the master current setting. Both samples
are cooled down to the desired temperature for a measure-

ment. The reference sample temperature is then maintained

M3

near this temperature by controlling the mixing chamber
temperature with a temperature controller and a conductance
bridge which is monitoring a thermometer (Th1) mounted near
the reference sample. The G heater supply is obtained from
a mallory 1.u volt mercury battery. The heater resistance
is H K9 (1%) and the heater current IG is measured with a
DVM across a series 1 K0 (1%) resistor. The SQUID zero

is first noted with the current comparator slave current

set at zero and G heater current turned off. The G heater
current is then gradually increased to a value such that the
SQUID shows a reasonable e.m.f. (substantially larger than
100 times the noise) and then this e.m.f. is balanced by the
current I through the specimen, supplied by the current
comparator. The thermoelectric ratio G at this temperature

is then

 

G(T) =1) = 2 I . (2-12)
Q E=O IG x “000
The heater current is then reduced to zero and the SQUID
zero rechecked. The temperature T is determined with
thermometer Th2 near the cold end of the specimen. Since
the temperature gradient across the specimen is small,
one can take this T as the average temperature of the

specimen.

A“

2.3.3. Thermometry

 

The resistances of the germanium resistance thermom-
eters are measured with either a SHE conductance bridge
(Model PCB) or an AC resistance bridge in the U—terminal
configuration. The SHE conductance bridge, which uses very
low excitation voltage (10 to 100 uV) is accurate to better
than 0.5%. It is self-balancing and has a differential out-
put for temperature regulation purposes. Because of its low
noise operation, it is used with a temperature controller
to regulate the mixing chamber temperature. The AC resist-
ance bridge is based on the design of Ekin and Wagner.(u0)
Using much lower noise JFET op amps than those which Ekin
and Wagner used, we can operate the bridge with 10 uV
excitation and still get 1% accuracy (Ekin and Wagner used
10 mV excitation). This low excitation level reduces
self heating of the thermometer at temperatures below 0.1 K
to a negligible level. At temperatures above 0.5 K, 100 pV
excitation can be used for better resolution.

The temperature controller has the usual differential,
integral and proportional controls and twelve output power
levels are available for optimal temperature regulation at
various temperatures. The time constants of the differen-
tial and integral controls are adjustable, allowing tuning
for different temperature responses of the dilution re-

frigerator. Together with the SHE conductance bridge,

“5

regulation of better than 0.1% over the whole temperature
range (60 mK to “.2 K) is achievable.

The thermometers Thl and Th2 are Lake Shore Cryotronics
series GR—200A-30 germanium resistance thermometers. The
thermometer Th2 was calibrated against another calibrated
germanium resistance thermometer (type CR 50)(“l) which
was calibrated against a cerium magnesium nitrate (CMN)
susceptibility thermometer and 3He'vapour pressure scale.
The calibration is better than 0.5% relative to the ther-
mometer CRSO. A plot of the calibration points is given in
Figure 2-8. The calibration points are fitted to a poly-

nomial of the form

T = z tnnR with n = 9
n
Typical deviations of the calculated temperatures from the
calibration temperatures are better than 0.5%.
The calibration of the thermometers yum; later checked
against NBS superconductive fixed point devices<u3) and

found to be absolutely accurate to about 2%. (See Sec. 2.5.)

2.“. Sample Preparation

Potassium is very reactive. It corrodes in air very
rapidly. Therefore, in earlier experiments made by others,

the samples were either encapsulated or prepared in dry

(11-13)

oil. Only more recently have samples been prepared

“6

 

.Aomn<oomlmov ~25 LopoEoEpmnp pom w>pso coapmsnfiamo .mlm mpzwfim
2.02:.
0000— 000— 00— O—
1 — d a . q .
. . no.
. . . . .. —.O I.
e e x
. . . m6
1 —
.. .. m

 

 

 

“7

(1“) In order to avoid straining the sample,

in a dry box.
or adding oil contamination, we prepare our samples in free
hanging form inside a dry box. A special sample can was
designed to allow us to mount two nearly identical samples
for making resistivity measurements with the temperature
modulation technique and also for G measurements in the same
run.

The standard sample preparation procedure is as follows.
The twin samples are prepared inside the dry box under
clean argon atmosphere (see below) and mounted on the sample
holder. The can is then sealed and transported out of the
dry box to be tested before mounting on the dilution re-
frigerator. A sketch of the sample can is shown in Figure
2-9. Two pure silver wires are used as thermal links to
the sample. They are spotwelded onto OFHC copper pieces
which are secured onto nylon blocks with screws. Each
silver wire is fed through a 1/8" stainless steel tube on
the top flange with teflon spaghetti tube as insulation.
Eight electrical leads come from each stainless steel
tube. The tubes are sealed with Stycast GT2850 epoxy.
All electrical leads are superconducting wires (%“ mil)
with either copper cladding or CuNi cladding. Eight leads
are used as current and potential leads to the two samples.
Two others go to each of the G heaters at the end of each
sample. The current and potential leads are soldered onto

OFHC copper pieces. (For leads with copper cladding,

“8

.mpmmH wCHposo

Icoommasm .ma .pmdooo on pupae; oHoo Esfimmmpom .HH .memmn 0 .OH
.mmHaEMm Edfimmmpom .m .mmomHQ pmdqoo ammo .m .QAOQQSm coamz .5

.mnzu Hompm mmmacfimpm =w\H .m .wwcmam mmmpm .m .mpmmH Hmfiucmpoa one
unmmpso .= .xcfia awaken» w< .m .mcmoH Havappooam Ampmmn c .m .Lmnemno
mcfixfie on psmenomppm mom Ham: hwdaoo .H “hopaon mHQEmm Edfimmmpom

 

 

 

 

 

 

 

 

 

 

.mum mpswfim

“9

about 1 cm of the cladding is etched away before attaching
to the copper pieces, so as to obtain better thermal isola—
tion). The copper pieces for the potential leads are
secured onto 1/8" threaded nylon rods with GE703l varnish.
The copper pieces for current leads are secured onto the
nylon blocks by screws. The G heaters (Dale l/“ watt wire-
wound resistors) are inserted into a hole in the copper
block and secured with GE7031 varnish. The top of the
sample holder has a 1/2" diameter well with slots cut to
allow a snug fit onto the mating post at the mixing chamber
of the dilution refrigerator. A nylon collar is used to
squeeze this assembly tight at low temperature. An indium
O-ring (0.0“5") is used to seal the sample can.

The dry box gas handling system is sketched in Figure
2-10. Argon gas from the tank is passed first through an
oxygen reduction catalyst (BASF R3-ll)(u2) and then through
a moisture trap filled with SA molecular-sieve. A pressure
sensor senses the difference between the pressures inside
and outside the dry box. When the pressure of argon inside

drops to less than l/2" H 0 above the outside atmosphere,

2
a solenoid valve switches on, admitting more gas to keep
the pressure inside at about l/2" H20 above atmosphere.
This way air leakage into the dry box is minimized. Ac-
cessories can be transported in and out through the access

port without contaminating the clean atmosphere. The

catalyst used in the purifying system can be reduced and

50

.Eopmzm wcfiaocmn mow xon mam .oalm mhsmfim

33::— cot—v.33...

 

 

.m>_

U.— 00

F?
n. 23

 

 

 

 

 

 

 

 

 

 

 

 

 

E

 

 

at:

 

 

 

(m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:8 2.35 3

 

 

 

 

 

é“ fi “@III
to: «385 3

 

 

88/1 e Rryuul.
£33888 G
\

 

:2.
f a were
1 an... .3 S.
Q11 ‘
_ cl. efifl
e>.flnr..----._w .033
30.8. ebemmoa .
AX)“ . oliol.

a5...—
3

51

reused again by passing H2 through it at a temperature

about 200 °C to remove oxygen. Before making samples, the
argon atmosphere in the dry box is further purified by
circulating the gas through the catalyst and molecular-sieve
purifying system with a rotary gas pump for about two hours,
until a test potassium piece can stay bright for about a
minute. When the argon atmosphere is in an acceptably
clean condition, potassium in a storage vial can be heated
up to melt (about 70 °C) and poured into a stainless steel
sample press (Figure 2-11). K sample wires are extruded
from this press by pressing K out through a screw bolt with
a hole in the center. Different diameter wires can be made
with bolts having different size holes. The K specimen
wire is placed on one side of the holder and cold welded
onto the two copper cross pieces to which the current

leads are attached. The welding occurs easily if the copper
pieces are first fully file cleaned. The potential probes,
made of the same potassium wire, are cold welded to the
specimen and to the copper pieces to which the potential
leads are attached. The separation between the potential
probes is about “ m 5 cm. With a sample diameter of 3 mm,
the geometric factor % is about 10-2. The reference sample
is made in the same way and mounted on the other side of
the holder. The sample can is sealed inside the glove

box. The argon inside the can solidifies at liquid helium

temperatures, so the samples are in vacuum during the

stainless
steel

 

 

 

 

 

Figure 2-11. Sample press.

53

measurements. No oil is used in preparing the samples and
the samples hang freely except at the two end supports.
With this design, we can measure both % %% and G of both
samples in the same run.

After the samples are mounted, and the sample can
sealed, the can is transported out of the dry box. The
room temperature resistances of both samples are measured.
Using a resistivity of 7.19 uQ—cm at room temperature for
K,(1l) one can determine the % of the specimens. The
residual resistivities of the specimens are measured by
cooling the sample can to “.2 K in liquid helium. The
residual resistivity ratios, defined as R(R.T.)/R(“.2K),
are then determined. Table 2-1 shows the details of all the
samples. Three sets of 3 mm and one set of 1.5 mm samples
were made so as to have a first order check of size ef-
fects. (The resistive electron mean free path for our pure
K samples with p0 = 1.5 nQ-cm is about 0.15 mm.). Our 3 mm
diameter sample is the largest diameter, free hanging K
sample for which resistivity at low temperature has ever
been measured. The resistance of that sample at “ K is

7

about 10' 9. Thus with a measuring current of 50 mA,

in order to measure this resistance to 0.1 ppm, one needs

0'16 volts

a null detector with ability to resolve 5 x l
or better, which the SQUID null detector is capable of.
For the 1.5 mm sample, a quenched resistivity (quenched from

room temperature to liquid nitrogen temperature) and a slow

Table 2—1. Characteristics of the samples. (a) Pure

5“

potasium samples; (b) K-Rb alloys; (c) Samples
of Guénault and MacDonald (23) (for comparison).

 

 

 

 

 

 

 

 

 

 

 

 

 

(a)
Diameter A

Sample p(“.2K)(10'90—cm) (mm) I (cm) RRR
K1 1.88 (quenched) 1.5 3.91310‘3 3850
K1 1.50 (slow cooled) 1.5 3.91x10'3 “800
K2 1.73 (quenched) 3 1.61_x10"2 “170
K3 1.70 (slow cooled) 3 1.18310"2 M200
K“ l.““ (slow cooled) 3 1.199(10-2 5000

(b)

Sample pfg°2K) diameter A (10—2 cm)

Nom. at.% Rb (10 Q-cm) (mm) A RRR
2.2“ 387 3 1.18 18.6
0.83 1u2 3 1.0g 50.5
0.32 56 3 1.18 129
0.13 23.5 3 1.02 30“
0.05 9.97 3 1.23 720

(0)

Sample Diameter Length

At.% Rb (mm) (cm) RRR

2.1 0.2 10 26.6
0.9 10 “6.5
0.2 10 157
pure 10 5“0

 

 

55

cooled resistivity (2 hours to cool down from room tempera-
ture to liquid nitrogen temperature) were measured. The
results are given in Table 2-1. Typically, the room tem-
perature resistance of a sample increased about 5—10%
the first day after it was prepared, and an additional
1 - 7% in two weeks to a few months, as checked before and
after measurements in the dilution refrigerator system.
This is an indication of how the sample surface deteriorates
inside the sample can. Also, a couple of sample cans were
reopened after two weeks to a few months and the samples
visually checked. Only thin surface corrosion was observed.
In the initial check, the contact resistances of the
joints are also measured to make sure that the whole SQUID
circuit resistance is acceptable and that there is no
excessive Joule heating at the current lead junctions. The
SQUID circuit resistance measured is typically three times
the specimen resistance because the two samples and the
potential probes made of the same material are all in the
circuit. After all the initial checks, the sample can is
mounted onto the cryostat. The set up of the samples on
the cryostat is shown in Figure 2-7. The silver thermal
links are separately spotwelded to the corresponding Ag—Au
alloy links. The link of the reference sample to the mixing
chamber is through an electrically isolated assembly. This
assembly is made of two l/l6" thick OFHC copper plates of

surface area about 5 cm2, separated by cigarette paper and

56

varnished together with GE7031 varnish. Its purpose is to
break the ground loop between the two samples." All the
upper and lower heaters (Dale RS 1/2 “K0 resistors) are
first wrapped with annealed insulated copper wire for thermal
contact. The copper wire is soldered to a silver wire which
is then spotwelded in place to the Ag-Au alloy. The ther-
mometers are mounted inside copper block wells soldered to
the silver thermal link. Apiezon N grease is used to get
good thermal contact between the thermometers and the copper
wells.

The K—Rb alloy samples are prepared in the same manner.
The potassium and rubidium materials were obtained from
Mine Safety Appliances Ltd. (MSA). The lattice constants
and other relevant parameters of the alkali metals are given
in Table 2-2. The K-Rb alloys were made by first mixing
one gram of Rb with 20 grams of K to get a nominal 2.2“
at % master alloy. More dilute alloys were made by suc-
cessively diluting approximately 2 portions of pure K with
1 portion of the starting alloy to obtain an alloy of con-
centration about 1/3 of the starting alloy. A total of
5 different concentrations of K-Rb samples were made. The
sample details are given in Table 2-1. The residual
resistivity of the nominal 2.2“ at % master alloy was mea-
sured to determine the resistivity per at % of Rb which was
0.172 uQ-cm/at %. The concentrations of other alloys were

determined using this value in conjunction with their

57

Table 2-2.

Parameters of the alkali metals (after Ziman

(1)).

 

 

 

Li Na K Rb Cs
Lattice const.(K at
90K) 3.“7 “.23 5.20 5.62 6.05
Atomic radius (3) 1.71 2.08 2.56 2.77 2.98
6D (K) “00 150 100 52 5“
Fermi energy 8F, for
free electrons (eV) “.76 3.20 2.12 1.81 1.57

 

 

58

residual resitivities due to the impurities. Our RRR and
concentration values are roughly consistent with those of

(23) The homogeneity of the master alloy

Guénault gt 31.
(2.2“%) was checked during the G measurements of the two
2.2“ at % samples in the same sample can. Both samples gave

G values the same to within a few %.

2.5. Error Analysis

2.5.1. Temperature Determination

Our temperature determination was based on the calibra-
tion of thermometer Th2 (GR-200A-30 Germanium resistance
thermometer, referring to Figure 2-7) as described in Sec-
tion 2.3. The calibration of the thermometers was later
checked against NBS superconductive fixed point devices<u3).

In the temperature range between “.2K and 0.009K, seven
superconductive fixed point device transition temperatures
were checked. The transition temperatures and the tempera-
tures indicated by the thermometers were within about 2%,
as shown in Table 2-3. A new calibration of Th2 using the
NBS superconductive fixed point devices and the CMN sus-
ceptibility thermometer for interpolation was obtained and
fitted to a polynomial in the same way as the old calibra-
tion. The smoothness of both calibrations was checked
against the Wiedemann-Franz law using an Ag-Au wire. The

_KR_0R
Lorentz number L - 57 - TKT’ which involved T as well as AT

59

Table 2—3. Transition temperatures of the superconduct-
ing fixed point devices used to recalibrate
the thermometer and the corresponding tem-
perature indicated by the thermometer.

 

 

 

Fixed Point Device Ttransition Tthermometer
Indium 3.“l“ 3.“23
Aluminum 1.179 1.1696
Zinc 0.851 0.8“27
Cadmium 0.519 0.5132
AuIn2 0.20“36 0.200“
AuAl2 0.160“3 0.1569

Iridium 0.09913 0.0967

 

 

60

in its determination, was found for the temperature range
0.06 to “.2K to be within 1% of L0 using the new calibration
and within 2% using the old calibration. This means that
both calibrations are smooth.

In our measurements of %-%% = % %%, the temperature dif-
ference AT was determined by two methods. The first method,
using the Wiedemann-Franz law, as described in Section 2.3.1,

AT = L T' Here R can be measured,to better than 1% and
o

T can be determined absolutely to better than 2%. The
power generated from the heater can be determined to 1%
(limited by the accuracy of the resistance of the heater,
which could be off by a constant multiplying factor). How—
ever, in the case of imperfect thermal isolation, a certain
percentage of the heat does not flow through the Ag-Au
alloy, thus AT is less than %B%. The value of AT obtained
from this method was checked :gainst that obtained from

the direct determination using the thermometer readings.
The uncertainty of AT in the latter method is primarily due
to the error in the thermometer calibration, whereas the
percentage error in AT depends on the magnitude of AT and
the sensitivity of the thermometer. For T < l K, the
sensitivity of our thermometer is sufficient to resolve

AT to within a few percent for %;'N 20%. When there is
significant heat loss, the temperature of the sample as
indicated by the thermometer is only an upper bound of

the true sample temperature. Since annealed Ag wire

61

(RRR m 160 to 1600) is used as thermal link to the sample,
this upper bound should be close to the true value. Thus
to first order, the error in AT caused by any undesirable
heat flow can be neglected if AT is determined directly
with the thermometer. (A new sample holder with improved
thermal isolation had been made after some sample cans for
the pure samples were found to have significant heat loss,
and the data showed that this assumption was reasonable.)
In the next section, estimates of possible heat losses are

given.

2.5.2. Heat Loss

Possible paths for heat losses in the sample holder
include the electrical leads, the nylon supports and the
stainless steel feedthrough tubes. From the data obtained,
the superconducting leads with copper cladding were found
to cause significant heat loss, whereas those with CuNi
cladding yielded consistent results when the values from
the two methods of determining AT were compared. There-
fore, it is assumed that the heat loss caused by the leads
with Cu-Ni cladding is negligible. Below, thermal conduc-
tance estimates are given of the nylon supports and the
stainless steel tubes.

The thermal conductance of the 0.1% Ag-Au alloy is

given by K = LOT/R. The residual resistance R = 88 00

62

was used in the calculation, and thermal conductance values
at T = “.2 K and 1 K were calculated for comparison with
those of the nylon support and the stainless steel tube at
corresponding temperatures.

The heat conduction path between the two samples through
the nylon supports includes also two copper to nylon con-
tacts which were designed to have weak thermal contact.
However, we consider the worst case by taking into account
only the thermal conductance of the nylon block. The thermal
conductivities of nylon at “ K and l K are:(30) K(“K) =
0.125 x 10"3 W/ch and K(1 K) = 2.5 x 10‘5 W/ch. For the
feedthrough stainless steel tubes, the heat conduction
between the Ag thermal link and the sample can includes
also the epoxy seal. Again, we estimate only the thermal
conductance of the stainless steel tube. The thermal con-
ductivity of stainless steel at “ K and l K are(30):

K(“K) = 2 x 10'3 W/cm-K and Kux) = 1.5 x 10‘3 W/cm-K.
Below 1 K, the thermal conductivity of stainless steel
varies about the same as Ag-Au, therefore, the ratio of
the two is expected to remain the same.

The thermal conductances of the Ag-Au alloy, the nylon
block and the stainless steel tube are shown in Table 2-“.
One sees that the thermal isolation due to the nylon block
and the stainless steel only is not sufficient to allow us
to determine T's and AT's to within a few %. Therefore,

it is important to have very weak copper to nylon boundary

63

 

 

 

 
 

 

 

eo.mv m\3 muoaxe.a Rm.ov m\3 muoaxm.a xx: muoaxm.sm moa
eo.av xxx mIOme.H am.mv x\z muoaxm.m x\2 mIOHxHH.H go:
sauma x Aaocum seuwavx Acoaszv x Asanwav x e
A.m.mvx mmcacfieomv x AcoH com
.opfiz 3<Iw¢ on» mo
mocmuospcoo Hassonp on» on moapmh on» cam M H can m : pm was» Hompm mmma

IGfidpm on» cam

QAOQQSm coaz: esp .mpfiz 5<Iw< on» mo mocmpososoo adamonp one .zum canes

6“

contacts in series with the nylon block as well as reducing
the cross section of the nylon block. CuNi cladding super-
conducting wires should be used for electrical leads.

Also, thin wall stainless steel tube or CuNi tube should

be used to get better thermal isolation. A new improved
sample holder has been made and satisfactory results below

1 K were obtained.

2.5.3. Thermal e.m.f. and Johnson Noise

%.%%, the sample was under

Since in the measurements of
adiabatic conditions, self heating was present due to the
measuring current. Therefore, the measuring current must
be limited to a value such that the induced thermal e.m.f.
would not hinder the desired precision. We estimate the error
due to this induced e.m.f. as follows. The Joule heat pro-
duced due to the current I through the sample of resistance
R is Q = 12R. Considering the worst case in which a tem-
perature difference AT is produced by the heat current Q
through the entire sample, then Q = KAT. Since KR/T = LO,

we have

AT=€§T— or TAT=

ons-

The induced thermal voltage

65

- _ . <18--
AV - SAT - GLOTAT Go LO Lo - GQR .

The ratio of AV to the voltage across the sample is then

given by

Therefore, from measurements of G and R, we can choose I

such that AV/(IR) is about 1 x 10-7, the desired precision.
The current I, on the other hand, must be chosen so that the
voltage IR is larger than 107 times the Johnson noise.
Johnson noise is given by vn = /“kTRAf_. In our calculations
we used Af = 1. In Table 2-5, values of AV/(IR) are given
forifluel.5 mm pure K sample (K1) and two K-Rb alloys (2.2“%
and 0.83%) at temperatures T = 0.1, 0.5, 1.0 and 3.0 K for
the currents used in our measurements. Values of vn/(IR)

are given for the same samples at l K. (We were limited

to a maximum current of 50 mA using the current comparator.)

2.5.“. Uncertainty inpO and Sample Contraction

In order to determine the values of the coefficient of
the T2 term for the K-Rb alloys as well as the "pure" K
samples absolutely, one needs to determine 00. For the
alloys, the temperature dependent part of the resistivity

below “ K is less than 1% of 00. Therefore, we could take

66

 

 

 

 

 

Table 2-5. The values of AV/IR at temperatures 0.1 K,
0.5 K, 1.0 K and 3.0 K and the values of
Vh/IR at 1 K for K-Rb 2.2uz, K-Rb 0.83%
and pure sample K 1.
AE Vn (H )-l/2
Sample I (mA) I (°K) IR IR Z
K-Rb 0.1 3x10"7
2.2u% 20 0.5 3.3x10'7
1.0 1.67x10‘7 6.5x10'8
3.0 1.32::10'8
K—Rb 20 0.1 1.25::10‘7
0.83% 0.5 9.1:{10-8
1.0 5.2x10‘8 1.0x10‘7
3.0 5.2)th-8
-10 -8
Kl(l.5mm) 50 0.1 9.5x10 7.6xlO
0.5 2.85::10'8
1.0 6.65x10‘9 2.ux10'7
3.0 3.8x10‘8

 

 

67

p0 = p(u.2 K). In the determination of p(u.2 K), the
geometric factor A/R was determined from the room tempera-
ture resistance in conjunction with the standard pure K
room temperature resistivity. The error due to using this
pure K resistivity instead of the actual alloy resistivity
at room temperature could be as much as 5%. Also, since

no correction was made for the variation of room tempera-
ture, the error in A/l due to the thermal coefficient in
resistivity could be as much as 2%. And the change in A/2
due to surface deterioration was estimated to be about 5%.
The absolute resistance of the sample at room temperature
and that at 4.2 K were measured to within 1%. Including
the change in A/z due to sample contraction from room
temperature to “.2 K, a conservative estimate of the un-
certainty in po is about 10% for the alloys. For the pure
samples, p(u.2 K) - p0 is about 10% of po. Also, pO

changes in time due to annealing of the sample at room
temperature. Including the error in A/z as described above,
as well as the variation in p0 due to quenching of the
sample from room temperature to 4.2 K during the residual
resistivity determination, the uncertainty in p0 is about
20%. However, since we are measuring % g%, the error in

QC does not affect our accuracy. Only the multiplying coef-
ficient of the appropriate power law is affected.

Another possible error is the change of the geometric

factor A/2 due to thermal expansion of the metal at different

68

temperatures during measurements. The linear thermal ex-
pansion coefficient a varies like the specific heat<uu).
For K, a = 83 x 10-6/K at 300°K. At T/eD = 0.01, a is
estimated to be about 6 x lO_9. Therefore, the changes in
A/£ below 1 K can be neglected for our desired 0.1 ppm

precision measurements.

III. THEORY

In this chapter, we review electronic transport theory,
confining ourselves to models suitable for K - a mono-
valent metal with cubic symmetry and a spherical Fermi
surface. The basic theories of resistivity and thermopower
are described and the thermodynamic consideration of the
thermoelectric ratio is discussed.. The present theories
of resistivity and thermopower for K and K—Rb are then re-

viewed.

3.1. Basic Transport Theory

Electronic transport in metals can be described by the

(1).

Boltzmann equation

. 3f

+ + k

-1; . v f - vk - v r = ‘(T) , (3-1)
scatt

where fk is the electron distribution function, E the wave
3f

vector, In?) the rate of change of the distribution

+_1+
function due to scattering and vk — Svkek (8k is the

scatt

electron energy). The distribution function fk in equilib-

rium is given by Fermi-Dirac statistics:

0 _ l
k - (gk_uy/kT 3 (3'2)
l+e

 

69

70

where u is the chemical potential. Likewise for phonons

not in equilibrium,

an

."’ =_—£- _
—v Vrnq 3t )scatt , (3 3)

with equilibrium distribution function n3 given by Bose-

Einstein statistics:

 

1
“Q = th/kT ° (3—u)

Assuming that the phonons are in equilibrium and restrict-
ing overselves to an external electric field E and tempera-

ture gradient 3T only, we have

Walr-
u
:flm
L'IHv

Defining
- O— -
— rk - fk - - ¢k ———-, (3 S)

for elastic scattering we have

71

Bfk

- 3t )scatt

r{<rk-r§)-(rk.-r§,)}ofi' dk'

k!
= fi& f(¢k—¢k,)Pk dk' , (3-6)
where
k' ' k'
pk dk = fk(l-fk.)Qk dk , (3-7)
v
and Q: is the transition probability of the electron from

state k to k'.

If we have the solution fk to the Boltzmann equation,
then the current density and the heat current density are
+ +

J = fe v fk dk (3-8)

5 = 1(e-u)$ fk a? . (3—9)

For small deviation from equilibrium,

f -f0

_ 3:5) = k k = gk (3-10)
at scatt 13k) TZk) ’

and since

72

afo
+ o = k
kak aenv’
Bfo
+ 0-__E_€_-y_+_+ -
vr fk - Be [ ( T )VT Vu] , (3 11)

we have the linearized Boltzmann equation

 

3fo f -f°
- _§€_1<_ i? - [fig-Bow) + e<E - $73)] = k 1‘, <3-12>
1(k)
0 Mi . . _ .. a;
rk = fk — Te—Hkh - [ELTH-vrrnefi - gm . <3-13)
The current density is then
+ + +
J = fe v fk dk
_ _l_ * EA _
_ “"3 ff e v fk hv d8 (3 1“)
1 + afo

 

If e v {- jfi§&(i)$ - [5%E(-VT)+e(E - %§)J%é}de,

where the integral fdA is over constant energy surface 8.
Likewise, the heat current density
0 v
-+ _ .
Q = 13 fr(e-u)§r’{- 3—51('12)'6-[-€—.f“—(-$T)+e<r - -—1-J-)]g£}de.
)4," h E e V
(3-15)

 

73

Defining the transport integral

 

o
+
f’ = 3- Ifr(k)? $(e-u)“(- 35—) 95 de , <3-16)
n 3 86 v

[Nth
+
+ V

and replacing E - —— by E, since the measured E is actually

+
E - %§-, we obtain the macroscopic.transport equations

3 = e E; . E + g-fii - (-VT), (3-l7a)
25 = e Kl - E + %.i?2 - (47%). (3-l7b)

From the definition of the electrical conductivity tensor,

Oi

mun»

, (3-18)
VT=O

we get

03

= e2? . (3-19)

7“

Similarly the thermal conductivity tensor

 

?= —+ > - 1? K1161
- ' '— "' —— :
Iv": 3:0 T 2 Y0
and the thermopower tensor
4+ E 1 we 1
S E {FM = 57f (KlKO ) ’
and the thermoelectric ratio tensor
3 4+ 1
f1?
At low temperatures, K2 >> , so the ratio
K
O

This is the Wiedemann-Franz law. In the case when

scattering predominates and for a metal with cubic

(3-20)

(3-21)

(3-22)

(3-23)

elastic

symmetry

75

such that the tensors reduce to scalars.

 

2 2
L = L = E—-E-= 2.U5 x 10-8 V2K-2
O 3 2
e
«4+
Kn can be evaluated as
++ _ aro
Kr1 - f <I>(€) (— '57:)“
2
2 (e)
_ w 2 3 ¢n
Be e-u
where
+ + +
me) = 1 f mm V(e-u)n 5%

Hw3h

The integral is over constant energy surface s.

., (3-2Ua)

(3-2UD)

76

3.2. Resistivity_p

For a metal with cubic symmetry, the conductivity tensor

reduces to a scalar,

(3-25)

e2 I ' 2
= __§_. Fermi T(k)V dA + 0(kT) .
Aw h surface

+
A(k), the mean free

+
Thus one needs to determine 1(k)v
path. In the nearly free electron model, 0 is simplified to

621'
(3-26)

with
(3-27)

neff 3

T can be determined from

77

 

 

k k k'
- —) = _, = f(g "g v)Q dk' : (3'283)
3 scatt 1(k) k k k
i.e.,
1 gk' k'
+ = f(1 - -—-)Qk dk' . (3-28b)
T(k) gk

In the case of a spherical Fermi surface and elastic scatter-

ing,

Fill-J

= fQ(e)(1 - cose)dA, (3-28c)

where 6 is the angle between E and k', Q(e) = QE' is a func-
tion of a only and the integral is over the Fermi surface.
In general, one cannot solve the Boltzmann equation
exactly, so the variational method, first used tnr Kohler
and Sondheimer, is used. In the variational principle, the

Boltzmann equation is expressed in the form
X = P¢,

where

78

- 1 Pk' dk'
P¢k - ET- f(¢k’¢kt) k

and

”11+ .5... + + 6..
X = - -a—€-V ° ['TIT—(‘VT) + 8(E - ?)J

The solution ¢ is the trial function that gives a minimum

value of

<¢,P¢>
2 ,
{<P,X>}

where

<¢,X> E f ¢kadk .

Applying this method to resistivity, one finds(l)

l 2 k' v
EFT- ff(¢k-¢k') Pk dk dk

o
ark

2
I fev¢n 3;— dkl

 

(3-29)

of.

Nearly all the calculations discussed below were based on

this equation.

79

3.2.1. Resistivity Due to Electron-Phonon Scattering

For electron-phonon scattering, assuming that the phonons

are in thermal equilibrium and considering normal processes

(1)

only, it can be shown that

 

T 5 6D
where
e n
Jn = f D z 2 dz -2
C) (e -l)(1-e )

and GD is the Debye temperature. At high temperatures,

6
D N T
T >> 6D, J5 m (17) , so pph a 5—. At low temperatures,

D

T << 9D, J5 is a constant, so pph

« (6%)5. This is the
T5 law from Block theory.

In the above discussion, only e - ph normal processes
were considered. The Umklapp processes are different from

the normal processes in that there is a reciprocal lattice

vector in the momentum conservation wave vector equation
E' — E = a + E . (3-30)

The normal processes correspond to 3 = 0. Schematic

80

drawings of the electron-phonon U- and N-processes are shown
in Figure 3-1. For a U-process to take place in a metal
with a spherical Fermi surface that is completely inside

the first Brillouin zone and not touching the zone boundary,
one must have phonon momentum with wave.vector at least

amin' The probability of a phonon with a wave vector

+ (-hwamin/kT)
qmin is proportional to e ‘ . Ziman predicted for

the U-processes a contribution to the resistivity of the
(1)

form

U a e-fiwqmin/kT = e-6*/T
pe-ph '

This contribution, however, should decrease much more rapidly
at low temperatures than the contribution from the N—pro-
cesses which have no 3min requirement.

When the phonons are not in equilibrium, there is phonon
drag. It will be shown in Section 3.5.1 that for K, the
phonon drag effect largely cancels out the normal T5 term,

leaving a dominant Umklapp term.

3.2.2. Resistivity due to Electron-Electron Scattering

For electron-electron Coulomb scattering in a simple

metal, assuming an isotropic relaxation time, the N-processes

 

 

Fermi __

surface Bri Ilouin

zone
H.

 

 

 

 

a, Normal processes k, -k=q

 

 

 

 

 

 

 

b, Umklapp processes k'— k = q + G

Schematic diagrams of the electron—phonon

normal and umklapp processes for spherical
Fermi surfaces that do not touch the lst

Brillouin zone boundary.

VINUI'G 3"]...

82

do not contribute to any resistance and the U-processes

(1)

can be shown to produce a resistivity

a kT 2
pe-e (a)
This T2 term is expected to dominate at sufficiently low

temperatures, since the phonon terms decrease much more
rapidly with decreasing temperature.1\similar T2 term can
be obtained for transition metals. In the presence of
electron-impurity scattering, which is isotropic, pe_e
is not affected. Therefore, the coefficient of the e-e
scattering T2 term is expected to be, to first order,
independent of the residual resistivity p0. In the case

of an anisotropic relaxation time, such as in the presence
of anisotropic electron-dislocation scattering, e-e scatter-
ing N-processes could produce a resistivity that is sample
dependent and proportional to T2.(17) Also when there is

electron charge modulation, a lower power law could be

possible.(l5) More details are given in Section 3.5.2.

3.2.3. Resistivity Due to Electron—Impurity Scattering

For elastic scattering of electrons by impurities one

gets a temperature independent residual resistivity be. In

(18) (19)

the case of inelastic scattering, Koshino, also Taylor

83

2

showed that at low temperatures, a small T term linear with

00 could be produced. More details are given in Section 3.6.
In the presence of multiple independent scattering mech-
anisms that can be characterized by relaxation times, T1,

the resistivity can be expressed in the form

3.3. Thermopower S

3.3.1. Diffusion Thermopower

For diffusion thermopower, we have

s = 511.1%. (3-31)
By substituting in
2 (12
KO = ¢(u) + 16- (RT)2[d-;_§¢(€)]e=u ,
K1 = 362- (kT)2[-932- ¢(€)]

de e=u

814

(145)

and keeping only lowest order terms, one gets

2

 

 

2
S = ;?_keT{d£nfife)} = eLOT{d£n§fe)} , (3_32)
€=u e=u
where
w2k2
L0 = 2
3e

This is the famous diffusion thermopower equation, first

derived by Mott.(u5) In the case of an isotropic relaxation

 

time T,
2
e 2 2
o = ——-— '* e TVA e
3 ft(k)vdA = = ————— AA . (3-33)
I” 1" 121311 1231*.
So
1 3A 1 31

Thus Sd is sensitive to the energy dependence of the Fermi
surface and to that of the mean free path at the Fermi sur-

face. Ziman called S the most sensitive electronic transport

85

(1)

property. Depending upon the structure of the Fermi
surface, %2 and gg are usually positive. For a spherical

Fermi surface, we thus expect to have a negative diffusion

thermopower and as T + O,

= -|eILO egg: <3-35)

-l/2

where g = 1 for T a E and g = 3 for r a E3/2. In

general, with an anisotropic 1, one has to evaluate

e2

Uw3fi

 

0(8) = Q: T(k)vdA

and apply Mott's formula (Equation 3-32) to get Sd'

3.3.2. Phonon Drag

 

The above discussion is under the assumption that the
phonons are in thermal equilibrium. This is the case at
high temperatures when the phonon-phonon interaction is
dominant or when strong interactions between phonons and
impurities or dislocations, etc., exist. However, at lower
temperatures when phonons are not in equilibrium, phonon-

electron scattering would drag electrons along with the

86

phonon flow; thus extra thermopower is created. We now
have to solve for the phonon distribution function n

from Boltzmann's equation

6 3mg)
-V o n = _ —
r q 3t scatt

In the relaxation approximation, one can define for phonon-
phonon, phonon-impurity, phonon-boundary scatterings, etc.,

relaxation times Tp,p’ Tp,I, Tp,b, etc., such that

an n -n0
in?) = — ?§——3 , etc.
p,p 9,9
We denote T as the relaxation time for scattering be-

pax

tween phonons and all particles except electrons,

T T T T
p,x pap p’I pgb

 

 

 

 

+ ... . (3-36)

The thermopower due to phonon drag SP). for N—processes only

should then be(5)

87

 

 

P
S C p’e a ijx
+ +
g Pp.x Pp.e Tp.x Tp.e
-l
> 6 a T
For T D’ Tp,p
eD/2T
T < 6D, Tp,p « e

At very low temperatures in a sufficiently pure sample the

boundary scattering T will dominate. At low tempera-

 

 

p.b
tures,
Tpix, + 1 ’
T << 6D, Tp’e << Tp,x, Tp,x+Tp’e
then Sg can be expressed as
C T ' C
v p x v
s = - ’ = - . (3-37a)
N + N
8 3 leTTp,x Tp,e 3 IeI
where Cv is the heat capacity.
6 /T U x
T‘ 3 D x e
c = 9 N k (-—> f ——————— dx (3-37b)
v o 9D <3 (ex-1)2

88

The Debye integral goes to a constant at T << 0D. So Sg

is proportional to T3 at low temperatures and negative.

The phonon drag thermopower in K due to U-processes is

_ *
expected to be of the form e 0 /T as suggested by Bailyn(25)

(22)

and by MacDonald (1960). For a spherical Fermi sur-

face, this contribution is positive as can be seen from
Figure 3.1. The change in electron momentum Ak in the U-
processes is approximately opposite to the phonon momentum
6, thus electrons are moving in the opposite direction to
the phonon flux, causing a positive Sg. Similarly, for

N-processes, the change in electron momentum is parallel

to 3, thus giving a negative S By employing a variational

g'
method, Ziman, as well as Bailyn, derived a simple rule for
a complex Fermi surface involving both U- and N-processes:

If the chord q passes through occupied region, the contribu-

tion to SE is negative, if it passes through an unoccupied

region, the contribution is positive.

3.3.3. Impurity

For allo s .
y , Tp,e is no longer much less than Tp,x

The contribution to SE due to N—processes must be modified

with the factor

89

With the Debye approximation for the phonon frequency dis—

persion, it is shown that(5 )

 

u -x
s = 3 5 (1)3 reD/T '13:“? WW
g e 6D (3 (l-e )
T
where f(x) = T pi:
19.1 13.6

For alloys, the diffusion thermopower SD is affected

by different scattering processes of different relaxation

times Ti. If the resistivity is expressed as
m* 1
9: 227:291’
neffe i i i
then

 

(3-38a)

90

with

—-——) - (3-38b)

This is the Nordheim-Gorter rule.

3.“. Thermoelectric Ratio G

 

G was first introduced by Garland and Van Harlingen.(6)

From Equation 3-22

1
G = e .
K2
Substituting in
2
3 ¢ (8)
K1 = 1— (kT)2 [-—-l§—— ,
Be €=u
2
2 3 o (6)
K2 = "— <kT)2 [—32—- ’
6 as e=u
one gets
i
a = e(3—%°?(El) . (3-39)
e=u

91
8.
Therefore G - L T.
o

This is analogous to the Wiedemann—Franz law, and is true
when elastic scattering is dominant. For metals at very
low temperatures, this condition is satisfied, so one can
get S from measurements of G; G is measured with both
electric current density J and heat current density a
flowing under uniform electrochemical potential ¢ = %+V,

in contrast to S which is measured under the condition

3 = 0. Garland gt a1.(6) showed that from thermodynamic

considerations, the change in the irreversible entropy

production rate AS and that in the equilibrium entropy

irr

production rate ASO are given by

as as 2

- irr irr l J
Asirr - ~§Ef—)V¢=O ( at; )3-0 f(ar) , (3-U0a)

and
As = {-6-3 > - (-v J >
o S V¢=O S 3:0
1 + +

= — T(uTVT°J), (3-u0b)

the Thomson coefficient.

U1

where “T = T %% i

92

AS is then Just that due to Joule heating and AS0 due

irr

to Thomson heat. Since

2
J__ + -+ +
o + SVT . J = 0 when E = 0,

and at low temperatures, S 2 AT, so

Therefore, the Joule heat is simply absorbed by the Thomson
heat, i.e., there is no extra thermoelectric effect in the
measurement of G due to the flowing electric current. A11

discussions of S can then be applied to G using the equation

For alloys where there are multiple scattering mechan-

isms, the diffusion thermoelectric ratio

This is analogous to the Nordheim-Gorter rule, but is not

restricted to the validity of the Wiedemann-Franz law.

93

Denoting by pa the resistivity due to impurities and by

on the resistivity oftflmepure metal,

.

G = p_p = G + jkL—— (G -G ) . (3-Ul)

 

 

i = 3; should give a straight
0a Ob 00

the diffusion thermoelectric ratio

So a plot of G versus
line with intercept Ga,

due to the impurities in the metal.

3.5. The Resistivitygof K

As described in Section 1.1.1, the resistivity of K
above 1.5 K showed an exponential behavior characteristic
of electron-phonon Umklapp processes, as reported by various

groups with high precision measurements. However, at tem-

2 behaviors inconsistent with simple

1.5

peratures below 1.5 K, T
electron-electron scattering theory,as well as a T power
law,have been reported. Below, the theories that describe
well the exponential behavior are given. Various theoreti—

cal models that have been proposed for the resistivity of

K below 1.5 K are also reviewed.

9“

3.5.1. The Resistivity of K Above 1.5 K

The exponential behavior in the resistivity of K can be

understood by the scattering of electrons by non-equilibrium

(“6)

phonons involving N- and U-processes. Kaveh and Wiser,

(47) (48)

Orlov and Frobose calculated the phonon resistivity

for K. Orlov(u7) solved the Boltzmann equations for elec-
trons and phonos, taken into account both N- and U—pro—
cesses, for pure alkali metals. The non—equilibrium nature
of the phonons results in a greatly reduced contribution to
the resistivity due to N-processes. The resistivity he

obtained is

pe-ph W C‘—§'yg e , (3-U2)

with 0* m 0.2 6D. His result fits well with experimental

(12)

data of Gugan, of Ekin and Maxfie1d(11) and of van

Kempen, gt a1.(13) Van Kempen et 1. fitted their data to

the form

0 = p0 + AT2 + BTn e'e/T,

and obtained n=1.0 and 6=l9.9i0.2 K. Kaveh and Wiser<u6),
using the variational method, including the non-equilibrium

phonon (phonon drag) and N- and U-processes of equilibrium

95

phonons, also derived fkn’ potassium a resistivity of the

form
p(T) = pN(T) + ou(T) - og(T)
+ o(T) ,
T+0Yu
with
2 -th/kT
[pu(T)] = AT e . T < l K (3-A3a)
e-ph
-hw /kT
[pu(T)]2 = BT7/2 e 1 . T m 3-6 K (3-u3b)
e-ph

and y m 1. pg(T) is the phonon drag resistivity. The
second form of pu(T) fits well the experimental data in the
temperature range with fiwl/k = 8°K. Another proposal by
Frobose(u8) also fitted well with experiment. He included
isotrOpic electron-impurity scattering and N— and U—pro-
cesses of e-ph scattering with phonons of isotropic spectrum.

The result is

-e/T
p = A <§>3/2 e (1 + 22> , <3-uu)

e-ph 0

96

 

with B = 0.98, e = 17.5 K.
Since
do
——R—e' h = ————“1‘/2 T‘”'5 e‘e/T (1+2.u8 1),
d(T5) 59 9
he plotted
T14 5 c195
1 d(T ) 1
g(—) = zn ( ) vs -
T 1+2.u8 g T

and got a good fit to the data of van Kempen gt gl.(l3) In

general, the form

describes well the resistivity of K down to about 2 K.
Kaveh gt al.(u9) later proposed an explanation as to why
different parameters n and 6 could give nearly equally

good fits. They considered the resistivity in the form

pe£_ph(T) = 4: 0(w,T)dm. (3-u5)

97

p(w,T) is the contribution to resistivity from phonons of
frequency m. In terms of p(w,T), the mean phonon frequency

S(T) is given by

$(T) = -—-4: wp(w,T)dw/€fp(w,T)dw

58‘

Q? wp(w,T)dw/pe_ph(T)

of)”

 

_ -1 a _
- Toe-ph<T7 3(T_1) (Tpe_ph(T)) . (3 M6)

For pe_ph(T) = ATn e‘e/T, one gets
S(T) = e + (n+1)T . (3-u7)

So, as long as 0 + n is constant, the mean phonon frequency

S(T) remains the same. This explains the relation between
different a and n's used in previous analyses.

3.5.2. The Resistivity of K Below 1.5 K

For temperatures below 1.5 K, there are discrepancies

between experimental results and simple theory. According

98

to the theory of isotropic e-e scattering, Lawrence and
Wilkins calculated the resistivity for simple metals and
predicted for K a term AT2 with A = 0.17 pQ-cm/K2. This
term should be comparable to the phonon contribution at
about 2°K and the coefficient A is independent of residual
resistivity p0. However, data of van Kempen, gt a1.(l3),
showed an unexpected highly sample dependent coefficient of

2

the T term in their attempt to fit the resistivity to the

form
p = p0 + AT2 + BTn e‘e/T

from 1.7 K to 1.1 K. Levy, 32 31. (1979),(l6) also reported
a sample dependent T2 term for K between 1.1 to 1.u K.

They attributed the sample dependence of the coefficient

to electron-dislocation scattering, according to a theory
proposed by Kaveh and Wiser.(l7) Kaveh and Wiser considered
the effect of anisotropy on electron-electron scattering.

In the anisotropic limit, such as for a sample in which the
dominant scattering mechanism is anisotropic electron-dis-

location scattering, the e—e N-processes produce a first

N,ani

e-e (T), which is about an order of magnitude

order term p
larger than the second order Umklapp term pg_e(T). Their

result may be summarized as follows

99

_e(T) + pg_e(T), isotropic limit (3-48a)

U pN, ani
N_ :“1 (T) = (T),

(T) + pe_e anisotropic limit

(3-u8b)

ee(T)->o

Denoting p0I and 00D the contributions to the residual
resistivity resulting from electron-impurity scattering and
electron-dislocation scattering, respectively, they pre-

dicted the coefficient of the e-e T2 term to be

p
(— COD 2 (1 + —°—¥-)‘2 . (3-u9)
00 DoD

Ac:

This explains fairly well the sample dependence of the co-
efficient A for data of Levy, gt gt., of van Kempen, gt gt.,
and of Rowlands, gt_gl., assuming the T2 power law is ac-
curately obeyed. 0n the other hand, Rowlands, gt gt.
(1978),(1u) had extended their measurements down to 0.5 K
and found a possible deviation from the T2 behavior, with

a best fit of the form AT1'5 from 0.5 K to 1.3 K. Bishop

(15) provided a possible explanationfkn’this

and Overhauser
apparent Tl'5 dependence with the theory of electron-
phason scattering. Phasons are collective excitations cor-

responding to phase modulation of a charge density wave (CDW).

100

(50) argued that when the charge density in a

Overhauser
metal is not uniform, a deformable Jellium model can be
used in which each electron experiences a sinusoidal po-

tential with the Schrodinger equation

2 + +
(55 + vO + a cos(Q'Y))¢k = E(k)¢k' <3-50)

Charge modulation occurs if the periodic potential is large
enough to sustain a density modulation. This is charac-
terized by a wave vector 5, which is incommensurate with
the reciprocal lattice. Electrons are scattered with the

wave vector equation
+ ‘ + + +
k' - k = q i Q . (3-51)

So a CDWéUmklapp effect occurs and enhances large angle
scattering.
Boriack and Overhauser<51) derived the electron-

phason interaction to be of the form

e¢ = g 2 ¢qfcos[(§+a)°7 - wqt]

q

V

- cost<6-E)-? - wth, (3-52)

101

where ¢q’ w E are magnitude, frequency, and wave vector

q,
of the phason, respectively. With this potential, they
obtained the resistivity, using the variational method
and assuming the Fermi surface to be rigidly displaced in

k-space by the electric field, as

- - 5 (ii. .111” 91
pe¢(T) — A(/2 T/G¢) J5(/§_T_) + B(®¢) Ju(T)
+ c (g: 2 J2 (%$) (3-53a)
where
zndz

 

Jn<x) =4? (3-53b)

(eZ-1)(1—e'z)

is the Bloch-Gruneisen function. With different param—
eters G¢, each term individually fits well the data of
Rowlands, gt_gt., but differs when extrapolated below

0.5 K (Figure 3-2).

3.6. The Resistivity of K-Rb

In 1960, Koshino(18) first showed that a T2 dependent

term should be expected at low temperature due to the

102

 

 

 

 

 

 

 

 

 

‘ 250 .- . 1
zoom/35
A IW4
5
£5 lESC)_'_~k32
J: ‘\'rL5
«)0 '
E2 '00" 0.2 0.4
E
- :5 £5()'-
f.
2
33 ()u=---m- | .—
0: l
I I
l l J
0.0 0.5 - LO |.5
TEMPERATURE (°K)
Figure 3-2. Plot of resistivity vs T for the data of

sample K2C of Rowlands gt_gl.(lu) The fitting

curve with Ju is shown. In the inset,

extrapolations of Tl's, J2, Ju, J5 fits to

lower temperat%§gs are shown. (After Bishop
.)

and Overhauser

103

inelastic scattering of electrons by impurities. Using
the variational method, with the transition probability

+ +
P(k',k) given as

l6w3Z2eu giE-E' 2
q d0) {p2+|'12_'12'|2}2

 

[n6(E'-E-fiw) + (n+1)5(E'-E+hm)] (3-5“)

where Nx is the number of impurity ions, Z the excess
charge of the impurity ion, d the density of the solvent
metal and p the screening constant of impurity potential,
the resistivity is obtained as

2

a z x T2 J <3-55)

pinel 2

where

2.“ for T + 0
2 in T...
T D °

So, at high temperature, the additional resistivity is
proportional to T and at low temperature, it varies as T2
and is linear in the impurity concentration. Later, Taylor

(1962)(19) pointed out that the impurity potential is

10“

displaced by the thermal motion of impurities and thus the
resistivity due to elastic scattering is reduced. Koshino
(1963)(l8) responded by calculating the reduction in
resistivity due to electronic scattering to 2nd order and
showed that at high temperature, the additional resistivity
due to inelastic scattering is cancelled out by this reduc-
tion, but at low temperatures, this reduction only reduced
the magnitude of the impurity concentration dependent T2
term by a factor of 2 from his previous estimate. Taylor
(196“)(19) then recalculated this term. By considering
first order processes in the inelastic scattering and zeroth
and 2nd order processes in the elastic scattering only,

and assuming the impurity to have an atomic mass M close to

that of the solvent, he obtained an expression for the in-

elastic resistivity

2 2 2
= W h kax 2 _
pinel DOT , (3 56)

3
2M k 6D

where Kav is an averaged wave vector close to 2kF (kF =
Fermi wavevector), M the host ionic mass and 9D the Deybe

temperature. He estimated for a highly impure specimen

~ -2 T‘ 2 T‘ 5
D - 10 (55 00 + 500 (a; 09 + 00,

105

where the T5 term is the normal Bloch-Bruneisen lattice
resistivity.

For K, the T5 term is not found, because of the phonon
drag effect. With Rb impurities in K, the closest atomic
mass one can get for the host metal, it would be a good
test to see if the predicted pinel term is observable.

Frobose and Jackle(55) calculated for K and obtained

_ -5 2

(20)

Kus and Taylor, using a different approach, obtained

for K-Rb the coefficient of pinel as
2

= 1.25 x 10’5 p T

pinel o

3.7. The Thermopower of Pure K

 

The thermopowers of the pure alkali metals were thoroughly

investigated by MacDonald gt gt, between 2 and 20°K in

(21) and below 2°K to 0.1 K in 196o.(22) To explain

1958
the experimental data between 2 and 20 K Ziman in 1959(2“)
calculated the thermopowers of the alkali metals at low

temperature using the variational principle. In Ziman's

notation, the lattice thermopower

c P

1 L IL

—— (-——)'——- (3-57)
Na 2Nk PLL

(DIP?

106

where

P - fff( ) qu' dk d+ d+
IL " ¢k-¢k' ¢q R q k:
and
p = fff(¢ )2 qu' dk dq dk' + other terms
LL q k '

By considering both normal processes and Umklapp processes

of the phonon drag effect and also with an anisotropic

Fermi surface distortion factor a, which is progressively
increasing from the lighter to the heavier alkali metals,

he was able to explain with fair agreement the data for Na
with e = 0 and K with s between “ to 7%. However, the large
positive values of Rb and Cs were not explained until later
(1961) when the e-phonon interaction was treated sufficiently
within the Ziman framework by Collins and Ziman.(52)
In Ziman's theorv, the contributions from the N— and U-
processes are not separable. And later, from deA experi-

(7 )

ments, the Fermi surfaces are found to be spherical to
1 part in 1000 in Na and K and only about 1% in Rb and Cs.
So the necessary Fermi surface distortion (m7% in K) in
Ziman's theory is not supported by experiment. On the other

hand, Bailyn(25) in 1960 treated this problem by taking

into account the non-equilibrium effect of the phonon

107

distribution and including both N— and U-processes. His
phonon drag thermopower is expressed as
[Jq] 3N

- 1 +.+ + + + .+ __O_ _
sg - - 5113qu E a (Jq.q+K)fi(q+K) vqw 3T (3 58)

where q is the phonon wave vector, J the polarization, K

the reciprocal lattice wave vector, NO the phonon equilibrium
distribution and a is the fraction of the phonon—phonon inter-
action among all other processes not involving e1ectrons,such
as boundary scattering,impurity, etc. No Fermi surface
distortion factor is included in his theory. 8 can be

g
separated into

sg = s; + SE“, (3-59a)
with
8%" = - 3NT5T3 g hq°qu-%gg = - 3%; (3-59b)
and
+ 1 + + + + + 3N0
sg = - §WTEng Kio a (Jq;q+K)h(q+K)-vqw TFF’ (3-S9c)

8g. is the normal phonon drag term. For Umklapp processes,

108

q+K is nearly opposite in direction to qu, so the contribu—
tion is positive in 88+.' With the inclusion of both longi—
tudinal and transverse polarizations of the phonons in the
U-processes, Bailyn was able to approximately explain the large
positive thermopower in Rb and Cs between 2 and 20°K.

The data for K and Na between 2 and 20°K could also be crudely
explained from the sensitive balance between normal and
Umklapp processes. MacDonald, gt g;.,(22) in 1960 reported
their thermopower measurements from 0.1 K to 3 K. They
measured the absolute thermo-electric force E of the alkali
metals and derived the absolute thermopower S = dE/dT.
Following Bailyn, they fitted their results below 3 K to

the empirical form

-*
s = AT + BT3 + Ce 9 /T,

where the AT term is the diffusion component

Ce1

Sdiff “ TE? ’

Ce1 is the electronic heat capacity, the BT3 term is the

normal phonon-drag component

 

109

and the exponential term is due to the Umklapp processes,
i.e., the positive 8g+ term. For K, they obtained A

t + 0.5 to -1.o V/K2, B m — 0.15 to - 0.30 V/K” and 9*

m 21 K. In 1975, Orlov<u7) calculated the transport prop-
erties of pure alkali metals by solving the Boltzmann equa-
tions for the electrons and phonons, including the Umklapp
processes. He obtained for Rb and Cs below 2.5 K, the posi-

tive thermopower

k (9D)2 k6*2 -6*/T

s rt Al FT — —— F(T) e (3-60a)

T TEF

which can change sign at lower temperature, and for K,

the negative thermopower

k kT

T' 2 _
EF 3 (——> 1 . (3 60b)

This describes well the data of MacDonald, gt gt.

3.8. Thermopower of K-Rb

In 1961, Guénault and MacDonald<23> reported their
experiments on thermoelectricity in K and its alloys below

3 K. Following the theoretical expressions of Klemens and

110

of Sondheimer,(53) they used as a first approximation to

the phonon drag thermopower

 

 

u -x
g e 6D o (l-e )
_ 22
where x kT
and
P
f(x) 0: 2:9
Pp,x + Ppae
T
= p,x (3-62a)
+
T0,): T10,6

is the fraction of phonon-electron scattering in all scatter-
ing of phonons such as phonon-phonon, phonon-impurity, etc.

When phonon-impurity scattering is dominant,as in alloys,

p,e

l
m Tp,I « S—E (Raleigh scattering limit) and T
q

T
p,x

a l/vq, so

1 l
f( ) = —--— = -—-—j§ (3-62b)
x 1+qu3 l+(§)x

111

In the case when electron-phonon is dominant, f(x) + l, and

(1) = l Clatt
g 3 Ne

power. In the presence of impurities, Sél) is quenched to

we have S , the total normal phonon drag thermo-
a smaller value expressed by Equation 3-61.

In their analysis, GuEnault and MacDonald obtained
qualitative agreement with data with appropriate values of ¢.
For quantitative agreement, they used a second approximation,
allowing differences between longitudinal and transverse

polarizations of phonons in the form

2 l l 2 l
s; ) = § sé ) (0L,¢L) + § Sé ) (0T,¢T) . (3-63)
Assuming Sg was negligible below 0.5 K, they determined
the value of Sd' The data below 3 K, after subtracting

Sd, were then fitted to S with appropriate values of

(2)
8
¢L and 0T, with 6L = l30°K and 9T = 90°K. The Umklapp
phonon drag was taken to be negligible below 3 K. The
relaxation time for phonon-impurity scattering was ex-
pected to be proportional to AMX/MO, where AMx is the mass
difference between the impurity and the host with mass Mo'
However, from the values of ¢, Na and Cs were found to
scatter more strongly than expected from the mass dif-
ferences, probably due to greater misfit of the atomic
volumes of Na and Cs in K.

The diffusion components were anomalously positive for

K-Rb and K-Cs. From a Gorter-Nordheim plot for K-Rb, they

112

8

obtained sdiff = 2 x 10‘ V/K at 3 K. Also, as indicated

from the G-N plot, no appreciable change in the Fermi
surface took place even up to 25% Rb in K. From the ex-

pression of Sd

1 3X
+f‘E] ’

3A
a _
e-u

l
s = — |e|LOTEK3— (3-614)

d

we see that there is anomalous energy dependence in the
scattering mean free path. In order to have a positive
Sd, it is required that the relaxation time t a E'2.
Later in 1968, Thornton, Young and Meyer(26) calculated the
thermopowers of alkali-alkali alloys and obtained satis-
factory agreement with the results of Guénault and Mac—

Donald. They expressed the change in diffusion thermo-

power AS due to addition of impurities as

S 1 + iL. , Ag alne ° (3 65)
Ap F

Ap is the corresponding change of resistivity. As T + O,

Ae: >F
8F

 

S + AS m - lelLOT (3-66)

113

To calculate (A€)F, they used the phase shift method and

expressed A0 in the form

AD 3 - 12n3hN 2
‘6’) " mu— ZQIYZ-l "' Yfil (3'67)

where

l o
-l; (e2inl — e2in1)
21
The phase shifts n: and n% are phase shifts determined

from experiments. An approximate expression was derived

for A€)F:

z<22+l>lv2|k 53E lvll

z(22+1)ly2|2

 

(Ag)? approx = 2 -

Using the phase shift data given by Meyer, gt a1.,(56)

the correct sign of Sd was obtained.

IV. THE EXPERIMENTS AND RESULTS

4.1. Resistivity for Pure K

 

The experimental procedure is as described in Section
2.3. Five sets of resistivity (%»%%) measurements were
made on the pure samples. Two sets of measurements were
made on the same 1.5 mm sample (Sample Kl) with the
sample being cycled back to room temperature for two weeks
in between the two runs. In the first run for the sample
(Kla), measurements were made from u.2 K to 0.3 K using
the temperature modulation technique. In the second run
(Klb), measurements were made primarily below 1.6 K. The
temperature modulation technique was used down to 0.18 K.
The temperature sweeping technique was used from 0.25 K to
0.07 K to determine the resistance ratio C as a function of
T, and the data were then differentiated to obtain %-%%-.
One set of data was taken on one 3 mm sample (Sample K2)
from 0.2 K to 0.8 K using the temperature modulation tech-
nique and from 0.07 K to 0.25 K using the temperature
sweeping technique. Another 3 mm sample (Sample K3)
was measured using the temperature sweeping technique in
three temperature ranges. 0.08 K to 0.19 K, 0.08 K to 0.H2 K

and 0.27 K to 0.U6 K. A third 3 mm sample (Sample KM)

11“

115

with improved sample holder, was measured from 0.37 K to
1 K using the temperature modulation technique and from.
0.088 K to 0.h3 K using the temperature sweeping technique.

For temperatures below 1.5 K, the values of AT obtained
using the Wiedemann—Franz law and those from direct therm-
ometer determination (AT') were compared. In Table “-1,
the ratios AT/AT' at different temperatures for the 1.5 mm
sample (K1) and the 3 mm samples K2 and K4 are shown. One
sees that the deviations are typically 20% to 35% for samples
K1 and K2. This means that there was significant heat loss,
so we can only consider the data obtained using the Weide-
mann—Franz law as qualitatively valid data. For T < 1.0 K,
the data obtained using the measured AT's are considered as
reasonably close to the actual results (especially for the
3 mm samples), as described in Section 2.5, and justified
in more detail below. For Sample K“, a new sample holder
with better thermal isolation and a silver thermal link
with much higher conductance were used. Heat loss was
reduced significantly. For all our samples, our quanti-
tative analysis is based on the data below 1.0 K using the
measured AT's.

In Figure H-l,‘% %% data using the Wiedemann-Franz law
from the first run on the 1.5 mm diameter sample (Kla)
are plotted on a logarithmic scale against % in the range
u.2 K to 1.2 K. Above 2 K, we found the usual exponential

behavior. Similar results extracted from the resistivity

116

 

 

 

 

Table “-1. The ratios of AT from the Wiedemann-Franz law
to AT' from direct measurements for the pure
sample K1, K2 and K4.

Sample T (K) AT/AT' Sample T (K) AT/AT'

Kla (1.5 mm) 1.4H8 1.H9 K2 (3mm) .2016 1.07

1.322 1.46 .2U99 1.08
1.212 1.37 .2627 1.11
1.107 1.50 .3070 1.2M
1.018 1.h2 .3u98 1.15
0.903 1.h1 .U003 1.21
0.820 1.38 .h588 1.31
0.739 1.37 .5115 1.3M
0.657 1.35 .5A66 1.26
0.585 1.32 .6127 1.26
0.515 1.3” .6972 1.28
0.M68 1.3N .7933 1.23
Klb (1.5 mm) 1.018 1.N2 KN (3mm) 0.37”“ 1.01H
0.5129 1.2M 0.U632 1.018
0.M671 1.29 0.5295 1.019
0.u288 1.27 0.6707 1.022
0.3069 1.19 0.8205 1.038
0.3336 1.2a 0.8650 1.051
0.3750 1.22 0.9U58 1.05u
0.2539 1.19 1.0217 1.068
0.2192 1.16
0.1983 1.16
0.173“ 1.18

 

 

117

 

   

<- [ii- 8: Maxfield

' 41v. Kunpen etal

5
T

 

 

 

Figure “-1. Plots of log (% g%) vs % for sample K1.
Similar results extracted from data of
van Kempen et al.(13) and of Ekin and Max-
field<ll) are also shown for comparison.

118

(11) and of van Kempen gt a1.(13)

data of Ekin and Maxfield
are shown for comparison. We see that the exponential de-
pendences of the data from different groups are similar.
Data below 1.6 K using the Wiedemann—Franz law for the two
runs of the 1.5 mm sample (K1) are shown in Figure “-2. One
sees first that the two sets of data differ from each other
significantly for temperatures above 0.4 K (i.e., due to
cycling of the sample back to room temperature, and at the
time of the measurements, we had no way to determine how
much the residual resistivity had changed.) and that the
exponential term is rapidly diminishing in importance

below about 1.3 K. Figure “-3 and Figure N-H show data
below 1.1 K obtained with the two different AT determination
methods for the samples K1 and K2 respectively. The general
shape of the data remains the same for both methods, which

2

suggest a temperature dependence lower than T and higher

than T for the resistivity of the 1.5 mm sample. This is
qualitatively in agreement with results of Rowlands gt a1.(lu)
which showed a possible Tl'5 dependence. For the 3 mm

sample (K2) a T2

dependence cannot be ruled out due to the
larger fluctuation of the data. The open squares are

results from differentiating data from the temperature
sweeping method. Good agreement is observed between the

two different methods. To have a closer look at the tempera-

ture dependence of the resistivity at lower temperatures,

we plotted the resistance ratio C vs T below 0.25 K, as

119

 

 

4 .- -
3 .-
ldp
—— It -
9°" ’ "1.:
2 -
(16 31‘) .:.. ‘2.“-
l "' ,‘.~
0 1 L 1 1 L 1 L 1

 

 

Figure “-2. Plots of %-%% vs T below 1.6 K for the two

runs on K1 (Kla and Klb).

120

.Mmm.o Sodom Aevo wcfipmfipcmmoMMHo Song omcfimpno dump ohm
mohmsvm ammo .3mH NcmpmlcnwEmcmfiz on» Eopm B< wcams osm B< popSmmoE
a

mean: sumo anon wcfizonm .Hx oHQEmm mom ma soamn B m>.wm_fl mo mpoam .ml: madman

3 no 9.0 to «.o o
4

 

q _ d d 4 o
or”.
own
no 0 ”a. o .l fiv—
_su_ h—II>) o duo 0 ..MD. mum
52.253. .... -Tm- v

. . .wmm
_

. ...c
. 3.38:. 55

 

 

 

121

mommsom coco

.xmm.o zoamn Aevo wefipaapcmaoeefio edge emeadpno dude mam

.zmH NcmhmlzcmEmUmHE osp Soap B< ucfimz cam B< venomwoe

a
mafia: mono anon wcfizonm «NM mamemm pom Ma zoaon B m> %m A mo muoam .al: ohswfim

o.—

O. v .p
«.0

«.0

o

 

 

0.0 To
_ q

—

O O

.... ...: as.
2 5:. 1r .

O

.0

o lf/ll hh‘

. . 3.33... 5:.

0

.fi

0
' co

 

 

o

122

shown in Figure 9-5 and Figure “-6. For both the 3 mm (K2)

2 fit below

2

and the 1.5 mm (K1) samples, we see a possible T
about 0.2 K and the data gradually turn away from this T
dependence toward a lower power above 0.2 K. The coef-

2

ficients of this possible T term for the samples determined

from data below 0.2 K are:

2.61 x 10‘13 Q-cm/K2

A(Kl)

A(K2) 2.57 x 10‘13 Q-cm/K2.

Due to the uncertainty in QC (i20%) as discussed in Section
2.5, the uncertainties for these values are about 20%.

In order to determine whether a single power law for
the resistivity is possible up to about 0.5 K, we measured
another 3 mm sample (K3) using the temperature sweeping
technique. A new thermometer calibration (checked against
NBS superconducting fixed-point devices) was used in parallel
with the existing calibration to check whether the devia-
tion from T2 is possibly due to the temperature calibration.
Figure 4-7 shows C vs T and T2 for the set of data in the
temperature range from 0.08 K to 0.U2 K using the old and
new temperature calibrations. One sees that both calibra-
tions yield similar results, and that the deviation from T2

behavior is not due to the temperature calibration. The

new calibration is used in the following graphs. Figure “-8

.Aee m.Hv Hm madame soc mmm.o Sofioo me one B n> o co wooed .muz ossmHm

c. I
mad do «3 ..o
_ q 4

A

 

mOA. o
. 1

low—

0

 

 

3

. . . Loo.
. . .. u
1. End _ n .. Susilmoo. .o

 

 

 

.AEE mV mm mademm mom xmm.o zoaon NB 6cm 9 m> 0 mo mpon .ml: mpdwfim

A o. F
mad «6 m3 1 ..o .
d d .

 

no.0 o
. a

O

 

 

 

 

‘Or-
In
x?

......”
N
o

125

.on soap

unpnfiamo manpmmodfiop 30: van A.V soapManHmo enduwpoQEop UHo wcfim: numb

anon wcfizonm AEE mv mm oHQEmm hog Mmz.o soaon

m

 

 

 

 

a one B n> o no noofid .su: magmas

3.:
to no «.o ..o - o
.. .. ..omk
h .. .. o ... ... Omflk
.. ....1 ones
.. .. . .1 u
.. ..omoaaod
Ll - P p P b b _ b -
N.ocau o. o

126

.mpcoEoLSmmoE mo mpom oops» mcfizonm ASE my mm oHQEmm you 9 m> 0 mo mpoam .wlz madman

m

nomxom _

Lawns
. . 1r 0 o L
Omsk .. «... . .. . iommh
T .. ..v 1

omwkr. .TJ . LOQON

one? . .88

 

 

127

shows C vs T2 in three temperature ranges, 0.08 K to 0.19 K,
0.08 K to 0.u2 K and 0.27 K to 0.46 K. Since the reference
sample was kept at different temperatures for these three
sets of measurements, we can only compare the temperature
dependences or slopes of these data. They are in excellent

2

agreement, as can be seen from Figure “—8. The T behavior

is not observed for the whole temperature range (80 mK to

H60 mK). The data below 0.25 K are plotted against T and

T2 in Figure u—9 for two sets of measurements. A T2 be-
havior is possible at the lowest end of the temperature
range, with A(K3) = (3.2:0.3) x 10-13 Q-cm/K2, similar to the

data of samples K1 and K2. A least square fit was made on

the set of data in the range from 80 mK to 420 mK to de-

termine the best single power law TN. N was found to be

1.910.03. Figure u-lo shows plots of c vs T2, T1°9 and
T1.8

In Figure 4-11, plots of C vs T, TL9

and T2 for sample
K“ are shown from 88 mK to N30 mK. One can see that the
behavior is similar to the results of Sample K3 (Figure U-7
and Figure N-lO). Since the heat loss problem was prac-
tically solved below 1 K for Sample KA, and the results

are similar to the data of Sample K3 where there was sig-
nificant heat loss, our assumption that the measured T's
were close to the actual T's for Sample K3 verified for

this temperature range. In Figure u-12, the results of

% %% 0f samples K3 and KM (obtained by differentiating the

128

.mpcosthmmoE no open
03p wcfizonm .AEE mv mx oaaemm mom m mm.o zoaon me can a .m> 0 mo mpoam .ml: opswfim

 

 

 

 

 

C: ... A .
mad 0N0 m —.0 90 no.0
q A q o h.+
. u 1 own
1 005
. . 1 000
v o o o+.h .1 “V
p _ _ e 1 §k¢omd
«.23 m a _ o

129

e .m.He . e n> o oo wooed .oau: enemas

.mm oedema too xma.o sodoo m.H m

a... S sup
0’ 0

a- 1 «.31 cons

 

— q

0—x0N

000k

coon
u

0000fimo .0

 

 

 

 

130

.:x mHQEMm ho0 mm:.o BOHmD B .m.HB «NB m> 0 mo mpoam .HHI: mkdwfim

110

may

0.:

 

 

w

5

NA. —A. 0

 

 

 

000m

000m

130

.:x oedema toe xme.o soHoo e .m.He .me as 0 do wooed .HHI: enemas

VA.

0A0

0.:

~20 00 0

 

 

1W

4

 

 

 

000m

000m

131

one mHOQE>m Como

.Aevo mnenefionenenneo none oenfieeoo eneo

Q

 

.zn one mm .mn eoaoeee non me nofieo a e> w no wooed .mHu: eneman
3:.—
o._ «.0 06 Yo ad 0
. T 3 . _ _ T 4 0
inc.

 

 

10¢

 

 

132

temperature sweeping data, using the new temperature cali-
bration) and the 55% data of Sample KN (obtained from the
temperature modulation method, using the new temperature
calibration) are shown together with the %-%%-data of Sample
K2 (using old temperature calibrations). One sees that

the results are similar. A straight line passing through

2 law cannot be fitted to the

zero that corresponds to a T
data in the entire temperature range shown to within the
uncertainty of our data. However, the data of Sample KN
from 0.3 K to l K are consistent with a T2 law. For Samples
K3 and KN, the resistivities p(N.2K) were determined during
the measurements with a newly added Sn-In standard resistor
which was mounted on the l K pot. This Sn-In standard
resistor turns superconducting when the l K pot is pumped
down to 1.3 K, so no deterioration in noise and sensitivity
takes place. In this way, the value of p(N.2K) for a
particular run can be determined accurately (aside from the
uncertainty in A/l, which is a constant multiplying factor).
Since the temperature dependent part of the resistivity of

K at N.2 K is about 10%, we can correct for pO accordingly.
From the possible straight line fit to the data of KN from

2

0.37 K to l K, the coefficient of the T term is found to

be

A(KN) = (2.7i0.27) x 10'13 Q-cm/KZ.

133

This is consistent with results of samples K1 and K2
derived from data below 0.2 K.

To summarize, we have measured %-g% on 3 mm diameter
and 1.5 mm diameter pure K samples down to 70 mK. Reason-
ably good data were obtained for Sample KN below 1K. The

2 dependence down to 0.37 K. From

data show a possible T
0.N2 K to 0.08 K, deviations from a T2 dependence were
observed and the best fit to the data was with T1'9. The
data for other 3 mm samples K2 and K3, for which significant
heat loss were found, show similar behavior when determined
using measured AT's. The data for the 1.5 mm sample (K1)

2

show larger deviations from a T dependence below 1 K.

However, since there was significant heat loss, this result

2

cannot be solidly confirmed. Possible T dependences were

found below 0.2 K for all the pure samples.
In Section N.5, our data below 1.0 K for the pure
samples are compared with currently available theoretical

models.

N.2. Resistivity for K-Rb

We measured %-%% on five dilute (3 mm diameter) K—Rb
alloys of concentrations 2.2N at. %, 0.83%, 0.32%, 0.13%
and 0.05%. We obtained the usual exponential behavior for
T > 2 K for all samples. This is illustrated in the log

(% %%) vs %~plot for the 2.2N% and 0.32% samples in Figure

13N

N-l3. We also calculated %»%% using the measured AT below
1.N K and compared with those obtained using Wiedemann-
Franz law. The deviations between AT using the two methods
are shown in Table N-2. For those samples (2.2N%, 0.83%
and 0.32%) with Cu-Ni cladding superconducting leads, devia-
tions are less than N% in the region where the thermometer
is sensitive enough so that relatively accurate AT can be
determined. For the 0.13% and 0.05% samples with copper
cladding superconducting leads, the deviations as shown for
the 0.13% sample are similar to those of the pure sample
where the same kind of leads were used. As discussed in
Section 2.5.1 and Just above, the measured AT should be
reasonably close to the actual AT of the sample. Therefore,
the measured AT's are used in the analysis for T < 1.3 K.
No significant change occurs in our data when the new tem-
perature calibration is used, so data using the old tempera-
ture calibration are analyzed in the following.

We fitted the data for the 2.2N%, 0.83% and 0.32%
samples from 0.1 K to N.2 K, assuming a resistivity of the

-0*/T

form 0 = pO + AT2 + BTp e , where the exponential term

is assumed from results for pure K. Hence

19.22_1_d
p dT p d
0

2A B P-l-0*/T 0*

2— +-— P-—

DOT DOT e ( T)

135

 

 

 

 

Table N-2. The ratios of AT from the Wiedemann—Franz law
to AT' from direct measurements for the K-Rb
0.13%, 0.32%, 0.83% and 2.2N% alloys.
Sample T (K) AT/AT' Sample T (K) AT/AT'
K-Rb 0.13% 0.33N5 1.10 K-Rb 0.83% 0.N7N6 1.0N
0.N010 1.1N 0.606 1.0N
0.5120 1.17 0.650 1.0N
0.5877 1.11 0.717 1.0N
0.6N03 1.19 0.8395 1.0N
0.6983 1.19 0.8917 1.00
0.8016 1.19 1.113 1.07
0.950 1.22 1.2Nl 1.08
1.092 1.28 1.338 1.09
1.25N 1.17 1.508 1.08
K-Rb 0.32% 0.3872 0.99 K—Rb 2.2N% 0.5362 1.03
0.N7N5 0.98 0.618N 1.03
0.5788 1.08 0.6888 1.03
0.6N71 0.99 0.8225 1.0N
0.7201 0.99 0.9829 1.03
0.81Nl 0.99 1.1NN3 1.13
0.9212 0.98 1.3268 0.986
1.017 0.99
1.155 1.01
1.336 0.98

 

 

136

 

 

.3211

I .
2.241

 

1111114

 

1° 0.2

Figure N-13.

0.4 0.6 0.3
1 J. ‘
1- ( x)

1 d
Plots of log <3 5%) vs F}. for K-Rb 2.211%
and 0.32% samples.

137

We obtained the best fit with p l. The values for B and

0* obtained from the least square fit are shown in Table N-3.
One can see that B is roughly independent of po and 9*

is approximately the same as that of the pure sample. The
fitting curves are shown for the 0.32% specimen in the tem-
perature ranges 0 to N.2 K, 0 to 2.5 K and 0 to l K are
shown in Figure N-lN. Below about 1.3 K, the exponential
term becomes small compared to the other terms as in the
pure samples. Like the pure sample KN, good straight line
fits passing through zero can be seen below 1.3K for all

the K-Rb alloys, as shown in Figure N-15. This implies that

p is proportional to T2 with p in the form

o = 00 + AT2, T < 1.3 K,

The slope of the straight line is equal to 2A/po. Table N-N
lists the values of 2A/pO and A for all the samples de-
termined with measured AT's. The values for the 3 mm pure
specimen (KN) obtained from an approximate T2 fit below

1K are included. A plot of A versus pO is shown in Figure
N-l6. A good linear relationship between A and p0 is ob-
tained. So we can express the AT2 term in the form

2 _ 2 S
AT ‘ Ainel po T + AxT ’

where S 2 2. The first term on the right hand side is

138

Table N-3. The parameters from the best fits to the term
-ex/T
BTe for the K-Rb 0.32%, 0.83%, 2.2N%

 

 

 

samples.
0 p0 (Q-cm) B(0-cm/K) 0* (K)
‘7 u 8 ‘9
0.32% .56x10 .9 x10 19.6
0.83% 1.N2x10-7 N.5x10'9 19.32

2.29% 3.87::10‘7 6.N6x10-9 18.75

 

 

139

Table N-N. The coefficients A of the T2 terms for the K-Rb
alloys and pure K sample KN.

 

 

 

C pO (Q-cm) 2A/pO (K-Z) A (0-cm/K2)
2.2uz 38.7x10“8 1.78::10‘5 3N.Nx10-l3
0.83% lN.2xlO-8 1.95xio'5 13.85
0.32% 5.60x10'8 2.1131(10'5 6.80
0.13% 2.35x10-8 3.55::10'5 1.17
0.05% 9.97::10‘9 6x10‘5 3.0
0 (KN) l.NNx10-9 111.6x10'5 2.7

 

 

1N0

 

<5
01
safe.»
V”
l I

..-
0"
T

 

 

 

 

 

 

 

\
O
1
\
L.‘
T
r—

 

 

li f
\

A
O:
C”
II”
F

 

 

 

THO

Figure N-lN. The fitting curve for 99 data of K-Rb
2

0.32%, assuming 0 = 00 + AT + BTe'e*/T.
Shown in three temperature ranges: (a) 0
to N.2K, (b) 0 to 2.5K, (c) 0 to 1K.

0* = 19.6K.

0H4
Q.

1N1

 

//’
//

 

 

() I l 1 All. 1 11. 11 1

l
. 0 0.4 0.8 1.2 1.6
' mo

Figure N-15. Plots of 8'88 vs T for T < 1.3K for K-Rb

alloys 2.2uz, 0.83%, 0.32%, 0.13% and
0.05%. Typical error bars are shown for
the 0.13% sample.

1N2

.maoaam
nmlx can now on unseewm coupoao Ego» NB man no ¢ ucoHOHmmooo one .mal: opzwfim

A EYO .159 Van

V m N
_ _ _ A O

O

 

 

 

 

1N3

consistent with the impurity concentration dependent T2

term predicted by Koshino-Taylor. From Figure N-l6, we

have for the resistivity of K—Rb alloys below 1.3 K,

p = 00 + 8.3 X 10-6 00 T2 + 2.2 X 10-13 T2 9-Cm.

In order to determine the uncertainty in Aincl’ we used

the equation

2A

X
M ‘ E— 00 +2Ainel

A plot of M vs 5; should yield a straight line with slope
o

2Ax and intercept 2A1ne1' This way, the independent errors
in M and p0 can be used in determining the uncertainty in

Aincl' Figure N-l7 shows such a plot. The results are

6 -2

A (8.510.26) x 10' K

inel =

Ax = (2.15:0.32) x 10‘13 0—cm/K2

The result of Ainel is in good agreement with the approxi-

(18)

mate theoretical values predicted by Koshino and by

(55)

Taylor (m10-5)(19). Frobfise and Jackle calculated this

T2 term for pure potassium and obtained Ain 1 = 1.8 x 10"5 K-

(20)

2

Also, Kus and Taylor calculated for K-Rb alloys and

lNN

o
rm. @2583 @8539 Adam. ....v z oaoam one .NHI: @81me

.. On

3. :LZUWCNCC w.
MW 3 .V N O

_ _ _ _ _ _ _ T A e

 

 

 

 

 

1N5

obtained

_ -5 -2
Ainel - 1.25 x 10 K
The agreement between experimental results and theoretical

results is good.

N.3. Thermoelectric Ratio for Pure K

The experimental procedure is described in Section 2.3.2.
The G data fortfluepure samples are shown in Figure N-18
along with data for the K-Rb alloys which we will discuss

later. The data up to N.2 K were fitted to the form

_*
2+9 e/T

G = G0 + AT T e .

We obtained 9* = 23:2 K. This is consistent with the um-

klapp processes "scattering temperature" obtained by Mac-
l.(2l,22).

Donald gt Also, this is in rough agreement with
the result for pure K. The fitting curve is shown for the
1.5 mm sample in Figure N-19. In Figure N-20, data below
1.1 K are plotted versus T and T2. One sees that G can be

expressed as

G = G + AT 3 T < 1.1 K

1N6

 

.xm.s nofieo nsofifie omum one one we one an eefioeee non eneo o

0:...

n ¢. 0. N —
d1 -

 

O O
O C
‘ ‘
o a
O O O
N 1
1 {I 4

 

 

 

.wHI: mpswfim

 

 

1N7

I E
o u 3 w I

+ B< + ow n o wcfiesmme .Hx oHQEmm mo open 0 map pom o>n50 wcfippfim one .mHIz opzwfim

m
AMY?
O

 

«1-
(‘0

N
A

__‘F

 

 

 

1N8

.x N.H zoaon mm one HM moHQEMm how me one B m> 0 mo mpoam

.omu: ennwfin

 

 

 

 

 

 

 

1N9

From the graph, we get

- 1
GO - - 0.03i0.03 V

A = - 0.30:0.01 -l;— .

V-K2
Since we are in the elastic limit, the thermopower can be
determined by S' = GLOT and compared with results of Mac-
Donald gt al. In Table N-5, the results for our samples
are shown. The results of MacDonald et_al. are also shown
for comparison. They fitted their data below 3 K to the

form
-*
S = A'T + B'T3 + C e /T

Since 8' = GLOT, we have A' = GOLO, B' = ALO. Because of
the difficulties involved in separating the diffusion term,
normal phonon drag and umklapp phonon drag terms from their
data, their values of A' and B' are of high uncertainty.
Since we can observe directly the normal phonon drag term

from our data (T2 dependence of G), our uncertainties in A'

and B' are much smaller.

150

 

 

 

Table N-5. Comparison between results of our data and the
results of MacDonald et a1. (22) for the pure
K and of Guénault and—MacDonald(23) for the
K-Rb alloys, assuming
8' = GLOT
_ 2 Q -0*/T
- (G0 + AT + T e )LOT
_ *
=A'T+B'T3+C'e9/T
This Work MacDonald, et a1.
19"8V +
Pure K A (10 K5) -0.07N-0.07N +0.5 to -l.0
B'(10-8 XE) -0.735i0.025 -0.15 to -0.30
K
0*(K) 23:2 N21
RRR N800 mSOO
Diameter (mm) 1.5 or 3.0 ‘30.15

free hanging

encapsulated in glass

 

K—Rb

Sdiff (10‘8 K1)

imp 3.53
T=3K

 

 

151

N.N. Thermoelectric Ratio for K-Rb

In Figure N-18, the G data for all the K-Rb alloys are
shown together with those for the pure samples. In Figure
N-2l, the G data for the alloys only are shown in enlarged

scale. Assuming the form

as described in Section 3.N and Section 3.8, one sees that
phonon drag is quenched more and more as the impurity con-
centration increases. The diffusion term G0 is positive.
The G data below 1 K are shown in Figure N-22. In Figure
N—23, the data below 1 K for the 2.2N%, 0.83%, 0.32% and
0.13% alloys and that for the 1.5 mm sample (K1) are plotted

vs T and T2. One sees that the normal phonon drag com-

N

ponents Gg are roughly the same for the alloys and the pure

sample, although the temperature dependence of G:

2

is no

longer T for the alloys. The values of the diffusion

term G0 are obtained by extrapolating the data to T = 0
and are given in Table N-6. A Gorter-Nordheim plot is shown
in Figure N-2N. The relationship

0

+—E(G

G = G
o imp 90

pure - Gimp)

is roughly observed and the diffusion thermoelectric ratio

152

 

 

 

 

Table N-6. The diffusion components GO of the pure

samples K1 and K2 and the K-Rb alloys.

95'2“ 0 (l) 1 lo8 1
Sample (10 Q-cm) 0 V 0(N.2K) Q-cm
Kla 0
1.5 6.67

Klb -0.0310.03
K2 1.73 -0.06 5.78
K—Rb 0.05% 9.97 +0.N35 1.0
K—Rb 0.13% 23.5 +0.N66 0.N26
K-Rb 0.32% 56.0 +0.N7N 0.179
K-Rb 0.83% 1N2 +0.N76 0.070N
K—Rb 2.2N% 387 +0.N85:0.3 0.0258

 

 

153

 

 

.4 -
.2 '— OAO. o.
' .0AD .0
L- '.
O 4. j1-.ir.1 ”.1.
1 .. ‘ ° 2.24%
G V — . .

O
A
P
O A ‘
32 ‘
A
A. ‘
° 0
O
r- ' o o

 

 

 

UK)

Figure N-21. G data below N.2K for the K-Rb alloys.

15N

.enofiae omun on» non mm.H soaoo e n> o no wooed

.mmue onnmfin

 

 

 

 

155

 

 

'. AAA I l I I
. A A A T
4 *- <- . 2.24%
317 ° “ . it
1 ° ° ° o l _
«VZF ° 0 " .83
o D O .0 o J
4 _ ° 13 T a
<- . ° 0 .32 j.
.3 - . “V
4a.. 17‘," v D.
.4 b . ‘7 v T

 

 

 

 

2

Figure N-23. Plots of G vs T and T below 1K for the
K-Rb 2.2N%, 0.83%, 0.32% and 0.13% alloys
and pure sample Kl.

156

.va Ho.oees.o+

 

IIFIT

 

 

 

F
on
o

 

.zmlz mpswfim

157

characteristic of Rb impurities in K is found to be

H

Rb in K
1 _ .. -8 V
The pseudo-thermopower S — GO LOT - 3.53 x 10 K at
imp
T = 3K can be compared with the result from data of
Gruénault and MacDonaldC23)
' -8 v
S - 2 x 10 - .
diff Imp) T=3K K

Since they neglected the effect of phonon drag below 0.5 K
in determining their diffusion thermopower, and from our
data, phonon drag is not negligible even at the lowest
temperature, the discrepancy between our results for the
diffusion thermopower is understandable. To illustrate, we
consider G = G at 0.5 K, then

0

s G L T = 2.9 x 10‘ K at T = 3K.

diff = o o

The discrepancy could be reduced if the appropriate con-

tribution of phonon drag is taken into account.

158

N.5. Discussion and Summary
N.5.1. Resistivityifor Pure K

We have measured the temperature derivatives of the
resistivities (% g%) of three 3 mm diameter and one 1.5
mm free hanging pure K samples down to 0.07 K. (Previous
reported measurements on K were only down to 0.5 K).

The results are given in detail in Section N.1. In brief,
our data below 1 K for the 3 mm samples (especially KN)
show a possible T2 dependence from 1.0 K to 0.N K, but
there is complicated behavior below 0.N K. In the range

1.9i0.03

from 0.08 K to 0.N2 K, best fits with T were

2

observed, and below 0.2 K, possible T fits were again

found. For the 1.5 mm sample (K1), the temperature de-
pendence was found to be lower than T2 from 1 K down to

at least 0.2 K. Below 0.2 K, the data are consistent with
a T2 dependence.

In this section, we compare our data for the pure K
samples below 1 K with currently available theoretical
models. Below 1 K, the contribution to the resistivity
due to electron-phonon scattering becomes negligible small.
From data above 1.5 K, we obtained the component of

Del-ph’ and below 1 K this is at least three orders of

magnitude smaller than the total temperature dependent

part of the resistivity (at 1 K, lg—g—élflmq x 10‘7, to

be compared with % %% W N x 10-”). So our results are

159

not affected by the phonon effect. Currently, there are
several theories for the resistivity of K below 1 K, as
described in Section 3.5.2, namely, isotropic\e-e scat-
tering, anisotropic e-e scattering, and electron-phason
scattering. We compare our data with these theories as

follows.
(a) Isotropic e—e Scattering

For simple isotropic e-e scattering, the contribution
to resistivity is expected to vary as AT2 and the co-
efficient A for a given metal should be independent of
other scattering mechanisms. For K, Lawrence and Wil-
kins(3) calculated A = 1.7 x 10‘13 0-cm/K2. For the data

2

of our pure samples, such a T law is not able to describe

our data over the entire temperature range up to 1 K.

For the 1.5 mm sample (K1), the temperature dependence of

the resistivity is lower than T2 and higher than T, except

below 0.2 K. For the 3 mm sample (KN), the data from

2

0.37 K to 1 K are consistent with a T law, with a co-

efficient.
A(KN) = (2.7eo.27) x 10‘13 Q-cm/K2

Similar qualitative behavior was observed for samples K2

2

and K3. There are deviations from a T dependence below

0.NK. For the 3 mm samples K3 and KN in the temperature

160

range from 0.08 K to 0.N2 K, the best fits to the data

1.9i0.03. 2

are with T A T fit is not possible to within

the uncertainties of the data. However, below 0.2 K,

from plots of c vs T2, the data for K1, K2 and K3 show pos-

sible T2

dependences. (There are insufficient data to make
such a plot for Sample KN). The coefficients were de-

termined to be

A(Kl) = (2.61:0.5) x 10"13 0-cm/K2
A(K2) = (2.57:0.5) x 10'13 0-cm/K2
A(K3) = (3.2:0.3) x 10‘13 0-cm/K2.

The coefficients A(Kl), A(K2), A(K3) and A(KN) of the pos—

sible T2

terms are consistent with the theory of simple
isotropic e-e scattering, i.e., they are of the right
magnitude and roughly sample independent. However, there
are presently no satisfactory explanations for the ap-
parent Tl'9 dependence for the 3 mm sample and the dif-

ferent behaviors of the 1.5 mm sample and the 3 mm samples

above 0.2 K.
(b) Anisotropic e-e Scattering

Kaveh and Wiser(l7) proposed a theory considering
the effect of anisotropy on e-e scattering. When the
dominant scattering mechanism is anisotropic, for ex-

ample, electron—dislocation scattering, then the e-e

161

N-processes produce a T2 term which is dependent on the
residual resistivity due to this anisotropic scattering.

With this theory, they explained why the data of Levy,
(l6) 1.<13)

t 1.

and of van Kempen, 23 showed sample

dependence when the T2 law is assumed to be accurately

2

obeyed. However, the possible T terms from the data of

our pure samples do not show significant sample dependence
as in other's results. Even though for the 1.5 mm sample,
our data show changes in magnitude for the two runs with

the sample being cycled back and annealed at room tempera-

ture for two weeks in between, the changes are small, and

2

the T law was not well obeyed. Since we were measuring

.1.
p
(we were unable to determine the change in no at that time).
2

g%, these changes could be due to the change in 00
The coefficients of the possible T terms from our data
do fall within the range of values predicted by Kaveh and
Wiser. Since we have no systematic way to determine the
ratio poI/poD (see Section 3.5.2) for our samples (they
are free hanging therefore relatively stress free and the
changes in pol/00D could be small for our samples), we
cannot make a rigorous test of this theory. More high
precision measurements on samples with controlled

pol/DOD are necessary to clarify this. The deviations

2

from T behavior are not predicted by this theory.

162

(c) Electron-Phason Scattering

Bishop and Overhauser(15) proposed the theory of
electron-phason scattering to explain the possible Tl°5
behavior of the data of Rowlands gt a1,(lu) (see Section
3.5.2). For our 1.5 mm sample (K1), the data appear to show
similar behavior below 1K. In an attempt to fit our data

2

0
for K1 assuming p = p0 + AT J2 (fig) and using the same

"phason temperature" 8 (= N.85 K) as. obtained for the

0
data of Rowlands et a1., we obtained only a fair fit, as
shown in Figure N-25. In order to have a better fit,

6¢ was varied until a best fit was achieved. The best
value of 0¢ = 6.5 K. Figure N-26 shows the fitting curve
with 0¢ = 6.5 K. However, a better fit is inevitable when
there are more parameters. And for the 3 mm samples, a

T2 fit is possible from 0.N K up to l K, thus a similar
kind of fit with J2 (z?) would not be appropriate. Since
the sample diameter of the samples of Rowlands, gt 91.,
was 0.8 mm, and the results of our 1.5 mm sample showed
closer similarity to the results of Rowlands et_al. than
to that of our 3 mm samples, such a deviation from T2
could possibly be associated with a size effect. Measure-
ments on more different diameter samples would be neces-
sary in order to sort out this possibility and to ascer-

tain whether the theory of electron-phason scattering is

appropriate for the resistivity of K.

oe

 

0.8... w
5.5 4:93 No N.2 + on u a 9.253.. J: 223 5. .6 Boo hymn. no“. «:3 9.33.. of. mm-.. .953“.
. . . 0:... . .
0. mo 00 to N o o
_ 4 _ a _ A _ a A 0
.. A
.. J m
m. J u—
Tomi
L 2
.. ....m c
Q _

 

 

 

 

16N

.nm.o u so noes
..e\oev me an + oo u o enanneee .nH soaeo fix no eoeo we m_ou had seen .emus enemas

Av: .—
0.— 0.0 0.0 V0 «.0 . 0

_ _ _ _ _ _ 1 . . 0

 

 

 

 

 

165

(d) Conclusion

 

From the above discussion, we conclude that the T2
law due to simple e-e scattering provides a better expla-
nation for our data below 1 K than other available
theories, even though none provides perfect explanations.
(More extensive measurements as suggested above are neces-
sary in order to ascertain the validity of the various
theories.) Although there are deviations from a T2 law,
for which no satisfactory explanation is presently avail-
able, the dominant scattering mechanism that is respon-
sible for the temperature dependent resistivity is best
described as isotropic e-e scattering. From the results
of the K—Rb alloys, as shown in Section N.2, the T2
dependent resistivity due to inelastic electron-impurity
scattering is at least an order of magnitude smaller than
the T2 term in our pure K samples below 1 K. So the in-
elastic electron-impurity scattering only contributes a

2

small correction to this T term in the pure metal. The

2 law could be due to the size effect

deviations from the T
associated with electron-boundary scattering, but more
systematic measurements are necessary to clarify this

point.

166

N.5.2. Resistivity for K—Rb Alloys

We have measured %-%% on five dilute 3 mm diameter K—Rb
alloys and found a dominant T2 term below 1.3 K that is
linear with residual resistivity. Below 1 K, the electron-
phonon resistivity becomes at least three orders of mag-
nitude smaller than the total temperature dependent part
of the resistivity, so we were able to study the dominant
scattering mechanism separately. Below 1.3 K, we obtained,
for the K-Rb alloys,

0 = 0o + (8o5i0-26)X10-600T2 + (2.15:0.32)x10-13T29-cm.

The term (8.5i0.26)xlO-6p0T2 is best described by the

theory of inelastic electron—impurity scattering. The
coefficient is close to theoretical values. We believe
that this is the first time the Koshino-Taylor term is
unambiguously being observed. The other T2 term (2.15

0.32) x 10'13 T2 Q-cm was found. This T2 term which is

independent of pO could be associated with e-e scattering.

N.5.3. Thermoelectric Ratio for K and K-Rb

We have measured thermoelectric ratio on free hanging
pure K and dilute K-Rb alloy samples down to 80 mK with
higher precision than that which has been attained before.
For the pure samples, the thermoelectric ratio was found

to be of the form

167

0 = 0 + AT2 + g

-93
e /T

From the data up to N.2 K, we obtained for the umklapp
phonon drag term 9* = 2312 K. We were able to separate

2

the diffusion term GO and the normal phonon drag term AT

below 1.1 K unambiguously and obtained

—_ 1
GO - 0.03:0.03 v

A = -0.30:0.01 —3;§ .

V-K

Our results when converted to thermopower are consistent

(22) which showed rather

to the results of MacDonald, et_al.
large uncertainties in the diffusion term and normal
phonon drag term.

For the K-Rb alloys, we found the positive diffusion

thermoelectric ratio characteristic of Rb impurities in K

to be

H

G0 = +0.N8i0.01 — .

Rb in K

.<

This is consistent with the results of Guenault and Mac-
Donald<23> when converted to thermopower and if the phonon

drag contribution is properly treated. Below 1 K, the

168

normal phonon drag term is roughly the same as in the pure
metal but the T2 behavior is no longer accurately obeyed.
The normal and umklapp phonon drag terms are quenched more
and more as the impurity concentration increases, similar

to the data of Guénault and MacDonald.

REFERENCES

10.

11.

12.

13.

1N.

15.

REFERENCES

J. M. Ziman, Electrons and Phonons. (Oxford University
Press, London, 1960)T9

 

J. H. J. M. Ribot, J. Bass, H. van Kempen, R. J. M.
van Vucht and P. Wyder, Phys. Rev., to be published.

W. E. Lawrence and J. W. Wilkins, Phys. Rev. B7, 2317
(1973). ‘—

Thermoelectric Power of Metals (Plenum Press, N. Y.,
1976) F. J. Blatt, C. L. Foiles, D. Greig and P. A.
Schroeder.

R. D. Barnard, Thermoelectricity in Metals and Alloys
(Taylor and Francis Ltd. London, 1972).

J. C. Garland and D. J. VanHarlingen, Phys. Rev. B10,
N825 (197N).

M. J. G. Lee, Phys. Rev. 118, 953 (1969).

D. K. C. MacDonald, G. K. White, and S. B. Woods, Proc.
R. Soc., A235, 358 (1956).

G. G. Natale and I. Rudnick, Phys. Rev. 167, 687 (1968).

 

J. C. Garland and R. Bowers, Phys. Kondens. Materie 9,
36 (1969).

J. W. Ekin and B. W. Maxfield, Phys. Rev. B3, N215
(1971)-

D. Gugan, Proc. R. Soc. (London) A325, 223 (1971).

H. van Kempen, J. S. Lass, J. H. J. M. Ribot, and P.
Wyder, Phys. Rev. Letters 31, 157N (1976).

J. A. Rowlands, C. Duvvury and S. B. WOOdS, Phys. Rev.
Letters NO, 1201 (1978).

M. F. Bishop and A. W. Overhauser, Phys. Rev. Letters,
N2, 1776 (1979).

169

16.

17.

18.

190

20.

21.

22.

230

2N.

25.
26.

27.

28.

29.

30.

31.

32.

33.

170

B. Levy, M. Sinvani, and A. J. Greenfield, Phys. Rev.
Letters, N3,1822 (1979).

M. Kaveh and N. Wiser, J. Phys. F: Metal Physics, 10,
L37 (1980).

S. Koshino, Prog. Theor. Phys., 2N, N8N (1960); Ibid,

25, 10N9 (1960); Ibid., 39, N15 T1963).

L. Taylor, Proc. Phys. Soc., g9, 755 (1962); Proc.
Soc. A275, 209 (1963); Phys. Rev. A135, l333‘(l96N).

3W. Kus and D. W. Taylor, J. Phys. F, to be published.

SJ'J’U

’11]

U

K. C. MacDonald, W. B. Pearson, and I. M. Templeton,
Proc. R. Soc. A, 2N8, 107 (1958).

D. K. C. MacDonald, W. B. Pearson, and I. M. Templeton,
Proc. R. Soc. A256, 33N (1960).

M. Gruénault and D. K. C. MacDonald, Proc. R. Soc.
A, 26N, N1 (1961); Ibid., 27N,15N (1963).

ED

q

M. Ziman, Phil. Mag., N, 371 (1959).

3

Bailyn, Phil. Mag., 5, 1059 (1960).

E. Thornton, W. H. Young, and A. Meyer, Phys. Rev. ,
166, 7N6 (1968)

C. Uher, C. W. Lee, and J. Bass, Phys. Letters, 61A,
3’411 (1977).

F. J. Blatt and H. G. Satz, Helv. Phys. Acta 33, 1007
(1960).

B. Llewellyn, D. MCK. Paul, D. L. Handles, and M.
Springford, J. Phys. F: Metal Phys., 7, 25N5 (1977).

0. V. Lounasmaa, Experimental Principles and Methods
Below 1 K (AcademIE’Press, London and New York, 197N).

 

H. London, Proc. Int. Conf. on Low Temp, Phys. (Oxford,
p- 157, 19517}

H. London, G. R. Clarke, and E. Mendoza, Phy. Rev. 128,
1992 (1962).

P. Das, R. DeBruyn Ouboter, and K. W. Taconis, Proc.
9th Int. Conf. on Low Teyp, Phy_. (Plenum Press, London,

9. 1253, 1965).

3N.
35.

36.

37.

38.

39.

N0.

N1.

N2.

N3.

NN.

N5.

N6.
N7.
N8.
N9.

50.

171

G. Frossati, J. Physique C 39, 1578 (1978).

O. V. Lounasmaa, J. Phys. E: Sci. Instrum., 99, 668
(1979).

J. E. Zimmerman, P. Thiene and J. T. Harding, J. Appl.

M. P. MacMartin and N. L. Kusters, IEEE Trans. Instrum.
and Meas. IM-15, 212 (1966).

D. Edmunds, W. P. Pratt, Jr. and J. A. Rowlands,
Rev. Sci. Inst. to be published. Current Comparator
#9935, available from Guildline Instruments, Ltd.,
Smiths Falls, Ontario, Canada K7A NS9.

J. A. Rowlands and S. B. Woods, Rev. Sci. Inst. N7,
795 (1976); J. F. March and F. Thurley, Ibid., 99:

616 (1979).

J. W. Ekin and D. K. wagner, Rev. Sci. Instrum. 9;,
1109 (1970).

J. L. Imes, Ph.D. Thesis, Michigan State University,
197N. G. L. Neiheisel, Ph.D. Thesis, Michigan State
University, 1975.

BASF Catalyst R3-11, available from Chemical Dynamics
Corporation. Hadley Road, P. O. Box 395, South Plain-
field, New Jersey 07080.

National Bureau of Standards, Superconducting Reference
Materials (SRM 767, 768).

See for example, M. J. Sinnott, The Solid State for
Engineers, John Wiley & Sons., Inc., p 295, 1961.

 

 

N. F. Mott and H. Jones, The Theory of the Properties
of Metals and Alloys, Clarendon Press, Oxford, 1936.

 

M. Kaveh and N. Wiser, Phys. Rev., 99, N053 (197N).
V. G. Orlov, Sov. Phys. Solid State, 96, 2175 (1975).
K. Froste, Z. Physick B 99, 19 (1977).

M. Kaveh, C. R. Leavens and N. Wiser, J. Phys. F:
Metal Phys., 9, 71 (1979).

A. W. Overhauser, Adv. Phys. 91, 3N3 (1978).

51.

52.

53.

5N.
55.

56.

172

M. L. Boriack and A. W. Overhauser, Phys. Rev. B17,
N5N9 (1978)-

J. G. Collins and J. M. Ziman, Proc. R. Soc. A, 26N,
60 (1961).

P. G. Klemens, Handb. Phys. 1N, 198 (1956). E. H.
Sondheimer, Canad. J. Phys. :9, 12N6 (1956).

B. D. Josephson, Phys. Lett. 9, 251 (1962).

K. Frobfise and J. Jackle, Proceedings of EPS Study
Conference, C-lN, 1977.

 

A. Meyer, C. W. Nester, Jr., and W. H. Young, Advan.
Phys. 16, 581 (1967); Proc. Phys. Soc. (London) 2g,
NN6 (1967).

APPENDIX

APPENDIX A

SQUID

(Superconducting Quantum Interference Device)

The basic working principle of the SQUID is based on
the Josephson effects. Josephson (1962)(5u) first pre-
dicted for a system consisting of two superconductors
separated by a very thin insulating layer, the electron
'cooper pairs' can tunnel through the layer, yielding a flow
of superconducting current across the layer at zero voltage.
This is the d.c. Josephson effect. The supercurrent density

JS is given by
JS = JC sin(AGJ) ,

where JC is the maximum current density, determined by the
properties of the Junction, and AGJ is the phase differ-
ence between the phases of the wavefunctions of the cooper
pairs in the two superconductors. If a voltage V is ap-
plied across the Josephson Junction, then the phase change

A03 is related to V by

1714

or
d(A0 )
____1_.= 3:
dt 2" V h
Thus A0J would change in time and so JS would oscillate

with frequency

f = v %§ = N83.6 v (MHz/0V) .
This is the a.c. Josephson effect.

Later, experiments confirmed his predictions and
also other weak links between superconductors were found
to have the same effects.' They are called Josephson Junc-
tions. Figure A-l shows the principal types of Josephson
junctions.

In a SQUID, the Josephson Junction is part of a super-
conducting circuit (Figure A-2a). The total phase change
A0 along a loop through the Junction and the bulk super-

conductor with total flux through the loop ¢, is

A¢o = - 2%5i'¢ + AGJ = 2Wn .

Writing

Induum contacts

V /

175

 

 

E -Suoerconducfor~|00 nm
0 /w

 

cup-r Superconducting

 

A Gloss substmfe

mock wlfh~ 2 nm Oxide

Crossmg strip

 

 

 

 

r? I; ”#05:

 

\

Gloss substrate

Fone neck
10sz
Superconducting
" metal fulm

~IO' 6m muck

 

 

  

Figure A-l.

\Ot'dvzed Nb
wnre~|mm OIO

Drop of lead-tun
solder

J D'SC bonmv
~'OOnm copper etc

 

 

I
/or ~ IOnm of
/ sermconduc for

 

 

 

 

2mmmo
ND wNe
sharpened to~2ym pom?

Nb post

(d)

Hannah 0‘ ND
/wnre or solder covared
copper wue

The principal types of Josephson Junction.

a) Tunnel Junction.
ductor barrier Junction.

d) AdJustable point-contact.
'SLUG'.

b) Normal or semicon-

c) Dayem bridge.
e) Solder drop

f) Crossed wire or hair pin Junction.

(After Giffard §£.al., Prog. Quan. Electr.,
N, 301 (1976)).

176

Is

Josephson
junction

 

.— 2 —.1L
AG)l —2nn+ 2nq) , (Po-20

. o
’5 :1c sm(A®i)

SCREW

NTT A
LOCK NUT} 0 05C LE

 

F1

 

 

 

 

Figure A-2. a) The Josephson Junction in a superconducting
loop. b) Single Junction, rf biased symmetric.
SQUID of Zimmerman, Thiene and Harding .

177

The quantity

'6-

O
¢'=¢-2}-A6J=n¢o

is the fluxoid and 90 is the quantum flux or fluxon. This
is London's theory of fluxoid quantization in the super-

conductor. Now

JS JC sin(AeJ) = JC sin(2nn+2w $9)

0

JC SiH<21T gfi) 0

With an external flux ¢ext’ the total flux through the

hole inside the superconductor

¢ = ¢ext + LJS

3L
+ LJC sin(2n ¢o) .

¢ext

(30))

From the derivative (after LOunasmaa

178

one sees that if 21rLJC/cbO >’ 1, then do/doext changes sign
from positive to negative through infinity. ¢ is thus a

multi-valued function of ¢ A plot of ¢/¢o vs ¢ext/¢o

ext'
is given in Figure A-3. The operation of SQUID could be
understood by considering the response of o to the external

flux ¢ We will consider a SQUID magnetometer, which,

ext'
in a modified form, is used as our null-detector. The most
widely used SQUID magnetometer is a single Junction, radio-
frequency biased SQUID (Figure A-2b). An inductor coil

er is put inside the SQUID hole. er is driven by a rf
oscillator at its resonant frequency (f0 = 19 MHz) of the
er Crf circuit. er is loosely coupled to the supercon-
ducting ring inductance L through mutual inductance M.
External flux ¢ext coupled to er through the superconduct_
ing ring and changes the effective impedance of er. Then
the change in voltage Urf across the resonant circuit,

which is a function of ¢ can be detected. Referring

ext’

to Figuresjbfi3and A-N for 2nLJC/60 = 5, if the d.c. ¢ext

is first biased at X (o = 00), then by increasing 0

ext ext

from the rf source by increasing the rf level, the flux
would move up along the curve XX', thus Urf would increase
up to Al (in Figure A-N), corresponding to point R (in
Figure A-3. Further increase of the rf level would cause
the Josephson Junction to become normal and a flux Jump
from point R to point S, dissipating the additional energy,

so no increase in Urf is possible. The flux in the super-

179

 

 

 

 

 

 

I V I T I T I
3 0 '-
anJc/¢°.1
2 5 ~ 3\
S\
2 0 *-
O
S.
6
\ 1S '- I -<
& ‘ /
: /’I’ ‘ R
' Ill ..... - w X
9' ’ z
1 r- Y
0 ---- H 1” x o .4
""""" It’l
/ /
O I, .’ 4L
[I
0.5?“ ' l . q
¢'¢Qlt/"l
X/ ,’ P
z
. ’1’ .........
0 ,p ’ 4
I l l l 1 L l l
0 0.5 1.0 1.5 2.0 2.5 3.0
¢efll¢o

Figure A-3. ¢/¢O as a function of ¢ext/¢o in three cases:
21rLJc = ¢o’ 300, and 500. Vertical lines with
arrows correspond to discontinuous changes of
one flux quantum in the fluxoid ¢' and to some-
what sma11§S)changes in the flux ¢~ (After

).

Lounasmaa

180

 

 

UH

 

 

 

 

RF LEVEL

Figure A-N. The staircase pattern. The voltage Urf

across the tuned erCrf-circuit is shown as a
function of the rf level. At the plateaus
the quantity plotted is Urf(max), the voltage
Just before a dissipative transition occurs.

(30).)

(After Lounasmaa

181

conducting ring would follow the hysteresis loops until

the rf level is increased to a level in which there is more
energy than that which is dissipated through the hysteresis
loops, then the flux would again move along S and T, Urf
would start to increase again (along A3 to Aé). A stair-
case pattern of Urf against the rf level thus results (Fig-
ure A-N). If while keeping the rf level fixed, a modulat-
ing field which moves the starting point to the right is
present, flux Jump would occur at smaller rf level, thus

Urf would be smaller. When the starting point is moved

to Z (cpeXt = g 90): a minimum Urf would be reached. Further
increase to point W (which is equivalent to W') results in
higher Urf' If a low frequency modulating field that sweeps
the starting point between X and Z (peak to peak ampli-

tude is ¢O/2) is present, Urf would oscillate from Urf max

to U at the frequency of the modulating signal. Thus

rf min
a triangular pattern in Urf appears (Figure A-5). The
amplitude of this triangular pattern can be detected by a
phase-sensitive detector. A typical SQUID set up is
schematically shown in Figure A-6. Now, external flux
experienced by the SQUID causes the triangular pattern to
shift horizontally. If the output of the PSD is integrated
and fedback to the SQUID to counteract the external flux,
the triangular pattern would be locked on. (The SQUID is

now in a locked-on mode). The feedback voltage is then

proportional to the external flux.

 

“VIV'VN

UN ("‘01,

 

 

 

Figure A-5. The "triangle" pattern. The maximum voltage
Urf(max) vs ¢dc for different values of the
rf level. (After Lounasmaa<30).)

183

 

 

 

 

 

    

 

 

 

 

 

 

i9MHz' 50
C (35C
RF
7ir'
level RF
.AMAP
Tune :> I
J
C]
i 1 feedback
A integrator
[TVAA
g—{D
r-‘x --------------------- 1

4 K environment

 

 

 

 

L____--_-___“.§MQ§§__J

Figure A-6. Typical rf SQUID electronics.