LIBRARY

Michigan State
University

 

This is to certify that the

thesis entitled

ELECTRON TRANSPORT PROPERTIES OF METALLIC TIN
IN HIGH MAGNETIC FIELDS AND AT
LIQUID HELIUM TEMPERATURES

presented by

John Arthur Woollam

has been accepted towards fulfillment
of the requirements for

__ Ph-D- ._degree in.____¥_Ph SiCS

 

Major professor

V

Date 26 January I967

 

0-169

ABSTRACT

ELECTRON TR WSPORT PROPERTIES OF METALLIC TIN
IN HIGH MAGNETIC FIELDS AND AT
LIQUID HELIUM TEMPERATURES
by
John Arthur Woollam

The high field thermal magnetoresistance and de Haas-
van Alphen type oscillations in the thermoelectric emf, are
used to study the Fermi surface of tin.

It is demonstrated that the thermal magnetoresistance,
using lock-in-amplifier techniques, has a much larger signal
to noise ratio than conventional electrical magnetoresistance
yet provides the same Fermi surface topology information.

The thermoelectric emf is shown to be especially sen-
sitive to the quantization of energy levels. Quantum oscil-
lations observed at 4.20K, are as large as 4/uV/OK at 1.2OK
and 20 k gauss in metallic tin. These originate from the
35 section of the Fermi surface, and their angular dependence
is studied. Higher frequencies are also seen.

Oscillations from the 35 section are observed in the
thermal magnetoresistance, and superimposed on a strongly
field dependent monotonic background.

In contrast with conventional Fermi surface investiga-
tion techniques, the thermal magnetoresistance, and the
thermoelectric effects can be studied in one experiment.

The possible effects of magnetic breakdown are observed simul-

taneously in both quantities.

ELECTRON TRANSPORT PROPERTIES OF METALLIC TIN
IN HIGH MAGNETIC FIELDS AND AT

LIQUID HELIUM TEMPERATURES

By

John Arthur Woollam

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1967

Acknowledgments

 

I would like to thank my thesis advisor, Dr. Peter
A. Schroeder, for his supervision, and eSpecially for his
help in the stereogram and data interpretation work.

The assistance and advice of Dr. Frank J. Blatt
is gratefully acknowledged.

I am grateful to the National Science Foundation for

financial support.

ii

 

 

I.

II.

III.

TABLE OF CONTENTS

History of Quantum Effects

The Theory of Quantum Effects

A.
B.
C.
D.

E.

F.

Quantization of Energy

The Degeneracy of the Levels
Oscillation in Free Energy
Oscillations in the Magnetization

Factors Effecting the Amplitude
of the Oscillations

1. Experimental

2. The Effective Mass
3. Temperature

4. Collisions

5. Electron Spin

Quantum Effects in Transport Phenomena

Quantum Theories of Thermal Magnetoresistance

and
A.
B.

Thermoelectric Power
Quantum Theory of Thermal Magnetoresistance

The Horton Quantum Theory of Thermoelectric
Power

The Zil'berman Quantum Theory of Thermoelectric
Power

Absolute Magnitude of the Thermoelectric
Oscillations

iii

IV.

TABLE OF CONTENTS, Continued

High Field Thermal Magnetoresistance and
Thermopower Tensors

A.

History of the Thermal Magnetoresistance
Tensor

Theory of the Thermal Magnetoresistance
Tensor

The Effects of Magnetic Breakdown

Theory of the Thermoelectric Power
Tensor

Kohler Corrections

Experimental Apparatus and Measurement
Techniques

A.
B.
C.
D.
E.
F.
G

Sample Preparation

The Apparatus

The Measurement of Voltages

Pole Pieces and Magnetic Field Homogeneity
Thermometry

Recording the Measurements

Field Modulation

Experimental Results

A.

B.

C.

D.

Quantum Oscillations in the Thermoelectric
Emf and Thermal Magnetoresistance

Quantum Oscillations in the Thermoelectric
Emf when Rotating at Constant Field Strength

Origins of the Oscillations and the F-S of Tin

Thermal Magnetoresistance in Rotation

iv

TABLE OF CONTENTS, Continued

VI. Experimental Results, continued

E. Field Dependent Structure in Magneto-
resistance

F. Correlation of Oscillation Amplitudes
with the Field Dependent Peaks in
Magnetoresistance

Figure Page

 

1

\OCDNONU'I-P—‘WIU

[p A) IO A) In A) +4 F1 +4 F4 +4 F‘ +4 I4 F4 +4
U1 4: \x In #1 <3 \0 03 ~q Ox U1 -: \w n) +4 o

9
12
33
35
A2
43
1+3
48
49
50
55
58
6O
66
67
72
73
75
76
78
80
81
83
84
86

List of Figures

Description

 

Cylinders of energy states

Oscillations in 5n, AU, and 6M

Zinc W(H) and P(H)

Chart of field dependence S(H), (H)
Undulating cylinders type Open orbit

Effect of Open orbits on high field S(H)
Effect of Open orbits on high field S(H)
Sample axies and coordinate system

General view of apparatus

Sample holder

Marginal oscillator

Field homogeneity plot

BIOCk diagram of measuring equipment
Constant AT circuit

Differentiator circuit

Bessel functions

Roots of bessel functions

Oscillations in thermoelectric emf 4.2OK
Integers vrs l/H at peaks

Simultaneous plot of W(H) and S(H) oscillations
Higher frequencies in emf oscillations
Angular dependence of periods

S oscillations in 1800 rotation H=2l k gauss
S oscillations in slow rotation. Higher freqs.

Sece at peaks vrs integers

vi

 

Figure Pagg
26 87
27 9O
28 92
29 93
3O 95
31 96
32 97
33 99
34 100
35 101
36 102
37 103
38 105
39 107
40 115
41 119
42 120
43 121
44 122

List 9: Figures (Continued)

*

Description

 

Summary of W(H) and S(H) frequency data
Free electron Fermi surface of tin

Cross section of 3 5 section of F-S. Weisz
W(H) 21 K gauss 360 rotation

Stereogram of open orbits in tin

4(a) section of F-S. Supports aperiodic
open orbits

Types of orbits

W(H) in rotation for various H's

Same as figure 33, crystal tilted
Electrical P(H) in rotation for various H‘s
Magnetic breakdown orbits. Young

Magnetic breakdown orbits. Young. Weisz
Inner section stereogram. Our data
Correlation of high field W(H) and S(H)
oscillation amplitudes over wide range of
angles. Shows angles of low oscillation
amplitudes.

Intermediate superconducting magnetic state
Temperature measuring bridge

Automatic angb recording device

Pressure regulator

Carbon resistor calibration

vii

Appendix

 

ngg Description

114 Intermediate supersonducting magnetic state

116 a-c wheatstone bridge output correction

118 Magnet sweep control

119 a-c wheatstone bridge circuit diagram

l2O Circuit diagram of automatic angle recording
device

121 Helium bath pressure regulator

122 Carbon resistor calibration

viii

 

 

elec

51,.

V- .1

I History of Quantum Effects

 

At low temperatures and in pure metals and semi metals,
electron mean free paths become long. Under these condi-
tions electrons in high magnetic fields can execute many

cyclotron orbits before being scattered. Thus,

w T >> 1 (1)

where w ='%§ is the cyclotron frequency and T is the aver-
age time between electron collisions. m* is the effective
mass defined by:
h2
-)(- _
mij ’ ‘17—'— (2)

d E
dk dk

i J
Under the conditions of equation (1) de Haas and van Alphen
experimentally discovered that the magnetic susceptibility
is an oscillatory function of the magnetic field strength
witrla frequency proportional to-%. (1) In 1933 Peierls
:showed that the magnetization of a free electron gas should
‘be oscillatory in-%. (2) Soon after that, Landau found an
explicit form of the magnetization as a function of field
strengthb) and Shoenberg and Landau“) recognized the
effect of the periodic field of the crystal lattice by
introducing a generally anisotropic effective electron

mass. JMeanwhile, much experimental de Haas - van Alphen

work was underway(5) on the multivalent metals.

 

 

 

.1.

I

In 1953 Onsager applied the Bohr-Sommerfeld quantization

rule(6):

I<5-§r)-dr=(x+%)h (n

h

where p'is the usual momentum and A the magnetic vector

potential. (3) is the condition for the quantization of

magnetic flux in units of hc/e, and shows that the oscil-

latory diamagnetism reflects Specific geometrical features
th

of the Fermi surface. The l——-maxima in magnetization

counted from-% = 0 occurs when

l _ he

(4)

mud

where A is an extremal area of intersection between the
Fermi surface (F-S) and the family of planes,

'5 o'H = constant. Lifshitz and Kosevich(7) expanded these
ideas, and a rigorous treatment will be given in the sec-
tion on the theory of the oscillations.

A large amount of both experimental and theoretical
Fermi surface work was done in the 1950's. Lifshitz, Azbel
and Kaganov(8) were able to give explicit expressions for
the galvanomagnetic tensor in high magnetic fields for
various Fermi surface conditions. Magnetoresistance and
Hall effect thus became effective tools. Azbel and Kaner(9)
were the first to theoretically consider the resonant
absorption at the surface of a single metallic crystal

when the field was in the plane of the metal. These cyclo-

tron resonance experiments are useful in determining

.
o'AQ‘.
_-.

 

Eu“

4,,
.mt

IE.

.AJ‘H
«v _'

'e-n
\
h

v"

on"
'- T

effective masses at different points on the F-S.

Cohen, Harrison, and Harrison(lo)

developed a theory
for the attenuation of sound by electrons in a metal,
showing the attenuation periodic in -% and the resultant
period prOportional to the extremal wave vector 5,
Acoustic attenuation exhibits both this ”geometrical reso-
nance", and de Haas - van Alphen type oscillations,so
measurements can give both a calipered dimension and a cross
sectional area of the F—S. In addition, the attenuation is
sensitive to open orbit directions, so the method can be
a very effective technique for studying F-S's.

Accounts of the work done in the 1950's are very

(11)

nicely given in The Fermi Surface which is the only

 

book explicitly about Fermi surfaces. A good portion of
the work reported is on the noble metals, copper, silver,
and gold. Until then, quantum effects had not been
observed in thesemetals, since they are monovalent and have
rather large sections of F-S. Work on zinc, aluminum,
bismuth, lead and tin are also reported.

A good recent summary of the de Haas - van Alphen effect
and a review of quantum effects in studying F-S's is given

by D. Shoenberg in LT9(12).

This presents a complete
bibliography of experimental work done before September of
1964, giving 141 references. Band structure calculations
are also summarized by V. Heine and he lists 55 references

to calculations on specific materials.

- a-‘

 

 

a:

1+

In view of the rather rapid advance of de Haas -
van Alphen,magnetoacoustic attenuation, Shubnikov - de Haas
effects, etc., it is not surprising that peOple might look
for quantum oscillations in the tranSport prOperties. Oscil-
lations periodic in-% have now been seen in practically
all observable prOperties. A good deal of this effort was,
and is still being done by the low temperature group at

Louisiana State University(13"l6)

where they attempt to

relate both bulk and oscillatory phenomena using the rela-
tions resulting from definitions of transport coefficients
and the Onsager relations(l7). However, the first quantum

oscillations in the thenmxflectric power were seen by M. C.

(18)
13)

Steel and J. Babiskin in the semi metal, Bi. Since

then, Grenier, et al( saw oscillations in another semi
metal, Sb. They postulate observing oscillations in zinc
but cannot tell if this is due to oscillations in the quan-
tities used to calculate the thermopower from their
measured prOperties(14). The isothermal thermopower is
defined as S = 9i§¥£l.. Our observations in tin have both
positive and negative emfs and since dT can't be negative,
the oscillations are undisputably oscillations in the ther-
moelectric power. We believe that this is the first defi-
nite observation of quantum oscillations in the thermo-
electric power of a pure metal. Furthermore, the above
authors looked only with the magnetic field along symmetry

directions of the crystal. We have observed how the

frequency changes for the field in various non-symmetry

directions, and have

observed the oscillations in continuous rotation at fixed

field.
Quantum oscillations in the thermal magnetoresistance
were seen in several metals by Grenier, et al(13—16).

(See also LT9 reference 12, p. 802)

Theoretically, quantization of tranSport phenomena
is far less understood than is the de Haas - van Alphen
effect, fundamentally because the de Haas - van Alphen
effect is due to the diamagnetism of conduction electrons
in thermodynamic equilibrium. TranSport phenomena necessar-
ily involve electron scattering and non-equilibrium thermo-
dynamics.

The Zil'berman(19) thermopower theory uses the Boltz-
man tranSport equation to treat the lack of equilibrium.
The effects of quantizing magnetic fields on the classical
Boltzman equation were not treated. The Kalashnikov<2®

and Horton<21)

thermOpower theories use the density matrix
formalism to account for the quantization imposed by the
magnetic field. Hewever, these theories use the equili-
brium equation of motion for the density matrix. For a
discussion of this limitation see the paper by Adams and
Holstein(22).

Thomas will soon publish: "Magnetic Corrections to
the Boltzman Equation" in the Physical Review<23). How-
ever, no thermopower calculations were made.

The existing theories will be discussed in greater

detail later.

A.

6

II The Theory of Quantum Effects

 

Quantization of Energy(24’25)

 

Consider a freeelectron in a .uniform magnetic field

parallel to the z axis. Use the magnetic vector potential

in the Landau gauge

73:: (O: HX: 0)

(5)

The appropriate momentum conjugate to T, when a magnetic

field is present is
‘5 = m7 + en

The Hamiltonian is then

2
H=——('16-e'f\')

The Schrodinger equation becomes

h2 2 ieHxh aw _ e2 2

 

_ _____. I 1 =
EVIL m 637 2m XV+EU

Now let

I = I(x) ei(kyy + kzz)

(9) in (8) gives

 

O

2 (hk - eHx)2 h2k
2s VZWX) '{ y 2m "(E ' Tu?) W)

O

(6)

(8)

(9)

ewe-I

 

 

.
'C»

This is the equation for the harmonic oscillator in i(x)

centered on x0, where

Xo z'TEH (11)

with a natural frequency of

_ eH

The quantum mechanical treatment of the harmonic oscillator

(26)

(H.O.) problem is given in many texts so we can write

down the eigenfunctions and eigenvalues as

 

i(k y + k z)

W = e y z X H.O. wave functions of (x-xo) (13)
heki 1

E: 2m +(l+-2—)ha) (14)

wherecm is given by (12).
(19)

I

211 berman solved the Schrodinger equation for the
eigenvalues and eigenfunctions allowing for the presence of
the periodic lattice potential and found the solutions

could still be written in the form of (13) and (14).
(6)

Onsager applied the Bohr-Sommerfeld quantization
Condition
{snowman (15)
to get
1 2weH

C

.'

l
.

V-

T?

a

.‘f

Rb

CO

where A is the cross sectional area in k Space perpendicu-

lar to the field direction. Equation (16) relates the

area to the energy given in equation (14) and will be impor-

tant in the explicit expressions for the quantum oscillations.
Equations (14) and (16) show that there is quantiza-

tion in the x-y plane in k Space and electrons condense

onto cylinders centered along the z axis as shown in

figure 1.

The Degeneracy of the Levels

 

xO must fall within the specimen so

-§Lx<xo<%L (17)

X

where the electron is contained within a box Lx’ L , L .

y 2
Equation (11) then says that
eHL
l X 1 CH
The number of allowed values of ky is thus
eH _
LxLy PET'T degeneracy for fixed kZ (18)

The energy between states of neighboring n values is hm

so the number of states per unit energy range is

m
1.1. ———§-= density of states (20)
x 5’ 27m

This is the same as the density of states in zero magnetic

km uncouth m

.82: £8820
.555 "I 67% stages. 60:93 :o .«o ooze—55 23 pops: mundane E Sexton 1.3.5
.53... 32.5% 35 “En—9&8 05 E 523on $98 o3... mo :omadumeccac 25.

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a ossewm

,-n
0

field so the average density of states is unaffected by
the field. The field condenses many states into highly
degenerate cylinders as pictured in Figure 1.

Now consider a slab in k space perpendicular to the
field direction, of thickness okz. The number of allowed

L

. 2
values of k2 in 6k is-§? bkz and the number of allowed

2
values of ky for fixed kz is given by (19). Thus the
total number of states for a fixed l in the slice 6kz is

the product or

 

eékz
degeneracy = LxLyLz E??? H (21)
E LXLyLz i H
which defines the degeneracy per unit volume for the LEE
level as
degeneracy/vol = g H (22)
where
66kz ( )
§ = 23
Ayah

Oscillation in Free Energy

Assume that the electron gas is at absolute zero tem-
Perature. The Fermi energy can be shown to be nearly
constant as the field varies.

All the l levels in the slice okz are filled when

fi2k2

E; 5 EF _._2ޤ > (x +-%) Mn (24)

 

 

1 an? 15:! III. I. j
-. rt
IL

11

l
and all levels above X are empty. If there are on elec-

trons per unit volume in the slice 6kz then

6n = i'éH + §H (25)

since from (22) §H is the degeneracy/unit volume of the

l
l £21eve1 and 1 = o is a filledlevel.

6n: (i'+1) in (26),

i
Thus an increases linearly with H until l coincides with
I
the fermi level. Then all electrons on the X.ED level

empty out. When this occurs,

(x'+ 7:.) m» = E; (27)

i
and since i is usually very large compared to %3

, mEF 1
A =73?)— n (28)
Thus on is a periodic function of«% with period
he
P =
“ng (29)

SO the pOpulation oscillates about 6no with amplitude
1
géii. This is shown in Figure 2.
The energy in okz is
I
5 A l hekg
U0 = «5H fiwZO. + 2) + Ono 2m (30)

 

m

 

 

AU

 

 

 

1 SM

M5 M; is b M":
'<—— H Increasing

l2

13

Evaluating the sum and using (26) gives

1 has fine on k2h2
on = ——————‘Z + ° 2 ( l)
o 2' §H 2m 3

 

For a slightly different value of H but with the same 1'

6n k2fi2

1 z
5 = ___.5 _______ _
U -§ §H n + 2m + (6no 6n) EF (32)
Call the difference between (31) and (32) AU. Then using
(26) and (27) get

2

AU =.% :2 [ 6n - 5n0 ] (33)

 

which is the change in energy of the whole Fermi surface
due to the slice bkz. It is also a periodic function of
'% with the same period as the number of electrons per unit
volume in the slice ékz, as given in (29). This variation
is shown in Figure 2.

Since the total number of particles in the system
remains constant,the cannonical ensemble is appropriate
andAIIis then the free energy per unit volume. Any other

physical quantity can be calculated from it provided it is
an equilibrium prOperty.

D. Oscillations in the Magnetization

 

For the thermodynamic system considered

BAU

2-1'V1'

.fl" TWA’D' '

In _

 

 

3
_1
CL

LIA»!
humidos

Whez

NOW

and

_eh dbn

6M = mg (on - ono) 171—— (35)

This neglects the effects of scattering from lattice vibra-

tions; that is, the sample is at absolute zero.

53—ng = (x'+ 1); '5 (X'+%)§ = :23 (55)
y
-E

6M = ‘HE (5n - 6no) (37)

The variation of 6M, AU and 6n with field is shown in
Figure 2. 6n - one varies between i -%§H so the range of
6M is 1% g Elf" Thus IBM is periodic in %I with a period
given by (29). Being a periodic function it can be ex-

panded in a Fourier series.

C?

6M = E 6kz Ap sin pX (38)

F=l

where x is dictated by the periodicity as

 

I
m
21rEF
= - -Tr<x<'n'
x ehH (39)
Now 7 w
2
I 6M sin px dx = 6kz E AD I sin px (40)

-w p -w

.‘ l '

‘. o 1
hOd S.
M :
.n‘.‘.
Uub

 

Now I”

So

 

 

h
‘ U.

15

which gives Ap as

Ap:(1)p:hEF

P

(41)

Now sum (38) over all possible k to get

Z

 

ks 22
M7133 iii édkz .131; an [10%; (RF. 2:5] (42)

but

9.15 << EF in most metals, so (43)

the integrand oscillates rapidly as a function of k2 except
I
for kz = ~O. When k2 = O (24) gives EF= EF and (42) becomes

a Fresnel integral and has the value

E eh 21rmeF
M ___ r113 (egH 2211} sin (_THJI- g) (42+)

 

Now for k2 = O (16) and (14) give

hA
" Egg (45)
SC)

.1.
M = E; (geyz {3411” 3142;; .. g) (246)

Thus, the period of the oscillation gives a direct measure

16

of the cross sectional area of the Fermi surface in a plane
perpendicular to the field. This area is measured at a
stationary section, in this case k2 = O, that is,a maximum
or minimum in the area. The period.of the oscillations

is

2we
P = TEA (47)

Factors Effecting the Amplitude of the Oscillations.

 

1. Experimental

If the peaks are separated by AH then the field must
be uniform over the sample to within at least this amount.
The sample must be an ordered lattice free from impurities
so that many cyclotron orbits can be completed. For rea-
sons apparent below,the temperature must usually be below

4.2° Kelvin.

2. The Effective Mass

The effective mass defined by equation (2) must be
small. The smaller this is, the greater the oscillation
amplitude. The 36 section of the fermi surface of tin,
for example, has a ratio of effective mass to real mass of
.09. In some materials this is as small as .003. The
effective mass is directly pr0portiona1 to the radius of

curvature of the fermi surface at an extremum.

3- Temperature

At temperatures above absolute zero the states near

ta 5‘

 

 

where

«r. 3

lower

dozi:

must

I.
LTO:

 

 

 

17

the Fermi level will be partially pOpulated instead of
empty or full. The thermal energy kT must be small com-
pared to the separation between Landau levels tan
Dingle(28) makes a complete analysis of the effect on the

magnetization oscillations and finds the amplitude multi-

plied by

Lp = si:H xp (48)
where

xp = 272p 7% (1‘9)

This factor damps higher harmonics to a greater degree than
lower harmonics. Therefore, the oscillations appear sinu-

soidal rather than spike shaped since the fundamental is

dominant.

4. Collisions

If F'is the average time between collisions then this
must be long compared with the time to complete one cyclo-

tron orbit. Collisions introduce an amplitude factor of

2V
ex ——
p we (so)

5. Electron Spin
The effect of Spin is to double all the levels so far

ctiscussed. The new set of levels contributes the same
fixequency but the effect is to introduce an amplitude

faetor
cos (pram*/m) (51)

 

e I;
3:
Where

4-:
biota

IL)
V

 

18

where p is the order of the harmonic in the sum in equation

(45). g is a "spin splitting factor".

F. Quantum Effects in Transport Phenomena
Pippard<24) extends the theory presented above to

get the expression

2
amplitude N0 = H2(9—%-‘E‘——5) amplitude g4; (52)

where No is the density of states. This shows that the
density of states is an oscillatory function periodic in
% with the same period as the magnetization oscillations.
Fundamentally, it is the oscillatory density of states which
gives rise to the oscillations in all the observed quanti-

-9—%%—£— is evaluated at the extre-

ties. In (52) the term
mal area of the F—S. (52) also shows that higher harmonics
will be emphasized in the density of states oscillations
because of the derivative of magnetization with respect to
field.

Dingle<28) makes a direct calculation of the density
of states by a different method which again shows it to
be an oscillatory function of-%.

If the average time between collisions (r) is prOpor-

tional to the reciprocal of the density of states, then
(52) becomes

 

amplitude T
'i.‘

oz' [Ho

2
(W) amplitude 37M, (53)

H q
1‘-
acre:

 

 

(.1
KO

In simplified models of the electrical conductivity

0 = TEFT' (54)

It is later shown that the electronic contribution to the
thermal conductivity is proportional to the electrical
conductivity. Thus the thermal conductivity also depends
on T.

The diffusion thermOpower as calculated from the

Boltzman equation is

 

2 2
s = 3%? (533%qu ___ EF (55)

Using (54) this gives

s~% Si. (56)

BE

Thus, both the thermal conductivity and the thermoelectric
power depend on T. (53) then shows that they are oscilla-
tory functions of-% with the same period as the magnetiza-
tion oscillations. The oscillatory thermopower, thermal
conductivity, and magnetization should yield the same cross
sectional areas of the F-S.

The above is a very approximate argument and should
not give the correct oscillation amplitudes. PrOper
expressions are found by including quantization in the ini-
tial calculations, rather than substituting a quantum con-

dition into an equation derived not assuming quantization.

 

'4'? 1
AA&

A 7.11.“

 

 

 

 

20

III Quantum Theories of Thermal Magnetoresistance and

 

Thermoelectric Power

 

In this section we quote some of the results of

Horton's Ph.D. thesis(21), but rely on the calculations

(22)

by Adams and Holstein , and Zil'berman(19) for physical

insight into the origin of the quantum oscillations in
(

transport. The Kalashnikov 2O) theory of thermOpower
oscillations, based on the Adams and Holstein paper, is

not presented.

Quantum Theory of Thermal Magnetoresistance

 

Horton(21)

calculates all of the tranSport coeffi-
cients in the quantum oscillation region and gets an equa-
tionikncthe oscillatory thermal conductivity R and an
equation for the oscillatory electrical conductivity 3.
These equations show that the Wiedemann-Franz law is valid
when scattering is by impurities. Near 10 Kelvin where
most of our eXperiments were done, impurities are the
dominant scattering sources.

The theory of Adams and Holstein<22>

gives a better
understanding of the physics involved than does the Horton
theory. We, therefore, present the Adams and Holstein
theory for quantum oscillations in the electrical conducti-
vity and use the Wiedemann—Franz law to relate these to

the electronic part of the thermal conductivity. This is
Justified within the limits of the Horton theory:

~' W2k2 ~

1
Kik:3 e2 T Oik (57)

 

for t:
The 0:
brim:

should

which
the CC

Oscil:

one

Steinl

 

the C,

not f

 

21

Adams and Holstein(22)

calculate the transverse gal-
vanomagnetic effects for the case of weak scattering. They
use the density matrix formalism and arrive at an expression
for the electrical conductivity in quantizing magnetic fields.
The only limitation of this theory is that it uses equili-
brium quantum thermodynamics, whereas transport theory
should treat the lack of equilibrium. Its virtue is that
it is a quantum mechanical treatment.

The equation for the electrical conductivity found by
Adams and Holstein is a product of an oscillatory part and
a term containing the scattering. Seven types of scattering
mechanisms are considered and it is the scattering terms
which determine the magnitude and temperature dependence of
the conductivity. The oscillatory part arises from the
oscillatory density of states which they find is

I
1

n(E) = 2 (2W)-2z———1—— (58)
X + 6

i=0

6 is a function of field and has values between zero and
one. Whenever a Landau level passes through an extremum
of the F-S, 6 = o and (58) diverges. Since Adams and Hol-

stein find the transverse conductivity to be

Oxx = constant X n(E) (59)

the conductivity diverges also. The infinite conductivity

occurring periodic in-% would be observable if it were

not for the fact that the Landau levels are broadened by

22

collisions.

Adams and Holstein make a Fourier expansion of (58)
and (59) since n(E) and Oxx are periodic functions. They
also calculate the effect of collisions, and the effect of

thermal broadening. The result is

I
1
,V E __gyga EWTMflYTa> ‘] (212 - V) 6
0 exp T cos 0
xx ( arr [sinh 21:pr PH 2 ( )
p=o

where T is the average time between collisions. The expo-

 

nential term thus accounts for the effect of collisions, the
bracketed term for the effect of thermal broadening. These
are the same temperature and collision corrections which
appear in theenuation for the oscillatory magnetization (48-
50).

Thus the electronic part of the thermal conductivity
and, therefore, its inverse the thermal magnetoresistance,
are oscillatory functions of-%. The periodicity P, of these
oscillations is the same as the magnetization oscillations

(47) and they determine the same cross sectional areas of

the Fermi surface.

B. The Horton Quantum Theory of Thermoelectric Power

Horton(21) uses the density matrix formalism and assumes
scattering by impurities only. His final expression is in

explicit form and includes the higher harmonics:

 

where

This
he pr

ics :

 

Coeff

 

 

r/J

23

 

 

l
hm E '5 p
,c 152( Fl {-12 212..”
S 1 ‘ 8 2’ RT A3 Sin(PH '1)
W P
p
(61)
1
452 hm§E(-1)p (gflp 11')
+ -—— A cos -
8w? EF p 4 PH '5
p
where
_ _ ‘Y _
A3 — 2we (1 Y)
A4 = 7T2e-Y (2"Y) (62)
2
2w kT
Y = hm

This agrees within a constant, to Zilberman's theory(19) to
be presented. However, Zifberman drops all higher harmon-
ics in (61).

The ratio of the coefficient of the sin terms to the

coefficient of the cos terms is

2EF
3wkT (63)

RF is usually a few electron volts and RT at 1° Kelvin is
8.6 X 10"5 e.v. so the ratio in (63) is roughly 104 and
(61) becomes

hm E p
1 2 F (-12 .. 2
8 ~ 1 + T5? T. E p e Y(1-Y)sin(pH—Trp - g) (64)
p

UV‘ $.a

 

 

‘A

 

It is difficult to make a calculation of absolute magnitude
of the oscillations from Horton's theory because the term
multiplying the bracket in (64) depends on the transition
probability between initial and final scattering states.
This, in turn, depends on the matrix elements of the scat-
tering potential. However, an estimate will be made from

I
Zilberman's final equation.

1
The Zilberman Quantum Theory of Thermoelectric Power

(19)

 

Zifberman calculates the eigenfunctions and eigen-
values of the Hamiltonian for an electron in the periodic
field of the lattice. The energy eigenvalues are essen-
tially the same as for the electron when not considering

the presence of the lattice (13,14). The eigenfunctions

_ i l ~ ~ -
wk nk — const. X sin[a(x+y+z)Jexp{i(le+K32)}®n(y yo)

1 3 (65)

differ from the eigenfunctions found with no lattice

potential present (13,14), only by the presence of the

sin term. When calculating matrix elements the sin aver-

ages to é-since it is a rapidly oscillating function.
Zifberman next calculates the current through a plane

at y = o where H is in the z direction and the current is

along y. This current Jy equals the number of transfers

across the plane per second times the electronic charge.

Here V

..

'1 (in l
D- ‘J‘C
A

defect

to the

contai
for th

arid 5U

b2
and f

f
Nhere

1
99h

 

 

25

2
I 6(E. E )

J . _
y 3 klnk3 xlmx3

constant X I
kl

{ ij 'vklnk3;xlmx

x n,m 2 3

(66)

fk1(Enk3)[l-fxl(EmX3)]- fxl(me3)[l-fK1(Enk3)j}dk,dx,dk3dx3

Here vklnk33xlmx3 is the matrix element of the perturbation
produced by elastic scattering by impurities and lattice
defects. The square of the matrix element is prOportional
to the transition probability between the initial and final
states. The term 6(Eklnk3- ExlmXB) accounts for elastic
scattering, that is, conservation of energy. The term
containing the distribution functions gives the probability
for the initial state to be full and the final state empty
and subtracts the contribution from the-y direction from
the +y direction contribution.

Zilberman then assumes the change in the distribution

function comes both from the presence of an electric field

and from a temperature gradient:

ark aE ark
_ - hc 6T l o l
fxl _ fkl + (xl-k1)é-H 3? w— + 6—3, W] (67)

where E0 is the Fermi energy. The scattering potential is

V=vO:a('5—15) (68)

P

where the sum p is over lattice impurity sites. The matrix

SHOW C
Engineer.

( w
9.717]
O

 

 

elements of (68) are then calculated using (65), and (66) is
then evaluated. The calculation is simplified by using the

(21). This introduces sums over

Poisson summation formula
harmonics of cosine terms and is the origin of the quantum

oscillatory effects. The oscillatory effects arise because
of the sum over quantum levels h and m in (66). The results
show quantum oscillations in the electrical conductivity in

agreement with the theory of Adams and HOlstein:

 

3
2'
2 kTE
=.%§E (eF +-3§9)E§E§ + .EhmEo - 80g 01 e-Ycos(%% ~-§)]
(hm)?
2 5T 16 207 a % 3 2 (69)
+ k 1%?[T 71-2 +104“) " T m E02 e-Y(l-Y)sin(-P—E - g9]
where Y = 2W2;§% and F is the electric field.

The thermoelectric emf oscillations are found by letting

 

Jy = o in (69), and dropping small terms. The result is
l
eF _ _ 8Eo _ 2 8T 1r2k2T[l_ 9 hm 15”” Eo)2 -v(1 )
_ '3y_ 'E'Ey'_—E;_ lBfi'E—'+ 4w*kT’ e "Y
2v v
x sin(-I—,-H - Ir) J (70)
where
an 5T #2 kT -2 hm '2 -Y 27 w (71)
3-5—- = k637[:- 6- 'E-O— + 7T2(1"Y)'11‘1FE; e Sin(fi - 1)]

I
(70) differs from Zilberman's final equation by the factor

27

72k2T. However, the factor w2k2T is necessary dimensionally

as well as being a direct result of letting Jy= o in (69).
Because of this, (70) and (71) have common terms and a more
simple relation results than was given by Zilberman:

1

e
2 2 18(hmE )
_ 28T1rkT3 9 hm o -v
9F"3‘B‘§"E;_[4'T66F;+41rk’l‘ e (1”)

X Sin(%% - I19]

 

(72)

which predicts the same period P (47) and phase of oscil-
lations -.£ as the Horton(21) theory (64). Thus, quantum
oscillations in the thermoelectric power yield the same
cross-sectional areas of the Fermi surface as the magnetiza-

tion and thermal magnetoresistance oscillations yield.

D. Absolute Magnitude of the Thermoelectric Oscillations

To estimate the oscillation amplitude we use

hm = 1.4 X lO~2 e.v. at 21 k gauss for an effective mass
of .09 m0 appropriate for the 36
orbit of Tin. (73)
kT = 1.1 X 10-4 e.v. at 1.20 Kelvin (74)
E0 = 5 e.v. (75)
This makes

y = .16 and e"Y = .85 (76)

 

steel

W111 1

and Q7

This
tions
Only

 

28

In (72) the term containing the sin is much larger than the

other two terms in the bracket and (72) becomes

 

1
2'
s = 122k (§§) e Y(l-y) sin(%% --g) (77)

where S is the absolute thermoelectric power. Neither
Zilberman nor Horton consider the factors effecting the amp-
litude of the oscillations as was considered in II,E for

the magnetization oscillations. However, the experimental
and effective mass factors will be the same. (See II E, l.
and 3.) Most likely the Spin factor (51) will be the same,
since this effects only the Landau level structure.

Adams and HOlstein found that the oscillations in elec-
trical conductivity (60) were damped by the same (48,49,50)
temperature and collision factors as the magnetization
oscillations were damped. It is difficult to Justify intro-
ducing identical factors for the thermoelectric power oscil-
lations. However, when (48) is evaluated at 21 k gauss and
1.20 Kelvin it effects the amplitude by less than 2% so (77)
will be evaluated not considering the effects of temperature

and collisions.

s ~'.4uV/OK sin(%1 - 4) (78)

This is a factor of 10 smaller than the observed oscilla-
tions. This is not too surprising since Zilberman considered
only one type of scattering mechanism and the scattering

potential (68) is probably much too simplified.

29

It is interesting that (77) predicts the oscillation
amplitude to be independent of temperature. Higher tempera-
tures will, however, broaden the Landau levels and reduce

the oscillation amplitude.

IV High Field Thermal Magnetoresistance and Thermopower
Tensors

History of the Thermal Magnetoresistance Tensor

The significance of high field electrical magneto-
resistance was first pointed out by Kohler(29) and by
Chambers(30). It was Lifshitz, Azbel and Kaganov (LAK),
and Lifshitz and Peschanskii(31) who essentially put the
theory in final form. They showed that the behavior was
independent of the collision process, being governed only
by geometrical features of the F-S. They also assumed that
quantization due to the magnetic field was not significant.
This makes high field magnetoresistance a very effective
tool for studying F-S tOpology.

An extensive reference to theoretical and experimental

papers through 1960 is tiven in The Fermi Surface by

 

Chambers(11). Fawcett reviewed both the theoretical and
eXperimental results through April of 1964(32).

The thermal magnetoresistance tensor is defined as

Wik(fi) from:(33)

a
6% .__ ' 7’1ka - wika (79)

‘where Qik is the heat flux vector, and Jk is the electrical

otnmrent vector, which is zero in our experiments.

Thus;,

8
6g_=—w.q (8o)

 

T; L
Sn; L:

, .2
y d-
3'0“
wni ‘

 

El)

 

P93!

31

If Qk has components along the sample axis (y direction)

 

 

 

 

only, then
B?) [wa Q)
3% = ' wyy Q‘y (81 )
6%) my a)

If the only temperature difference measured is along the
y direction, then

— -w o (82)

For constant power and sample length, this gives

AME) ~ wyym) (83)

which is used to experimentally plot w as a function of

field strength and direction.

The thermal magnetoresistance, W(H), was treated theo-

retically by Azbel, Kaganov and Lifshitz<3u). They showed

that for low temperatures, where impurity scattering domi—

nates, the Wiedemann-Franz law (57) is obeyed. Therefore,

the electronic part of the thermal magnetoresistance tensor
is ccmmfletely analogous to the electrical magnetoresistance

terfisor. The thermal magnetoresistance will supply the same

F-E; topology information as does the electrical magneto-

reSiStance .

.11

sista

T
Jul

.8

LIV

32

The first experimental work on thermal magnetoresis-
tance at 4.20Kelvin was by Shalyt(95) in 1944 on Bi. In

1950 Hulm<36)

studied single crystals of tin to 1500 gauss,
but did not rotate at fixed fields, and his work was inci-

dental to the study of superconductivity. His samples were
rather impure so that 1500 gauss was not large enough to

reach the conditions of reference (31):
mT >> 1 (84)

as defined by (1). In 1953 Mendelssohn and Rosenberg<37)
studied Zn, Cd, Sn and T1 as a function of field strength
to 18.5 k gauss, but for some reason only the zinc crystal
was studied for the field along more than one direction.
In Cd, Sn, and T1 they don't even specify the field direc-
tion. Since the tin sample was only 99.996% pure, perhaps
it made little difference what the field direction was.

P. B. Alers(38) was the first to observe the aniso-
trOpy of the thermal magnetoresistance by rotating at
fixed field. By also measuring the electrical magnetore—
sistance of zinc he was able to observe the behavior of
the Lorentz ratio (see equation 57)

1 72k2
L(H) = 3 73—2—— (85)

as a function of field and angle. His results are shown
in Figure 3.
The Lorentz ratio of single crystals of lead and
(39)

indium in transverse fields was studied by Challis, et al.

They did both rotations and field sweeps at 20 Kelvin in

fields up to 20 k gauss. Although they find a 353

3F

33

Figure 3

iiiiitiillillllLign

 

 

Ll'llllliJJJllllllll

q

 

 

-120 «a «so ~30 0 Jo
9

Zinc.

Thermal

Elufncal

3. 5 OK 12.3 KGauss

“Fe wor

will vie

electr;
AZ:

ficients
scatteri

valid f0

 

 

34

H
:0

variation in % in rotation, the W(H) varies by 500% and
p(H) by 65%. With the large variation in each quantity it

is not surprising that their ratio is not quite constant.(12)
The work demonstrates that the thermal magnetoresistance

will yield the same information about the F-S as does the
electrical magnetoresistance.

Azbel, Kaganov, and Lifshitz(34) also predict the
Wiedemann-Franz law to relate the Hall and Righi-Leduc coef-
ficients. This is true only in the low temperature, impurity
scattering region for compensated metals, but is generally

valid for uncompensated metals. A metal is uncompensated if

nA = nh - ne + o (86)

and compensated if nA = o. The Righi-Leduc effect is the
thermal analogy to the Hall effect. A transverse temperature
gradient is measured when heat flows in a sample transverse
to a magnetic field.

We conclude this section by stating that very little
has been done on the thermal magnetoresistance in single
crystals, and that it shows promise as an alternative to the
electrical magnetoresistance for studying F-S's. We have
demonstrated its effectiveness in a reasonably pure sample
of tin. It would be interesting to further check its use-
fulness by looking at one of the AuSn or AuAl2 crystals of

Longo, Sellmyer, and Schroeder(40).

B.

Theory of the Thermal Magnetoresistance Tensor

The Azbel, Kaganov, and Liftshitz(34) theory shows
that the Wiedemann-Franz law is valid at low temperatures.
The thermal magnetoresistance is completely analogous to
the electrical magnetoresistance. Therefore, we need only
consider the electrical magnetoresistance which was exten-
sively studied by LAK(31). Good treatments of the theory
41) 43)

are presented by Ziman( , Chambers(u2) and Sellmyer(
and need not be presented here. Fawcett<uu) summarizes the

results as shown in Figure 4.

Figure 4
Type of orbit and Magneto- . ThermOpower
state of compensation resistance
I. All closed and uncompensated ~'O (saturates) HO
ne # nh
H2 , 1
II. All closed and compensated «i (quadratic) H
n = n.
e n
III. Open in one direction ~H2 cos2 a H0
IV. Open in two directions 'VHO (saturates) H0
V. Singular field-direction ~HO (saturates) HO

'3, or in our case a, is in the plane perpendicular
to H making an angle a with the Open orbit direction.

Tin is a compensated metal having orbits open in one
direction, and orbits open in two directions, so II, III,
and IV would apply. (possibly also V) To demonstrtate the

origin of the field dependencies listed in the chart,

36

consider the tensor for III.

~H2 ~H ~H

p =
13 ~H ~H° ~H° (87)
H—>«=

The magnetic field is in the z direction and perpendicular
to the sample axis. The open orbit is along the x-axis.
Thus, if the Open orbit is along the sample axis, the magne-
to resistance-%£-measured would be proportional to H2. If
the open direction is perpendicular to the sample length,
the magnetoresistance saturates. For Open orbit directions
making an angle a with the sample axis a similarity trans-
formation of (87) is made to rotate the Open direction into
the sample direction. This is the origin of the cos2a term

in III.

The Effects of Magnetic Breakdown

Blount(u5) found that when high magnetic fields are
present in metals having relatively small energy gaps, there
is a finite probability for electrons to Jump the gap and
follow new orbits. This is called magnetic breakdown(l%7w9)
and the effect was seen in several metals with hexagonal

close-packed structure<50758).

Blount found the probability of breakdown to be

P = exp(-.::) (88)

37

where

H0 = Egg-1:9- (89)
where A is the energy gap, EF is the Fermi energy and K is
a constant approximately equal to 1.

Magnetic breakdown can have a pronounced effect on the
magnetoresistance. Complete breakdown has the effect of
eliminating Bmagg reflection. Open orbits on the F-S can
become closed orbits and closed orbits can become Open.
Suppose the angle a in Figure 4 III is near 5. The open
orbit will cause saturation of the magnetoresistance. If
magnetic breakdown is present this Open orbit can become
closed at higher magnetic fields. The magnetoresistance
would become quadratic in H.

Magnetic breakdown has a pronounced effect on quantum
oscillations also. The oscillations must originate from
closed orbits on the F-S. If these closed orbits become
Open because of magnetic breakdown there can be no oscil-
lations. Since the breakdown probability (88) does not
vary sharply with field there is usually a mixture of Open
and closed orbits. The effect is to severely damp the
oscillations but not completely eliminate them. Because of
(88) they are more heavily damped at higher fields where
breakdown is more complete.

Magnetic breakdown in tin will be discussed in a

later section.

Az':
theory 1
Thomson
(3314), f
electric
cients.

The
vector 1

their so

where B

+‘n
Hue OHS a
Coeffici

PElatiOn

 

 

38

D. Theory of the Thermoelectric Power Tensor

Azbel, Kaganov and Lifshitz<54) extended the LAK<31)
theory to include the heat conductivity tensor and the
Thomson coefficient. Bychkov, Gurevich and Nedlin(59),
(BGN), further considered the theory to include the thermo-
electric power, peltier coefficient and the Thomson coeffi-
cients.

The heat current vector is a, and the electric current
vector is 7. In general, 3 and a are defined in terms of

their sources as

a .
_ ik B 1
J1 "‘T‘ Ek + bik‘EE; 'T (90)
C .
_ ik O 1
Q1 “ ’T“'Ek + dik'dx; 'T (91)

a 1 .
where Ek and-EEE- T' are the generalized forces required 1n

the Onsager(17) relations. Thus, in a magnetic field the
coefficients are functions of H and the Onsager symmetry

relations hold:

aik(H) = aki(-H) (92)
dik(H) = dki(-H) (93)
bik(H) = ck1(-H) (94)

Rather than use (90) and (91), new coefficients can be

.e
i...
.1
a. .

.nu

.-€I‘€

-
gur-

fi

3...,

/\

A.
w u
a».

 

 

W

“evils‘

¢

and

defined such that:

-l 6T

Ei _ ik Jk + aik'dig (95)
5T

qi z Bik 3k ‘ Kir'a£; (96)

where 01k is the electrical conductivity tensor, Kik is

the heat conductivity tensor, aik is the thermal emf
tensor, and Elk is the Peltier coefficient.

The equations relating aik’ bik’ cik and dik determlne

the relations between 9 K

“ik’ a.- and Rik given in (95)
and (96).

ik’ lk

Rewriting equation (95) gives

 

-1
a.- b -a.
_ lk _ 3k if
Oik "VT— and aik ~ *‘it“““' (97)
and rewriting equation (96) gives
-1 (dik ' ciflafmbmk) 0
8ii: = 012 afik and Kik = 2 (90)

T

Using (92), (93) and (94) the following relations hold:

Oik(H) = Oki(-H) (99)
Kik(H) = Kki(-H) (100)
TaiK(H) = Bki(-H) (101)

coeffic.

I. r
CIJC 3.1d

then E11
”ha

i..E

are:

 

O
‘IOP Clo.

fi
0‘ elecf

40

Notice that if Bki is known as a function of H that aik

is also known as a function of H from (101)
To calculate 81k equation (98) says that c1k and a1k

must be known. If we calculated aik directly then b d

ik an
a1k must be known. From (90) and (91) notice that c1k and

a1k are found when only E is present, whereas to find bik

there must be a temperature gradient present. Thus Bik is
calculated,and from this aik found.

The expression for a1k was worked out in detail for
various F-S geometries by LAK and Lifshitz and Peschanskii(31)
for the electrical conductivity. BGN(59) calculate the cik
coefficients using the method of LAK, et a1. Thus, knowing
01k and a1k (98) and matrix multiplication give Bik' (101)
then gives aik'
These tensors for the high field thermoelectric power

are:
Vxx vyx(-Yo) vzx

_ 1 _
“1k ‘ T Yovxy Vyy Vzy (102)

 

\'-Yovxz “Yovyz vzz

for closed F-S's with nA % o that is, with unequal numbers

of electrons and holes. Here YO is defined by

"Ill (103)

_<
o
n
(0)3
4|

 

 

les, 4;

0

V
H
«l

41

and viJ are independent of field to first order. Vxx and

v are defined b
yy y

w2k2T2 d
vXX 1' Vyy = T a-é- £N<De - nh) (104)

The diffusion thermoelectric power calculated from the

Boltzman equation is

2.
1r k T dflno
3e dE EF (105)

S:

 

O is defined from simple theory<4l) as

 

 

 

 

_ ne T q

a - m (10C)
If gar-E. E = 0 then (105) becomes

F

s = W2k2T dfln n (107)

3e dE EF

If ne = the number of electrons and nh = the number of
holes, then (107) is

S _ W2k2T dfin (ne - nh) (108)

7 3e dE EF

and

v = v = ST (109)

XX yy

2+2

aik is therefore the thermoelectric power tensor in high
magnetic fields.

The diagonal coefficients in (102) are independent of
yo and therefore independent Of magnetic field strength.
This applies only when there are no Open orbits and nA # o.

For a compensated metal nA = o and BGN get

-1 -l -1
-YO Vxx "Yo Vyx 'Yo sz
_ 1 -1 _ -l _ -l
Gik - T’ -YO ny Yo vyy YO sz (110)
sz vyz sz

The x and y diagonal terms are linear in magnetic field,
whereas the diagonal terms in electrical and thermal magneto-
resistance are quadratic in field. In (110) z is again the
direction of magnetic field so the transverse magneto
thermOpower is determined by either the xx or yy terms.

BGN treat the same three (III, IV, v in Figure 4,)
types of open orbits as LAK did. As an example of orbits

Open in one direction BGN take the "corrugated cylinder":

Figure 5

W
‘~.___,SI"7_“-.___,z"—_“~§__,,/i

Agair

W“,
- U

pe

43

Again take H as along the z direction, and define

Y
O
‘1 :51??? (111)

where 6 is the angle between the cylinder axis and the plane

perpendicular tO the field direction as shown in Figure 6:

Figure 6

e
H

'2‘:
<—

 

 

The x axis is taken as in the plane, passing through both

H and the cylinder axis as shown in Figure 7:

Figure 7

X

 

 

 

Therefore a is the angle between the cylinder axis and the

x'direction.

44

BGN get the following thermoelectric power tensor:

v (-n) -Y_1v (-n) -Y'lv (-r)\

 

xx 0 yx o zx
a _ l, _ _ _ -
“ik — T Yovxy( n) vyy( n) vzy( N) (112)
-YOvXZ(-n) vyz(-n) vzz(-n) j

and when n-avm vik becomes a constant. From (111) we see
that n —> °° when 9 is near 0 or 1r independent of the value
of Y0, since Yo is always finite.

Thus, the thermOpower saturates (HO) when H is perpen-
dicular to this type of open orbit direction.

In (87) the xx and yy terms were H2 and H0. In that
case the Open orbit was along x. If the Open orbit was
along the sample axis direction, H2 was Observed. If the
open direction was perpendicular to this, H0 was observed.
A rotation of the coordinate system gave an H2cosec
dependence as summarized in III of Figure 5.

For the thermoelectric power tensor the situation is
not the same. Notice that the xx and yy terms in (112) are
identical. Saturation would be Observed as long as e 9" 0, 71'
independent of the angle the sample made with the open
orbit direction. No rotation of the coordinate system is
necessary. Therefore, 22 0032a term is present as in

case III for the magnetoresistance.

115

For the two other types of orbits (IV and v in Figure
4) BGN get

(113)

em;
<<
<<

1k —

 

where the v11 are constants at high fields, and saturation
results. The magneto thermoelectric power tensors are

summarized in Figurezl.

Kohler Corrections

 

In an actual crystal a crystallographic axis of high
symmetry is not always along the sample axis and the
tensoral relations defined above have to be modified.
Kohler(6cn discusses this in detail. He shows that the

thermopower is

 

I
i

-l E - E

X ('1! i)
E = E cos2$ + ElsinEO+( " sin2o cos2o
A I! 2 l 2
sin c +'i£ cos
H

where Iland 1_mean parallel and L.to the [110] axis
o is the angle between the sample axis and [110] axis. The
crystal used for our experiments had the [110] axisllp from

the sample axis and the Kohler corrections are negligible

46

since:
cos 11° = .982 cos2 110 = .963
sin 11° = .191 sin2 11° = .036
. . _ 1 .r‘ r . 1. . [é
So even if EH _ 7 EJ_tne measurel E¢’IS within 7, of

E<b ‘ Ell

A.

47

V. Experimental Apparatus and Measurement Techniques

Sample Preparation

 

In an attempt to Obtain a sample with a major symmetry
axis oriented along the sample axis, seven samples were
grown. They were in the shape of a long cylinder and
were grown in a Bridgeman furnace(61). The first sample
(SnI) was grown in a graphite crucible from 99.9999% pure

tin and resulted in a resistance ratio

0300

= 31,000 (115)
°4.2

The sample had a two fold axis 520 Off the sample axis.
Therefore, little work was done on it and the results not
analyzed. The remaining samples were grown in glass tubes
internally tapered by pulling them slowly out of hydro-
fluoric acid. The tapering allowed the finished crystal
to be easily removed from the tube. The tin was etched
with two parts glycerin to one part HNOB, and washed in
distilled water. Tin has a very low vapor pressure, and
the glass and tin were extensively outgassed.

The crystal selected for most of the eXperiments
(SnII) had a resistance ratio of 30,000. The [001] axis
was 790 from the sample axis and the [110] axis was 860
from the sample axis. The sample was mounted vertically
for the August through September data. On October 7th it
was mounted so that [001] was 16°, and the [110] was 70

from the horizontal plane. 'H was rotated in the horizontal

4E
plane. The geometry is shown in figure 8.

Figure 8

 

 

.} _< a
16°orll°j 7 °r +
(M x

'H is in the x-z plane

B. The Apparatus

 

The apparatus was designed to place the long cylindri-
cal sample transverse to a uniform magnetic field. It was
also necessary to have the sample in vacuum and to have one
end of it in thermal contact with the liquid helium bath.

A general View of the apparatus is shown in figure 9 ,
and in figure 10 the details of the mounting of the sample
in the vacuum can is shown.

The cross hatched part in figure 10 was made of
c0pper because of its high thermal conductivity. This,
along with the fact that liquid helium could enter the
inner helium container to which the crystal was directly
attached, assured that the end of the sample was close to

bath temperature. The long section between the

Figure 9

 

To High Vacuum

 

Quick Coupling
(Yo High Vacuum \
F

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

E _‘l
I)

To Helium To Helium
Pump . Pump ri—

0

'5

E

5’:

Helium
”VAN Level W

 

 

 

 

 

 

 

 

 

 

 

\h‘f/ \\

/
\

\Maqnet
flognet

//

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 10

i

 

 

 

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50

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51

copper holder and outside of the dewar was made of thin
walled stainless steel to reduce the heat flow to the helium.

The region around the sample was maintained at high
vacuum by pumping through the long stainless support tube as
shown in figure 9. An oil diffusion pump, backed by a
mechanical pump, kept the vacuum at 2 x 10'"5 mm Hg. as mea-
sured at room temperature near the diffusion pump intake.

The region surrounding the sample was cryOpumped to probably
10"°mm Hg .

One of the most useful features of the apparatus was
the veeco "quick" coupling which allowed the entire apparatus
to be raised or lowered. See figure 9. The sample was
thus centered in the magnetic field before each run. Since
the "quick” coupling was a vacuum seal the region above the
helium bath could be pumped tOCLl3 mm Hg, correSponding to
1.0080 Kelvin. The ultimate vacuum varied between runs
depending on whether another dewar was simultaneously being
pumped. The lowest possible temperature was never higher
than 1.400 Kelvin.

The "quick" coupling was useful in case of a vacuum
leak in the apparatus after the transfer of helium. When this
happened, the apparatus was removed, and the leak fixed. It
was then placed back over the dewar in the raised position
and slowly lowered through the coupling, back into the
bottom of the dewar. It could be lowered in about twenty

minutes with a loss of less than a liter of helium. Lowering

52

the apparatus through the quick coupling meant that no
"snow" collected inside the dewar. See figure 9 .
Temperatures intermediate between 4.2OK and 1.00K

were reached and maintained, to within 2 x 10-3 0

Kelvin,
by pumping on the helium bath and maintaining a constant
pressure above the helium with a regulator described in

the appendix.

The Measurement of Voltages

 

The voltages were measured continuously as a function
of field strength, and orientation at fixed field. The-g%
voltage pickup in the loop formed by the two lead wires was
kept to a minimum by non-inductively twisting the wires.

An increasing field out of the paper induces a current in
each tiny loop, and each loop cancels the effect of its
neighbor. The tightly wound leads present a very small
area to the changing field. Being tightly wound, the two
wires were in good thermal contact with each other at all
points along the wires.

The emf leads were made of NbZr wire which remains
superconducting in the presence of our maximum field of
21.5 Kgauss. They were spot welded to the sample to elimi-
nate thermoelectric voltages generated by solder connec-
tions.

The NbZr wires were prepared by repeatedly coating
them with Ge 7031 varnish which was thinned 4:1 with

50—50 alcoholtoluene. The wires were then baked for

for eight hours at 45°C. The resulting insulation was
better than 20 Meg ohms after twisting the wires together.

The leads were wound around the sample. The sample
and each lead formed non-inductive loops even though the
sample was a "straight" wire. The sections of lead wire
wound around the sample were in good thermal contact with
it. The NbZr was therefore well below its 11°Ke1vin
transition temperature.

As a consequence of the winding procedure the %%
pickup voltage was very small. When the field changed at
the rate Of l Kgauss per second the pickup was ~l uV.
Since sweep rates were generally less than 300 gauss per
minute or 5 gauss per second, the pickup was about
5 x 10-9 volts. This is comparable to the amplitude of
the smallest oscillations observed and was therefore accep-
table. With even more care, for example more windings
around the sample, this could probably be reduced to 10-9
volts.

NbZr was chosen for lead wires, because it remains
superconducting in high magnetic fields. The thermopower
of a superconductor is zero. Therefore, it was hoped that
the measured thermOpowers were due to the tin alone.

M. C. Steele, and J. de Launay<€39 showed that a
thermoelectric potential exists between a superconductor
and the same metal in the normal state. They found the

empirical relation:

54
with

E0 ~10"7 volts

Thus, at 1°K, E could be ~10‘7 volts. With the well

twisted wires, in good thermal contact, they would both go
normal at the same place. The ~10-7 volts generated in

each lead tended to cancel each other. Voltages measured
when no temperature gradient was across the sample, were
about 10"6 volts. Part of this came from the incompletely
cancelled ~10-7 volts in each lead. Another possible source
of this voltage was the thermoelectric emf generated in the
normal part of the wire. When a large temperature differ-
ence (up to 3000 Kelvin) existed across the wire of

inhomogeneous composition, emfs were developed.

Pole Pieces and Magpetic Field Homogeneity

The pole faces for the Harvey Wells magnet had a 6"
diameter and a 2-%g"gap. They were aligned parallel to
each other using an NMR probe. Alignment bolts were
changed to make the NMR signal sharpest. The marginal
oscillator used is shown in figure 11. A 60 cycle modula-
tion coil was used along with a proton (water) resonance
source.

New pole faces were designed and built from 1010
steel. These had a 4” diameter and a l 767 gap. This gap
allowed two %Efl aluminum sheets to be inserted in back of

the pole faces. This insured that the pole faces could be

removed in the future. A Spectro chemical analysis Of the

55

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1010 steel showed:

.12% carbon

.35% manganese

IA

.01% phosphorus
.O25% sulphur
.33% silicon

Analysis was done by Allis-Chalmers research laboratories.
The saturation magnetization was 21.0 kgauss. Using experi-
mental data from Magnion Corporation the maximum field was
estimated at 30 kgauss. The Olofsson Corporation of Lansing

ground the faces of each pole piece parallel to within

___i‘...
10,000

The 30 kgauss pole faces were designed for use with the

inches. They were ground flat with the same accuracy.

Hoffman model HLMR-ll dewar. Unfortunately, the dewar would
hold helium for only two hours after consuming seven liters
of helium. Pumped to 10K it would probably last less than an
hour. The dewar was leak tested and no leaks were discovered.
It was concluded that contact was present across a vacuum
Jacket.

The 30 kgauss system mentioned above has great poten-
tial for a number of experiments. Higher frequencies in the
thermOpower quantum oscillations would be resolved. Combined
with the field modulation technique (to be described) it
would greatly increase the effectiveness of the oscillation
studies.

The present experiments were done using the 6" diameter

57

pole faces with 3” gap, with a glass dewar. The field
strength as a function of distance from the pole face
center is shown in figure 12. Data were taken with a
Rawson rotating coil gauss meter. These plots were impor—
tant because the field must be uniform over the dimensions
of the sample. If the field at one point on the sample
differs from that at another by more than the separation
between two oscillation peaks, the oscillations cannot be

(

Observed. 63) The separation between peaks is given by

H2 P (116 )

where P is the period of the oscillations. Figure 12 shows

that the field at 201cgauss varies less than 40 gauss over

-1

1 inch. Thus 1.0 x 10_7 g is the smallest period obser-

vable. At 18 k guass this is “'7 x 10-8 g.1 over 1 inch.
For tin the dominant period along [001] axis was
7

~'5.7 x 10- g-1 and was easily resolvable with emf probes
about one inch apart. Oscillations of period 2.5 x 10"7 g’1
were Observed. To Observe even higher frequency oscilla-
tions the probes must be placed closer than 1 inch apart

to get better homogeneity over the sample.

Thermometry

 

Temperature gradients across the sample were measured
using an a-c wheatstone bridge and carbon resistors as tem-

perature sensors( 6A). The relative geometry is shown in

Figure 12

2|

 

20.5-

_

Q.
9.

 

no
338 x 32... 252?:

|lfi\/\\/| _ AWIgw
7.

_
5.
9

h

7:

 

l6.5L
l6

INCHES FROM CENTER

58

figure 13. The resistors were mounted directly over the
emf leads and the heater was wound with manganin 350/foot
alloy wire. In order to make magnetic fields produced by
current in the heater negligible, six feet of wire was
wound clockwise and the remaining six feet counterclock-
wise. The leads to the carbon resistors were Evenohm NO.
36 wires( 65). Evenohm has the feature Of changing resis-
tance by less than 1% in going from room temperature to
4.20 Kelvin. The resistors were mounted over insulating
cigarette paper using GE 7031 varnish.

It was important to keep the insulation between leads,
and between leads and ground, very large because the carbon
resistors reached 20,000 ohms, and a large resistance
could not be measured accurately if a shorting resistance
of comparable magnitude existed. USually this was greater
than 10 Meg ohms, and was accomplished by having the
Evenohm wire doubly Formvar insulated. The resistors were
ground flat on both sides to aid making good thermal contact
with the sample and reduce the heat capacity. This allowed
them to follow temperature changes rapidly.

The a-c bridge and supporting electronics is repre-
sented schematically in figure 41. Essentially, an off-
balance d.c. signal of the wheatstone bridge is caused by
a temperature variation of the carbon resistors. This

amplitude modulates the a.c. carrier signal from the

 

 

 

 

 

 

 

 

 

 

 

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61

oscillator. The modulated signal is amplified outside the
helium bath by an amplifier tuned to the carrier frequency.
A second output from the oscillator, in phase with the
carrier, is mixed with the modulated carrier. The mixer
functions as a double—pole double-throw switch being

driven at the oscillator frequency. The output from the
mixer is the amplified d.c. off-balance signal from the
bridge. This is further amplified by a conventional d.c.
amplifier. The tuned amplifier, oscillator, mixer and

d.c. amplifier are contained in a single "lock-in amplifier"
manufactured by the Princeton Applied Research Corporation
and designed for variable frequency Operation from 15 CpS
to 1500 cps.

It was found that at low temperatures where carbon
resistance values were large, and at higher frequencies,
the capacitive reactance was significant. This distorted
the true resistive output. For this reason the bridge was
Operated at 23 or 17 cycles per second. A new bridge was
designed having variable capacitors in the bridge arms.
(See Appendix) In practice it was difficult and time
consuming to find the Optimum capacitance to introduce,
so low frequencies and low initial value carbon resistors
were used.

Known resistances of .l, l, 10, and 100 Ohms were
introduced into one side of the bridge. Thus a recorded
differential variation in resistance was calibrated. The

absolute value of temperature gradients was thus known,

once each carbon resistor was calibrated.

The resistors were calibrated against the vapor
pressure of the helium bath using mercury and oil mano-
meters, and a Mcleod gauge.

The carbon resistors follow an empirical formula

quite well:(65 )

K _ B M
loglOR+W—A+T (1.1”)

This is usually approximated by

03
A
(.1
F-J
«1
v

loglo R = A ‘1' 7f

So
dR -B -
R— : gr GT (118)

and for small changes, dB is prOportional to dT.

Temperature gradients were sometimes too large to
Justify using (llfh. Each carbon resistor was therefore
calibrated using (ll ), and temperature gradients measured
knowing the resistance of each resistor.

The output of the lock—in-amplifier could be made pro-
portional to the resistance of either of the carbon resistors.
It could also be made proportional to AR, the difference in
resistance between the two resistors. (See Appendix). When
changes in the resistance were small (llfh was valid, and
the lock-in-amplifier output was prOportional to AT. This

was always true when looking at quantum oscillations.

63

As discussed earlier(83 ), AT is prOportional to the y
component of wik’ where wik is the thermal resistivity
tensor. The output of the lock-in-amplifier was therefore
proportional to the thermal-magnetoresistance. It could be
plotted continuously as a function of angle or field
strength.

Since at low temperatures the thermal conductivity by
electrons dominates over the conductivity by the lattice,
the magnetic field dependence of the resistivity is com-
pletely analogous to the field dependence of the electrical
resiStivity. The thermal method has the very significant
advantage of being free from field pickup in probe lOOps,
and other noise problems inherent in d.c. electrical sys-
tems. The very narrow band pass amplifier of the a.c.
method allows recovery of signals buried as low as 40 db
in noise.( 66)

There are two main disadvantages of the thermal method.
The first is that the system must be under a vacuum.

Vacuum leaks are very troublesome. The second disadvan-
tage is that the carbon resistors reSpond slowly. Rotations

at fixed field and field sweeps had to be done slowly to

insure that details were recorded.

—j§253311ng the Measurements

Tfinree transport prOperties were measured. These were:

the tkuirmal magnetoresistance, the thermoelectric emf, and

64

the electrical magnetoresistance. Section V,C described
the geometry for voltage measurements. Section V,E des-
cribed the method of temperature gradient measurement.

The thermoelectric emf was measured between the
twisted leads described in section C and resulted from a
temperature gradient across the sample. A heater wound on
the end of the sample as described in C, provided the tem-
perature gradient. Thermoelectric measurements were made
with a constant heat input.

The thermoelectric emf was measured by a Keithley 149
nanovoltmeter and recorded on a Mosley x-y recorder or a
strip chart recorder.

Electrical magnetoresistance voltages were measured in
the same way. In this case a constant d.c. current was
provided by a Princeton Applied Research constant current
supply. This maintained a constant current independent of
load changes.

The magnetic field was swept linearly in time by an
apparatus designed by J. LePage and a schematic is shown in
the appendix. At times it got out of control because it
was difficult to keep the output of the regulator zeroed.
It would be worthwhile taking the output of the regulator,
amplifying it and applying this to the input of the inte-
8Pat0r. This would make regulation semi-automatic and
PGIieve the Operator to take care of writing numbers on

65

A schematic of the measuring apparatus is shown in
figure 13.

The thermal resistance and the thermoelectric emf were
recorded simultaneously using the dual pen strip chart
recorder. Field values were read directly from the Rawson
gauss meter and recorded on the chart paper. Recording the
thermal resistance oscillations and the thermoelectric oscil-
lations simultaneously permitted a determination of their
relative phase.

The Moseley x—y recorder was useful for observing
the fine structure in data because its y axis is much
larger than the y axis of the strip chart recorder. It was
very useful for recording the oscillations occurring when
the field was rotated at constant magnitude. However, the
x-y recorder was not used for field sweeps when looking at
oscillations because the oscillations needed to be spread
out to be resolved.

The x-y recorder was used when looking at the thermal
and electrical magnetoresistance in rotation at fixed field.
It was also used at fixed angle to determine the gross
field dependence Of these transport prOperties. In this
case the x axis was driven by the output of the Rawson
gauss meter.

In rotation, angles were read off a pointer on the
magnet and recorded by hand on the graph. Much Of the
labor of this method was later eliminated by recording the

angles automatically every degree on one pen Of the strip

66

chart recorder. The problem with this method was mentioned
earlier; the strip chart was not wide enough to reveal all
the desired details. A schematic of the automatic angle
recording apparatus is shown in figure 42 .( 67)
A method for measuring themOpower directly was tried.
The output of the lock-in-amplifier was used to control

the bias on a power amplifier containing the heater current

as shown in figure 14.

Figure.14

‘ saolscm pic
50 K Heal-er)

AT é Bail”? I:

When the temperature difference across the sample tried to

 

LIA

 

 

 

 

change, a voltage was applied to the transistor bias which
increased or decreased the heater current. This did not
work in earlier experiments because the sample temperature
responded too slowly to the heater current changes. The
reSponse was fastest at loKelvin when the emf leads were
close together. With a bit of effort this could be made to
work. Changes in'H would have to be quite slow . With
AT a constant, 3 = 9—§%£-is prOportional to the emf output.
For looking at higher frequency oscillations, a diff-

erentiator was made. If the signal was

211Fl 2vF2
6 = Alcos-—H—— + Aecos-5H—— (119)

67

and 6 was electronically differentiated then

8 O
-%% =-%%-3% = constant X-g% (120)

2TrF1 2WF 2WF 27rF2
=Al-I;2—-Sln H +A2?Sln H

 

 

If F1 >> F2 the F1 oscillation amplitude would be enhanced

by the ratio of F1 to F2. Higher frequencies would thus ,

be emphasized. The circuit for this was:

 

Figure 15

t-lifiw
3 l
Em W?— IM

L‘D‘fl $014+

The lkO resistor and .002 uf eliminate the differentiation

 

 

of very high frequency or pulse type signals.

A
(-1
(\3
1-1

v

(121) gave an output signal large enough to drive the
chart recorder when sweep rates were about 200 gauss per
minute.

Because other experiments occupied most of the avail-
able time, differentiation was tried only once. The signal

was rather noisy but one should be able to increase the

68

signal to noise ratio by increasing the value of the 1K

resistor.

Field Modulation

 

The thermoelectric emfs in tin turned out to be very
large. In fact, on one field sweep it was necessary to P'

use the lOu volt scale on the Keithley voltmeter. Had

these voltages not been so large, field modulation would ( _

 

have been used to Observe them. Field modulation makes
use of lock—in—amplifier (LIA) techniques to extract
signals much below the noise level. Considerable prepara-
tion was made towards using this method, and this progress
will be described.

The LIA output in field modulation thermoelectric
experiments is proportional to the derivative of the emf
with respect to field and is therefore ideally suited for
studying quantum oscillations. These oscillations are
nearly sinusoidal in H and the derivative is also sinu-
soidal with the same frequency.

The modulation coils used in preliminary experiments
were designed in the Helmholz configuration and mounted
against the coils in the Harvey-Wells electromagnet. The
a-c impedence when mounted in the magnet at 20 k gauss was
16 Ohms. This value was found to be rather independent of
field, as expected. Modulation was achieved using the
oscillator output Of the LIA amplified by a MacIntosh

30 watt power amplifier. The 16 ohm coil impedance was

69

matched by using the 16 Ohm MacIntosh output terminals.
Modulation amplitudes were measured using a pickup coil
and oscilloscope. The peak to peak amplitude at zero d.c.
field was 115 gauss. The amplitude was 125 gauss at 3
kgauss and decreased slowly as the field went up, reaching

lOO gauss at 14 kgauss. Modulation was most effective Ij
below the saturation field of the magnet core.

The oscillating field creates a fluctuating thermoelec- l_

 

tric emf in phase with the field. This emf is then mixed
with the pure oscillator frequency and the LIA output is
QEEEE. In addition, the oscillating field induces a.c.
voltages of the same frequency in the emf leads. These

-%% oscillations would appear in the LIA output if it were
not for the fact they were 90 degrees out of phase with the

modulating field. If the field oscillated as

H6 cos mt G22 )
then

d dB - i
«5% = AaE-= HbmA sin mt (12))

and the cosine and sine are 900 out of phase. A is the
cross sectional area and can be kept small by twisting the
emf leads as described in section 0.

Stark(€%3) showed that large amplitude field modula-

tion was useful in separating de Haas - van Alphen

7O
frequencies. Quantum oscillations vary as
sin 21F- (124)
and with a modulating field this is

(125)

 

. 2vF
S in (H+Hoc os mt)

H6 is the amplitude of the modulation field. This can be

written as
~ [- 27TF HO 1 - r
._ Sin L T (1 - fi— COS (1)13) _. (120)

Seely(69 ) shows that this is

2vF 2vF n
JO sin T + 2 finer); (-l) J2n(6) cos 2nmt

.. (127)
- 2 cos 232, (-l)nJ (6) cos(2n+1) mt
H 2n+l
:0
where
QWH F
O n
5 = —-;Fr' (120)

If detection on the LIA is done at the frequency m, then
all thatremains of (127) is

27H F
2vF o -
- 2 COS GT) J1(T) COS mt (129)

 

71

Seely( 09) shows the zeros of the n32 Bessel functions Jh.

The first zero of the n = l Bessel function is for

6 = 3.882. Therefore

_ H23.882 (1,

Ho - 2v F

will make (129) equal to zero! For example, if H0 is 60

gauss at 13 k gauss there will be no LIA output from the

 

36 section of the F-S. However, oscillation frequencies
from other parts of the F-S will still be present and can
be studied without interference from the large amplitude
36 oscillations.

Field modulation therefore has two distinct advan-
tages over d.c. measurements. The use of LIA technique
greatly improves signal to noise ratios. By picking
modulation amplitudes, certain frequencies can be elimi-

nated in order to study others.

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VI Experimental Results

 

A. Quantum Oscillations in the Thermoelectric Emf and
Thermal Magnetoresistance

 

The tin single crystal refered to earlier as SnII
was mounted with two different orientations with respect
to the plane of rotation of the magnetic field as shown
in figure 3. In position I the [001] axis was 110, and
[110] axis 40 out of the plane of H. In position II,
[001] was 16°, and [110] 70 out of the plane of rotation
of F. The crystal was in position I from August through
October 7, and in II from then on.

A field sweep in position I is shown in figure 18.
The magnetic field was swept slowly from 14.5 k gauss to
19.5 k gauss while a constant heat flow was maintained in
the sample. The oscillations are in the thermoelectric
emf, and recorded on an x-y recorder. The sweep was made
with the field fixed at 220 from the closest approach to
the [001] axis. The oscillations are periodic 111%? as

shown in figure3£3. n is an integer corresponding to a

peak position at field H.

emf amplitude ~'sin-%% (131)
which is a maximum when
-%% = 2vn (132)
The measured period was found from:
1
d(-—-)
P'= —a%}-= 5.4 x 10.7 gauss-1 (133)

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®H mHSMHm

 

 

Figure 19

 

 

 

 

77
Using (132) this means that at 19 k gauss:

n — 1 «'100 (134)

— 5.4 x 10‘7 x 19,000

 

Therefore, lOO quantum levels remain below the Fermi
energy at 19 k gauss, and the approximations of n >> 1
used earlier are Justified. Notice that the oscillations
are very large even at a temperature of 4.2OK. At 19

k gauss the thermoelectric power oscillations have an
amplitude of about 1 av per OK.

Many sweeps were made at various angles. Usually the
thermoelectric emf and the thermal magnetoresistance were
plotted simultaneously on a strip chart recorder.

Figure 20 shows a typical field sweep for crystal position
I. The upper half is at higher fields than the lower, and
shows that the oscillations are not quite sinusoidal,
indicating that higher harmonics of the fundamental are
present. The oscillations are not Spaced evenly because
it was difficult to keep the sweep rate constant and to
monitor both oscillations simultaneously. The bottom half
shows the thermal magnetoresistance oscillations diminish-
ing in amplitude more rapidly than the thermoelectric
oscillations.

The thermal magnetoresistance oscillations are super-
imposed on a large slowly varying field dependent term as
is apparent in lower half of figure 20 . The thermo-

electric oscillations, however, have little monotonic

 

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d -«Q»¢3o§.~u 0.9.13; airflow {A < ;
om opsmflm

79

field dependence and the oscillations are positive and
negative. In this particular sweep the oscillations in
thermoemf were seen from 20 k gauss to 4.5 k gauss and a
total of 370 oscillations observed. In figure 20 the
thermal magnetoresistance oscillations were not observa-
ble below 7 k gauss. For field directions where the
thermoelectric oscillations were very weak, oscillations
in the thermal magnetoresistance could not be seen.

Most oscillatory quantities are superimposed on large
monotonic field dependent backgrounds. Such is the case
for the thermal magnetoresistance oscillations. The
thermoelectric emf oscillations are unique in being
alternately positive and negative. This factor makes
period and amplitude determination relatively easy, and is
an advantage of this technique.

Figure 21 (b) shows emf vs. H for H 180 from the
closest approach to the [001] axis. For this direction
a higher frequency oscillation is clearly visible.

Higher frequency oscillations were seen in a range of
.i 16 degrees from the closest approach to [001]. The
reason for this will be explained later. Only one fre-
quency was seen in the thermal magnetoresistance.

Figure 22 shows how the oscillation periods varied as
a function of the distance from the closest approach to
the [001] direction for crystal position II. Both the

low and high frequency oscillations agree with the data

 

 

80

m _

mg

mmsoo x
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moocooo

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Ow QON

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3
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82

found in pulsed field de Haas - van Alphen effect by Gold
7

and Priestley('o), and the field modulation experiments

of Stafleu and de Vroomen<7l). The origin of these

oscillations will be discussed in a later section.

Quantum Oscillations in the Thermoelectric Emf when
Rotating at Constant Field Strength

Figure 23 shows quantum oscillations when the field
is rotated through 180° at 21 k gauss. The direction
marked 0 on the graph correSponds approximately with the
direction of closest approach to the [001] direction.

Figurezylshows the same rotation oscillations at a
slower speed and over less than one half of the angle of
figure 23. At 6': 240 degrees a higher frequency oscil-
lation is clearly visible. Details of this part of the
rotation are displayed in figure 21(a).

Oscillations occur in rotation because the cross
sectional area of the F—S perpendicular to the field is
dependent on field direction. At certain orientations
there are more cylinders occupied by quantized electrons
than at others. In rotating from one direction to another
cylinders pass through the F-S causing oscillations.

amplitude ~'sin 27 ggil ( 133

 

13 a maximum when

PH

 

 

 

 

 

 

 

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mm osswfim

 

 

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is...
:m ohswfim

Since the argument of the sin is a large number, the sin
is a rapidly oscillating function of f(0). How rapidly
these oscillations occur at any given field, depends both
on P and on f(9).

From figure 24 the angle of peak positions is found.
A plot of sec 9 at peak positions against corresponding
integers is shown in figure 25, to be a straight line,
indicating that

f(@) = sec 9 (

*— I
\N'
v

This agrees with the angular dependence of period as shown

in figure 22. A summary of the data is given in figure 26.

Origins of the Oscillations and the F-S of Tin

 

In figure 27 the free electron Fermi surface of tin
is shown.( 76 In zone three the 5 orbits have a period of
5.7 x 10-7gauss'1 with H along [001]. This agrees with the
corrected best curve in figure 22. In addition, this
section is nearly a cylinder. The cross sectional area of
a cylinder varies as sec 9, where 8 is the angle between
the cylinder axis and the perpendicular to the area. Figure
25 verifies that this sec 9 is followed and the low fre—
quency oscillations therefore originate from the 56 section
of the Fermi surface.

A recent band structure calculation by Weisz changes

the free electron picture drastically, but the 56 section

pl.l8

'I.l6

800 o
-I.I4

L'|.l2

-|.l0

LLoe

"LOG

-I.O4

4.02

 

Figure 25

August 30

 

255
255
76°

76°

87

Figure 26

Summary of Frequency Data

 

([1101 axis 4°, and [0011 11°, out of plane of H)

 

 

degrees from Measured 7 -l
resistance minimum Period X 10 g Comments
up 220 5.3 i .3 used x-y
recorder
up 22° 5.3 i .5 "
O u
up 23 5'1.i .2
O u
up 23 5.1 i .2

 

340°

0

Sgptember 15 ([110] axis 4°, and [001] 110, out of plane of'H)
down 180 4.4 i .2 oscillation in
thermal magne-
toresistance
down 180 4.9 i .4 thermo emf

340

September 18 ([110] axis 40, and [001] 110, out of plane of H)

 

200.5°

up 24.50 5.2 + .1

September 25 ([110] axis 4°, and [001] 11°, out of plane of H)

249

- 550 4.8 + .1

 

October 10

 

237

241

255

261

215

230

October 26 (

mud

degrees from
resistance minimum

 

down 20.5

down 17

down 30

up 5°

down 430

0
down 28

 

240

240

245

250

256

260

down 160
down 160

down 11°

down 60

zero

up 40

[1101 axis is 7°

(‘1

measured

period X 107

 

”-9.i .2

5.0

l+
\»

5.3

|+
L.

5.5

I+
L.

3.7

l+
04

5.66 i .05

4.4 .4

I+

5.3

|+

.3

5.45 + .03

5.64 + .04

( [1101 is 7° and [0011 is 16° out of plane of'fi)

Comments

strong oscillation
is thermal resis-
tance

two frequencies
present, no resis-
tance oscillatials

no resistance
oscillations

no resistance
oscillations

no resistance
oscillations

and [0011 is 16° out of plane

of'H)

no thermal resis—
tance oscillations

two frequencies,
no thermal resis-
tance oscillations

large thermal
resistance
oscillations

large thermal
resistance
oscillations

 

265

2773

27755

283C)

up 140

up 140

up 190

up 24

2.9 + .2

5.48 i .08

two frequencies,
no thermal resis-
tance oscillations

two frequencies,
no thermal resis-
tance oscillations

strong mixing of
frequencies

no thermal resis-
tance oscillations

strong thermal
resistance
oscillations

(158/

F1 gure 27

i D?!

 

 

 

   
     
 

-L--OC--‘
I
l

I

0
d n-b0-
’
’l
'

’....o .

é.

 

 

 

 

 

 

  

?--..
a.
\
fl

4"

o
—
o‘
,0

 

 

 

 

 

 

nee 4 at) elem.

nee 6: element

The free-electron Fermi surface in the reduced-zone scheme. The principd
dimensions of the zone ere:

1‘1,.2_",1~x..l..21,rW-Llial-i'f-2 23 ,WH-E 31' .
a 2 a. 2 c a a a a
whma-5-821mde-3481.

Reference 70

KO
}_.3

is virtually unaltered. However, the higher frequency
oscillations of period 2.9 x 10"7 g-1 were designated

by Gold and Priestley as originating from the 5m orbits.
The Weisz calculation shows these orbits no longer exist,
and that this period originates at the upper end of the
36 section. A cross section of this piece of the F-S

(72 )

is shown in figure 28, and in three dimensions will
appear like a dog's bone. The diagram is pr0perly inter-
preted by removing zone 2 from within the zone 3 section,
for both the TX and the XL planes.

The higher frequency oscillations seen in rotation
in figure 24 and in figure 21 (a) cannot be assigned to
the upper end of the dog's bone, because the frequency in
rotation is too rapid. If the f(9) for this section is
sec 8 then the correSponding period P is .9 x 10"7 gauss-1.
This roughly agrees with the ”D” periods seen by Gold and

Priestley. Not enough data was available to make a defi-

nite assignment.

Thermal Magnetoresistance in Rotation

 

Quantum oscillations in the thermal magnetoresistance
were not seen in rotation since the monotonic field depen—
dence was so large. A 3600 rotation is shown in figure 29.
Notice that each maxima and minima is reproducible every
180°, which is consistent with the existence of a 2 fold
axis of symmetry close to the crystal axis. The signal to

noise ratio is very large and drift is nearly absent. The

 

~ .- _ F - :1.
F 1:?11111: _,-. ~

 

 

 

 

 

 

 

I I I I I I I IJIF‘
|.5"" ""
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ZONE 6

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.2—

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(I l0) Reference 72

 

 

 

not 0} on!

x45

 

 

 

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mm opsw.m

c4

electrical magnetoresistance will later be shown to have

a much lower signal to noise ratio and drift is present.

The advantages of lock-in-amplifier techniques are evident.
Tin is a compensated metal and the magnetoresistance

has an H2 dependence for field directions perpendicular

to closed electron orbits (refer to the table of figure 4 ).

This behavior is exhibited generally in the region between

1000 and 2000 in figure 29. For field directions perpen-

dicular to orbits Open in one direction, the magnetoresis-

tance is

p ~'a + bH2 cos2 a (138)

a is the angle between the current vector 3 and the Open
orbit direction, In our experiments'j is 4° from [1101 and
aperiodic open orbits occur for field directions in a two
dimensional region around [001], as shown in the stereo-
gram in figure 30. As the field moves into the two dimen—
sional region a increases and is 900 at the angle of closest
approach to [001], giving a resistance minimum.

All open orbits in tin are on the 4(a) hole section of
the F-S. This is shown in figure 27, and a topologically
equivalent surface is shown in figure 31. ,As is evident,
the surface is very complicated, and the open orbits on it

are "aperiodic"

in nature. An electron on an aperiodic Open
orbit does not repeat the same path in succeeding zones, as

shown in figure 32.

 

7'!“ TA
Iigure 3O

   

M ' "0

 

MW

Reference 73

ms mocomommm

§S

§s§

 

\g
Hm oaswam

Figure 5°

R0
[00‘] 95:: T N
//

HLO E
,’ORBIT OPEN
I/ORBIT

    

‘ I/i/////i//////

.//////

[no] /. ’4
(6) I (01(2)) PLANE
IN —
. "B°°]'Ax's PERIODIC OPEN ORBIT
CLOSED ELECTRON mo _____—-——
HOLE ORBITS

EXTENDED 0R3”

   

/(C) , . (d)

3 NEAR (010) -PLANE
EXTENDED ORBIT

I NEAR [foo] - AXIS
.WT

Reference 4”

E. Field Dependent Structure in Magnetoresistance

 

The most unusual feature Of both the thermal and
electrical magnetoresistances is the appearance of two
pairs of peaks within the two dimensional region of
aperiodic Open orbits. For crystal position I, the first
pair is at -15° and +19°, the second at“: 32°. These
peaks are shown in figures 33 and34 for various field
strengths. Figure 33 is for crystal position I, and
figure 34 for position II. Figure 35 shows the electrical
magnetoresistance for position II.

The inner pair of peaks occur for angles where a is
large and cosea small, characteristic of open orbits.
Notice the inner pair of peaks for both crystal positions
rise up at higher fields. This would occur if the Open
orbits of this region became closed at higher fields.

This behavior was seen by Young<74) in the electri-
cal magnetoresistance and attributed by him to magnetic
breakdown between the 4(a) and zone 3 sections of the F-S.
(See figure 36) These sections would be connected if it
weren't for Spin orbit coupling which introduces a small
energy gap of ~'0.003 Rydbergs as calculated by Weisz(?2).
The concept of magnetic breakdown was discussed in section
IV. The 0.003 Rudberg energy gap corresponds with
HO 3 4,000 gauss.

Young's interpretation is not easy to visualize, but
is related to the angle 8 shown in figure 37. Magnetic
breakdown peaks occur at the angle 9, where orbits pass

near the point x. The range of 9's permitted in his

 

 

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Thermal Magnetoresistance
25 Sept.Dete

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8200 x i
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Magnetoresistance

Thermal

IO October Data

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Electrical

2 January Data

ad

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103

 

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x E coo... oeoN fie 2: .3335. 2.2a 6:»

[\
I“

ogswflm

104

argument is

10° < e < 20° (159)

and the peaks observed in our experiments fall within this
range.

An extensive study of magnetoresistance in tin was made
by Alekseevskii, et a1, so we attempted to correlate
our work with theirs. The stereogram of their work is
shown in figure 30. The portion of interest is within the
two dimensional region Of Open orbit field directions
shown in figure 38. The inner pair of squares with rounded
edges mark maxima and minima field positions found by
Alekseevskii, et al. The great circles correSponding to
our field sweeps are marked by the angles of [001] and
[1101 out of the plane of rotation of H for the two
different crystal tilt positions.

The 11°, 4° curve predicts peaks at -l4°and +l6° and
minima at.: 22°. Peaks were Observed at -15° and +l6° and
minima at —20.5° and +20°.

The crystal position II curve (16°, 7°) predicts peaks
at -7° and +10° and measured to be -16° and +19°. The mea-
sured angles are extremely dependent on crystal tilt angle
in this region, so an error of.i 2° in x ray orientation
could greatly change this agreement. Minima are predicted
at -19°, +21° and observed at.i 22°.

The outer peaks shown in figuresjfl+ and A35occur very
rouEhly for field directions along [210] directions but

this could not be definitely established. The unusual

 

 

 

ea

 

 

 

 

 

 

 

>\

 

 

 

 

 

105

feature of these peaks is that they disappear at higher
fields and suggests that magnetic breakdown might be
involved here, also. Their disappearance at higher fields
would correSpond with closed orbits (H2 behavior) changing
to Open orbits (H2cosga behavior). The 4(a) section of
the F-S of tin is very complicated and aperiodic orbits

on it are even more difficult to picture.

Correlation of Oscillation Amplitudes with the Field
Dependent Peaks in Magnetoresistance

 

The most striking feature of the thermoelectric
power quantum oscillations is that they nearly disappear
inside of.i 22° and reappear within_i 4° of the angle of
closest approach to [0011, as shown in figure 50. This is
the region of the inner magnetoresistance peaks which be—
come more pronounced at higher fields.

The oscillation amplitude reduction is consistent with
the presence of magnetic breakdown between zones 3 and 4(a)
since at breakdown fields not all electrons would complete
cyclotron orbits on the 36 section of the F-S. It is the
completed cyclotron orbits which contribute to quantum
oscillations.

Young's description of which orbits are involved in
breakdown does not eXplain the oscillation amplitude reduc-
tion. Young's and our data are consistent with each other

but not with his interpretation. All of the data agree

 

 

 

 

107

———_————_

 

 

 

 

mm shaman

oo . I a bMun. I how I cowl oo.¢l

J. .
_

1 _
_ mount.nE<
_ .23.. 2 32:.

.H. _ cezfizouo
_ .0 03.025
_
_
_

as

coagfluecozcoes

IL

 

108

with the concept that the rounded squares in the stereo-
gram Of figure 38 are due to magnetic breakdown. This
idea was not suggested by Young and could be further
studied by repeating the present experiments for various
tilt angles.

We have demonstrated that the combination of thermal F]

magnetoresistance and thermoelectric emf quantum oscilla-

 

tions is an effective tool for F-S studies. Being able
to measure both quantities simultaneously is a distinct i
advantage. The thermoelectric emf is shown to be very

sensitive to the quantization of energy levels.

10.

11.

12.

13.

14.

150

16.

17.

18.

References

 

. W. J. de Haas and P. M. van Alphen, Proc. Amsterdam 33,

1106 (1950).
R. Peierls, z. Phys. g; 186 (1953).

See apoendix to: D. Shoenberg, Proc. Roy. Soc. A 170,
541 (1959). ———

. D. Shoenberg, Proc. Roy. Soc. A 175, 49 (1940).

. J. A. Marcus, Phys. Rev. 11, 539 (1947).

L. Onsager, Phil. Mag. $3, 1006 (1952).
I. M. Lifshitz and A. M. Kosevich, JETP 29. 730 (1955).

I. M. Lifshitz, M. Azbel, and M. I. Kaganov, JETP 21
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Y. M. Azbel, E. A. Kaner, JETP 2, 772 (1956).

M. H. Cohen, H. M. Harrison, W. A. Harrison, Phys. Rev.
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The Fermi Surface, W. A. Iarrison, M. B. Webb, eds.
John Wiley and Sons, Inc., N.Y. 1960.

 

Low Temperature Physics LT9, J. G. Daunt, D. 0. Edwards,
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C. G. Grenier J. M. Reynolds, N. H. Zebouni, Phys. Rev.
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C. G. Bergeron C. G. Grenier, J. M. Reynolds, Phys. Rev.
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C. N. Rao, N. H. Zebouni C. G. Grenier, J. M. Reynolds,
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J. R. Long, 0. G. Grenier, J. M. Reynolds, Phys. Rev. 140,
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See: J.P. Jan, Solid State Physics 5, Academic Press,
N. Y. 1957, p. 1.

M. C. Steele and J. Babiskin, Phys. Rev. 8, 359 (1955).

109

 

19.
20.

21.

22.

23.

24.

25.

26.

27.

28.
29.
30.
31.

32.
33.

34.

35.
36.

110

G. E. Zil'berman, JETP 2, 650 (1956).

V. P. Kalashnikov, Fiz. Metal. Metalloved 11, 651
(1964).

P. B. Horton, Ph.D. Thesis, Louisiana State Univer-
sity (1964). Unpublished. University Microfilm #64-
8804.

E. N. Adams, T. D. Holstein, J. Phys. Chem. Solids
10, 254 (1959)

R. B. Thomas Jr., Phys, Rev. received 28 April 1966.
To be published.

 

A. B. Pippard, The Dynamics of Conduction Electrgns,
Gordon and Breach, N. Y. 1965.

 

 

C. Kittel, Quantum Theory 9: Solids, John Wiley and
Sons, Inc. 1963, Chapter 11.

 

R. M. Eisberg, Modern Physics, John Wiley and Sons,
Inc. 1961 p. 256.

 

H. M. Rosenberg, Low Temperature Solid State Physics,
Clarendon Press, 1963 p. 352.

 

R. B. Dingle, Proc. Roy. Soc. 211, 500 (1952).
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R. G. Chambers, Proc. Roy. Soc. A 238, 344 (1956).
I. M. Lifshitz, Y. M. Azbel, and M. I. Kaganov, JETP
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M. Lifshitz and V. G. Peschanskii, JE TP 8, 875
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E. Fawcett, Adv. Phys. 13, 139 (1964).

C. G. Grenier, J. M. Reynolds, J. R. Sybert. Phys,
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M. I. Azbel, M. I. Kaganov and I. M. Lifshitz. JETP
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s. Shalyt, J. Phys. (U.S.S.R.) g, 515 (1944).
J. K. Hulm, Proc. Roy. Soc. 204, 98 (1950).

37.

38.
39.
40.
41.

42.
43.

44.
45.
46.

47.
48.
49.
50.

51.

52.
53.

54.

55.
56.

111

K. Mendelssohn and H. M. Rosenberg, Proc. Roy. Soc.
218, 190 (1953).

P. B. Alers, Phys. Rev. 101, 41 (1956).
L. J. Challis, J. D. N. Cheeke, P. Wyder, Reference 12.
J. Longo, D. Sellmyer, P. A. Schroeder, To be published.

J. M. Ziman, Principles of the Theory of SolidsL Cambridge
University Press, 1964. p. 258.

R. G.Chambers, reference 11.

D. J. Sellmyer, Galvanomannetic Investigationo _f he

Fermi Surface of AuSn. Ph.D. Thesis 1965, Michi “a State
University.

E. Fawcett, reference 32.
E. I. Blount, Phys. Rev. 126, 1656 (1962).

M. H. Cohen, L. M. Falicov, Phys. Rev. Letters 1, 231
(1961). .

A. B. Pippard, Proc. Roy. Soc. A 270, 1 (1962).
L. M. Falicov, P. R. Sievert, Phys. Rev. 138, 88 (1965).
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M. G. Priestley, L. M. Falicov, G. Weisz, Phys. Rev. 131,
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R. W. Stark, T. G. Eck, W. L. Gordon, F. Moozed, Phys.
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R. W. Stark, Phys. Rev. Letters 9, 482 (1962).

A. R. Mackintosh, L. E. Spanel, R. C. Young, Phys. Rev.
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A. S. Joseph, A. C. Thorsen, Phys. Rev. Letters 11,67
(1963)

Lawson, Gordon, reference 12, p. 854.

Higgens, Marcus, Whitmore, reference 12, p. 857.

 

57.
58.

59.

60.
61.

62.
63.
64.
66.

67.

68.
69.

70.
71.

72.
73.

74.

112

R. W. Stark, reference 12, p. 712.

R. C. Young, Phys. Rev. Letters 15, 262 (1965). also
Phys. Rev. To be published.

Y. A. Bychkov, L. E. Gurevich, G. M. Nedlin 19, 377
(1960).

Kohler, Ann. Phys. 45, 196 (1941).

Bridgeman furnace temperature controller designed by
D. J. Sellmyer.

M. C. Steele and J. de Launay, Phys. Rev. 85, 276 (1951).
D. Shoenberg, Trans. Roy. Soc. A 255, 12 (1962).

J. J. Lepage, Ph.D. Thesis 1965, Michigan State University.
R. D. Moore, 0. C. Chaykowsky, Modern Signal Processing

Technique for Optimal Signal to Noise Ratios. Princeton
Applied Research Corporation Technical Bulletin 109 (1963).

 

I want to thank Andrew Davidson for designing and con-
structing this.

R. W. Stark, private communication.

Seeley, Electron Tube Circuits, MoGraw—Hill Book Co, Inc.
New York 1960. p. 603.

 

J. B. Ketterson, Y. Eckstein, Rev. Sci. Ins. 31, 44 (1966).
A. V. Gold, M. G. Priestley, Phil. Mag. 5, 1089 (1960).

M. D. Stableu, A. R. de Vroomen, Physics Letters gg,
179 (1966).

G. Weisz, Phys. Rev. 149, 504 (1966).

N. E. Alekseevskii, Y. P. Gaidukov JETP 19, 481 (1960)
and JETP 13, 770 (1962).

N. E. Alekseevskii, Y. P. Gaidukov, I. M. Lifshitz, V. G.
Peschanskii, JETP 12, 857 (1961).

R. C. Young, Phys. Rev. Letters 15, 262 (1965) and Phys.
Rev. to be published.

See also R. J. Kearney, A.B. Mackintosh, R. C. Young
Phys. Rev. 110, 1671 (1965).

 

75.

76.

113

E.A. Lynton, Superconductivity, Methuen and Co. LTD.
London 1962, p. 58.

F. H. J. Cornish, J. L. Olsen, Helv. Phys. Acta g5,
369 (1953).

 

114

Appendix

 

Intermediate Magnetic State

 

Figure ”O shows a plot of the thermal magnetoresistance
against fields below 600 gauss. The sample is at 1.4OK.
Below 150 gauss the entire sample is superconducting with
complete Meissner effect. Between 130 gauss and 260 gauss
there appear ”tubes” of alternate normal and superconduct-
ing regions(Y5). The bump appearing in the resistivity
was satisfactorily eXplained by Cornish and Olsen using

a model of conductivity by the alternate tubes. The sharp
corner at 260 gauss is at the critical field H6 for tin at
this temperature. In very pure materials, the thermal
conductivity below Hé/2 is purely from the lattice so the
experiment presents an interesting way of studying pure
lattice conductivity, involving almost no phonon-electron

collisions.

 

Figure 40

Thermal

115

 

 

Magnetore sfsian ce

 

 

 

600

ISO 200 260 300 Field In Gauss

IOO

 

116

Bridge Correction

 

LfK

 

 

 

 

 

 

 

 

 

 

 

 

 

(R+Rl)(R+R2)
E = 11 [1°4K + 2R+R1+R2 1 i1 = i2 + 15
12(R+R1) = 13(R+R2) AV = V1 - v2 = R(12 - 13)
R+R1 R+Rl
AV = R 12 - :2 = R 1 - 12
R+R2 R+R2
R
AV ”'RiRg (R2 ‘ 31) l2
R+R (R+R )(R+R )
E = 12 + 12 l 1.4K + 1 2
R+R2 2R+R1+R2
(R - R ) E
AV = R 2 1 - 6 E R - R
R+R2 R+Rl 4 (R+Ri)(R+Ré) 2 l
1 +-—7——- 1. +
RTRE 2R+R1+R2
AV = RE 6
4 (R+Rl)(R+Rl+A)
(2R + 2R1 + 5) 1' +(2R+R1+R1+A)
K is a constant
K6 .. .
AV = (2R + PR +‘67 6 —-o to 6K max1mum

1
R ~'5K Rl ~'3K

 

117

Bridge Correction Table

 

 

amount to be added to
6 = (R - R ) k ohms bridge output to get
1 2
true linear output

 

 

 

1 6% E)
2 11% ‘
5 17%

1 22% r
5 2575

6 5576

above percentages calculated for R = 5K ohms

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