t—M‘mwmw

[J LIBRARY
i
1

  
 
   

khan-gen 514.6

I... University

This is to certify that the
thesis entitled

QUANTUM OSCILLATIONS IN THE PELTIER
EFFECT IN ZINC

presented by
Harry Joseph Trodahl

has been accepted towards fulfillment
of the requirements for

____fl1°_D°_ degree mm

   

Z/U/ ) Hi: 5/

Major lprolessor

V

Banal/[(11181 /(’/f /JZ'§7

J

0-169

ABSTRACT

QUANTUM OSCILLATIONS IN THE PELTIER
EFFECT IN ZINC

by Harry Joseph Trodahl

Peltier measurements have been employed to study quantum
oscillations in the thermoelectric power of zinc single
crystals. The three lowest frequencies, due to the a, a
and v orbits, were observed at temperatures ranging from
10K and 4.50K and in fields to 22 kilogauss. The amplitudes
found were between .01 and l uv/KO.

An attempt has been made to correlate the results of
these measurements with an expression derived by Horton.

As his calculation is valid only for the case of a free
electron sphere we have heuristically extended his results

to include metals with complex Fermi surfaces. The theory
and measurements agree only on the form of the temperature
dependence; neither the absolute amplitude nor the field
dependence is correctly predicted by the theory. It is
suggested that the primary source of the disagreement may

be found in the rather simplified scattering potential assum-
ed by Horton.

QUANTUM OSCILLATIONS IN THE PELTIER
EFFECT IN ZINC

By

Harry Joseph Trodahl

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Physics

1968

Acknowledgments

 

I would like to express my thanks to Dr. F. J. Blatt
for his interest and guidance throughout this work and
to Dr. P. A. Schroeder for many helpful discussions. I
am very grateful to my wife, Judy, for her assistance in
proofreading and typing the thesis.

The financial support of the National Science Foun-
dation is gratefully acknowledged.

(ii)

II.
III.

VI.

VII.

VIII.

IX.

Table of Contents

Introduction

General Theory

Quantum Oscillations-general theory

Quantum Oscillations in the Thermoelectric
Power

A. Introduction

B. Horton Theory

C. Extension to a Complicated Fermi Surface

Nonoscillatory Magnetothermoelectricity

Experimental Approach

A. Seebeck Measurements at Low Temperatures

B. Peltier Measurements, general

C. Thermal Contact Considerations

D. Double Junction Technique

Experimental Apparatus

Cryostat

Pumping System

Measuring Electronics

Magnetic Field Electronics

weenie

Temperature Control and Measurement
Operation, Data Reduction

A. Introduction

B. Point by Point Methods

C. Continuous Detection

Samples

A. Zn 1

B. Zn 2

Results

A. Zinc Fermi Surface

(iii)

10

17
17

32

33
40
42

1+6
1+8
49
60
63

6A
64
66

59
69

7O

XI.

C)

(iV)

Data Interpretation

Zn 1

Zn 2

l. The a Oscillations

2. Monster Neck (8) Oscillations
3. Monster Arm (y) Oscillations

Conclusions

A.

Modified Horton Theory in Zinc

B. Auxiliary Data
C.
D

The a Oscillations
The B and y Oscillations
1. Absolute Amplitude

2. Field Dependence, Apparent Dingle
Factor

3. Temperature Dependence

. General Conclusions

Nonoscillatory Seebeck Coefficient

7O
72
75
81
81
87

96
97
102

103

101.l
107
108
108

\OCDNmU‘l-EWTDH

[.4
O

11.
12.
13.
14.
15.
16.

17.

18.
19.
21.
22.
23.
2A.
25.
26.
27.
28.

29.

'2

/ o

31.

Figures

. Thermocouple circuits

Simple Peltier Junction
Peltier Junction between anisotropic metals
Landau levels in a free electron gas

. Two dimensional one OPW Fermi surfaces

Curvature of a one OPW Fermi surface

. A30) vs. l

Experimental Peltier Junction

. Thermal Contact diagrams
. Double Peltier Junction

Cryostat

Measuring electronics

Thermometer bridge

Bridge to cryostat leads

Circuit diagram of control electronics
Control circuit switch positions
Current control sections

Mode 1 control logic

Mode 2 control logic

Magnetic field electronics

Field sweep control

Typical recorder trace, mode 1
Typical recorder trace, mode 2

The zinc Fermi surface

8 oscillations in Zn 1

{38‘ vs. T, Zn 1

Thermal damping of the B oscillations
[Sal vs. H, Zn 1

S vs. 9, Zn 1

S vs. T, Zn 1

(V)

77
78
79

32.
33.
34.
35.
36.
37.
38.
39-
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.

56.

57.
58.

59-
60.

(vi)

Field rotation, Zn 2
a oscillations, Zn 2
[S l vs. 6, Zn 2
[Sa I VS. T, Zn 2
f vs. 6, Zn 2
r88 I vs. 9, Zn 2
[3: l vs. H, Zn 2
[3: | vs. T, Zn 2
vs. a, Zn 2
|E§| vs. 8, Zn 2
Typical v oscillations
[Syl vs. H, Zn 2
le| vs. T, Zn 2, 329 from <0001>
lel vs. T, Zn 2, 350 from (0001)

_a vs. 9, Zn 2

f6 vs. 9, Sn

436‘ vs. 8, Sn

{Sél vs. H, Sn

Seebeck coefficient of Pb

Seebeck coefficient of Ni

Pb vs. Ni in a magnetic field

Seebeck coefficient of Fe

Seebeck coefficient of W,low temperature

Seebeck coefficient of w, intermediate
temperature

Thompson heat correction
Wheatstone bridge

Low temperature resistanc of a carbon
resistor

Magnetoresistance of Manganin

Sample alignment

80
82
83
84
85
86
88
89
9o
91
92
93
94
95
100
114
115
116
118
119
120
121
122

123
127
132

133
136

139

:

Tables

. Fermi surface curvature

Comparison of theoretical and experimental
amplitudes

. Apparent Dingle relaxation times
. Revised amplitude estimates

(vii)

102

104
104
105

II.
III.
IV.

VI.
VII.

Appendices

 

Sn Measurements

Mrasurements on Polycrystalline Materials
Thompson Heat Correction

Thermometer Linearity

Manganin Resistance

Kelvin-Onsager Relations in a Magnetic Field
Effects of Sample Misalignment

(viii)

113
117
126
131
135

137
138

I. Introduction

 

If one constructs a closed loop of two dissimilar
conductors and heats one of the Junctions a current is gen-
erated which can be detected with a compass. This was the
first experiment, performed by J. Seebeck in 1821, in which
the presence of thermoelectricity was indicated.1 His ex-
periments were followed in 1843 by Peltier's discovery
that a current flowing through a biconductor Junction could
produce or absorb heat.2 These effects are appropriately
known as the Seebeck and Peltier effects.

The Seebeck effect is more easily treated if one con-
siders the voltage causing the current to flow. This is
measured by breaking the circuit as shown in figure 1a or
b and measuring the potential difference across the break.
It is possible to define a function SAB(T) for which

rT2

v = - JTlSAB(T)dT (1)
SAB(T) = - BV/BT. (2)

Here V is the measured voltage when no current is flowing
through the loop and T1 and T2 are the Junction temperatures.
Furthermore, it is possible to define a coefficient S(T)
for each conductor if the coefficient used in equations
1 and 2 is defined to be

SAB(T) = SA(T) - SB(T) . (3)

S(T) is called the absolute Seebeck coefficient or the
thermoelectric power of the material.

At the Junction between two conductors the Peltier heat
is proportional to the current flowing through the Junction,
and the Peltier coefficient H is defined as the proportion—
ality constant,

(1)

(2)

A.
I I w... I
. .

Thermocouple circuits

Figure 1.

a = HABI . (4)

The sign convention used is that the Peltier coefficient

is positive if the Peltier heat is evolved when the current
is in the positive direction, from conductor A to conductor
B in figure 2. As for the Seebeck coefficient, one may
define a Peltier coefficient for an individual material.
The coefficient in equation 4 is then given by

HAB(T)==HA(T)-HB(T) . (5)

There is a third thermoelectric effect, the Thompson
effect, first predicted by William Thompson in 1882. Along
the length of a conductor which is both supporting a tempera-
ture gradient dT/dx and carrying an electrical current I
a reversible heat is evolved per unit length given by

dQ/dx = u(T)I(dT/dx) . (6)

u is the Thompson coefficient of the metal. It is interest—
ing to note that this is the only thermoelectric effect
which can be observed in a single conductor.

That the three thermoelectric effects are closely re-
lated has been known for a long time and the derivation of
these relations has been one of the primary tests applied to
the various treatments of non-equilibrium statistical mechan-
ics.3’uCa11ed the Kelvin-Thompson relations after William
Thompson (Lord Kelvin) who was the first to suggest them,
these relations are

TS (7)
T(BS/8T) (8)

T1121
H

in which T is the absolute temperature. The most elegant
derivation of these equations is based on the Onsager re-
ciprocal relations which demonstrate that the Kelvin relat-

4 7

ions are a direct consequence of microscopic reversibility. 7

llll ll. ll. lllil ll Ill l lull l l I t A l l
I ll Iiillllll! III“ III l
I I

(4)

WWW rm?nmm
“(639 19:41 (‘0 _ ..
/ .1”? ”’43-’11 al§\‘mgn
.1 I

3..

6V‘

 

 

‘tlllllltl‘Itlllll .

II. General theory

The results we can obtain from transport effect cal-
culations are equations expressing currents as functions
of appropriate driving forces. In a conductor we are con-
cerned with both electrical and thermal currents.

3'= LEEFe + LETGT (9)
fi = LTEFa + LTTGT (10)
3 = electrical current density

U'= thermal current density

T = temperature field

@ = u - eV = electrochemical potential

u = chemical potential

V = electrical potential

The quantities Lii in general are nondiagonal tensors in
an anisotropic material. The use of'fim rather than -é§V
as a driving force permits uS to apply the Onsager reci—
procal relations, ’

-- TLET- <11)

However, due to the temperature dependent term in the elec-
trochemical potential, the identification of the various
measurable quantities is difficult. The only easy one to
identify is the electrical resistance, measured under the

condition that VT = 0,
p = LEE(-l/e), (12)
The Seebeck and Peltier tensors are simply defined by

S = p ° LET (13)
n = LTE ~ 9 . (14)

(5)

(6)

The combination of equations 11, 13 and 14 results in the
first Kelvin relation,

n = TS . (15)

It remains to be shown that, at least in isotropic
materials, equations 13 and 14 lead to the quantities de-
fined in the introduction. Because of the terms LEE?“ and
LTfifin in equations 9 and 10 this is not a trivial problem,
but the proper identification can be made. In anisotropic
materials a measurement of the "Seebeck coefficient" involves
a rather complicated combination of the elements of the
Seebeck tensor as well as the elements of most of the other
tensors, Lii' However, we are here concerned only with
the Peltier effect, i.e. the reversible heat produced at a
Junction between two conductors.

We extract from the paper by Domenicali 4the following
four equations which represent the total of all heat pro—
duction per unit volume in a conducting region.

Joule qJ =ifjpiJJiJJ (16)
Thompson qT =i§JuiJJJaT/axi (18)

Bridgeman qB =i?3“135J3/5X1 (19)

The Thompson tensor, p, is given by
n = T 5(H/T)/5T . (20)

In Peltier measurements we are concerned only with revers-
ible heat production, i.e., the heat production that is
reversed by a reversal of the current direction, so that
we may neglect the Joule heat term. We shall also neglect

the Thompson heat term by requiring that temperature grad-

(7)

ients be small; more will be said about this later. Only
the Peltier and Bridgeman terms remain and it is quite simple
to show that

qB+qP=€- (n5) . (21)

The total reversible heat production from these two terms
inside a volume, V, bounded by the surface, s, is given by
Q =l £7" (Hj)d'r =l (r13) ° fids , (22)
P+B V s
by the divergence theorem.

The experimental arrangement is reasonably well approx-
imated by figure 5. The Junction between conductors A and
B is in general a complex structure consisting of solder,
crystallites of A and B, and impurities. We would like to
calculate the reversible heat in the region of the Junction,
inside the dotted surface in figure 3. If we assume that
this surface crosses the conductors sufficiently far from
the Junction that the currents are only along the lengths
of the samples, equation 22 results in
e =(j°n-i-i°n-?)I . (23>
P+B A A A B B B
Thus it is the difference in the appropriate diagonal terms
which is measured.

Perhaps the most common method used in transport calc—
ulations is through the Boltzmann equation, which can be
solved to obtain the electron distribution function f(k).
One may then apply the equations

elv‘(k)r(k)d3k (26)
- .[e(k)'\7(k)f(k)d3k (27)

3

C“
I

and identify the tensors Lii by comparing equations 26 and

27 with 9 and 10. In equations 26 and 27

(8)

 

 

 

.. _\ A
x/’ \\ ’é
_l <--
" _ ._ raw;
’g\\ .7 . [’4 B
'//1/VZ"‘— .’ ’ ,

[a] (2 . unit vectors

Figure 3 . Peltier Junction between
anisotropic metals

(9)

k=wave vector
V(k)=velocity=3ke(k)
6(E)=electron energy

The combination of the Boltzmann equation with an assump-
tion that there exists a unique relaxation time T(k) at
each point in momentum Space leads to the familiar logarith-
mic derivative formula for the thermoelectric power.

s =<r2/3)<k2T/e>{atloe o(e)]/Be}€=€f. (28)
in which 0(a) is the electrical conductivity as a function

of energy 6, T the absolute temperature and k the Boltz—
mann constant.

*
III. Quantum Oscillations - general theory-

The state of a free noninteracting electron gas con-
tained in a three dimensional square well can be described
by a Sphere in single particle momentum (k) Space the in-
terior of which has an electron in every state while the
exterior is empty. The boundary between the filled and
empty regions, called the Fermi Surface, is not infinitely
sharp; there is a region that is a few (kT) of energy thick
in which the available electron states are only partly filled
or empty.

In the absence of a magnetic field these states are
distributed thngpghout k Space with a constant density.

The applicationAa field redistributes them, with the same
average density, onto the surfaces of cylinders aligned
parallel to the magnetic field (see figure 4). It should
be noted that an electron in an energy eigenstate does not
remain stationary on one of these cylinders. It will orbit
in a circle about the circumference with a frequency given
by

we = eH/mc . (29)
The radii of these cylinders increases with magnetic field
and as the outer one passes through the Fermi surface the
electrons in those states fall into lower levels. This
process repeats as each Landau cylinder passes through the
Fermi surface, giving rise to an oscillatory magnetic field
dependence of the internal energy and the density of states
at the Fermi level. Similar oscillatory dependence could

* The material in this entire section is covered more
completely in many solid state physics texts, for example
see Kittell. See also Pippard.

(10)

(11)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4,(after Rosenbergg) Landau

levels in a free electron gas. ’

(12)

in principle be found in any electron related measurement
and has been observed in the magnetization (de Haas-van
Alphen), the electronic free energy (magnetothermal oscilla-
tions), the resistivity (de Haas-Shubnikov) and in an im—
pressive list of more exotic effects including most of the
tranSport coefficients.

The exact solution of the problem of a free electron
gas in a magnetic field is not difficult. One finds that
the oscillations are periodic in the inverse of the magnetic
field with a period of

A(l/H) = eh/efmc (30)

where Sr is the electron energy at the Fermi level, m and e
the electronic mass and charge respectively.

Electronic states in real metals are affected to vary-
ing degrees by the periodic potential of the crystal lattice.
The free electron states of energy e and momentum k are mix-
ed with states of (nearly) the same energy and with mo-
mentum differing by one reciprocal lattice vector G, defined

by

‘- .5 .5 2
G - nibl + n2b2 + njb3 (,1)
n = integers

.31: 32x53/(2W(a2x 53) ‘ a1) and cyclic permutations
3i: the basis vectors for the periodic lattice.

In the free electron approximation the energy is proportion-
al to k2 so that mixing can occur only between states for
which

122 s (12 + 6F. (32)

The energy of states near the plane given by equation 32
is shifted in such a way that an energy gap appears between
adJacent states on opposite sides of the plane. More impor-

(13)

tant here is the fact that an electron executing an orbit

in the presence of a magnetic field will be reflected from

k to k +'G upon reaching the Bragg plane (given by equation
32). This can give rise to several orbits which may differ
significantly from the free electron orbit; several possibil-
ities in a two dimensional lattice are shown in figure 5.
Obviously if the various sections of'k space defined by

the Bragg planes are moved by the appropriate reciprocal
lattice vectors G, k.+ G is adJacent to'k and a diagram
representing the orbit in'k Space appears.

This method of obtaining the perturbed Fermi surface
is commonly called the one 0. P. W. (orthogonalized plane
wave) scheme. The logical extension would be the use of
some form of perturbation theory to calculate the energy
shifts near the zone boundary, which approach is the basis
of orthogonalized plane wave calculations.12 For sufficient—
ly small perturbations the effect of such a calculation is
a simple rounding of the Sharp edges of the Fermi surface.

In a magnetic field the orbits of electrons in k Space
are altered by the Bragg reflections; they are no longer
circular. However, the orbits are quantized and quantum
oscillations, periodic in the inverse of the magnetic field

strength, do occur with a period given by13

A(1/H) = 2e/ncAO . (33)

In this expression A0 is the extremal cross sectional area
of the Fermi surface in the plane perpendicular to the mag-
netic field. The periods given by equations 30 and 33 are
equivalent only for the case of a free electron gas.

Quantum oscillation amplitudes are affected by the
detailed form of the dispersion relation 6(k) near the
orbits involved as well as thermal and collision broadening
of the Landau levels. Brailsfordlugives the following

expression for the oscillatory term in the free energy.

(l4)

 

 

 

 

 

 

 

Two dimensional one O.P.W. Fermi surfaces for

a) square lattice, extended zone scheme
b) square lattice, reduced zone scheme
c) square lattice, repeated zone scheme

 

 

 

 

 

 

- /
><

d) rectangular lattice, extended zone scheme'
e) repeated zone scheme; note the open orbits

Figure,5

 

(15)

2kT(eH/2th)3/€ cos[(vhc/eH)A o-2wvv-w/4]

osc 2 L
ztfa A/Bk: )k k0 v V3/2 sinh(21r2 va/hwc)

 

exp(fl' ) (34)

CDTCO

H = Hz= magnetic field strength

k = quantum momentum along the magnetic field

A = A(kz) = cross sectional area of the Fermi
surface in the plane perpendicular to H

k = kz at the point at which (BA/akz)=0

wb= cyclotron frequency = eH/m*c

m*= cyclotron effective mass = (h2/2f)[aA/5€Jk =const=k
_ z o

T = oscillation relaxation time
Y = phase factor depending upon the electron dis-
persion law

The Shift of the Landau energies due to impurity scattering
has been omitted from equation 34, although Brailsford
includes it. The collision broadening factor exp(-1rv/wc T O)
was first derived by Dingle 15 for a free electron gas. For
uncharged impurity scattering (6 function Scattering centers)
the relaxation time appearing here is Just that which appears
in the resistivity formula

To = TR . (35)

(16)

If a sufficiently large magnetic field is applied to
a metal an electron may simply ignore a Bragg reflection
plane and continue along the free electron Fermi sphere.
This requires a discontinuous Jump in i at the plane, but
of course the energy remains the same. The probability

of such a process occurring is16

p: e‘Ho/H (36)

Hdwmceé/heef (37)

€g= the energy gap between adjacent points on
opposite sides of the Bragg plane

6 = free electron Fermi energy

f

m free electron mass

Magnetic breakdown, as this is called, can obviously give
rise to an incredible number of new orbits.

IV. Quantum Oscillations in the Thermoelectric Power

A. Introduction

Theoretical expressions for equilibrium properties
such as the magnetization or entropy of a free electron
gas are relatively easily obtained from the expression for
the free energy (equation 34) through the application of
appropriate thermodynamic derivatives. This is impossible
for the non-equilibrium tranSport phenomena. Another diff-
iculty is encountered if one attempts to use the Boltzmann
equation to calculate the transport tensors; the transverse
diagonal terms of the velocity matrix are zero in the Lan-
dau representation.

There have been a number of attempts to calculate the
oscillatory terms in the transport effectsl7"23
the density matrix formalism or a modified form of the
Boltzmann equation. These calculations have been made under

using either

a variety of assumptions about scattering mechanisms and
dispersion laws, though none are directly applicable to the
thermoelectric power oscillations in a complex metal.
The only results presented in a form which can be easily
extended are those obtained by Horton.23
B. The Horton Theory

The approach used by Horton is to solve the quantum

mechanical equation of motion for the density matrix p

b = «3; [m] (38)

in whichtK is the system hamiltonian and [ ] indicates the
commutator. Neglecting electron-electron interactions he
writes the hamiltonianIK as the sum of single particle hamil—
tonians, which in turn are written as the sum of a perturbed

(17)

(18)

and an unperturbed term. The former contains only the
electron scattering terms while the latter contains all
terms involving the magnetic and electric fields and can

be used to obtain eigenfunctions and energy eigenvalues

in the free electron approximation. Using these as the
basis, Horton calculates the matrix elements of the scatter-
ing hamiltonian assuming a Sparse and random distribution
of delta function scattering centers. The entire hamilton—
ian matrix is then used in equation 38 along with an assumed
initial density matrix and the resulting equation solved
for time infinity. The expectation values of the electri-
cal and thermal current densities are then obtained from
the equation

<Q> = Tr(Qo)- (39)

There are a number of further ingredients in the cal-
culation. Fermi—Dirac statistics are used, the energy
shifts and line broadening due to collisions are neglected,
and it is assumed that an electron is allowed many orbits
between collisions, i. e.,

0001’ >> 1 . (’40)

The calculation is made only for a free electron gas.
The results of this calculation are presented in terms

of the elements L of a 6 x 6 tensor which can be related

13
to the tensors of equations 9 and 10 as follows;
eLEE TLET
L = . (41)
eLTE TLTT

The magnetic field is restricted to the z direction and
with the help of symmetry arguments and the Onsager relations
the thirty-six elements are reduced to nine.

(19)

/L?2(H) L12(H) o L25(H) L15(H) o
-L12(H) L22(H) o -L15(H) L25(H) o

o o L33(H) o o L36(H)
L25(H) -L15(H) o L55(H) L45(H) o
L15(H) L25(H) o -L45(H) L55(H) o

o o L36(H) o o L66(H)

(42)

 

The only elements needed here areL22, L12, L25, and L15.

ce~r a) -l
1:412:11?— L 1- T2 [1 +nghfcz Z( -—-)V A 2(X)COS b

1‘ +00
7‘6 EEVE—Ez (—) A30.) sinb j} (43)

C

L — 5" 2291-1313 12£®Ff #733,” A30.) sin b

15 6-H6f(1“2+w2)1‘ 10w2

1424/?)lhwc . {—12" ‘1
+ 2 cf 6 M3" A4(*) °°s bl (44)

 

 

L =fl2£§(fl)2._£_ r1_15€' WCfLiquufiinb
25 3 m e:f (I‘2+uo:2 L 81r2 kT VW

+3.57%? :5}: 5, %)VA4(X) cos b 1 (1+6)

(20)

 

n = number of carriers in the (spherical) band
2
2w kT
X = Vt—EEE—) (47)
2W3
_ f w
b - «th ) - 1; (1+8)
A2(x) = 2x csch x (49)
A3(x) = —W csch(x)[ l — x ctnh x] (50)
W2 2
Au(x) =-§ csch(x)[2ctnh x - 2x ctnh x + x] (51)
r =-% = scattering probability per unit time (52)

Since the calculations are restricted to a free elec-
tron gas the results are not suitable for comparison to
experimental results. To illustrate this one may look at
the resistivity tensor Lgé/e. Neglecting the relatively
small oscillatory terms one obtains the classical result

p0 RHH O
P = -RHH pO 0 (53)
O O

951%.. =25. (54)

RH: J— . (55)

Thet;ransverse resistance (p11 or p22) is independent of

(21)

field in this model, dramatically untrue in real metals.
We cannot expect meaningful results by simply applying
equations 13 or 14 to equations 43 through 52.

However, it is instructive to look at the zero field
limits of the transport coefficients and compare them with
Boltzmann equation calculations for free electrons. This
limit for the resistivity (equation 55) is quite familiar.
In this limit equation 13 results in

1T2 2

_ k T
- Beef (56)

 

which is a standard result for impurity scattering in a
free electron gas.

C. Extension to a Complicated Fermi Surface

In the absence of a rigorous treatment of oscillatory
transport phenomena in complex metals an attempt has been
made to extend Horton's results. Although the methods
used in this extension are by no means rigorous they do
appear reasonable. By equations 13 and A-26,

(L

311= p11(LET)11 + p12 ET)21 + p13(LET)51 (57)

H) = TS H) . (58)

“11( 11(‘

The coefficients (LET)ll and (LET)21 are to be identified
with L25/T and LIB/T from equations #6 and 44. In doing

so we find that for a free electron gas the former is
larger than the latter by a factor of order wow for a free
electron gas and that the relative magnitudes of the oscill—
atory terms are almost the same. If p11 and p21 were equal,
then, the oscillatory contribution to Sll from the second
term would be much smaller than that from the first. Al-

(22)

though for a free electron gas p12- wchll’ giving almost
equal contribution from the two terms, 'we may find a more
favorable situation in real metals. This must of course
be Justified in some way for any particular case (see sec-
tion IX). For the present, however, the last two terms
will be neglected.

If we restrict our attention to the oscillatory terms
in 811’ we may write

311: pllLQB/T (59)

in which the oscillatory terms in p11 are to be neglected,
a very good approximation in most cases, although again in
need of Justification under any given conditions. We shall
regard p11 as an experimentally determined function of mag—
netic field while an expression for‘E25 will be obtained
from an extension of equation #6.

In a real metal the electronic states from which a
given set of oscillations arise are those near the periphery
of an extremal cross section of the Fermi surface. With
these orbits is associated a cyclotron effective mass mf
and a de Haas-van Alphen frequency f given by the inverse
of equation 33. Here we shall assume that these are the
important parameters, and shall rewrite equation #6 in terms
of these. In so doing we have treated the small section
of the Fermi surface as a "free quasi-electron"sphere which
is uniquely determined by m*and f.

The cosine term in equation 46 can be neglected to
order kT/ef, an excellent approximation in metals. In the
high field limit (wCT>>l) the remaining oscillatory term is

125:2)??? n EL? 3: %)VA3(X) sin b . (60)

7w

OR)

In order to alter this as suggested in the preceeding para-
graph we require the following substitutions;

(23)

 

   

m - m (61)
* 2/2
1 2m 5 /
n —e-— ( f) 62
672 h2 ( )
we —-> eH/m*c (63)
hwcf
of H (64)

Upon performing these,

..._2_ .02 1/2k'1‘m"f
125- 611-2 ( hj) $5726: %)V A3()\) sin b (65)

in which b is now defined by
b =(2wfv/H) - w/u . (66)

In fact, the oscillations do not come from a Spherical
surface, and an amplitude correction is required, We might
expect that the electrical current caused by a temperature
gradient is proportional to the number of effective carriers
involved. This is equivalent to assuming that the tensor
elements LiJ from different bands or parts of bands simply
add to produce the resultant L13. Applying this idea to
the oscillatory portions only suggests that the oscillation
amplitude should be proportional to that range of kz values,
dkz, for which the cross sectional area of the Fermi surface
is within some small limit, (AA )0, of the extremal area.

We therefore expand AA in a Taylors series,

(24)

2
BA 1 a A 2
(AAf)O= (5E)k dkz+-§(5—E2 )k dkz + ...... (67)
z o z o
The first term is zero at an extremum.
f 2 2 1/2
dkzs L2(AA )O/ (o A /okz)k ] (68)

O

This is to be compared with the same parameter for the Sphere
used to derive equation 65, which must now be multiplied by

:(a2A/5k2 z)f qefs 11/2

2 2
(a A/dk Z)k0

 

in which erfs = free quasi electron Fermi Sphere, for which
the appropriate derivative is easily Shown to be given by
2w.

An estimate of the value of the area derivative can
be obtained under the proper simplifying assumptions. The
most reliable method utilizes de Haas van Alphen frequency
data as a function of field angle. For a perfectly cylindri-
cal Fermi surface, the frequency as a function of angle 9
measured from the cylinder axis would be given by

fc(e) = fC(O)sec e . (69)
We define, for some real frequency f,
6f(9) = f(e) - f(O)sec e (70)

where e is measured from the field direction of minimum fre-
quency, and expand assuming circular symmetry,

(25)

one) = .1924. 893+ ...... . (71)

The second derivative of the area is then given by

2
a A My
(———) _ s-—T97 (72)

from purely geometric arguments.

Another estimate can be obtained by assuming a one
O.P.W. Fermi surface. A small neck in such a structure is
drawn in figure 6. For sufficiently small pieces of the
Fermi surface the cross sectional figures bounded by abc
and a'b'c‘ will be approximately similar, allowing one to
write

A(kz) = A(ko) (E; )2 (73)

where

km: the minimum Fermi momentum measured from the
central axis as shown in figure 6

kmo= km at the point kz= k0.

The area derivative is then given by

0/

2A 2A(ko) (52km)
3133- mm) a: “o

(74)

(26)

 

 

 

 

RF:

6’

Figure 6. Curvature of a one OPW Fermi
surface

(27)
The last factor in equation 74 can be estimated by recall-

ing that the curvature of the Fermi surface is that of the
free electron sphere in the one O.P.W. approximation.

52

 

l( km ) l =1 (75)
52kz ko RFE
2 .

B A
2 _ o 0 FE

Bkz k0,e-O

in which RFE is the radius of the free electron Sphere and
is given by

RFE= (31T2no)1/3 . (77)
Approximating the area by
2
A g wkm , (78)

which is an underestimate, we obtain

lifll e- Efl , (79)
5k: RC, 9 =0 (371.2110) 1/3

no= total electron density.

Written in terms of the de Haas van Alphen frequency,

2 1/2
Lg. = ”EVE; f . (80)
dkz ko,e=o (3W2n0)1/3

(28)

The factor [(52A /ok:)k ]-1/2 can be found in the
standard expression for the free energy (equation 34).
Another factor found there which will be used without Just-
ification is the Dingle factor inside the summation. In-
cluding both of these factors,

 

I: =%(2§)1/2 kTm’fl‘f x
25 6 n3 [(52A /ak:)k 31/2H3/2

. -1)V
6' (If? A3()() [sin b][exp(€)f,r)]

(81)

we must obtain an estimate of the relaxation time T;
two methods of doing so are immediately apparent. The first
is to use equation 54 and the zero field resistivity. Of
course, since we are here using the free electron approx—
imation and assuming that T is independent of‘k, this method
may be used only in one O.P.W. like metals and in the low
temperature impurity scattering regime. Furthermore m0 and
n must be the free electron mass and the total number of

o
electrons in the metal.

mo (82)

 

'We might also use the value of n,obtained from the
Dingle scattering factor. This method has the advantage of

(29)

providingthe local relaxation time on the part of the Fermi
surface giving rise to the oscillations but has the dis—
advantage that TD is rather more difficult to extract from
the data.

If there are N identical sections per Brillouin zone
presenting a given cross sectional area 'we must multiply
equation 69 by N.

The transverse resistivity p11 in equation 59 can
take on a variety of forms. One of the most common and
the only form considered here is

A 2
911 = 90 + 02(3.J)H (83)
h = unit vector along H
J = unit vector along 3

It is generally true that, when this is the behavior of p11
the second term completely dominates the first, above about
ten kilogauss at liquid helium temperatures. Using only
the second term,

- 5¥§ ce 1/2 km*fN (8,3) 1/2
311— W (:13) 02 H X

 

[(52A /6k§)]l/2T

v
3): (ff—$7) A3(l)[exp(i—E¥D)] sin b

(84)

There is a very interesting variant of equation 84.
By applying Kohler's rule to equation 82 'We find that

(50)

p2(h,3) depends upon residual resistivity in'a very special
way, namely that

92(h23) =m (85)

0o

where a(h,3) is not a function of the zero field resistivity.
Putting this into equation 84, coupled with the substitu—
tion for T given in equation 82,

1/2 2 A
> “keno (fibml/eamm x

[(aeA/ak§)kO]1/2 °

'0
(D

\N

(g = 545
11 6%? (h

§%)VA3(X)[eXp(-£-:TD)] sin b (86)

in which T no longer appears.

The summation in equation 86 contains all of the os-
cillatory dependence. The relative harmonic content is
a rather complicated function of temperature and magnetic
field. The function A3(l) is plotted in figure 7. The
maximum value is very nearly unity.

(51)

K .m> AKVM< .N. GMSWHW

_ _
m.

3%..

 

V. Nonoscillatory Magnetothermoelectricity

In view of the information concerning Fermi surface
topology which is contained in the high field magnetoresist—
ance of Single crystals we might ask what effect this top- -
ology has upon the thermoelectric power. Bychkov, Gure-
vich and Nedlineuhave made qualitative calculations; their
results indicate that the diagonal terms of the Seebeck
tensor should saturate at high fields except for the case
of a compensated metal with no open orbits. In this case,
the transverse diagonal elements Should be linear in mag-
netic field.

(32)

VI. Experimental Approach

A. Seebeck Measurements.a§ Low Temperatures

 

Typical Seebeck coefficients of metals in the liquid
helium temperature range are so small that the direct mea-
surement of a thermoelectric emf requires the measurement
11 volts. This would be diffi-
cult under any circumstances but the problems are compound—

of a voltage as small as 10'

ed in this case by the temperature gradients supported

by the measuring leads. These gradients can cause variable
thermal voltages on the order of a microvolt or more. A
number of devices built to cope with this problem have been
reported,25’31the most sensitive of which is capable of
measuring Seebeck coefficients of about 5 xlO-gv/Ko at about
20K.

B. Peltier Measurements, General

The sensitivity of carbon resistance thermometry at
liquid helium temperatures suggests that a measurement of
the Peltier effect may be more sensitive than a Seebeck
measurement. The simplest experimental arrangement would
be that shown in figure 8. The two samples A snd B are
suSpended from a heat sink at temperature TO and a current
I is passed through the Junction. The temperature difference
aT will consist of a Joule term proportional to 12 and
Peltier and Thompson terms proportional to I.

AT=-% [RI2 + n1] + g1 (87)

Here K is the thermal conductance from the Junction to the
bath, R is the Junction resistance plus some fraction of the

(55)

(24>

 

Figure 8. Experimental Peltier Junction

(55)

sample resistances, and g is a constant involving the Thomp-
son coefficient. For the presentvwashall neglect this term,
requiring that it be small. To this order, the Junction
temperature change caused by a reversal of current direction
is

8T -—- AT+I - AT_I = 3%(2111) . (88)

The addition of a known amount of heat, Qt = r12, to the

Junction causes a temperature change of
- l 2
oTr = K(ri ) . (89)

Solving equations 88 and 89 for H,

2
(90)

cflm
F38

told
FH*

U = H - U =
A B r

Being a reversible effect, the Thompson heat also
contributes to ST. The difficulty in the treatment of this
effect is that it is distributed along the length of the
samples and its effect on the Junction temperature depends
upon the temperature distribution along the samples. We
will assume that a fraction n of this term is effective, i.e.

ATth = moth/K . (91)

By equation 6,

ATth = ”IKE—1T (HA‘ )J-B) : (92)

and using the Kelvin relations,

(36)

d[SA- SB] I(QT)

th= NT d'I' K ° (93)

 

aT

Ifthe Seebeck coefficients are slowly varying functions of
temperature,

(n - H )
moths“ AK B 1%?) . (91+)

 

This must be kept small in comparison with HI/K, result-
ing in the restriction that

alts

<< l/n . (95)

It is experimentally Justifiable to require that

(95a)

el‘fa
IA
0
.t‘

for measurements accurate to about one percent.

The aT referred to in equation 95, the total tempera-
ture difference across the samples, is in most cases domi-
nated by the Joule term,

AT- = RIe/K . (96)

J

Combining equations 88, 95 and 96, we obtain

(57)

i2 2521.22... 5. ~04 (97)
T RK(6TH)2

which puts a lower limit on the measurable Peltier coeffici-
ent.

6T
n22. 173%) («72—0)2 x 25 (98)

Here 6T0 represents the smallest temperature change that

can be measured accurately. This is, of course, a function
of temperature, thermal stability, and thermometer circuitry.
It is an empirical fact that we are able to measure changes
of about 10"5 Kelvin degrees to a few percent accuracy at
20K, so that equation 98 becomes

)HL2 (2.5)(-B-¥)1/2 X lo"5 . (99)

The most favorable conditions for this measurement would
include a superconducting reference metal B, so that the
electrical resistivity and thermal conductivity of this

arm can. 'be neglected. If the Junction resistance is small,
we may write

R amoAL /a (100)

in which L, a are the length and cross sectional area of

(58)
the sample. If the thermal conductance is due only to the
thermal path through A,

K = kAa /L (101)

where RA is the thermal conductivity of the material.
Recalling that

pk = LOT (102)
L0 = Lorenz number,

we see that

HZ 2Lcl/2 x 10'5. (105)

The Lorenz number for impurity scattering is Just the class-
ical value, given by

8

Loa 2.4 x 10’ MKS units , (104)

n.3 5 x 10"9 v (105)

which is equivalent to a Seebeck coefficient of about 1.5
x lo‘3 nv/K° at 2°K.
It is obvious that the above calculation concerning
the Thompson heat correction is by no means rigorous although

it does provide useful guidelines. It is not difficult to
formulate the problem in more detail and in fact solutions
can be found under certain simplifying assumptions. Some
of this work has been done and is discussed in Appendix III-
Very little new and useful information can be obtained
beyond the fact that the Peltier coefficient measured corre-
sponds to that at some temperature between To and To+ aT;
the exact temperature depends very critically on the assump-
tions. However, even this fact is dependent upon smoothly
varying Peltier coefficients and small temperature differences.

The reference material must be chosen with some care.
One would expect that a superconductor would be ideal, as
it possesses no electrical resistivity, high thermal re-
sistivity and zero Peltier coefficient. In the zero field
measurements a superconductor was in fact used (see Appen-
dix II). Unfortunately in high magnetic fields one would
have to use hard superconductors, which have a number
of undesirable qualities. It is very difficult to obtain
a good electrical contact to these materials, as no known
solder will wet them. Furthermore they are all very stiff,
and using them in the confined volume inside a magnet dewar
presents technical difficulties. Nevertheless, these diff—
iculties could be overcome if necessary. In the measure-
ments made on zinc, however, the quantum oscillations were
of primary concern. These are, of course, unaffected by
thermoelectric heats generated by the reference. Further-
more the size of these effects are such as not to require
maximum sensitivity.

The use of normal metals still requires an optimization
of the measurements by adJusting the size of wire to be
used. One must chose this in such a way that the electrical
and thermal resistance are at least of the same order of
magnitude as that of the crystalline sample in high magnetic

(40)

fields. It is also desirable that the Peltier or Seebeck
coefficient of the reference is known, although this does
not affect the amplitude of the Single crystal quantum
oscillations. Since a severely cold worked metal has a
more predictable low temperature behavior in its Seebeck
coefficient (see iron data in Appendix II), it may be ad-
vantageous to use such a reference material.

C. Thermal Contact Considerations

There are a number of requirements on relative thermal
paths which assure that the constants K appearing in equations
88 and 89 are identical. In particular, placing the ther-
mometer and the calibrating heater onto the same block which
in turn is in contact with the Junction may produce erroneous
results. This situation is Shown schematically in figure
9a. If the thermal resistance W1 is not much smaller than
W2, the measurements would be in error by the factor (W2
+ Wl)/W2. However, if the Situation can be represented by
figure 9b, the temperature differences measured at the car-

bon resistor are

 

 

W W1W5+ W
216 1.
M3 =qJ (W1+ W2 ){ W3(W1+W2) I (106)
Wl+fiw2+ W3 + W4+ W5+ W6

for heat qJ produced at the Junction, and

 

 

w ”W W6 1
T"‘h qh(wl+l W2 ){w3(wl+ Wé) J (107)
w + w + w + “4+ “5+ w6

1 2 2

(41)
lfie‘lflfdl‘j't: Tue-m.» 9M [5712?]

[:7 (nu/:71 a N]
var,

//HEQT7-5'/;k/L/

 

 

 

 

 

 

 

 

 

 

[AZ-aria W [Jun/c 7704/]

J.
\M, w, _
TELRMOMETA'RJ

W: W

/// H541“ /$IN/K////

We

 

lM M lwfltnrm
\‘2— rfiéfiMAL

    
 

V, {'ow'put'TO R
cflRBON
Rem-rel? (Ex 6 #4 Na 5
3 £45 )

Figure 9 . Thermal contact diagrams

(1+2)

for a heat qh produced by the heater. These differ only
to order W5/W6.

Figure 9c shows an experimental arrangement that can
be represented by 9b. In this case, W5 is the thermal re-
sistance through the thermal conductor between the Junction
and the heating resistor r, W6 consists of the high resist-
ance leads to the heater as well as any exchange gas in
the system; W5 can be much smaller than W6 with little
difficulty. In practice it is found that exchange gas
pressures up to 50 microns ch) not affect the measurements.

D. Double Junction Technique

There are a number of advantages to the double Junction
system shown in figure 10. The center of sample A is anchor~
ed at temperature To’ as are the lead wires B. The car-
bon resistors Cl and C2 constitute two arms of a Wheatstone
bridge, and if the response of this thermometry system
is linear (see Appendix IV), the bridge output v is given by

v = Gl(ATl - AT2) + C2 (108)

in which Gl represents the bridge sensitivity, G2 an adJust-
zible bridge offset, and ATl’ AT2 are defined as in figure
L10. Neglecting the Thompson heat,

{(R1 R2) 2 (l l ) rlii r213"
v = G -—— --— I + HI-— +-— +———— +-—-——'> + G ,
1 K1 K2 K1 K2 K1 K2 J 2

 

 

(1+3)

T.

+u‘
nu

 

 

Y a...» 6.51% waun.

 

 

\_.
\

 

 

 

\\\\7

\_\\\

Figure 10. Double Peltier Junction

all'l‘lll‘l'ul‘li‘lq'lllill.lqvl‘ I:

 

 

 

 

(44)

where 11 and 12 are the currents through the heating resist-
ors r1 and r2. All other symbols have meanings analogous

to those in equations 87 to 90. Reversing the,&Amp1e curr
rentresults in a change in v given by

1 1
5"" = G1{2"I(K’1 + 1(2)} (110)

For calibration we start with 11:10, 12= o and switch to

il= O, i2=io. The resulting change in bridge output is

r I"
5Vr = G1{1§(K%'+'K§)} ° (111)

Evidently if r1: r2= r,

ON)

1 r 6v"
“T2 57% . (112)

If the carbon resistors have differing sensitivities
one must replace equation 108 with

v = G11(AT1) - G12(AT2) + G2 (113)

but this has no effect on equation 112.
Besides the obvious doubling of sensitivity obtain-
able by this method it has the further advantage that the

 

 

 

 

 

(45)

Joule temperature differences of the two arms tend to
cancel (see equation 109). This is particularly useful
for continuous measurements in a changing magnetic field,
in which the magnetoresistance causes large changes in
the individual Joule terms.

 

a “'|"‘ ll: J 1 ‘II 1. I 1

 

 

VII. Experimental Apparatus

A. Cryostat

Figure 11 is a diagram of the lower portion of the
cryostat. The sample chamber was evacuated through stain-
less steel supporting tubes. The proJection at the
base of the sample holder was soldered through a hole at
the bottom of the enclosing can during the runs but other-
wise the sample holder was supported only by the twelve
pin electrical connector at the top. This arrangement allow—
ed the removal of this section when mounting new samples,
while still assuring rigidity in Operation.

The ten mil sample current leads were run directly
from the connector to the ends of the sample and were either
Pb of six-nines purity or four-nines silver, severely cold
worked. All other leads went through a secondary connector
at the terminal boards. Seven mil copper leads were run
from the connector through holes in these boards, and held
in place with G.E. 7031 varnish. In order to limit heat
conduction down these leads they were wrapped around the
supporting rods. The ends sticking out of the terminal
boards were then clipped to about 3/16 in. and the coating
stripped off to provide a soldering "post".

The carbon resistors were attached to the sample with
G.E. 7031 varnish using one layer of cigarette paper as
an insulator. Care was taken to place the two resistors
on the same side of the crystal in order to minimize inter-
ference from transverse effects. The leads from these to
the terminal boards were three mil manganin wire, twisted
together to minimize pickup noise.

The calibrating heaters were coils of manganin wire

(45)

 

 

 

(47)

u
l

f‘.’.
,1-
( .-

L l,\.'

.
r
r4

“1:." '

_ Cilia-Ion ,r-rtf

‘ ‘\ TERMIN

 

 

 

\~.

 

 

_ . a .
M . .. l
1“ . . i - I!
l m
m .
_~ . 1,: I- 31)..
_ h l I: I ‘Jilil ! $11 ..:.Il:.v‘l."l‘||lull..bll.lll.‘il.l pill“.

 

/’
I
l
i
{.
1

E
CE

 

\

Vpr.'§lj:{:: '
\

POLE FP-

mnm.

 

Figure 11 . Cryostat

 

 

(48)

on 1/2 in. lengthsof approximately 50 mil c0pper wire,
potted in G.E. 7031 varnish. The c0pper form was soldered
directly to the Junction. About two or three feet (70-
100 Q) of manganin wire was used, one inch (3%) of which
was used as leads to the terminal board.

As only the room temperature resistance of the heaters
was measured the resistance ratio and magnetoresistance of
manganin were needed. These measurements were made and
are discussed in Appendix V.

The samples were in the form of a rod about 1/2 in.
in length and 3/32 in. in diameter. They were soldered to
the support with Cerrolow 117 solder.

The dewar fit between the pole faces of the magnet as
shown in figure 11. The magnet could be rotated about the
axis of the dewar. Obviously the current in the sample
always flowed transverse to the magnetic field direction,
to the accuracy of the sample alignment (Appendix VII).

B. Pumping System

 

There was very little to this "system". It consisted
of a mechanical vacuum pump, a valve, and a thermocouple
gauge. Although a diffusion pump was available in the
system it was used only for purposes of'leak detection.

An exchange gas pressure of about ten microns, mea-
sured at 40K, was needed in the sample chamber to provide
a thermal path from the sample to the liquid helium bath.
After this was admitted the system was closed off and left
unpumped for the remainder of the run.

(49)
C. Measuring Electronics

A block diagram of the measuring circuit is shown in
figure 12. The bridge-phase sensitive detector 1 (PSDl)
vuusthe basic thermometer circuit and the output from PSDl
was Just the v referred to in equation 108. In the simplest
mode PSD2 was omitted from the circuit and the output from
PSDl charted by the strip chart recorder. A standard switch-
ing sequence then produced a square wave-like trace on the
chart from which the ratio EVE/6vr was obtained. PSD2
was used for detection at the switching frequency. When
this was in use only the sample current was switched so
that the output from PSD2 was proportional to 2HIII/K.

A more extensive description of the Operation will be given
in the next section.

Only the control box and bridge were not commercially
made. Identification of the other units are as follows:

Sample current supply-~Princeton Applied Research
model Tc-602R constant
voltage supply, modified
as suggested in the manual

PSDl--PAR model .1134 with model CRll-A preamplifier
PSD2--PAR model HR-8 with type C preamplifier
Recorder—~Texas Instruments ServoRiter II

Figure 13 is a circuit diagram of the bridge used in

this work. It is a standard Wheatstone bridge with a number
of Special features which facilitate its use in low tempera-

 

 

 

(50)

» AIIIII "1.
F
a
1,
—
a
. a!
I... _..
:IL IN \I .
. n. a... lay
M . ! I 0.!
4“ .II A! a‘/
w A an (I.
win no r
1“ pl» 5
.
_
_
, 1,. . 1

_a_ .sa: ::+y

 

1 . .
. . _
II!“ t“. O - vlul‘vonrt‘i'l I. {I ‘l
_ . F 1|
1 o . .
.
u
i :N .
_ I . _
z.) _ i ,
_ P
“ _ nflv_ _
. 2. . u .1 _ _
a .z 1 a . _
.. ML 0. . . . r.“
A A

M WW m. 911.- I. . .o .f...
1 .. f. i _ .5.
1 r .. ., .1 a
. / o I. . ._ _ ..\._
. fl ob . m _;
W 0. H . th... “FL
1. ._ -.
i flu . a r- a..- a
_ .. 1 9 4.. 1).! . -3! ill.

_

m _

_ .

- .
.p
o
f a;
v v.
o
.r;
,l
. I/ _

 

a
2.

 

II‘ :0 I l I OI. [I ['1 Iwmd
,. V
i! ....... i , I . 1-5 .0: -1 1
_ _. F.
.
. . _ .
u 1 m T.
m 1 f
1 . . nu
w . 1.1 O» .0 o
H. . m ...\ I).
n .. .1 n H F. .
_. {a _ 1 C 5 J
1.: 1.3, U .f.
s) l... .r... .. t _ .r 1'»
‘llfv' «9.4. _ luv“ .
w ./I h /.a or— +1.
? H _ z ,o
H o .JIIYWQ. 0/ v.”
5 i M. C. .-
u I. (u an”
. n I
_ l _. C
.I a 1r I- III!

 

Figure 12. Measuring electronics

 

 

 

(51)

ture differential thermometry.

The carbon resistors were connected to J3 as shown in
figure 14a. The reason for using four leads becomes appar-
ent when considering figure 14b, which is a diagram of the
bridge in a calibrating mode. This lead arrangement added
the lead resistance to both arms (R01 and RS) so that at
balance RS=RCl.

The total resistance of the ratio arms of the bridge
could be selected as 2K, 6K, 20K,60K or 200K. For purposes
of standardization either the ratio arms or the cryostat arms
could be replaced by 5K, 1 per cent resistors. There was
also a provision for changing the resistance of one of the
bridge arms by .1, l, 10, or 1000, an input attenuating
network, and an out of phase balance network. The latter
was used to eliminate the out of phase signal arising from
interlead capacitance.

The control panel contained all of the switching cir-
cuitry as well as the current source for the calibrating
heaters. Figure 15 is a complete circuit diagram.

The power supply section of the control panel was a
standard type of regulated supply, capable of a 10 volt
output but set to about 6.5 volts. The heater current
control was a simple potential divider—current limiter cir-
cuit with full range currents of about .01, .l, l, or 3 ma.
R1 was a 1353 Q, .2 per cent resistor used to measure the
heater current. The potential across R1 was fed directly
to pins 5 and 6 of connector 01’ A schematic of this por-
tion of the circuit is shown in figure 17a.

The sample current was passed through the reversing
switch shown in 17b. The relays were arranged as shown so
that one was off for both current directions, thereby re-
ducing the load on the internal power supply. This current
was measured by measuring the potential across the 3.784 0

 

 

 

 

.2
.r3

owofigp pmumsoshozs .MH madem

 

Lat. loan 2. “an.
have V 3...... 3 u...
.238 I: z<.4
«In?! OIL; 3t f “4
be... a x “.5
.8& ”an
n... 1:... 3...... L95 in.»
2® ® ® g g 8.! #58 .3: .8 2a
b 1:» L: ”J

g u“ is» Q3533, .51: ._

3.9!» x22»- l.t|.- . ‘3 Zn!“ "ran
«3 ¢-§.u\uch "nu
Spot .In :9. =03“ .. um.

33.5;zss.x

\Ictu 41.. {.1 u to...“
:36- .asn :ld 0.”

on t...itu!.qu

 

 

axé‘K Lint i.u¢nz:e‘

a .~ .. 3:1 :6
n 3......

at! in. d u .IvL.3M

 

 

 

 

J.

 

 

 

 

 

 

 

 

 

AA
w‘v

 

wkn cl .cu

~\.K23 .uu
‘KK‘. oaV

otuxolo .uv

VK‘ .DV
.‘(i .qU
Xx! . .u
‘3 ~§£ I g?

gluttu

  

 

 

kJ‘ ‘3‘ 34.

m oQEmw 029k wk mum} 9T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\“.0

fitquh .2:
‘~.“O 02‘
$..1I. .v:
.5538! .1—
3;.41uze
4:. .J
1n u 2.-
1a! .J
On I. up.
1’: .J
12%.:
«no. .3
4.: .9

i Q... no.

1.0: . a.

v. 4 ”1.2:. i

43 ‘Gfimk

 

 

 

 

 

 

 

 

 

 

 

 

M: 3 mil
ma mg a nin
lead, ~ MAO.

 

 

 

 

 

#3 I?
.........- .._.__........_.__._.I
I
I
MI
AA AA “A I
I
I
01?: (we I
I
I
I
._.._...-.. _.....-._..__.._.____.._.I
)0
"Dare-£70,?
4 8 (I-.. D_
M M '2‘

Figure 14 . Bridge to cryostat leads

 

{I
‘I‘Ill
\I‘ ‘II

 

 

moaconuomao Honpcoo .ma onsmam

25-h .3196”. 53%. gm: guinea} o, Auk 53... u. u .70

. .cn . aux-Cam Siam,

- . ‘ K4
. .-....\uZuN. .>..H.~IZM..AH .1... 7-0.0
«0-6 5.... as- .o 12:... > .m saw-aw .t a

r"

 

mozzq .mm m x v

9‘

 

 

m
Km to #3 58d J...- ..
9 . O ) . .
rI . . . . .0 Com .4. :3- rd... a norm
4. . . . , .Bw magi mm. . “ET-f a... . .n dump}; .3

Sum
Save.

Q

MI

54

{LI .

I
wan-.4
H‘ c‘) .‘J Q.

 

\I

 

A

 

._
70mg Q \ .I- i .. I II. .mI. --I I-
T m 1...»...- . a . 1-31.1-2.
.awwmwn-z A. . Sign . wife
. ‘ .. I! .4 —l I a

 

 

-m-I--§¢-—T --. I

. --_—...

 

CTIIIflIilII III III II III II- 0. I III-I..I I-‘.Ot|. I“. III. I.

 

 

 

 

I ‘
I
I
”—4 ‘r-w
-——~II
./
>3
f
‘1 V
.4
\1
Q
9.
It.
3
C:
O.

 

 

 

 

SI . . A. . .lfil s _ . I H . ~vI. \- “
uEITm «Mm-sq T _ N .u( I.“ . .QN? WING ADV-“II N“ M . .«I w u . .. _ A. .H
K... x . v. Koo.»- ‘ o. . _ x. u: . h .

a:o2- ~«:.- _.«xwvm . x h. . . mag -snnr--; -vtsa

-.- 7L . I I I -. . i- I. z .- - --III.--I.I-. .- II I- - .- 2-15- -I.I---I.- ,I I...- .-I I III; .a m. _ . .v... _ _

-I .IIIIri-L . M E w “a (k . T“... .w «L .,

. v ~ tIIIII a I III IIoIII I‘ . III A... I,..Inn<./L P»... u
‘I IUI II ~ I -. I I. I.-.I.:IIIII III II..IIII.III|- .Io I “I l .-Ioo b. ._ I\w I II..-. .9—3 —;-WI A)! f
. - u. ..... . - s--. .. . Lip: ,. .- U.

I LT.

 

.
. . .. . .
.I - I- I. - . . . . ... .. 3 . \ 31...... . .
- . _. a...- “ . 344.wa- .N... . x.- .-2.........- .. _
I I. . h .. 1""I.lv.III.IIII-III’§IIIIIII-III .. .lo .1 .o. IIII.’ I): \v .1“ . . .1...
_ a was?!- .-uj. ...+.I..-.--...-.-..-.-........-.........-.-“Hm-Hm. 1...... ...o-.-..--.... i .. Q 3...... i
. . .. . . . .- -. III. .I-o

 

 

 

   

 

 

 

 

.
_ _ N) .- N—F. W- .A $ . _. .W . _ 1H 3... Is..I-. -. -
. 1(2).” I... J 0.2,. o ._ . .. ...+.II.-II I II. 3...... . .
IIA \ . a u . n _ . I . I. .I.l . ..p E... I...“ .
m ‘_N~f\.-vI~/(fi<)qll~1|v*\l‘ ‘1” m .. fi fi . . IT I! . .P-J A \I.O [\I‘ . 3) AI 5
I \mrw : . . . . u .H . ~ .4 1. VI . I. .. . ion-.5). .- ..... .. . ,L
. M 2M”: n.1\ . 1 . ~ . a]... h. . N 7 "4 .\o If \ 3. [J m. \I IV
V:_ ....~n.. x... ._ _ . __ . A. 3“ . .5 .. , ~ I! ., ....\ W. I _.I m
,\Xr\r.N1 f. . . . 2...<_C\ XONV x...- v. E- . .. _..I m
m I J \/ )xerIlI II. . a m . w w _. ~ .- J-TAV AV W .U- .lu I. _ m n . . 1.. “I... _ .sII F .- .m
I” . . u a . III..4.-.I.l.ul .I a .o... n I I Its .... I .IN
. ix -- i: . 1. . .. . . l- . . ex .iIIE 11-
,. l. I.-- . . M . I . . .v . .IlII-HIo I. .IIIII--. .i, . - I
.u JCA ...\:.. Q: O‘JJ/J 1.. fl .. P/0 m fl m . . .. . IV“ mum... ...~ H. . _ .I.II.II... ..2...I m 12,-- --.IIII.II«
_. .- . .- T-v. . .2.) g . F- In»... . . 1 . m .. . _ . -... -. .
. \1iy . x . I F . . . _. A N a v _ . . HIIIIII-I.“ .1 . a H II x. o s « l m .
. 9.1.... (2...- Id: : . .u .. _ I... . _. . _._ .. -1- _. .m. _ -
. ft... .. 2..-}... _ _ I. . .._ _ . Tit-21A,- -. .1 ‘ 4.1.1... .1 .. 1:31 .
* r! ..... ”BK .1. N1“ . v r In» a 4 . . . . 4.. . fix» .. \II . .. . . . - If. F a ’4‘. I. _ F .
h. . .. .RIIFII # /I‘ . w * )w-LI I \W}. N c I . (I (II. _ \Jn. . A G n .\
Y:( A.v . J _ /\ v. I/I, \.\I “Q ./ra \ / \sl .. I/ . . . . N. .a.. - I . H . I! ,
- . . .. I... -.. / -_....-L-._ 7 .21..- :
d‘wlIII.“ -W~ I\4\?—\L( ) .1?pm\.J -§ . w o . . _ . n . ‘I . . . ..‘ :U“. ‘. / ~ .~
: Y. . .Q: «iv/“l . ‘.I?o..\\/\)( 3'.) ‘ (I I '- oI ‘1 2.... ~ \I u H « f I _II‘. /.\ 'IIW. fii .a
n It . . ... g a. . . . c .. . ..
1.31» .r i . _. . . 3...... .17... . .x r . . LI... ...-. $3.6 .\/ “7......
I. . NLL 3 .x.-,)\ ”-1 I n .. 0-1 .II. II “7.6 aI . m . .... Z . w. I. Ix :2.
\ ’ . 1 . ‘ . . . . -. ,-I-.. I ..I .
_ rMKE. up -. _ A wit. ..3... .. . 1.. 5.3. . Tito . 4. x. -.:.. ._
K It . 1H .( . no .) .I . . I\. n“ v t . . \
V 1 . I. .
_ .lz: “A. «u _ v ..a_ _ \(FN fr»-
. _ n . a n . — .1va 327.9 N .
. . _.
k. .7. r u

 

 

Nun . Mum .>.\.

.I.III CI... II I II uuvI: III-c, . . .'. . .AIII .I... if.‘ -I I II In . I

 

 

 

 

(55)

 

S
3' 5;. 5% ‘Su (2‘
5..-

I. s, 5., 38 R2 37' Sq

5.- 52

 

Front panel

 

 

 

C, 6‘:

 

 

 

 

 

Back panel -

Figure 16. Control circuit switch

positions.

(56)

4.5 V
’ 500 K

50K

IOOJL 5K

500.“.

+ .-_.-._./’*
.....——.

 

 

Rel::§

' a
B3311 /‘ H“) '

v 0.27. v—-—-9+O R;

if" “u “1

 

 

L
r

- \l

 

u?

{0 Samr'fi

 

'—

b

Figure 17. Current control sections

(57)
resistor R2, appearing across pins 7, 8 of 01’ Either this
voltage or that across Rl (measuring heater current) could
be monitored at the front panel G.R. connector C4.

Control relays Re1--Re3 were all Magnecraft type W133—
MPCX2 mercury wetted reed relays with a response time of
about two milliseconds.

The switch control was made to operate in three modes.
The most complex, labeled mode 1, was a completely automated
mode in which the sample and heater currents were switched
in the sequence shown in figure 18a. Figure 18b is a block
diagram of the circuit used to accomplish this. The vari-
ous components of this network are identified in figure 15,
in which the ”logic" section includes the nand gates, the
or gate, and the relay drivers. The multivibrator period T
could be selected from the following: .02, .08, .2, .35,

1.0, 4.0, 10, and 20 seconds.

In mode 2 the sample current was switched automatically
in a square wave manner with a frequency of l/2T. The cir-
cuit used to accomplish this was Just a variation of that
used in mode 1 and is shown in figure 19. In this mode F.F.2
was removed from the circuit and Rel was under manual control.
The lead marked "to Cl, no. 2" was used as a reference sync.,
for it carried a 5 volt p.p. square wave of the same frequency
as the sample current. The control would not operate in
mode 1 if this lead was connected to an instrument. The
capacitance across the input terminals of most instruments
is enough to affect the triggering of F.F.2. This was re-
medied by placing a 10K or 100K resistor in series with
the instrument, as indicated in figure 19b.

In mode 3 both currents were under manual control. The
circuit used for this was the same in either case and is

 

 

1.:
Sample 0 |

 

 

 

 

 

 

 

 

 

 

 

Curran! ’7 27" 37" QT . 77'
"I"?
A
1. — . l
Hdafer 1 1 J 1 ‘
Cunard O T ? A v . : ; r I ;
a.
A ‘
_ it '
Header 2 '
Currn w} 0 . I I A - l |_

 

4

x .
Mdflwbrofor .. . .
00‘5“”

'P 2'? a? $7 ' j

 

 

 

 
   
 

 

a.
‘ Baby 3
- a
.1 Frizz“; HOP 1 Driver
Mulhvabrah .~ F. Relay 2
,Vonnb h. ’T' F‘ ”raver
5; Relay 1

 

Dnver
D“: or sch ~
F {0 C. ’pa'n‘fiz-

:D.— = "(ma 1““ = Or so“ + com/aria.~

 

Figure 18. Mode 1 control logic

mm

[M unit/IBM 7'0)? , N p
'8 ' F fl IRE? I

MAWAL
a cow/ea;

R21 O
73 [,ffi‘g V D" DRIVE?

 

 

 

 

 

 

 

 

 

 

 

 

lo-IMK
f r”- -~V ——————
.u
b 4%? “8.1 flag
‘ I
l :7
I
4- 6.5V ‘
WiH/o—q Q“ b.
C V . Q,,b,

 

Vncl
. SAMPLE [twain/7; pin 4
Hmrm amen/7; F1973

Figure 19 . Mode 2 control logic

(60)

shown in figure 19c. Closing or opening the switch reversed
the states of the appropriate relays. When the switch was
closed the currents could still be switched by shorting the
lead marked ”to Cl" to ground, which provided a method of
external control.

D. Magnetic Field Electronics

The magnet used was a Harvey Wells 6 in. magnet cap-
able of providing up to 22 kilogauss with a homogeneity of
about .01 per cent in the central one inch cylinder. The
magnet current was controlled by an adjustable internal re-
ference voltage to which could be added an external refer-
ence signal.

The magnetic field sweep was a second generation model
of a system originally designed by James Le Page.32 The
output of a stationary pickup coil placed between the pole
faces was compared with a reference voltage, the difference
amplified and integrated and used as a reference for the
magnet power supply. By adjusting the amplifier gain it
was possible to keep the field sweep rate proportional to
the reference to within about one percent.

In the unregulated mode the pickup coil was not used;
the reference was simply integrated so that the magnet current,
rather than the field was controlled.

The reference for a constant sweep rate was a constant
voltage, supplied internally. For any other sweep form
one would have to supply a reference external to the sweep
control.

The primary difficulty encountered with this circuit
was that the system would oscillate unless the reference
voltage was increased slowly from zero. This necessitated

' (61)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VoLTAGE,
- Snakes” _v‘ \é-V,
‘r——-‘—’-
\é TB
.PO'WWM- ”“‘ R£CORDER
DIV/DER
$4055M£r:R//
MIMA/ET / MRGAIET
Pu. c ' (0N1- ROL
FAG: - t ,
/// ,\ SW55)»
\ / Rein/P \\ CoNTROL
/I (EHL . L

 

 

A, _'|

Figure 21. Magnetic field
electronics

(62)

HeapCOO modem camam .mm onswam

ommmm xOHHQHHSM I

932 onflEw - ma. a.
<1 oom + - z a.
<1 0mm ..

In. ,

\N xww kex

3mm -mq
NOMHZN r Ha
nonazm u as

m on vomzdw m

aekuw§§n\
Mus tum

«\Vzl

 

 

  

11': m
. rm

 

 

 

 

 

,uqsonw mammdSo
pom 59500 90309:»...

         

 

 

 

 

(63)

cranking the sweep rate potentiometer to zero and back

up whenever a new sweep was started. It could be correct—
ed with an appropriately designed, RC controlled transistor
switch at the input to the sweep rate potentiometer.

The magnetic field was measured with a Rawson rotating
coil gaussmeter which had been calibrated against an NMR
probe. The difference between the potential divided out-
put and a constant voltage source was plotted by the strip
chart recorder described in the preceding section. The
source could be changed in steps equivalent to 2.26 kilogauss.
Due to loading of the Rawson meter in the network the follow-
ing equation was used to calculate the magnetic field:

H = 2.26 Kilogauss [N + x(1.03)] (11h)

N = number of (100 mv) steps applied from the
constant voltage source,

x = position of the pen tracing, in fractions of
full scale (100 mv).

E. Temperature Control and Measurement

Temperatures below 4.2OK were obtained in the standard
manner by pumping on the liquid helium bath. The pressure
above the bath was controlled through the use of a "Walker
manostat" and measured with either a manometer or a McLeod
gauge. The temperature of the bath was then determined
using the 1958 scale of the liquid Helium vapor pressure.
After the carbon resistors had been calibrated against the
vapor pressure, they were used as the basic thermometers.

VIII. Operation, Data Reduction

A. Introduction

 

The data desired in this study were a set of values
for the Peltier (equivalent to the Seebeck) coefficient
as a function of magnetic field strength and direction.
Generally, the approach employed was to change the magnetic
field continuously while monitoring the Peltier heat.

B. Point by Point Methods

This method of measurement did not require the second
phase sensitive detector in figure 12; the output from PSDl
was fed directly to the recorder. If operated in switching
mode 1 the recorder trace was similar to the one shown in
figure 23, the amplitude of the square pulses represent-
ing (6TH) alternating with (bTr). By comparing a 6TH to the
two 6Tr's flanking it, a value of the Peltier coefficient
was obtained. One could carry out this operation while vary-
ing the field to obtain results but the calculations would
be overwhelming. An improvement was affected by doing all
of the computational work on a computer but as the charts
still had to be read by hand the work was still so great
as to make the method impractical. It was hoped that a
digital recording system would be completed in time to use
it in this application but it was not.

Since the thermal resistance of the system showed little
field dependence other than a monotonic increase it was
possible to use mode 2, switching the sample current only,
to obtain meaningful data. In this mode the system had to
be calibrated by obtaining 5Tr as a function of field for
every temperature used, although the mesh of points used for
this measurement was much more coarse than that necessary for

(61+)

I ill . II I I . .

(65)

soapMpon m
«H woos .mompp AmUHOooh Hooflmza .mm onswfim

/

liili !' II! III!!! ‘! 'IIDP
m— 1. 2;: 2.34;. . :23; ,2, _. El... % éi:1.g . 3+.
- . . ILLII .I.I..II.L+1I.III$I+.IIL:I . .

if. T I .1.
., , . .< .l . O3...” ,. .
w . . ,_ -.1.1.II.§+.3 .II

         

 

L:=_ A ......

II.'II .b
. 1‘

 

l .;. m sea
. . #=.- ...-+-.-+ 3. -1 . ......

I IYIIIU 3. .II’SI... OIIII .. ..

.sga,i++:3+.3ii++Ii++.

 

 

 

.~ . 3 _..-;..3 , . . .. . 3 .. .23- . : . . .
vi W4...IIII IIII, ‘I "y|.|lll.§l.. Il-.II .IlIIll".I’I...rl1lI III I. (ll l l}.4.l..3l .MIIIOHUI. Ini‘v .3-+
, . .
r . .I. .I 3.1+... 11. I...) I .. 1 I .. I - I - -. I I I I. .
m. I IIII'IIII +1I Li {I fl IIOI II . ..- II...
. - -- . - --- .H..--- W: - ...:.+ +1.
VI 1 .I I. “I I III. | .IlllIIIIIIOTIIIIIIIII Ii. I. III. ..I..| I' I
w I. V YII'W‘ I! l 0+ l| III I It 7 + \} ... 3.. 1‘. .l I ll! lilkll II
. _ . L . til I l'l I .
TIC} ‘ .

 

(66) -

the measurements of 6TH. The reduction of such data was
still quite tedious, but the oscillations could be seen
directly; see figure 24. With such data we obtained de Haas
van Alphen frequency measurements as well as a limited amount
of amplitude information by inspection of the envelope of
the square wave. However, in order to measure harmonic
content we still had to do a great deal of manual data
handling.

Since the phase sensitive detector used had a Q of
25 in the input amplifier, the frequency at which it was
Operated was kept higher than about 25/T, where T is the
switching period. Typically, T g 2 sec., f = 23 cps.

C. Continuous Detection

This method of data aquisition was basically quite simi-
lar to the second point by point method described above, the
only difference being that the square wave form was detect-
ed. This was accomplished with the second phase sensitive
detector. The output from this detector was, of course,
the amplified envelope of the square wave, and was directly
proportional to n/K, in the notation used in section VI, B.
The scale (l/K) still had to be determined as a function of
magnetic field strength or direction. Perhaps the best
approach to this problem would be to repeat the magnetic
field sweep while switching the heater current at the same
frequency, thus obtaining a trace representing the scale
factor. The primary reason that this was not done was that
the required switching mode was not provided in the control
box. The methodINould assume that the thermal time constants
associated with the Peltier and calibrating heats were

(67)

newconpm uaofiw monooon oGHH Hanomwfio .aomzm w
«m moot noose» 9o690ooa Hwoamha .zm onswam

 

(68)

identical, or else very small.

The calibrating method used in this work consisted of
switching the heater current, either manually or via control
mode 1, while sweeping the magnetic field at a high rate.
This provided a rather coarse mesh of relative BTr values.
By then measuring the Peltier coefficient accurately at
one field, Ho’ the Peltier coefficient at any other field
can be calculated.

H(H) = VH(H°) 5Tr(H) ] x(H) (115)
Lx(HO) 5T (H )

 

where x(H) is the output from PSD2.

The highest switching frequency with which this method
has been successful is 2.5 cps. PSD2 must be tunable in
this frequency range. The detector used in this application
had a variable Q in the input amplifier, which was normally
set near 10, so that oscillatory signals of frequency great-
er than about .1 cps would not be passed through this stage.
PSDl must, of course, be able to pass 2.5 cps, which, with
a fixed Q of 25, required.a bridge frequency of greater
than 100 cps.

IX. Samples

Two zinc samples were used in this work. The prepar-
ation and characteristics are discussed in this section.

A. Zn 1

 

The first sample was cut from a slug supplied by Metals
Research Ltd. with a Servomet Spark cutter. The orientation
of the cylinder axis was within a few degrees of the [0001]
axis. The resistance ratio (DEOOOK/94.2°K) of this sample
was measured to be about 15,000.

E. Zn 2

 

The second sample was oriented with the [0001] axis
within a few degrees of perpendicular to the cylinder axis.
The resistance ratio was 5800. This sample was cut from
a rod grown in an optical zone refiner built and described
by J. C. Abele.33

(69)

IX. Results

A. Zinc Fermi Surface

The Fermi surface of zinc has been studied with a deter-
mination applied to few metals. As a consequence the Fermi
surface dimensions, effective masses, and magnetic break—
down fields are known in some detail.34"40 Figure 25 is a
sketch of the results of this work. In Spite of an apparent
complexity the Fermi surface is very nearly a one O.P.W.
surface, although deviations are relatively large for some
of the very small sections. The oscillations observed in
this work were due to the a, B and y orbits indicated in
the figure.

B. Data Interpretation

The frequency of oscillatory terms in any measurement
is a function of the field direction‘h. Thus,

”S = Sl(H,T) sinEié-m +15 . (116)

.1

One may observe this effect by monitoring the Seebeck coeffi-
cent while varying either the field strength or the direction.
Obviously either method will provide amplitude data. However,
the frequency information that can be obtained from the

two methods is quite different. Sweeping the field strength
in a fixed direction provides a frequency in that direction

(70)

(71)

 

 

 

( 000/)
I

 

fldlwulr. .

....\ apréWW/AWWMW.
\ v: “WWR.11&_‘MH .
$

 

u
&w
. c .6
.mwwm; .bsbah

 

(OI/0)

 

 

.Figure 25 (after Gibbons and Falicov34)

The Fermi surface of zinc

(72)

while from rotational data one can extract |df/dh|, i.e.

the change of frequency with field direction. A combination
of the two methods can be used to provide a detailed picture
of the absolute frequency as a function of field direction.

C. Zn 1

 

In this crystal the current was along the (0001) axis,
the magnetic field restricted to the basal plane. Due to
the six fold axis along (0001) one need make measurements
over only sixty degrees of field angle.

The only oscillations that could be observed in this
crystal were those due to the monster neck (a) orbits, which
could be observed for all field angles. Some of the data
are shown in figure 26. A pronounced beating is evident
for the last two angles, which are near the (1000) axis.

An inspection of figure 25 will serve to assure one that
this can be expected, for near the (1000) there will be
two sets of B orbits of nearly the same cross sectional
area. The frequencies of these oscillations were measured
with sufficient accuracy to identify the (0110) axes of
the crystal. As we were most interested in the amplitude
of the oscillations we picked one of these as the field
direction for most of the study. Such a choice both max-
imized the amplitude and minimized the beating.

Figure 27 shows the measured amplitude at 21 kilogauss
as a function of temperature. Although the error bars on
the data at 4.20 are rather large, the oscillations are
clearly evident. It should be noted that we have plotted
the total amplitude here, rather than the amplitude of the

 

(75)

 

 

 

Zn v0. Pb Banal Plan.
.2-
OOISO‘
Q'
Q
\. -
’
.33
In
41 l I l
.750 .000 .050 L500
\Mvoml
2.. O'IGZ'
5'
O I F-
\-
>
3:
In
1 l l 3 1*
.750 .000 .850 ~9°°
\Mvolts)

 

 

 

 

I l J 'T
.3_ .750 .000 .850 -900
\Mvoltt)
0-I80°
.2 —
§’
0
\l "
)
1.
5'
750 .300 050 .900
l l 1 L5

 

Figure 26. B oscillations in Zn 1

(71+)

 

 

I . . . . w

w N _ \
_, ,_

 

AWOOOVIh
A020vII
ox ._muI

 

 

(75)

fundamental component. That the two are quite different
is evident in figure 28, which shows the wave shapes at
two temperatures. A plot of the oscillation amplitudes as
a function of field strength is shown in figure 29. Such
data were collected at only one temperature.

Figure 50 shows the dependence of the background Seebeck
coefficient as a function of field direction at 1.32OK.
As this pattern was repeated every sixty degrees itvnusa
consequence of the crystalline structure rather than some
peculiarity in the physical arrangement. If one subtracts
the Seebeck coefficient of Pb, the value of S at (0110)
is O _-_l- .05 LIV/KO.

The temperature dependence of the anisotropy of S is
shown in figure 51.

D. Zn2

 

In this crystal the current flowed along the (1000)
axis. The magnetic field could be applied along any vector
in the (1000) plane, i.e. the plane defined by the <0001>
and (0110) axes.

Figure 32 is a diagram of the measured Seebeck coeffi-
Cent as a function of field direction at 19.6 kilogauss. The
proper inversion symmetry is observed about the <0001> and
(0110) directions except for a ten per cent discrepancy
about twenty degrees from the (0001) axis. The oscillations
extending to 500 on either side of (0110) can be associated
with the B orbits while those centered 550 from the (0001)
arise from the monster arm (y) orbits. The background See-
beck coefficient is almost constant except near the (0110)
axis where we find a pronounced dip.

i.‘ 111+

11"! ll

 

ii I IIIIIII. II
1" Ir

(76)

 

2?)? V5 . :5
' S J?- (o o o I)
‘pwwdka f/‘ (0110)
.2- //‘\ fi»”“\\2. 5 OK
- ,I’ \ _— /.J I I“... I“
\\\\’///// f/f\\ \\"#/// ,/’~ \\.
,"/ / ’ O I
/’ _ I. 32 II
' /- // h“;
\.\ / / \I
/ \
\
O 7“" 1 I r I *—
19.5‘ 20.0 20.5- 2/. 0

 

' H, #90055

Figure 28. Thermal damping of the
a oscillations

_ (77)

.047 J" ((700.1)
’Sy H- (0110)
@317) T: 024%

1

,.01- 11111

 

 

I

1'2 - 16 H M0)

 

Figure 29. lssl vs. H, Zn 1

 

~53 div/03V.)

.OS

(78)

 

\ / :r~ coco»
'4 H“ 20 KGOUSS

 

1L.—

J11..._......J
)0 O )O

M F36 3‘2"? T 33; “3 G L E

 

Figure 30. S vs. 9, Zn 1

f 4104/61
(000»

(79)

6/ ’9‘ 0/“,6

+5 (Man)

i "' (WM)
5/=27 A3:

 

...; ..., a. .
. _ . ms . .

I4.

. . III-II

.0“.

I . .a 0.

..: ..L‘M—I:k_
(Cu/4.0.70“!

.fi..
Mun-w PI.\ I
..H

I, S

s-sa .

«IIdIIIIIII 4 IIIIHIIIflII‘I.I<III.1IIIIII4.II.IJI4

:0».

0I.

m 2N .scfiscpcs cacam .mm magmas

.1 l - . .2 a 2.. i 1 l mu m e
a .H .§.H -..-3H

oIII .OII.| <I I I.II&I I. 4|.III r. I

u. n...

. .II.4..I..I¢. .IIL III III-

 

 

. IIIL

 

 

  

 

 

 

. L o . .. . a
. _ n u H . .
. 4 I o. I II VI III... » . . I «I .o I o I 73+. .NI 0I III . . H ..II+ I I I III I II IIIIIIIMII III. I L.I 1 I I 9 III! IH.0 0 0| III
. _ u . _ . . U _ 1 .
H 1 . m H w H . l . H 3 .H . . 3 _. _ . q. H i ,
. . . , I _. _ + . i a. n . l t.‘ I
I 0. I I9 I v . I I II I.0l OII 0 II? I I0: I . 0 II II I I0 III 0 I. 0 III I I I. In 1
. . _ _ . _ . . . . . i N
c . o . w c c n . .- o c m m . . I
H _ H . . . .
, - . , .. - l- - - .3 -......- .. .- -.a3- 4 -- . -- 2 . _ -.. - a..-.....2..:. --
,_ n a H i . I 1 . H l . a H U a . . a . _
_ . _ . _ _ . 1 . . . u . _
0 cl I . I I I w. w. » ... II I .I I+I..II|«. I. I.0I II . .3 .LI- IIIIIIYIIIoII I I. . I I II. II III . IL
. — _ i _ . a
. . _ . _ _ . . _ .
. m I. ._ h d . . . P . m N «I +I m m c » W a . + . w . .
I VI s ..P e i p w _ .r a h _ P b a n F
4 q < d 4 a 1 q I 1 . 4‘ («I d 1 1 .. . < 1 I M A. j
. . _ . . _ _ . M . H L _
.59 .... ”can . . H 163 5:... a 1%.. .. .33 .. . ..n-.
P 0 II 'I I 0 I I4 I0 I. II 9 I0 I. I. I. «I 4 ...... Ildl. ..0 I ...? . ...“.I I r IL. II OI III, I I 2%! IL II III: .II IIII.I . . I . .
H m _ . u . .
H a a . . . _ . . .3 . . _ . ..

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. a.
_ .
r. 3 . I 3 I .. m. . I ...
a 1 o . .- k H
_ . _ _ . .
- ., . -- . - -..- II..- ..-I. -- I - g .2 I
u _ H . _ n w _ . _
d . . l o . . 4 _ M. a u . _ p w
. , 0 . I I. Ifl - ..3 III I I: - 4 II M I+++ -.IHMII -IHT. II 4 III+_+ Ifi . .LII .I
. . . h H _
fl _. 4 .. — . c H .. n H L W — h . 9H e .
_ I0 I 4 . I w I f . If II. PI. I. . IFIIII 00 I fl. I .n II+II I. 0— I '14. 4+ w a LA I—
. _ . . ., _ n _
. . . . H . ... . 1 . H I . . 3 . .... a
_ _ . _ _ u _ a m o _ a _
. I a . P M I. . _ a s I . . .
_< I A DI 1 . IN. .. _‘ ~ I‘ < 1‘1 4.I dI 4
. . H . .
. a .. 6* .. . m H «‘2‘? H a 3... «fix .. m , .. .. .
-.-. I .H..+IL. - I-.+-- I ..L I - l HI I ,
. n . _ H 0 .
3. . . -. . I . . .
_ n ..
_ .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. .
I L .
e- . -4-
.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
m . .3 I I I ...... r - .. H . - II. . ...- I
r1 l N , tn
1 . M , _ i
3. ... . . H.-- I - I I. ». HI... .. w .. .I—I . ...I ..I ... II... . I I .I
l. . . , . _ I. . ,
w L - -.-. I- .I _ I I ...I: I. .. 9 I
r m l M w H M
l q 1 . u
H . l . . ... .. . .3 . - . ..... . I - .
. .. HI . . . . H
w. a 4 .. I LI - I T ..Lr. I - - I +.. -I. 4 .. :4. 2:... I II... .1
» LI » w p a a . w a _ _ . _ . _ n s k w s . s a w .m p w p I»

 

 

 

 

(81)

The a, B and y oscillations were all observed in this
crystal. We shall discuss them in that order.

1. The Q Oscillations

The needle (a) oscillations had the lowest frequency
observed, about 1.6 x lO gauss, as well as the largest
amplitude. A plot of typical data is shown in figure 33.
The oscillation amplitude wasa.sensitive function of field
angle as can be seen in figure 3#. Figure 35 shows the
amplitude as a function of temperature. The apparently
high values at the lowest temperature may well be due to
an error in recording the temperature.

The secondary peaks observed in the oscillatory See-
beck coefficient are due to Spin Splitting of the Landau
levels. This effect has been quite well documented35 and
will not be discussed here.

The needle oscillations have been observed in the
Seebeck coefficient by Grenier et. al.40
with ours where the two overlap.

Their results agree

2. Monster Neck (3) Oscillations

The B oscillations were observed over almost a 1000
range of field angles centered about the (0110) axis. A
single rather accurate frequency determination 33° from
(0110) coupled with the rotational data of figure 32 was
sufficient to obtain the frequency vs field direction plot
of figure 36. The amplitude at 19.6 kilogauss is plotted
vs the field direction in figure 37. Due to the obvious

(82)

m :N .mcofiflflomo a . mm 83mg

“‘1’

-.\“

‘04.”.

 

 

L..._.-__-_-.--
0’)

 

-——--—.- .v. .

 

”-..-.. ---”-..u—W O

lg“! (Hr-bl? arysmqg)

 

(83)

 

l_ 1 l. ,‘l
4.0 3° 20 lo
(000!)

Figure 34 ° lgor.‘ vs. 9 Zn 2

(84)

,. .. pdwam
m cm «a .m> .ww_ mm m

A m NH
... 3th .

h—O—F 'L—+—-—‘
#—

 

h'pfla—q H
P‘“"‘ F—+-‘.-——-1

0.0

..m .9

7%
«ma

1.3

 

(85)

m cm ‘0 .m> mm .mm mpsmfim

§

@T
\0
d.

I “ .‘a-c- -- ...-‘4“...-

<3
\r5
|

— A w“--O—-—-~-- --.I

10
0
4

u _ .Q,m4.

1

mwm- .

”mmzemho:
Q

Q

 

.zoqtow m Imunoka .0
mm.

a
/
/

x Qo>> hzmwmmm .J

x

N CN.«® ..w> _mm— .NLM QMSWHGH

 

 

.36 .o.» kw .q. x AS ”9

All... . _.
$8» 23‘“:ka Q .‘MQ _ . M
. , _ >
H
W w
... w
W w.
, 2.4
_.
) W
m M
_ a w

. 4

..r

v -- -‘r~-~r v-‘I‘u' ».‘

..—
Q
\

M, -J

khim‘mcx & - +
33% \wkmm T. _
Qoiv K,

.4
,.__...
- ’5
313:

M- -~~r~-.u-m~—w-. 9% .-

fix.

(87)

impossibility of measuring the amplitude of these oscillations
with the field directly along <OllO> (see figure 37) all
further studies were carried out 330 away from the <OllO>.

The observed amplitudes are plotted vs field strength in
figure 38 and vs temperature in figure 39. These results
include little harmonic interference, as the continuous
detection mode was used (see section VIII C) in which case
PSD2 acts as a low pass filter.

3. Monster Arm (x2 Oscillations

 

The frequency of the y oscillations is plotted against
field direction in figure 40. These results were obtained
from a field sweep at 155° and the rotational data. A
plot of the amplitude as a function of field direction follows
in figure 41.

lwflwasrather difficult to find a field direction in
which the beating of several y oscillations was absent. We
picked field directions approximately 34° away from the
<0001> for our measurements of the amplitude. The results
are shown in figures 43, 44 and 45. As there was beating
for both field angles we have plotted the average amplitude
in figures 44 and 45.

5U

W

Ho
-.10

-.05' .

 

 

 

(88)

. 1.48% J" <2 000) ,

n 1.76% ,
‘ 2,156/1/ [[7” {01.10)
v 4.22%

 

 

116 . H M61 ' 2'0

Figure 38. lsB| vs. H, Zn 2

(89)

N QNwa..m>

«Q \N

«:33 Essen
33» .294 .mm .l

«33» L.

a
_

m._ .mn. mhswfim

 

(90)

m QN mo .m> >m .0: mpswfim

 

 

 

00W. . ohm: 00m
«35» 433 0..» Seat MT
. . 202.:th Qummux
”a
a:
NO®NI%\IV%W

W

Amaaemeo\\

k

e

 

(91)

 

 

Ia! ..
. v _
.2- (Xi/.0) I ]- (/900>
//~ 2/ A7?
7— 2. /.6'°/{
.l-
I‘ . (DI/0)
T l ‘ I M9
0 o ' o I l
25 3o ‘35 40° 15° 50°

Farm DIRECT/0N ,flmezs 7'0 (000‘!)

Figure 41. - I'SYI ’vs. 9, Zn 2

(92)

, H—o—a—o»-——.—-V v7.

WWWWWWWWWWW

hauls. .
g ,_-. MW— 7, , . 3 . . a . g ._. .--, .

 

 

 

WWWWW/{WWW ' . r

v u-c - annuuodnomo-mw on"

Figure 42. Typical 7 oscillations,
continuous detection mode

 

(93)
ZLowcu‘c/ (0110)

' J'll (Mflfl)

kW

20

HOP)

 

/5 '

 

 

I'Sl vs. H, Zn 2

Y

Figure 43,

(94)

In

AHOOOV SOPH 0mm ..N SN «B..m> _>m._
~x<o\|h .V . m.

  
  

wk \W

«aloe 2.030%
«8qu 8.2.x eNm-\.\.
8de -h

.:z.o&3wfim

 

 

aaooov seem omn
.m es .5 a» _>m_ .3. 853a
«$55 a. m . N

. , u .

—5\ .

(95)

 

Qk \N
«33» sisaaow

«3on 83$. .hm. 4*
Koacxv «uh.

_
.
_
.
M

......—
._— —.
f.._....

“x.
\'

i

Q.

\_

-\ >.
’10
2U)
- ._ My--- -..)... ---.--..

‘

 

“Q
~\

X. Conclusions

A. Modified Horton Theory.In Zinc

The result given by equation 86 is valid only if the
relaxation time T is constant over the Fermi surface. If,
however, the relaxation time is not constant, it is evident
from the discussion leading to equation 86 that that equation
must be multiplied by n = TR/Tl’ where T1 is relaxation time
for the section of the Fermi surface reSponsible for the
oscillations and TR is given by equation 82. It must be
remembered, however, that the entire theory assumes that
the scattering potential can be approximated by a 6 function.
Subject to this assumption, the oscillations should still
follow the form

0V V T

a. v
S = S x(ll) expfigy ] A3(l) sin b . (117)
c D

The temperature dependence of the fundamental should then
be given by A3(x),where l is defined by equation 47. The
field dependence of the fundamental amplitude should be
proportional to

 

expELTTD] A5(l) . (118)
c

Evidently the parameter TD can be derived from the measured
amplitude [§(H)| through the use of the equation

(96)

(97)

~

r S H F l Fm*c l
I = — —— —
ln L %_{T%i J T ( e ) H + constant (119)
5 D
and the slope of a plot of in fi%§{%%i 1 vs. 1% . The

exact harmonic content one obtaigs from equation 117 is
probably somewhat unreliable for the factor (-l)v/ v may
depend on the shape of the Fermi surface. Nonetheless the
relative harmonic content should include the factor

expfiing] A3(l) . (120)
c

In zinc equation 86 becomes

2 E J

3’: 2.9 x lo37 nv/Ko { N(“fl/mo) le/2a }
V

2 2
(a A/akz)k0

(121)
All units used in the further evaluation of equation 121
must be gaussian. This equation must be multiplied by the

factor n defined above if the relaxation time is not con—
stant over the Fermi surface.

B. Auxiliary Data

The evaluation of the parameter_a requires a knowledge

(98)

of the magnetoresistance [Ag/p0]. This may well be a func-
tion of both the current and field directions.

Stark25 has made measurements suitable for application
to the Zn 1 measurements. He reports a magnetoresistance of

-%2 = 12 + [lO8/(KG)2] H2 . (122)
0

Thus,

4 2

1 = [1.08 x 10‘ Joe (125)

aZn

Unfortunately the exact value of po is not reported although
the residual resistance ratio (DBOOO/poo) is given as al—
most 50,000. The use of the resistivity at an elevated
temperature (eg. 30K) in the denominator of the resistance
ratio would of course reduce the ratio, i.e.

p o
.3199 < 50,000 (124)
T

We shall assume that the resistance ratio for the po appear-
ing in equation 123 is given by 35,000. As this may be in
error by as much as 40 per cent, the value of.a obtained
may be in error by a factor of 2,

aZn 1 = [1.08 x lO-u](p3000)2/(35,000)2 . (125)

 

 

{I ll 1
III I! ‘IIIIIII. lllll ll.ll
III

i
I ll {Izll

(99)

The handbook value for the room temperature resistance of
zinc is 7.0 uQ-cm56, giving

a = 4 x 10"48 gaussian units . (126)

Stark has also reported magnetoresistance data on a
crystal of the same orientation as our Zn 2. Unfortunate-
ly the value of_a derived from his measurements suffensfrom
the same uncertainty that is discussed above. Assuming a
resistance ratio of 35,000 and a room temperature resistance
of 6.7 uQ-cm, we can plot the value of_a as a function of
field direction (see figure 46). Of course_a is not defined
near the (0110) and <0001> axes where the magnetoresistance
saturates.

One of the assumptions made in the derivation of equation
86 was that the Hall field is small. In particular, we
must establish that

RHH

T << (DOT . (127)

Logan and Marcus39 have reported Hall effect measurements
on a zinc crystal with the current vector lying 8-% degrees
from the <0001> axis. For no field direction did they find
a Hall resistivity (RHH) larger than 1.5 x 10"9 n-cm in
fields to ten kilogauss. In such a field the resistivity
of our Zn 1 should be at least lo’YQ-cm, thus reducing
inequality 127 to

.015 << wcT . (128)

 

 

llllill‘ ii.‘ ‘lllll‘l‘.lalllll1l

(100)

J ll ( I00 :7,"

 

 

. a
'. /0— {/0'48650) V
A ll
/ \\/
J...
( 0/ 1'0) ‘ (502/)

Figure 46. 3 vs. 8, Zn 2

(101)

At higher fields inequality 127 should become even less
restrictive because the resistivity is proportional to H
and the Hall resistivity tends to decrease with increasing
field339.

The evaluation of the right hand side of 127 requires
a knowledge of the relaxation time T about which there is
a significant uncertainty (see section X, D, 2). The use
of the resistivity relaxation time defined by equation 82
results in an wcT of about 1000 in Zn 1 for the B orbits
at 20 kilogauss. As even the smallest apparent Dingle

2

relaxation time (table 5) does not imply an wcT of less than
1, it would appear that we are Justified in neglecting the
Hall term in the expression for'S (equation 57) in refer-
ence to Zn 1.

Hall effect measurements in a crystal of the same
orientation as Zn 2 have been reported by Borovikul. He
reports the ratio of Hall field to resistivity field for
all field directions and in fields as large as 25 kilogauss.
This ratio remains below .1 except when the field is near
the <0001> axis or the basal plane. Thus for this crystal

inequality 128 becomes
c

.1 <( w T (129)

Since the effective mass of both the s and the y orbits
is the same (.13 mo), the values of wow obtained using the
resistivity relaxation time are identical,

(102)

in Zn 2. Inequality 129 is thus satisfied. If we again
use the smallest apparent Dingle relaxation time, wcT be-
comes slightly larger than unity, thus introducing the possi-
bility of a 10 per cent error in equation 86 when applied
to Zn 2 calculations.

The curvature of the Fermi surface (BQA/aki)k can
be obtained from equation 72 and values of a extra8ted
from the data represented in figures 56 and 40 or can be
calculated through the use of equation 80. The results of
these operations are listed in table 1.

Table 1. Fermi surface curvature

eq.72 eq.80(N.F.E. Approx.)
(8211/8142)k ,a 1.1 .235
O

2 2
(5 A/akz)ko:Y 1'9 ~75

In all further calculations we shall use the results of
equation 72. In any event this factor has a relatively
small effect on the calculation (<50 per cent), as it enters
as a square root in equation 121.

C. The Q Oscillations

A calculation of the amplitude to be expected in the
a oscillations is all but impossible. Magnetic breakdown
occurs between the needles and the monster in fields greater

(103)
35 36

than about one kilogauss ’ , invalidating Hortons treat-
ment. The probability of breakdown depends on the oscillat-
ing density of states on the surface of the needle provid-
ing a complex mechanism for obtaining oscillatory effects
and giving rise to very large (vv 80 per cent) Shubnikov-

de Haas oscillationsBB’36 Furthermore a very large Hall
effect has been measured with the field along the <0001>
ax1355,36;

derivation of equation 86 which are valid in this case.

in fact there are few assumptions made in the

D. The_§ And_x Oscillations

 

1. Absolute Amplitude

 

The cyclotron mass of the electron in a s orbit is
.12 m0 if the field is directly along the <0110> and scales
approximately as the frequency in other field directions38.
The y orbits have roughly the same cyclotron mass (.13 mo)
when the field is directed along the axis of the monster
arms.38 For our amplitude calculations we shall use an effec—
tive mass ratio of.13 which should be accurate to about
20 percent. Table 2 gives the calculated amplitude factor
(i.e. everything outside the summation) obtained from equa-
tion 121 for the various oscillations observed, along with
the amplitudes measured at the maximum of the amplitude
vs. temperature curve. All amplitudes are calculated for
a magnetic field of 20 kilogauss. The last column is mean-
ing ful only if the Dingle factor exp[-wyu%3; is near unity.
The use of the resistivity relaxation time defined by equa-
tion 82 results in a value larger than .9 for this factor.
Evidently the ratios in the last column of table 2 represent
a substantial disagreement between the theory and the measure-
ments.

(104)

Table2. Comparison of theoretical and experimental amplitudes

J H (slth(4§o)

3 osc. <0001> <0110>

B osc. <1000> <0110>
+330
y osc. <1000> <1000>
+350

.0016

.002

.020

(SI

.10

.20

3'
(020) -+g+exp
K th

.05

10

2. Field Dependence, Apparent Dingle Factor

The use of equation 119 to derive an effective Dingle
relaxation time from the data presented in figures 29, 38

and 43 results in the values listed in table 3.

Table 3. Apparent Dingle relaxation times

J H T(°K)

<0001> <0110> 1-35

8 <1000> <0110>+33° 1:22
Y <1000> <0001>+33° (:25

column lists the resistivity relaxation times.

¢D(lO-13sec)

4.65:10

7.1.i10
2.5 1:30

5.35i30

4.5.130

The last

TR(lo-13sec)

2100

560

If the effec-

tive values of TD represent the real relaxation time of the

 

 

{lilillll‘l‘i‘lll=|.

(105)

orbits in question the Dingle factor becomes quite small,
approximately 0.1. This would reduce the calculated ampli-
tude estimates listed in table 2 by an order of magnitude.
It is interesting that if we also include the factor TR/TD
as suggested in section X.A.we calculate the amplitudes
listed in table 4, which are in much better agreement with
the data. The value of TD used in these calculations was

Table 4. Revised amplitude estimates

J H (81th(nv/K°) (S(exp(nv/K°)
a <0001> <0110> .072 .05
a <1000> <0110>+55° .02 .10
Y <1000> <0001>+35° .2 .20

5 x 10’13 sec., which is near the average of those listed
in table 3.

Unfortunately there is a very basic difficulty with
this argument. If we include the Dingle factor of .1 in
the calculation we must consider the effect it has on the
harmonics, namely a very strong damping, (.1)n+1 for the
n th harmonic. This prediction is clearly at variance with
the high harmonic content observed in the s oscillations.

There is a mechanism, the B—H effectue, which can give
rise to nonsinusoidal wave shapes in the presence of complete
collision damping of the harmonics. This effect arises
because the field seen by an electron inside a material is

B = H + 4wan (151)

(106)

where nm is the demagnetization factor. Since the magnetiz—
ation M is itself an oscillatory function of B, a nonlinear
equation must be solved. For relatively small changes in

B, this equation can be written

y = yosin(x + y) (132)

x = (32) H (133)
HO

y =33, (uremia) . (154)

The strength of the nonsinusoidal effects introduced in
this way depends upon the magnitude of yo; if yO is much
less than unity the effect becomes negligible.

An expression for the de Haas-van Alphen (magnetiza-
tion) oscillations can be obtained from the free energy
(equation 34) by the thermodynamic relation

M -- ($9.1. . (135)

From this one can find the following expression for yo:

3203kT(e/2th)3/2f2nmexp(-F/wcv)
y° IV(52A/Bk§); [sinh(2HkT/hwc)] 115/2
O

 

(136)

For the B oscillations in zinc,

(107)
yo = .003[exp(-1T/wc*r)l (137)

at 20 kilogauss and 20K. This is rather too small to give
rise to significant nonsinusoidal behavior.

The arguments concerning the relaxation damping of
harmonics are quite general.14 However, some theory is needed
- to extract values of the Dingle relaxation time TD from the
data. It is therefore more reasonable to put faith into
the harmonic argument, with the obvious conclusion that
the field dependence predicted by Horton's theory is not
nearly strong enough.

It might be noted that the possibility of extracting
an apparent TD does not necessarily imply an exponential’
field dependence. There was generally a rather large amount
of scatter in the data and another form of field dependence
might have appeared exponential over the rather restricted
field ranges used.

3. Temperature Dependence

In contrast to the field dependence and absolute ampli-
tude of the oscillations, the temperature dependence shows
at least qualitative agreement with Horton's theory. For
purposes of comparison the expected form of this dependence,
proportional to A3(1), has been drawn in figures 27, 39,

44, and 45. The vertical scale of A3 has been adjusted to
approximately agree with the data, while the horizontal
scale is of course defined by the cyclotron mass and the
magnetic field. All cyclotron masses usgg for this purpose

were those reported by Joseph and Gordon .

(108)

E. General Conclusions

Evidently much more theoretical work must be done be-
fore the amplitude of these oscillations can be understood.
The difficulty with the theory presented here is probably
in the scattering potential, as the form of this potential
may affect the Seebeck coefficient quite strongly. Further-
more the theory cannot be easily modified to include any
field dependent scattering.

More detailed experimental data may also assist in
sorting out the field dependence due to collision broaden—
ing from the intrinsic field dependence. In particular,

a study of the amplitude and the field dependence in crystals
of varying purity would be very helpful.

F. Nonoscillatory Seebeck Coefficient

Although the exact form of the field dependence of the
average Seebeck coefficient 3 was not studied in detail,
we did note two pronounced dips in 3. Both of these occurred
for the field directly along the <0110> axis, although the
current direction differed in the two cases. The theoretical
results of Bychkov et. a1.24 would predict that's is propor-
tional to the field strength for all field directions out
of the basal plane, but that it should saturate for field
directions in the basal plane where significant bands of
open orbits occur.35’36 This clearly is not in complete agree-
ment with the experimental results.

10.

ll.

12.

13.
14.

References

T. J. Seebeck, Abhand. deut. Akad. Wiss. Berlin, p. 265
(1822-25); Ann. Physik 6, p.p. l, 153, 253 (1826);
Ostwald Klassiker 19, (1895), as referred to in refer-
ence 3.

J. c. A. Peltier, Ann. Chim. Phys. 56, p. 571(1854);
L'Institut 2, p.p. 155, 265 (18 5). as referred to in
reference 3.

B. s. Finn, Physics Today_gp, p. 54 (1967).
C. A. Domenicali, Rev. Mod. Phys. 26, p. 257 (1954).

S. R. de Groot and P. Mazur, Non Equilibrium Thermo-
d namics, North-Holland Publishing Co., Amsterdam‘(l962),
p-po 33 ~375.

J. F. Nye, Physical Properties_gf Crystals, Oxford
University Press, London (1957), p.p. 215-232.

 

J. M. Ziman, Electrons and Phonons, Oxford University
Press, London (1960), p. 273.

N. F. Mott and H. Jones, The Theory.2f the Properties
of Metals and Alloys, Oxford University Press, London

'(1936, reprinted by Dover Publications, New York (1958),
p. 30 .

H. M. Rosenberg, Low Temperature Solid State Physics,
Oxford University Press, (1963), p. 352.

C. Kittel, Quantum Theory_g§ Ph sics, John Wiley and
Sons, New York (1963), p.p. 179-267.

A. B. Pippard, Low Temperature Physics, Gordon and
Breach, New York—(1962), p.p. 3-146.

W. A. Harrison, Pseudopotentials ih the Theory pf Metals,
W. A. Benjamin, New York (1966).

L. Onsager, Phil. Mag. 4 , p. 1006 (1962).
A. D. Brailsford, Phys. Rev. 149, p. 456 (1966).

(109)

15.
16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.
28.
29.
30.
31.
32.

(110)

R. B. Dingle, Proc. Roy. Soc. 506, p. 517 (1952).
E. I. Blount, Phys. Rev. 126, p. 1656 (1962).
E. N. Adams and T. D. Holstein, J. Phys. Chem. Solids

.19. p. 254 (1959).

E. M. Lifshitz and A. M. Kosevich, J. Phys. Chem. Solids

.3. p 1 (1958)

P. S. Zyryanov, Phys. of Metals and Metallography l8 ,
no. 2, p. 1 (1964 ‘""

P. S. Zyryanov and V. P. Kalashikov, Ph . of Metals
and Metallography.l§, no. 2, p. 8 (1964.

G. I. Guseva, Phys. of Metals and Metallography_l§,
no. 3, p. l (1964).

G. E. Zilberman, JETP (USSR)._9,p . 762 (1955), Eng-
lish translation in JETP 2, p. 650 (1956)

P. B. Horton, Quantum Transport Theory of Free Electrons
in a Strong Magnetic Field Ph. D. dissertation, Louis-
iana State University (1964).

Yu. A. Bychkov, L. E. Gurevich, and G. M. Nedlin, JETP
(USSR) 31, p. 554 (1959), English translation in JETP

10 p 377 (1950)~

. B. Pi gard and G. T. Pullan, Proc. Camb. Phil. Soc.
48. p 1 (1952).

A. R. )DeVrooman and C. Van Baarle, Physica._3, p. 785
1957

I. M. Templeton, J. Sci. Instr. 22, p. 172 (1955).
. M. Templeton, J. Sci. Instr. 32, p. 314 (1955).

I
A. M. Guenault, Phil. Mag. 1 , p. 17 (1967).
c. L. Foiles, Rev. Sci. Inst._58, p. 751 (1967).
J Clarke, Phil. Mag. 1 , p. 115 (1966).

J. J. LePage, Magnetothermal Oscillations and the Fermi
Surface of Beryllium, Ph. D. dissertation, Michigan State

 

Universityfi(l965).

\N
\N

37.

38.

39-

4o.

41.

42.
43.

44.

45.
46.
47.
48.

(111)

J. C. Abele, Fermi Surface Studies pf Aluminum and
Dilute Aluminum - M_gnesium Alloys Using the Magneto-
thermal Oscillation Technique, Ph. D. dissertation,
Michigan State University (1968).

 

D. F. Gibbons and L. M. Falicov, Phil. Mag. 8, p. 177
(1963)

R. w. Stark, Phys. Rev. 155A, p. 1698 (1964)

W. 2. )Reed and G. F. Brennert, Phys. Rev. 130, p. 565
19 3

R J. Higgins, J. A. Marcus, and D. H. Whitmore, Phys.
Rev. 151A, p. 1172 (1965).

. S. Joseph and W. L. Gordon, Phys. Rev. 126, p. 489
(1962). "'"

J. K. Logan and J. A. Marcus, Phys. Rev..§§, p. 1234
(1955)

C G 1Grenier, J. . nolds, and N. H Zebouni, Phys.
Rev. 129 p. 1088 (1963).

E. s. Borovik, JETP (USSR) 50, p. 262 (1956), english
translation in JETP 3, p 243 1956).

A. B. Pippard, Proc. Roy. Soc. 272, p. 192 (1963).
J. A. Woollam, Electron TranSport Properties of Metallic

Tin in High Magnetic Fields and at Liqpid Helium Tempera-
tures Ph. D. dissertation, Michigan State University

(1967)

A. V. Gold and M. G. Priestly, Phil. Mag..5, p. 1089
(1960).

M. Bailyn, Phys. Rev. 157, p. 480 (1967).
D. K. C. McDonald, Physica 29, p. 996 (1954).
J M. Ziman, E a P, p. 407.

F. J. Blatt and R. H. Kropschot, Phys. Rev. 118, p. 480
(1960).

49.

50.

51.

52.
53.
54.
55.
56.

(112)

M. Garber, B. W. Scott, and F. J. Blatt, Phys. Rev.
120, p. 2188 (1965).

V. Rowe, Phonon Drag Thermopower_in Cadmium, Zinc, and
Magnesium, Ph.D. dissertation, Michigan State University
1967).

F. J. Blatt, D J. Flood, V. Rowe, P. A. Schroeder
and J. E. Cox, Phys. Rev. Letters_1§, p. 595 (1967).

 

 

M. C. Steele, Phys. Rev._§1, p. 262 (1951).
D. Grieg and J. P. Harrison, Phil. Mag. 12, p. 71 (1965).
P. A. Schroeder and A. I. Davidson, private communication.

L. Colquittand H. Fankhauser, to be published.

Handbook of Chemistry and Physics, 30th edition, Chem—

 

ical Rubhér Publishing Co., Cleveland (1948), p. 1959.

Appendix I

.§2 Measurements

A single run was made on a tin crystal supplied by
John A. Woollam, NASA, Cleveland. The crystal had a resist-
ance ratio of about 30,000 and was oriented such that the
cylinder axis was 790 from <001> and 860 from (110). All
fields were applied in the plane perpendicular to the sample
axis.

Figures 47 through 49 represent the data collected
during this run. It will be noticed that the amplitude of
the oscillations in tin are very large. The data obtained
in this work appears to be quite consistent witfljthe more

conventional measurements performed by Woollam.

(113)

(114)

~22.
. p :3r7’
8
o Pages/v7 Worm O

6 44
{/0 qau55\ 6 p

. v I D 01.0 E a F

-23 . ‘ «msnr 9

~21?

~21

   
 

‘2. PRESENT WORK

 

’ 3T00 c.
A/vé-J FROM max) 40

Figure 4
._ 7 . f6 vs. 9, Sn

(115)

Moa.m «cm. qo ..m> _wm_ .m: mhswam

 

km com. .o.\ «33 6.31.
e H .HH H H _ 1 .
H H 4. _ . . , m

HS: H. \ «m.
H a. M
a
. l ....
7% L . \ M
\ _
\ m...
H
1%
stlw‘nmnw
mo Eyck
.xo&o\n\m\
”x.mw

 

'o >'°
'1»;
3b..

12

{3'

Gm.

(116)

Cm ..m

Ext \\

coov. ERG. oNN skamQN ..—
..Aaev £3?m.h.m\ exam. m .\ ... +

I

Imp—4
P-P—‘4

k.

.m> 3.2 .m: omswfim

----——4)— --

.l—_—.~—\..-....»r.._. -

<\J

.. -4

...- -—— -»-'-— -vm-

4

e—-—+—-1

90

. | 4‘.
_ .-"¢~‘-—a—~-~.~*——--‘-m— | —-

 

em in.

Appendix II

Measurements on Polycrystalline Metals

Equation 28 can be applied to a metal in the impurity
scattering region to obtain

2 2

Si =1}! '5'? ' (A’l)

H.103

If the same is done for high temperature phonon scattering,

s = 3LT . (A-2)

These terms define what is known as the diffusion thermo—
power of metals. At intermediate temperatures one finds
the so called phonon drag peaks in the thermoelectric power
of most materials. This effect has been studied in some
detail.45'50 There is also some evidence that a magnon drag
effect may be present in metals.51

The cryostat and measuring techniques used in this study
were essentially those used in the single crystal studies.
Since the samples were in the form of wires the carbon re-
sistors could not be attached directly; an intermediary
copper block was used. The best reference material was found
to be lead, which has a zero Peltier coefficient at liquid
helium temperatures in the absence of a magnetic field. The
electrical contacts were soldered.

The results of this study are shown in figures 50-55.

(117)

 

I‘”.. _~. .

(118)

r‘
L'

T (31"

9’.
.}

 

'3 I£;-€

‘zlAZ -

44-—

 

i

 

Pb in IOOO gcuss

 

Figure 50. Seebeck coefficient
'of Pb

(119)

WU

 

 

ROG“

 

:OT~

 

‘ _ Figure 51. Seebeck coefficient
}
of Ni

. (120)

53; /VQ' vs. 1/35
722/37

 

.o/-- I I

 

 

2
H M5)

Figure 52. Ni vs. PB in a magnetic
field

(121)

    

 

 

1
.|2- .
Fe vs Pb
.l0-
annealed
“.08—
x
O
\
>
'3‘06-
(0
.04-
.02-
// 1 l J I '
' l 2 3 4
- T(°K)

‘4“

Figure 53. Seebeck coefficient .
of Fe

.07

.05

‘ S(P-V/°K)

.03

.02

0|

 

(122)

1 ‘ 1
2 ,‘ 3
T no

Figure 54. Seebeck coefficient

‘ Of W, low temperature

 

(123)

   

 

 

 

 

.3 " _
W
'25 @553" ‘ 2000‘“
.l - ‘
’/’/////’ 25C) (je§]k(
_3|_
“:2 -

Figure 551 Seebeck coefficient
! of W, intermediate temperature

(124)

The Pb data were taken in a field of one kilogauss against
a variety of reference materials. These data agree with
results published by Steele.52

The Peltier coefficient was measured in two samples
of nickel. The results for both samples can be fit by a
sum of two terms,

—s = (.0075 uv/KO)T + aT3 . (A-B)

This is Just the form one might expect if diffusion and
phonon drag contribute. The fact that the more strongly
annealed sample shows a larger T3 term is consistent with
this interpretation. Greig and Harrison53 have reported
measurements which fall between the two curves in figure 51.

One set of measurements was made in a magnetic field;
the results are shown in figure 52. We find a relatively
large change in the thermoelectric power at the highest
fields.

Figure 53 shows the results in iron. The cold worked
sample displays a Seebeck coefficient which is strictly
proportional to the temperature as would be predicted by
equation A—2, while the annealed sample shows significant
deviations at the higher temperatures. It has been suggest-
ed that this type of behavior may be related to magnon drag.

The tungsten data show two interesting features, the
linear temperature behavior which does not extrapolate to
zero at zero degrees Kelvin and the positive curvature at
higher temperatures. The former is difficult to understand;
the Seebeck effect must eventually go to zero.

51

(125)

Schroeder and Davidson54 have measured the Seebeck
coefficient of tungsten down to 130K and these results are
shown in figure 55. It would appear that there is a peak
near 100K. Although this may be interpreted as a phonon
drag effect it has been suggested by Colquittand Fankhauser
that such behavior in the transition metals may be due to
electron—electron scattering.

55

Appendix III

Thompson Heat Correction

Consider the situation shown in figure 56. We wish
to calculate the effect that the Thompson heat has on the
Junction temperature T1‘ This requires a knowledge of the
temperature distribution along the samples.

We define the quantity

k<x.T) =‘K1(T>A(x) (A-u)

where K is the thermal conductivity and A the cross section-

al area of the samples. In the absence of any thermal con—

duction other than that along the samples, it is easily
shown that

d
k12+(%§

TEE—H i)= (Ii-5)
dx

The heat inputgmu~unit length dq/dx includes terms due to
Joule, Thompson, and Peltier effects;

f(XaT) ='%%%:; 12 + P%% -'%}a;
+ 6(X)(nl(Tl) - II2(Tl)) I

+ 6(x)ROI2 (A-6)

(126)

 

 

 

 

 

 

Figure 56. Thompson heat correction

(128)

where RO is the Junction resistance. In order to represent
the true situation somewhat more accurately we might make
provision for a heat loss -K ol(T )6(x) and a heater input

ri 2o(x) at the Junction.

We now need only solve these equations to obtain a
Thompson heat correction to T1. Under the assumptions that
material 2 is a superconductor and that material 1 is in
the residual resistivity region and that the cross section-
al area of 1 is uniform it can be shown that the correction
to T1 due to the Thompson effect is approximately given by

T T
(6T ) l l-——;——— r g(T)P“kT)TdT dT (A-7)
1 th= 2 T [TP‘KT)]2 {JT }
O 0
where
W dT
P =.a§ (A-8)
dH H
8 = -'% PET; - T11 (4-9)
0
k0= kl/T = const. (A-lO)

The complete solution of this equation is extremely tedious
and cannot be given here. One must solve for P through
equation A-6 and the boundary conditions

T(Ll) = To = T(—L2) (A-12)

(129)

dT dT _ 2
(ki dx)xt>o - ( 2'52 xieo —HI(T1) + ROI
+ r12 — K(Tl) (A-l3)
11m T = lim T (A-14)
X90 X90

The exact form of PM is strongly dependent upon such un-
known factors as the Junction resistance and exchange gas
conduction, and an exact solution is of doubtful value.
In the case that both of these effects are negligible, we
can approximate Pu’by the easily integrable form

a: (T-Tl)l/2
P = const x T . (A-15)

 

If we further approximate H(T) by a straight line between
T0 and T1’ 'we find that the apparent reversible heat at
the Junction correSponds to the Peltier coefficient at a

temperature given by

T = T — (T

H 1 - Tom - 3- %i) (we)

1 3

if (T — To) << T . (A-17)

1

Equation A-16 defines a temperature intermediate between
T1 and To' One can pick forms for P which include Junction
resistance and Junction to bath heat exchange but it is

I‘l'l‘ll‘l‘l‘ll‘l'll‘l‘lll'lll‘l‘

 

(130)

generally found that, if inequality A—17 is followed, the
Peltier coefficient measured correSponds to a temperature
between T0 and T1'

‘l‘lt‘l‘(‘l‘l(\[ll.1‘l

 

Appendix IV

Thermometer Linearity

The linearity of the thermometer system depends upon
the linearity of the bridge as well as that of the thermo-
meter itself. The output from a Wheatstone bridge (see
figure 57) is

TL“ ( 5 )n ( 8)
v = 2-———-— A-l
R3 + R4) n R3+ R4

if Rl to R4 are such that the bridge is in balance when

6:0.

A typical plot of R vs. T for carbon resistors is shown
in figure 58. This can be fit rather well by

R = Roe-a/T . (A—19)

Typically, a is a few degrees Kelvin in the Ohmite 1/8 w,
55 0 resistors used in this work. A change in temperature
from T to T + 6T results in a resistance change of

-% ) + ....J} (A—20)

8R = Roe““/T%19g [1 + 8T(-92—
T T

From equation A-20 we see that for bridge linearity we
must require that

(131)

{ll‘l‘l‘ll‘l 1‘

 

 

 

(132)

 

oVe

 

 

 

 

 

Figure 57. Wheatstone bridge

l“l‘lll‘l

 

 

/0

(133)

33 1“. 07m (TE

. ...»‘m .

 

 

\-1

JP 77°."

1!

Figure 58. Low temperature resistance
of a carbon resistor

 

[“ll‘l'l‘lll‘!

 

 

511(9 _
T2

(134)

-%) <<1,

(A-21)

or in the liquid helium range, where a/T is of order unity,

“BIS
A
A
[.3

(A—22)

Assuming that inequality A-21 is obeyed we can substitute

6R from the first term of equation A-20 into A18,

in which we have

R3 m R4

For linearity of
that

(1051‘
ET

<<

 

We must take the
inequality A-22,

assumed that

_ -a/T
Roe

(A-23)

(A-24)

v vs. CT in equation A-23 we must require

more restrictive of the two conditions,

as the basic restriction.

(A—24a)

 

'.“ll\("ll) ‘

 

 

 

Appendix V

Manganin Resistance

The resistance of a 56 Q coil of manganin was measured
at two temperatures and in fields to 20 kilogauss. The .
results are presented in figure 59 in which the vertical E)
axis has been normalized by dividing the measured resist—

 

ance by the room temperature resistance. For all tempera- r

 

tures and fields in this range, b ‘_
~512L§%- = .877 i .5 per cent. (A-25)
R(300 ,0)

 

(135)

 

(136)

 

 

A- 4.2 °K
4/“'_°‘°“ "’ ° A~~ ‘ A o- I.04°K
.880” I /°'__MO___°-____ 0.... ' A o\

‘ V“ \\ \.
JfiKZfiZlm % O \\\°\\‘\<\o\\>\u
8400-220) / m

i i

: r

I .‘Q

:3
33737;;
H

2.

J

.874]:

3 - 1 /‘o ‘ 2‘0
H (#6)

Figure 59. Magnetoresistance of

Manganin

 

 

. I 1 . I‘ll-'1'“

 

Appendix VI

Kelvin-Onsager Relations i .a Magnetic Field

 

 

In the presence of a magnetic field the application
of the Onsager reciprocal relations results in

n(H) = TS(-H) . (A—26)

(137)

 

 

. .‘.4|1.l'll ‘ 3

 

 

Appendix VII

Effects of Sample Misalignment

The theoretical discussion of section IV assumed that
the sample current was transverse to the magnetic field
direction. In the measurements this was true within only
50 and we must establish that misalignment of this magnitude
does not significantly change the theoretical treatment.

The misalignment affects the theory through changes in the
magnetoresistance, the Hall coefficient and the factors

L15 and L25. Upon a rotation of the current direction to

an angle 9 from the plane perpendicular to H (see figure

60) the magnetoresistance changes by the factor cosge while
the Hall coefficient changes by the factor cos a. For a

50 misalignment both of these factors are less than 1 per
cent, a negligible effect. The effect of the misalignment
on the factors L15 and L25 is rather difficult to assess.
Although it appears doubtful that these would be significant—
ly affected, we must consider this as a basic uncertainty

in the interpretation of the results. This type of misalign-
ment may have caused the lack of symmetry mentioned in connec-
tion with figure 32.

The possible misalignment of the crystal axes with re-
spect to the magnet field direction is a more serious prob-
lem. The ability to rotate the magnet allows field align-
ment to within about 0.50 in the X—Y plane (see figure 60).
However, the misalignment perpendicular to this plane may
be as large as 5°.

If the amplitude of the oscillations is found to be
insensitive to a rotation of the magnetic field it is reason—
able to assume that a further rotation by a small angle
m (figure 60) to the desired crystalographic direction would

(138)

 

 

“".“l“‘ ““[.(l

(139)

A Z) Mlle-NIT flown/01v

 

 

”XI:
I57 [/55 1”
0/ m: X')’ PAH/V5
ngPli‘
flat/5
Y
\P Dis/5’50
¢\J erzo
X DIRECT/01V

Figure 60. Sample alignment

 

 

. l1l‘lllll‘l".|lllll i!

 

(140)

not significantly affect the measurements. This argument
cannot be used on the a oscillations in zinc, which are
strongly angle dependent, and is of doubtful validity in
the case of the y oscillations which show a relatively
strong angle dependence. Possible misalignment of this
type introduces an uncertainty (less than a factor of 2)
into the absolute amplitude measurements while the tempera-
ture and field dependences are essentially unaffected.

 

..‘Ii‘l'l I