GALVNQWGMETE'C ENVESSfi-GATW
@133 THE FEM! SURFACE GF AuSn.

lesés far the Degree cf 9’51. D.
MICHiGAN STA‘E‘E UNWERSETY
[>73va 1. Seiimyar
39:65

THESIS

0-169

   
   
   

I r71 ‘0 fi "f if
'..’ A. In;

Michigan Stat.
University

This is to certify that the
thesis entitled

GALVANOMAGNETIC INVESTIGATION
OF THE FERMI SURFACE OF AuSn.

presented by
David J. Sellmyer

has been accepted towards fulfillment
of the requirements for

Ph 0D 0 degree in Phi? 8108

M

WV thor professor

Date M

ABSTRACT

GALVANOMAGNETIC INVESTIGATION
OF THE FERMI SURFACE OF AuSn

by David J. Sellmyer

The topology of the Fermi surface of the
intermetallic compound AuSn is investigated using
high-field galvanomagnetic measurements. The re-
sults indicate that AuSn is a compensated metal,
that there are Open orbits along the hexagonal
axis, and that there are open orbits along <10IO>
directions. A tOpological model of an open sheet
of Fermi surface, which is in agreement with the
above observations, is prOposed. Certain dimen-
sions of this model are deduced from the Hall co-
efficient.

The quantum chemistry and electronic structure

of AuSn is discussed in terms of valence bond theory

and simple band theory, i.e., a single-OPW model.

GALVANOMAGNETIC INVESTIGATION
OF THE FERMI SURFACE OF AuSn.

By

‘ \
\ ,

David J. Sellmyer

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics

1965

Acknowledgements

I sincerely wish to thank Professor P. A. Schroeder
for his interest and assistance in this research. In
particular, his help in the preparation of samples was
indispensable. I am indebted to Professors F. J. Blatt
and M. Garber, as well as Professor Schroeder, for pro—
viding the necessary support for the completion of this
research and for many informative discussions about various
aspects of this work.

Financial support by the National Science Foundation

is gratefully acknowledged.

Table of Contents

Page
1. Introduction 1
2. Theory of High-Field Galvanomagnetic
Effects in Metals 9
Dynamics of Conduction Electrons in
Magnetic Fields 9

Kinetic Formulation of TranSport Problems 12

High-Field Magnetoresistance and Hall

Effect: Closed Orbits 15
Influence of Open Orbits 19
Summary of High-Field Galvanomagnetic
Properties 23
Rules for Compensation in Intermetallic
Compounds 25
3. Preparation of Samples 28
Metallurgical and Sample Analysis
Techniques 28
The Achievement of a High Degree of
Order in Intermetallic Compounds 36
4. Apparatus and Experimental Technique #3
5. Results and Discussion 51
The Crystal Structure of AuSn 51
Magnetoresistance and Hall Effect 51
Topology of the Open Sheet of Fermi
Surface of AuSn 69
On the Electronic Structure of AuSn 74
6. Conclusions 78

iii

Table of Contents (continued)

Page
Appendix A 80
Appendix B 82
Appendix C 87
Appendix D 89
References 92

iv

O\U1~F-‘\Nl\)l—'
0

IO.
ll.
12.

13.

14.

15.

16.

Figures

Cyclotron orbits in a magnetic field
various types of closed and open orbits
Part of a cyclotron orbit

Rotation of coordinate system

Partial phase diagram of AuSn system
Phase diagram of non-ideal intermetallic
compound AlBl

Graph of resistance ratio vs. impurity
content for metals

Schematic diagram of apparatus and
circuitry

Diagram showing sample size, shape, and
potential probe placement

Sample holder and rotating apparatus
Crystal structure of AuSn

Direct and reciprocal lattices for
hexagonal crystals

Partial standard (0001) stereographic
projection for hexagonal crystals

Graph of Ap/p vs. magnetic field direction
for AuSn I.

Graph showing approach of magnetoresistance
to saturating and quadratic behavior

Log-log plot of Ap/p vs. BR for AuSn I

Page
10
11
16
24

37

38

41

44

44

45

52

53

53

54

17.

18.
19.

20.

21.

22.

A1.

D1.

Figures (continued)

Graph of AP/p vs. magnetic field direction
for AuSn IX.

Log-log plot of AD/O vs. BR for AuSn IX
Hall voltage vs. magnetic field direction
for AuSn IX

Stereogram showing current and field
directions for AuSn III and AuSn VIII
Graph of AP/D vs. magnetic field direction
for AuSn III and AuSn VIII

Topological model for the open sheet of
Fermi surface for AuSn (not to scale)
Schematic diagram of Bridgman Furnace and
electronic control circuits.

Simple compensated Fermi surface

vi

Page

59
6O

64

66

67

71

81
91

I:-

My

"

I.
II.

III.

BI.

BII.

BIII.

Tables

Page
List of semiconductors and metals 6
Summary of high-field galvanomagnetic
properties of metals 21
List of AuSn crystals grown 33
List of AuSn samples 34
Crystallographic angles between planes
(hlklll) and (h2k212) for arbitrary c/a 82
Crystallographic angles between (001) and
various planes for 1.10 S c/a S 1.90 8}
Crystallographic angles between (001) and
various planes for 1.10 S c/a S 1.90 85

vii

1. Introduction

The last decade has seen great advances in our under-
standing of the electronic structure of elemental metals.
The stimulus for the large amount of experimental and
theoretical work in Fermi surfaces was, to a large extent,
several developments which occurred at about the same time
in the late 1950's. These were: (1) the prediction by
Pippard(l) from the anomalous skin effect in cOpper, that
its Fermi surface touched certain Brillouin zone (BZ) faces,
(2) the explanation by Lifshitz(2) and coworkers of the
curious high-field galvanomagnetic effects in the noble
metals, (3) the observation by Schoenberg(3) of the de Haas-
van Alphen (deA) effect in copper, and (4) the realization
by Gold(4) that he could explain his deA periods in lead
on the basis of a free electron model sliced by zone boun-
daries. The exploitation of the nearly free electron
(single-OPW) model by Harrison(5), in particular, has im-
measurably aided the interpretation of tapographical ex-
periments on polyvalent metals.

The success of the high-field tOpographical eXperiments
such as the deA effect, high-field galvanomagnetic effects,
magnetoacoustic attenuation, and cyclotron resonance, de-
pends upon having long electron mean free paths. For this
reason the experiments are done at liquid helium temperatures

on very pure metals. Typically, the impurities must not be

-2-

greater than about 100 parts per million. This means that
the topographical experiments can be performed on random
substitutional alloys only if they are very dilute; in
favorable cases, however, dHyA oscillations can be seen
in alloys having as much as one atomic per cent solute.(6)
There is, however, another class of metals, 23g2_
metallic intermetallic compounds, which in.principle
satisfies the criteria for observing the highefield effects.
By definition, a binary intermetallic compound is a.mixture
of two metals which has a well defined stoichiometry and
atomic arrangement. That is, it can be written as AiBy
where x and y are integers and where the A and the B types
of atom each has a unique set of basis vectors in the unit
cell. If one could,in practice, prepare an intermetallic
with perfect order and with x and y exactly integers, then
the potential would be perfectly periodic as in the case of
a perfectly pure elemental metal, and the electrical re-
sistance would approach zero (apart from electroneelectron
collisions) as the temperature approaches absolute zero.
The work of Pearson and coworkers(7) at the National
Research Council in Ottawa, Canada, has shown that inter-
metallics can, in fact, be prepared with a rather remark-
able degree of crystalline perfection. That is, they can
. 9(295°K)/p(4.2°K),

which approach those for pure elemental metals. The recent

Ill

be prepared with resistance ratios R

-3-

realization of this fact led the NRC group to investigate
several intermetallic compounds with the deA effect.(8)
Although there has been a considerable amount of re-
search on certain semiconducting III4V and II-VI compounds,
there is relatively little known about the electronic
structure of metallic intermetallic compounds. The work
that has been done has been primarily crystal structure
determination(9), and the correlation of certain recurring
structures in intermetallics with electron concentration,
i.e., the HumeeRothery rules for electron compounds.(lo)
For example, the intermediate phases Aan, Cu3A1, and CuBSn
all have the body-centered-cubic structure and, assuming
the usual valencies for the individual atoms, all have
electron per atom ratios of 3/2. That the area of inter-
metallic compounds has remained relatively uninvestigated
is due mainly to the complexity of the chemical binding in
most intermetallics. In general, the chemical binding is
a mixture of ionic, covalent, and metallic binding. To give
Just one example of this, the compound NaTl is a reasonably
good conductor, has an cpen, valence-type structure, and
exhibits ionic charge-transfer in its NMR preperties.(ll)
Chemists were successful in developing valence theory
and in discovering the quadrivalence of carbon because there

exist many simple compounds such as Cflh and CH301 in which

the carbon atom apparently is attached to four other univalent

-4-

atoms. However, intermetallic compounds have resisted
such attempts to construct valence theories because a
given.metal will often have a different apparent valence
when it is in combination with different metals. In fact,
even in a given binary system, the two metals will have
different apparent valences in the several compounds they
form. For example, in the mercury-potassium system the
following compounds exist: KHng, KHgS’ KHg3, Kng, and
KHg.(12) In spite of such difficulties, Pauling has at-
tempted to correlate a large amount of eXperimental data
on interatomic spacings, magnetic moments, etc., with his
valence-bond theories of metals and intermetallic com-
pounds.(13)
Physicists, on the other hand, have had the most
success in understanding electronic and other prOperties
of those metals which in the atomic state have a known
number of s and p electrons outside filled shells. In-
versely, they have had the least success in understanding
the prOperties of the transition metals where it is not
known, a’ riori, how many of the atomic d electrons remain
localized and how many go into conduction bands. Inter?
metallic compounds are similar to transition metals in
the sense that it is difficult, in most cases, to predict

how many of the valence electrons go into conduction bands.

In fact, it is not even possible in many cases to predict

-5...

whether a given binary intermetallic will be a metal or
semiconductor.

In this connection, it is interesting to consider the
list of semiconductors and metals in Table I. It is seen
that for the group IV elements, the energy gaps decrease
until metallic behavior is reached, as we go down in the
periodic table. This behavior can best be explained on
the basis of the ”Phillips' cancellation theory”.(lu) If

|¢k> is the state vector of an electron in a crystal, then

2
‘%“ + V Mk) = Ekl‘hc) ’

where V is the periodic potential of the lattice and the
other symbols have their usual meanings. In the Spirit
of OPW theory, let l¢k> be constructed from a 'smooth'

part Ik> minus some orthogonalization terms:

”’18 = Ik) - Z lnk><nklk> ,
n
where <rlnk> = m;(r) represents the core levels. It

follows that the equation for lk) is

2
--Jg--m--+’\r+vR Ik>=EKIk>,

where VR is a non-local repulsive potential,

VR = ; (ELK-En) Ink><nkl .

Table I.

Position* Material

 

*

**

(2.4)
(3.4)
(4.4)
(5.4)
(5.4)
(6.4)

(3.5)
(4.5)
(5,5)
(6.5)

(3.4)
(4.4)
(5.4)
(6.4)

Position, (1,3), means 1
table.

C(diamond)

Si
Ge

Sn(srey)

Sn(white)

Pb

InP

InAs
InSb
InBi

MgESi
Mg2Ge
MgQSn
MgePb

Energy Gap (eV)**

List of semiconductors and metals.

Crystal Structure

 

th

6

1.12

0.75

0.08
metallic
metallic

1.30

0.33

0.17
metallic

0.77

0.55

0.25
metallic

row, 3

th

diamond
diamond
diamond
diamond
tetragonal
f.c.c.

zincblende
zincblende
zincblende
tetragonal

fluorite
fluorite
fluorite
fluorite

column in periodic

For compounds, (1,3) refers only to the position

of the second listed element in the compound.

Most of the energy gaps are taken from W. D. Lawson and
S. Nielsen, Preparation of Single Crystals, (Butterworths
Scientific Publications, London, 1958), pp. 241,242.

 

-7-
Furthermore, it can be shown(15) that

<r[V + Ver'> : V(r) [6(r - r') - Z<rlnk><nk1r'>] .
. cor

levels
But note that
2: <r4nk><nklr'> = 6(r - rt) ,
all - ,
states

so to the extent that the core levels form a complete set
of functions, the true potential will be canceled by the
repulsive potential VR in the region of the cores. This
is the basis for the remarkable degree of success enjoyed
by the nearly free electron model in polyvalent metals.
In the lower Z elements, therefore, there are fewer core

orbitals in which to eXpand the delta function, so that

the remaining effective potential (V'+-Vh) is a decreasing

 

function of the row number of the periodic table -- from C

to Si to Ge to Sn to Pb, for example. Thus the lower Z

elements in a given column will have stronger zone boundaries,
larger energy gaps, and greater energetic advantages for
forming valence—type structures rather than metallic structures.
These ideas are borneout quite well by the elements in

column IV of the periodic table - as Table I shows. The
purpose of this digression is, however, to point out that

we see from the table that the very same thing happens in

the IIISV and IISIY'intermetallic compounds, as we go down

-8-

in the periodic table. Such trends suggest that the OPW
approach might work in certain of these compounds as well
as in the elemental metals and semiconductors. To the
author's knowledge, however, there have been no OPW cal-
culations of band structure in metallic compounds. There
is evidence that the single OPW method does give a first
approximation to the Fermi surfaces of B' - CuZn(16) and
the fluorite structure compounds AuX2 where X = A1, In,
Ga.(17)

In summary then, we have seen that the whole area of
metallic intermetallic compounds is a little-investigated
one and that the possibility of determining the Fermi
surfaces of these metals has been demonstrated by the ob-
servation of the deA effect in several of them. These
facts have stimulated the present galvanomagnetic inves-

tigation of the Fermi surface of AuSn.

2. Theory of High-Field

Galvanomagnetic Effects in Metals

Consider an experiment in which a pure metal single
crystal is placed in a high magnetic field and the re-
sistance of the crystal is measured as the magnetic field

is rotated. It has been known(18)

since the 1930's that

a plot of the resistance vs. magnetic field direction
contained a remarkable amount of detail. This behavior
remained a mystery for decades until it was explained
theoretically by Lifshitz and coworkers(19) in 1956. In
this section we outline the theory of high-field galvano-
magnetic effects in metals to demonstrate the kinds of in-
formation obtainable from these effects. we follow the
Chambers<20) formulation of conduction problems as applied
by Ziman(21) to galvanomagnetic effects. The theory and
practice of these effects have been reviewed by Chambers<22),

Pippard<23), Baratoff(24), and Fawcett(25). Of these reviews,

the article by Fawcett is the most complete and up-to-date.

Dynamics of Conduction Electrons in Magnetic Fields

The equation of motion of electrons in a magnetic

field §,along the z-axis is

e n
E. = Tc (.Y;~ X E) (l )
where
_ -l .
‘XK, — h vge(§) (2)

-9...

-10..

This means that the electrons move on surfaces of constant
energy in k,Space and that the change in the wavevector

is normal to §,and normal to Xk which is itself normal

to the surface of constant energy. Thus k,is confined

to an orbit which is the intersection of a plane normal to
§_and a surface of constant energy. If fi'is the component
of k,in the x-y~ plane and 21the component of 2k in the x-y

plane, then (1) becomes

 

 

. e
E = "5(XLX Pa). (3).
Y
' The time required
v
to traverse a complete
K
“‘ orbit as in Fig. l is
X dK
dK ‘h 11 27F
1 ,S... dK =-£-. _-.-_ a _-._
e3 L T eB .‘a' mb
dA (4)
Fig. l

where mbis the cyclotron
frequency and dKlland dflare defined by Fig. l. The cyc-

lotron mass, me, is defined by

wb E eB/mcc , (5)

or, using (4),

m = eB = h d’cll
c who 2w v (5)

 

-11-

OPEN
I’ORBIT

 

I IN (010)-PLANE

I Ilfiooj-AXIS
CLOSED ELECTRON AND PERIODIC OPEN ORBIT
HOLE onmrs
OPEN ORBIT EXTENDED ORBIT
I \

   

% ‘5‘)
«3 (d)
APERIODIC OPEN ORBIT EXTENDED ORB”

(taken from E. Fawcett, Ref. 25)

Figure 2 Various types of closed and open orbits.

IIVJ

h‘

er

-12-

-1 1
Since vL = h d€/dKL ,

2 dK 2
_ h 11 _ h dA
mc I 27.jr de dKL — w de ’ (7)

where dA is defined in Fig. 1.

Fig. 2 shows, in an extended zone scheme, some of
the types of orbits possible for a hypothetical cubic
metal with a multiply-connected Fermi surface. For a
general field direction, there will be no cpen trajectories
and all the closed orbits will be electron orbits, i.e.,
they will enclose occupied regions in kfspace. Fig. 2(d)
shows an extended orbit which must pass through several
zones before closing back upon itself. Figs. 2(b) and (c)
define periodic and gperiodic cpen orbits as shown. In

 

Fig. 2(a) in which the field is along a symmetry axis,
there are no cpen orbits but there are closed hole orbits
which enclose unoccupied regions in kfspace. This is

called a singular field direction for reasons to be ex-

 

plained later in this section.

Kinetic Formulation of Transport Problems

Let fo(k,r)dkdr'be the equilibrium number of particles
in dkdz, When electric and magnetic fields, E and g, are
present, there are forces on the particles which change the

distribution to

rm) = roe) +s(§g) <8)

-13-

where g(k) is the (small) diSplacement from equilibrium;
we now assume no dependence on g, Now by (8),

Bfo

895,) = "5'5“ be. (9)

where A6 is the average energy gained by an electron from
the field E, The electrons are accelerated by E but are
scattered at a rate l/T where T is a relaxation time as-
sumed here to be independent of k, The effective ac-

celerating field is then

an“) = se““‘t"/T (10)

where the exponential factor is the probability that a
particle scattered at time t' will reach time t without

further scattering. Thus
I:

e [XkIt')'Ee'(t't')/T dt' (11)

-.,, t
e (;%)f%(tt)-§e‘(t't')/T dt'. (12)

This is the equation of Chambers. Now consider a new set

MM

and

8(Est)

of krspace variables. The point k'can be specified by:

(l) the energy 5 which, for any orbit, is constant
in a magnetic field,
(2) the component of k'along E, kz, also a constant,

and

-1h-

(3) a phase variable s defined by

dm = wbdt (13)

~ dK
.. is __.__.11
a _ (130er v, . (14)

This variable is chosen because it increases at a constant
rate m =lmb in a magnetic field, and m = 2r for a com-
plete circuit. We can therefore consider the 'out-of—
balance' part of the distribution function, g(k) to be

a function of (e,kz,m):

T

Bf n

s(€.kz.m) -= f—(———,€° s-r(e.kz.o")e (M We as" . (15)
C

Then the current is

e 1 . _ e l.
:1 = Z? gangs — I? gjyemzmmemzewedkzdcp . (16)

Substituting (15) into (16) and using
5—81?" ( > < >
lim = -5 e - e , l7
T-a-O e F

where EF is the Fermi energy, we obtain

27 m
2 m A n
- _ 1 e - C I! n -(m-m AA” T.
:1 - I113 Ff! dszmc-D: £(€F.kz:mydcp XIEFskch )9 0 (E8;
-m 1

Changing variables and using

i=g‘gs (19)

-15-

we have for the conductivity tensor in Cartesian co-

ordinates,
27 w
2 m
l e . c . - t
“is = mpjwz 5;]! dcpdm'vi(1<z.cp)vj(kz.cp-cp')e c. ”Do" (20)
o _

This is the Shockley tube-integral formula(26)

which is
applicable to conduction problems in both high and low

magnetic fields.

High-Field Magnetoresistance and Hall Effect: Closed Orbits

Let T be the average relaxation time and'fi the average
mobility of an electron in an orbit as in Fig. l. The
high-field region is defined by the condition

 

wbr = 'EB >> 1 , (21)

for all_the conduction electrons. This means, in effect,
that the electrons complete many cyclotron orbits before

they are scattered by collisions with phonons or impurities.
In the high-field region when only closed orbits are pre-
sent, we may use the fact that the velocities are periodic
functions of the phase variable o'. The range of integration
of m' from O to m may then be Split up into portions of
length 27. Thus if f(m' + at) = f(m'),

O

-16-

2V
[GIN/(0°? “WWW ___ Z e-2m/mcrfe- co 'c/cn 'r fICD')de
=0
2.", O
wCT -mfifln'r
a: aar' e c f(m')dm' . (22)
O

The conductivity tensor then becomes

W27
=? 2;] dkz 31%;- dq’dm'v1(kzm)v3(kz:¢'¢')e-m'flnc'r (23 )] '

With the range of m' limited, the exponential term can be
expanded in powers of l/mbr and the behavior of the mag-
netoresistance and Hall effect can be determined in the

high-field limit. Thus we use

e“°P'/‘”c" = 1021-“; £1 + . (24)

The zero order terms containing an x in the indices are

prOportional to

h dKll n
vx(¢)dop = vicosefi; Vi = fi—fdky = 0 9 (25)

where the symbols used are

defined in Fig. 3. Thus,

 

0

there is no term independent
of B in 013 if i or j equals

x. For the first order terms,

 

 

 

Fig. 3.

-17-

using (14), Fig. 3, and integrating by parts, we obtain

2t 27

h h
I I I = ._._. I I = __
ftp VX(..., )dcp mo] co dky(cp-cp ) 27 m. ky(cp) . (2:6)
0 0
Thus,

031:: 71]“ z 2'?’ a?" d‘pvy (q;)—21-ky(q))

_ 1 2 l .
._ 171:5 of dkz “’cmc fdkxky
ec

7939 Aez(k)dkz .=. Egg-v9 , (27)

 

where Ae(kz) is the area of a slice of the Fermi surface
in the x-y plane and Vé is the total electron volume in
krspace. When there are closed hole regions of Fermi sur-

face as well as electron regions, then another term must

 

be added to oyx:
oyx = 11? 93°- [[Ae(kz)dkz -]Ah(lcz)dkz ]
1 co co ne ' nh
= 11:13 -B— (Ve - Vh) = 'B— Q ’ (28)

where Vh is the hole volume in k-Space, 0 is the volume of

the unit cell, and n and nh are the number of occupied

e
electron and hole states, respectively, per unit cell of

the crystal. The Opposite sign appears in front of'Vh

because the hole orbit is traversed in the Opposite direction

-18-

as shown in Fig. 2.

From (23) and (24) it can be seen that terms of order

l/B in oXX and oyy vanish. For example

°xx(l]§) «fvxkydm «fkydky ._ o. (29)

The first non-vanishing term for these matrix elements is

of order l/B2. With similar arguments it can be seen that

the lowest order terms in Oxz and oyz are of order l/B,

and that 02 contains a term independent of E.

2
We can now construct the (asymptotic form of the con-

ductivity tensor in the high field limit.

 

 

-2 -l -l
/ ans aXyB aXZB \
— _ _ -l -2 -1
o — axyB ayyB ayzB , (30)
-l -l
\‘asz "ayzB azz /

where we have used the Onsager reciprocal relations and

where the aij are independent of field. From (28),

. (31)

Since our experiment consists of putting a constant cur-
rent through the sample and measuring the resultant vol-
tages, we must invert (30) to obtain the magnetoresistance

behavior. The gisymptotic form of the resistivity tensor

-19-

will depend upon the value of the quantity

nA s nh - he . (32)

If nA # 0, the metal is said to be uncompensated while if

 

nA = 0, the metal is said to be compensated. When (30)

 

is inverted in the two cases nA % 0 and nA = 0, we obtain

for the leading terms in the resistivity,

 

 

 

 

Q B
{ bxx - FOUR; bxz\
— _ a a
pn £0 — '55 n byy byz (33)
A A
\‘bzx bzy bzz)
2 2
{ cxxB oxyB cxz\
—- g 2 2
pnA=O — chB nyB Cyz (31]- )
\czx Czy czz/ ’

where the coefficients bij and c are independent of field.

13
Influence of Open Orbits

Consider a Fermi surface similar to that shown in
Fig. 1 and let the field be in a direction such that there
exists open trajectories, e.g. as in Fig. 1(b). Let the
direction of Open orbits be the 2 direction. Using Fig. 3,

-20-

we have for the zero order term,

 

dK
a _ h 11
Giy ny(cp)dcp -— fvxsins —-—m v
C .L
h .
_—_ - —_ dlc .
me I X

(35)

By inspection of Fig. 1(b), it is seen that this integral

is non-zero. Since an Open orbit in the i direction car-

ries current in the y and z directions, the yz components

of E'Will have terms independent of B. In this case (30)

 

becomes
-2 -l
{ dXXB deB dXz
3' A = -d 13"1 d d
open.x xy yy yz
-l
\‘dsz dyz dzz )

where the d are independent of B and where dx

13

 

(36)

is not

related to nA because the volumes are not defined when

there exist Open orbits. By inverting (36) we obtain

2
{ exxB exyB exz \
pOpenfi‘c = eyXB eyy eyz

 

where the eij are constants.

\ ezx ezy ezz /

 

(37)

-21-

When there are two bands of non-intersecting_0pen
orbits in different directions, it can be shown<27) that
the asymptotic form of the resistivity tensor has all
of its leading terms independent of field.

When the field is along a symmetry axis of higher
than twofold symmetry, Open orbits can intersect to form
closed orbits of character Opposite to that of the Open
sheet. For example, in Fig. 1(a), when B is along a four-
fold axis a z-axis, only h21§_orbits occur on the electron
sheet for -d/2 < kZ < d/2, where d is the diameter of the

arms. These field directions are called singular field

 

directions because a measurement of the Hall coefficient

will not give the true volume of the sheet as it does when
the field is in a general direction for which there are no
Open orbits. For a singular field direction, the Hall co-

efficient gives a measure of the effective volume of the

 

sheet of volume'V(28),

d/2
Veff =[i v- ABZ(1{z)d1CZ , (38)
- /2

where ABZ(kZ) is the area of the Brillouin zone for kz =
constant, and d is the minimum diameter along B of the
arms supporting orbits of character Opposite to that of
the sheet. Since there are no Open orbits at a singular
field direction, the resistivity tensor has the same form

as (33) except that nA is replaced by n i An where

A

 

 

 

.COfipooan pfipso :oeo one spas d cameo em weaxmsgm Op anSOfineanoe oemfie one CH we m. H
.hpfimcop pCOSMSO page Log UHOfim Ofinpooflo .O.H +
we so a as season-ease
em 2 m G I Arcades-Sea em 2 .aflawfim .>
28308:. 039
em 2 7m 2 Anon-333v em 2 E :25 SH
Hagan-2.6 ode
a wee a fin am 2 m 2 a «wee am 2 5 demo .HHH
as" .s
peerage-mace
am. 2 m 2 ass-seas am 2 as ease-e 3. .:
Em...-
We. a a I c a poaemeomaooes
em 2 m G Anceeaseewv em 2 was “some? :4 .H
360 :o- dado-mace o o e m
noboOmwMWm-Ea Omega-Sat consummmoaBoS-meg was flame momommata
a has zen .

 

 

 

 

.mfimpoe e0 moaunoaose oeuocmmsocm>fimw UHOfiMIemfig mo zsmEEsm

 

 

.HH manna

-23-
d/2
An = Zfiz. ABZ(kZ)dkz . (
- /2

\N
\O
V

Summary of High-Field Galvanomagnetic Properties

The field dependence of the galvanomagnetic pro-
perties of metals in the high-field limit is summarized
in Table II, taken from Fawcett's review article.(29) By
magnetoresistance, we will always mean transverse magneto-

resistance which we define as

A B - 0
where the resistivities are measured with the field per—
pendicular to the current. The longitudinal magneto-
resistance has a difinition similar to (40) except that
the field is parallel to the current density g. The
transverse-odd field is defined as RHB/J and the trans—
verse—even field is defined as RTEBe/J, where

pr(B) - oxy(-B)

R a (41)
H 2B

 

pr(B) + p (-B)

X
RTE 2B2y II‘ ° (#2)

 

be

7"

h.
y\s

 

(n

In Table II, in the case of open orbits in one direction,
Ap/p and RTE depend on the angle a between {,and i, where

x is the direction of Open orbits. This comes about be-
cause we must measure the voltages in the sample coordinate
system rather than in the system in which the Open orbits
are in the i direction. We therefore must make a similarity
g_ transformation of'? as
lg’ given by (37) with the

rotation matrix corres-

ponding to the rotation

 

a shown in Fig. 4. This

XV

leads to the angular de-
pendences shown in Table
Fig. 4
II.
Table II also points out the major qualitative dif-
ference between the magnetoresistance behavior of com-

pensated and uncompensated metals. For an uncompensated

metal, the magnetoresistance tends toward saturation for

2

a general field direction and is prOportional to B only

when there are Open orbits perpendicular to the field.

For a compensated metal on the other hand, the magneto-
resistance is prOportional to B2 for a general field dir-
ection and approaches saturation only when there are Open
Orbits perpendicular to the current direction. This means

Iflfirt Open orbits are somewhat more difficult to find in

-25-

compensated metals than uncompensated ones because the
sample axis (which determines the direction of g), must
be chosen such that it is normal to an Open orbit dir-
ection if open orbits are to be observed. In the next
subsection we turn to a discussion of the criteria for
the existence of compensation in metallic compounds and
the information about the quantum chemistry that might
be gained from a study of the state of compensation in

these compounds.

Rules for Compensation in Intermetallic Compounds

The experimental determination of the state of com-
pensation in intermetallic compounds will be useful be-
cause, in general, the complexity of the chemical binding
precludes a knowledge of the number of electrons in con-
duction bands. However, the success or failure of the
rules for compensation given below may indicate, for
example, that the Fermi surface of a given intermetallic
is consistent with a single OPW model or, alternatively,
that some of the 'valence' electrons have gone into
localized bonding orbitals.

Consider a perfectly ordered intermetallic compound

having 31 and s atoms of atomic number Z1 and Z2, res-

2
pectively, per unit cell. Let there be N unit cells in

—26_

the crystal, F full zones, and J hole zones.(30) Since
the total number of electrons must equal the total num-

(31)

her of occupied states, we have

slz1 + $2Z2} N = 2FN + neN + (2J - nh)N. (43)

Denoting the quantity in braces by nT, the total number

of electrons per unit cell, (43) becomes

-nA E ne - nh = nT - 2(F + J) (44)
Therefore if nT is odd, the metal cannot be compensated
and nA must be an odd integer. If nT is even, either nA

must be an even integer or the metal can be compensated

 

if nT = 2(F + J). However, compensation is to be expected
when nT is even because of "spilling over" arguments. For
all nonmagnetic metals which have been investigated, if nT
is even, the metal is, in fact, compensated.

If, in a metalliccompound, the core electrons lie
appreciably lower in energy than the 'valence' electrons,
(44) becomes
= (slV

+ S2V2) - 2(FV + J)

’nA l

(45)

th

where V is the'valence'of the i type of atom, FV is the

1
number of zones filled by the 'valence' electrons, and nV

-27-

is the total number of 'valence' electron per unit cell.
Eq. (45) should be most useful when the number of atoms
per unit cell is fairly small. For example, the equi—
atomic ordered alloy B'- CuZn has the CsCl structure so
that s1 = s2 = 1. If we assume Vbu = l and Vin = 2,

then n.V = 3 so that CuZn should be an uncompensated metal.
Ioreover, a suitable modification of a single ~0PW Fermi
surface for this metal, which is consistent with the mea-

sured dHVA periods,(32)

predicts hole surfaces in the first
zone, multiply—connected hole lenses in the second zone,

and no overlap into higher zones. This implies F 0

v:

and J = 1 so that nA = -13 this can be checked by a direct

measurement of the Hall coefficient in high fields.
The application of (44) and (45) to AuSn will be

discussed in section 5.

3. Preparation of Samples

In this section we discuss attempts to produce
single crystals of several magnesium intermetallics,
the preparation of AuSn, and comment upon the necessary
criteria for obtaining long mean free paths in inter-

metallic compounds at low temperatures.

Metallurgical and Sample Analysis Techniques

With minor variations, the procedure for making all
of the alloys was as follows: The pure metals were etched,
washed with distilled water, dried with a heat gun, and
weighed in the correct proportions to about 50 micrograms
on a Mettler balance. A small, high-purity graphite cru-
cible was then outgassed for several hours in a Lepel
induction furnace; the temperature was held above 1000°C
and the pressure was kept below about 10'” mm Hg during
this outgassing. The pure metals were then placed in the
crucible and melted together either under vacuum (pressure 5
10-4 mm Hg) or in about one atmosphere of argon - depending
upon the vapor pressures of the metals involved. In some
cases the alloy was chill-cast by pOuring it into a cool
copper or graphite mold several times. In other cases
mixing was done by agitating the melt mechanically and
by the action of the rf field. When the melt was not
poured, the sample was cooled, turned over, and remelted

about five times.

-28-

-29-

Two methods of producing single crystals were used:
zone refining and the Bridgman method. The zone refining
methods were similar to those described by Lawson and
Nielsen(33), with the heating element being either a
length of Kanthal resistance wire or a small rf coil.

The zone refined samples were in the form of rods of
diameter about 1/8” and length about 6”. The samples
were situated in a long graphite boat which, in turn,

was inside an evacuated pyrex or vycor tube. The heating
element was pulled along by a geared-down synchronous
motor at a rate of about one inch per hour.

In the Bridgman method the samples were enclosed in
evacuated, sealed-off, pyrex tubes or in graphite crucibles
which were inside evacuated, sealed-off, vycor tubes. The
sample holder was suspended by a chromel-alumel wire and
was drOpped atabout 3/4" per hour through a temperature
gradient in a double furnace. The temperature of each part
could be kept constant to $100; schematic diagrams of the
double furnace, its wiring diagrams, and its electronic
temperature control circuits are shown in Appendix A.

The first attempts to prepare single crystals were
made on several magnesium compounds: Mg2Cu, Mgsz, and
Mng. These compounds turned out to be fairly difficult
to prepare for several reasons. Magnesium has a relatively
high vapor pressure at the melting temperatures of these

compounds and therefore, was continually deposited out on

-30-

a cool part of the container. Furthermore, magnesium
attacks quartz so that the Vycor tubes had to be coated
with a carbon film to prevent deterioration of the tubes.
This made it very difficult to tell whether or not the
metal was actually molten in the graphite boat used for
zone refining. In the case of MgQCu, the alloy invariably
became coated with a black scum which could not be iden-
tified. Several ingots of Mg2Pb were prepared by melting
the two metals in about one atmOSphere of argon gas in

the induction furnace. Hewever, upon exposure to the
atmosphere for several hours, a shiny ingot of Mg2Pb

would turn into a pile of black dust. The compound Mng
which has the CsCl structure is similar to Mgng in that

it also decomposes upon exposure to the atmOSphere. But

by keeping the ingot in nitrogen gas or liquid nitrogen,

it was possible to enclose the sample in a quartz tube

and pass a molten zone through it five times. Two of these
zone refined samples had resistance ratios of 125 and 94.

A crude sample holder was made and the magnetoresistance

of the R = 94 sample was measured as a function of magnetic
field direction at a field of 21 kG. The magnetoresistance
was isotropic to within experimental accuracy'(~5%) and

had -a value of about 0.5. Possible reasons for this
isotropy are: (l) the sample may not have been a single

crystal; it was not checked by X-rays because of the

decomposition problem, (2) there may be no Open orbits

an/ i—

in Mng, (3) the field may not have been sufficiently
high that the electrons were in the high-field region.

One of the interesting questions one can ask about
the electronic structure of metallic compounds is: For
a compound to be an OPW metal, is it an unnecessary,
necessary, or sufficient condition that the constituent
metals themselves be OPW-like? If the answer to this
question is that it is a sufficient condition, then Mng
would be an OPW metal. Since the symmetry of this metal
is so simple (simple cubic), the single-OPW model was
worked out. The model gives small hole 'pockets' in the
first zone, a 'concave cube' hole surface in the second
zone, electron 'pancakes' in the third zone, and electron
'cigars' in the fourth zone. There are no Open orbits in
this model. Because of its simple crystal structure, it
would be interesting to investigate the Fermi surface of
Mng.with the leA.effect to see if it does, in fact, con-
form to the single-OPW model.

As mentioned above, the metallurgical problems as-
sociated with the magnesium compounds are formidable. In
particular, it is very difficult to cut a sample from a
large single crystal of a reactive material, orient it
with.X-rays, and mount it in a sample holder. For these
reasons it was decided to change to a more tractable alloy

system.

7.
"/2-

The gold-tin system is attractive because both of
the elements have low vapor pressures and are quite non-
reactive with air. The melting temperature of the com-
pound AuSn is 418°c,(34) which is well within the cap-
ability of the Bridgman furnace. Furthermore, AuSn has
the hexagonal, NiAs type structure with c/a = 1.278;
there are four atoms per unit cell in this structure which
is a reasonably small number compared with the 48 atoms
per unit cell of acompound like MgECu.(35) Other motives
for studying AuSn are that it has been studied by a variety
of tranSport(36) and magnetic(37) techniques as well as by
the deA effect.(38) '

After growing the single crystals by either zone re-
fining or the Bridgman method, the resistance ratios were
measured as a function of length along the rods. This was
done by attaching phOSphor— bronze current and potential
Spring clips to the sample along its length. A current of
about one amp was passed through the sample in both dir-
actions and the potentials read at T = 2950K. and T = 4.2OK.

Table III lists the various samples grown, their
approximate dimensions, the method of preparation, the
type of container, and the minimum and maximum or average
resistance ratio for each sample. The samples listed are
not all monocrystals; there are polycrystalline portions
on many of the Bridgman samples, especially near the
initially solidified end. Samples VIII-l, 2, 3 and IX-l,2

were grown in a graphite crucible having three thin slots

Table III. List of AuSn crystals grown.

 

 

 

 

32;. Dimensions Method of Container Rmin/Rmaxh
(in), Preparation or Rave

I 1/8 x 6 ZR(6)a GBb 12

1-1 1/8 x 4 BC PTd 36/60

II 1/8 x 4 B GCe 16/35

IIIf 1/8 x 1/8 x 6 ZR(6) GB 10/42

IV 1/8 x 5 B PT 14

'V l/4 x 5 B PT 19/26

VIg 1/8 x 6 ZR(9) GB 10

VII 1/8 x 5 B PT 13

VIII-l 1/16 x 1/16 x 4 B 00 38/54

VIII-2 1/16 x 1/16 x 4 B GC 39/60

VIII-3 1/16 x 1/16 x 4 B so 34/43

IX-l 1/16 x 1/16 x 4 B so 34/166

IX-2 1/16 x 1/16 x 4 B CO 46/130

a - ZR(n) = zone refined; n passes

b - GB = graphite boat

c - B = Bridgman method

d - PT = pyrex tube

e - CC = graphite crucible

f - part of sample II used as a seed crystal

g - part of sample V used as a seed crystal

h - R 2' p(295°K)/p(4.2°K)

Table IV.

List of AuSn samples.

 

Cut from Samplea _§‘_ ._TE __§i
I-l 60 7° 88°
III 42 19° 16°
v 25 20° 36°
IX-l 73 4° 60°
VIII-2 116 12° 88°

a - i.e. cut from sample designated in Table III.

b - m
c - 9

angle between plane of [0001] and J and {10I0(.
angle between [0001] and J.

-34-

_35-

cut down to its center. In this way it was possible to
grow three crystals with only one 'drOp' through the
Bridgman furnace. At the suggestion of W. B. Pearson,
samples VIII and IX were made slightly gold rich and
this seemed to increase the resistance ratio. Sample
VIII was 0.1 at.% gold rich and IX was 0.2 at.% gold
rich. For reasons to be explained below, the crystals
tended to have the highest purity (i.e. highest R) in
the same regions in which there were a lot of small angle
grain boundaries. For this reason it was necessary to
take four to six X-ray photographs to find a large enough
'single' region from which to cut a sample. The X-ray
technique used was the Laue back-reflection method using
a molybdenum target. The film used was Kodak type KK and
the exposure times were about 30 to 40 minutes. The sam-
ples were cut out with a Servo-Met spark cutter. The
samples on which liquid helium runs were done are listed
in Table IV with their resistance ratios and orientations.
In the course of orienting the hexagonal AuSn cry-
stals, it became necessary to calculate the interplanar
angles in order to construct a standard (0001) stero-
graphic projection. These angles are a function of the
c/a ratio for hexagonal crystals and have already been
published for magnesium, zinc, and cadmium.(39) Because
of the increasing interest in electronic and other pro-

perties of hexagonal transition metals and intermetallic

-36~

compounds, the crystallographic angles were calculated

on the M.S.U. CDC 3600 computer for 1.10 s c/a s 1.90,

in 0.01 steps.(40) These bounds were chosen because most
hexagonal elements and many hexagonal intermetallic com-
pounds have c/a ratios within them. If we exclude graphite,
c/a ratios of the hexagonal elements vary from about 1.14
for selenium to 1.88 for cadmium.(ul) Also, for example,
there are many intermediate phases of the NiAs type for
which 1.2 g’c/a g 1.8.(42) The calculation is described

in Ref. (40) and the results are contained in Appendix B.

The Achievement of a High Degree of Order in Intermetallic

Compounds.

Consider the schematic, partial phase diagram of the
gold-tin system in Fig. 5.(43) AuSn is a compound which,
in principle, is simple to prepare because if one solid-
ifies a melt which is slightly off the correct stoichio-
metry, the solid formed will have the exact stoichio-
metry over at least part of the solid. For example, if
one cools an alloy at the point B on Fig. 5, the compound
of perfect stoichiometry will begin to solidify out at
point C. The residual melt will then slide down the
liquidus toward the right until point F is reached. Then
the compound AuSn2 will begin to precipitate out. If one
solidifies a long thin rod of nominal composition Au Sn

1 1
from the bottom up, it is indeed found that the bottom end

-37...

 

 

 

 

Temp 0C

 

 

 

 

50.0 66

Atomic Percent Tin

Figure 5 Partial phase diagram of AuSn system.

-38-

Temp 0C

 

 

 

I
I
I
I
I
I
I
I
J

 

 

49.8 50.00 50.2

Atomic Percent B

Figure 6 Phase diagram of non-ideal intermetallic

compound AlBl.

—99_

has a more nearly perfect stoichiometry. That is, the
bottom end has a much higher resistance ratio than the

top end (See Table III, samples VIII and IX). Unfor-
tunately, the bottom end also tends to be polycrystalline
and/or to have many small angle grain boundaries because,
in the growth process, one of the crystals has not yet

had a chance to dominate over all the other crystals which
have started to grow. The preparation of a peritectic
compound such as AuSn2 would be rather difficult. Cooling
a melt at point D would result first in the solidification
of AuSn at E. Only when the melt had reached the composition

at F would AuSn begin to solidify at point G.

2
The preparation of high purity compounds of the type
we have called 'simple' is, in fact, more complicated than
the elementary discussion we have given above. The reason
is that the vertical line representing the solidus of an
intermetallic compound on a phase diagram is not, in gen-
eral, of infinitesimal width. Compounds can exist as
substitutional or subtractional (vacancy) solid solutions
between certain limits of homogeneity as shown in Fig. 6.
For example, the separation of the limits of homogeneity

is 0.5 atomic per cent for AuSn(44)

and 2.7 atomic per cent
for MgCu2gu5)

What magnitude of resistance ratio would one expect
for a compound having a phase diagram like Fig. 6? If

we denote the compound by A50ixB50 x’ then from Fig. 6,
3F

~40—

a rough estimate of an average value of x might be ~x0.1.
This corresponds to about one imperfection in 103 atoms.
From a consideration of the purities of pure metals and
their resistance ratios, one can induce the following order-
of—magnitude relation between resistance ratio, R, and the

amount of impurities, I, in parts per million present.
R ~ 104/1 , 1315.104. (46)

That is, four-nines (i.e. 99.99 at.% pure), five-nines, and
six-nines pure metals typically have resistance ratios of
about 102, 103, 104, respectively. If we assume that the
same approximate rule holds for intermetallics, we could
expect the compound in the above example to have R -.10,
Fig. 7 is a plot of (46). The points on the graph were
taken from a table of R versus composition in Jan et.al.
and it is seen that AuSn does qualitatively follow (46).
From Fig. 7 and Table III, it follows that the purest part
of sample VIII (R.~'l50) was within about 60 ppm of exact
stoichiometric prOportions. This is a rather remarkable
degree of crystalline perfection for binary ordered alloy-
a much higher degree of perfection than can be achieved

in metallic superstructures of the CuBAu or CuAu type.(47)
In this discussion we have neglected as being small, the

residual resistivity due to impurities in the original pure
gold and tin; the gold used was five-nines ASARCO and the

Resistance Ratio R

-41-

 

 

 

10

Figure 7

 

J

10

2 103 10

Impurity Content, I, in ppm

Graph of resistance ratio vs. impurity content

for metals.

-42-

tin used was six-nines COMINCO.

The process of lowering a sample through the Bridg-
man furnace to make a single crystal typically took about
fifteen hours. For the zone refined samples, each pass
took about four to six hours so that several days were
spent zone refining a single sample. Thus, with the pre-
liminary weighing, etching, outgassing, sealing off, etc.,
it normally took three or four days to prepare a single
crystal with the Bridgman method and about a week to do so
by zone refining. The determination of resistance ratios,
checking crystals for singleness, cutting to size, and
final orientation took at least a week for each sample.

It can therefore be seen that an appreciable amount of the
time Spent in this research was devoted to sample prepar-
ation.

Summarizing, we have tried to show in this section
that as regards the preparation of highly ordered metallic
compounds, the best chances for success lie with those
compounds having the following properties: (1) the sep-
aration between the limits of homogeneity of the inter-
mediate phase should be extremely small, and (2) the
intermediate phase should occur at an exact stoichiometric

composition.

4. Apparatus and Experimental Technique

The experiment consists of measuring the magneto-
resistance and Hall coefficient as a function of the
magnitude and direction of the magnetic field. Fig. 8
shows a schematic diagram of the apparatus and circuits
used to measure Ap/p as a function of B. The sample,
imInersed in liquid helium at 4.2°K, is held in a fixed
position in the field of a Magnion superconducting
solenoid having an inner diameter of 1.5" and a maximum
field of about 58 kG. The source of the sample current
was several large lead storage cells producing a current
that was held constant to one part in 104. A fairly
large sample current of about three amperes was used in
order to obtain transverse voltages of a readable mag—
nitude, i.e., of the order of microvolts. The small
DC voltages were amplified with a Keithley model 149
millimicrovoltmeter which fed the YAaxis of a Moseley
2D, X-Y’recorder. The noise level of this arrangement
was about 5 x 10"8 volts. For the function-of—field runs
illustrated in Fig. 8, the X-axis diSplacement was pro-
portional to the current (and therefore, to the field) in
the solenoid.

Fig. 9 shows the sample size and shape as well as
the points of contact.of the potential probes. Only five
of the probes, e.g. l-5, were actually necessary to com—

pletely determine the vector electric field, E, in the

-43-

—44—

Figure 8 Schematic Diagram of Apparatus and Circuitry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
I. I—
I
—>
r—fl
’ Keithley MOSELEY
Amplifier -——Y X-Y RECORDER
(L__. __
r' ‘7' '- ‘" — r — ‘1 I X
I \S \ I
I , SOLENOID
l POWER
| SUPPLY
I
L I
L— — —- —— —-- --I Superconducting Solenoid
«yli‘l';
Z Liquid He Bath 8 . I3 ..
k J_ a ”It
5 H‘s... IE_ 7
Figure 9 Diagram Showing Sample Size, ,
to 1+ ‘2‘"
Shape, and Potential Probe 0'5 ' \6 o
Placement 3

 

-45-

 

 

 

 

 

 

 

 

 

 

      

" Drive Shaft
Aluminum _ Spiral Gear
,Support,
Hollow Cylinder
Probe
Springs

 

 

 

 

 

 

 

Teflon Spacer

Teflon Busmng '

 

I—l/2"Cublc Lucrre Sample Holder

Figure 10 Sample helder and rotating apparatus.

-46-

sample; the three extra ones were included as a safety
factor against open circuits; as shown in Fig. 10, the
thin brass voltage probes were held against the sample
with metal springs and it was not known before the ap-
paratus was actually tested at liquid helium temperatures,
how reliable the contacts would be. In practice, the
probe spring method was wholly reliable: not one open
circuit was encountered in any of the runs.

In order to measure the galvanomagnetic prOperties
as a function of magnetic field direction, it is nec-
essary to rotate the sample in the longitudinal field
of the solenoid. Fig. 10 shows an exploded view of the
apparatus used to do this. The lucite sample holder in
Fig. 10 was designed so that the sample could be mounted
for transverse or longitudina1-to-transverse runs. In
the transverse runs glwas always perpendicular to B,as
illustrated in Fig. 10; in the longitudinal-to-transverse
runs, the sample could be rotated from a position where
J,was parallel to B,to a position where J,was perpendicular
to B, The spiral gear arrangement is similar to that used
by Thorsen and Berlincourt.(u8) for pulsed field dHVA ex-
periments. The drive shaft, which emerges from the tOp
of the cryostat through a "quick-coupling" vacuum seal,
is coupled to a linear potentiometer through which a

constant current flows. The voltage taken from this

-117-

potentiometer, which is proportional to the angle of
rotation of the sample, is fed into the X-axis of the
X-Y recorder for the 'rotation' plots. One complete
rotation of the drive shaft corresponds to a rotation

of the sample of 4.50. The misorientation due to back-
lash in the gears is estimated to be less than 1°. In-
cluding the error in the X-ray orientation of the samples,
the total uncertainty in orientation is estimated to be
i2°. In certain cases, where the passage of B,through a
symmetry plane of the crystal corresponds to an extremum
in the rotation plot, the uncertainty in orientation is
considerably smaller than 2°.

The rotation plots were always made with the solenoid
in the persistent current mode. The rotation and function-
of-field data were taken point-by-point using the point
plotting attachment of the X-Y recorder.

Current leads of No. 34 cOpper wire were soldered
onto the sample with In-Bi eutectic solder (22 at.% B1,
superconducting at 4.2°K(u9), melting temperature 72°C.)
The current leads were brought to the sample through a hole
on the axis of the left side of gear shown in Fig. 10.
Small No. 42 cOpper wires were soldered to the 0.014",
sharpened brass voltage probes. These wires were glued
to the lucite block with a general cement and brought out
a hole on the right side of the cylinder shown in Fig. 10.
The flexibility of the current and potential leads per-

mitted the sample to be rotated through 180° without
difficulty.

-48-

About ten liters of liquid helium was needed to pre-
cool the solenoid because of its large heat capacity; it
weighs about fifteen pounds. It was found that the best
way to precool the solenoid to liquid nitrogen temperatures
(770K) was to dip it into a dewar of liquid nitrogen for
one hour. It was then quickly withdrawn from the liquid
nitrogen bath and lowered into a stainless steel, double-
dewar. Immediately thereafter, the liquid helium transfer
was begun. The most efficient transfer pressure was ~5/8
pounds until the solenoid was completely immersed and ~3/4
pounds thereafter. The level of the liquid helium in the
dewar was monitored by four carbon resistors placed at
various heights above the bottom of the dewar. Each car-
bon resistor formed one arm of a resistance bridge which
was balanced when the resistor was immersed in liquid
helium. Each resistor was surrounded with styrofoam to
keep the cold helium gas from lowering its temperature to
4.2OK before it was actually covered with liquid helium.
In this way, a sudden drOp from almost full scale to zero
was noticed on a microammeter when a given resistor became
immersed.

In order to achieve fields above about 40 kG, it was
necessary to "train" the solenoid by quenching it once or
twice. By quenching is meant the sudden transition of the
superconducting solenoid from the superconducting to the
normal state with a moderate or high current flowing in

it. It was also necessary to retrain the solenoid upon

-2). 9-

changing the direction of current in it. The occurrence
of these quenches was unfortunate because about a liter
of liquid helium was evaporated for each quench and a)
considerable amount of time was wasted bringing the
field back up to a high value. The field could not be
taken from zero to a high value arbitrarily fast; it was
necessary to monitor the inductive back-emf across the

solenoid, vL = L dIS/dt, and to keep v < 1 volt for

L

B 3'40 kG and Vi 5,50 millivolts for B > 40 kG. This
meant that it took about one-half hour to change B from
0 to 40 k0 and about 1.5 hours to change B from 40 kG
to 56 kG. If the above limits on.Vi were exceeded, a
quench of the solenoid would result. Even if'V was

L
kept within the above limits, the solenoid would quench
until it had been trained as explained above.

Also with reSpect to the performance of the solenoid,
during one run, the liquid helium bath was pumped on to
lower its temperature to about 1.30K. This was done to
look for de Haas - Schubnikov oscillations in the magneto-
resistance. Unfortunately, however, because of the re-
duced temperature, the solenoid became very quench prone
even though it had already been trained at 4.20K. This
degradation effect has been observed before and although
the phenomenon is apparently not completely understood, it
seems to be connected with the effect of temperature on the

(50)

flux-jumping mechanism.

S50.-

As explained in Appendix C, it was necessary to
change the direction of the field in the solenoid to
determine the Hall voltages. Let the magnetoresistance
Ap/p and the pair of orthogonal transverse voltages‘V£
be considered functions of B and t where B > O (B < 0)
refers to the up (down) direction of field and t is the
angle of rotation of the sample. Let Ii be angles at
which there are dips or peaks in the rotation plots. Then,
in a typical run, the following quantities were measured
in the order given: (1) 09/9 (440 kG, I). (2) vt(+40 kG, l).
(3) Mo (+50 kG. I). (4) Vt(+5° kG. I). (5) Ap/p (B > 0.1),).
(6) Mo (B < 0. I2),t0 -40 kG. (7) vt(-4o kG. I).

(8) Ap/o (B < 0. I2)ito -50 kG. (9) vt(-50 kG, I).

(10) Ap/p or vt as functions of field at other interesting
angles Ii until the helium ran out. The run described
above is an ideal one; the function-of-field plots were
sometimes interrupted by the quenches previously referred
to.

Since the topological prOperties of the Fermi surface
determine only the field dependence of the galvanomagnetic
properties and not their absolute values, the solenoid was
not accurately calibrated. The magnetic conversion ratio,

MCR = 2.43 kG/Amp, was determined by the manufacturer to

an estimated accuracy of 10%.

5. Results and Discussion

The Crystal Structure of AuSn

Fig. 11 shows the hexagonal, NiAs structure of AuSn.
The black balls representing the gold atoms are on a simple
hexagonal lattice and the white balls representing the tin
atoms are on a hexagonal close-packed lattice. The space
group. P63/mmc, indicates that there is a six-fold screw
axis and also a glide plane along the hexagonal axis.

Fig. 12 illustrates the primitive translation vectors,
a1, a2, E3’ 2; in the direct lattice and the primitive
translation vectors, 21’ be, 25, in the reciprocal lattice.
Fig. 13 shows a partial (0001) standard stereographic pro-
jection(51) for hexagonal crystals. Both figures show that
all of the §,directions are normal to {1120} planes and the
1; directions are normal to {101:0} planes. Similarly, the a
vectors are along (1120) directions and the b vectors are
along <1OIO> directions. On Fig. 13 and on all succeeding
stereograms, the <1I20> directions are marked as dots and
the <10I0> directions are marked as slashes on the peri-

meters of the stereogram.

Magnetoresistance and Hall Effect

Fig. 14 shows the magnetoresistance as a function of
magnetic field direction at 50 kG, for AuSn I. This sample

had a resistance ratio of 60. The current direction, given

-51-

 

2N; no\o

 

 

 

n!_.m\N.M\_3 - cm

 

 

 

 

 

5. Aux-3036.3” =4

 

 

 

 

ass-Xe.-

 

 

 

H5010 84mm

 

 

-53..

 

 

 

(1120)
(1010)

 

 

 

 

[1120]

 

 

[1210]
[0110]

 

 

 

Figure 12 Direct and reciprocal lattices for hexagonal crystals.

 

 

1120
E53
0001
cab,3 'I2IO
22
0110
9.2
2110
1010
al
“’ b
~l

Figure 13 Partial standard (0001) stereographic projection

for hexagonal crystals.

-5h-

 

 

 

Au Sn I
B=50kO
1.5—
O
b
:0—
sale
Q.
<3
0.5—
J-' °
" <II o>
(l0l0)
0.05—I-r—Lr—J—r l lw—l—r—I—u—
90 -60 —30 0 so 60 90

MAGNETIC FIELD DIREC TION

Figure 14
for AuSn I.

 

Graph of Ap/p vs. magnetic field direction

-55...

by the cross on the stereogram, is 20 away from the basal
plane and 70 away from a.{1OIO} plane. The tOp rotation
curve is a transverse run and the bottom one is a nearly
longitudinal-to-transverse run. Both curves are essentially
symmetrical about point b, i.e., they do not cross at this
point. The small letters on the curves and stereogram per-
mit making a one-to-one correSpondence between the field
directions on the great circles on the stereogram and the
field directions on the rotation curves. Although the curves
were made in a point-by-point manner, for simplicity of pre-
sentation, no experimental points are shown on Fig. 14 or
other rotation plots. The data were taken quasi-continuously,
i.e., often at one or two degree intervals. Each point was
taken with the solenoid in the persistent current mode and
with the sample motionless. Consequently, the noise due to
emfs induced by varying flux through the measuring circuit
was negligible; the total noise was only of the order of the
thickness of the line in Fig. 14.

Let‘V1(B) be the voltage measured along the length of
the sample at a field B. Fig. 15 shows a linear plot of
AV 2 V1(B) - v1(0) = V1(O)Ap/p as a function of field for
two different field directions on two different samples,
AuSn I and AuSn IX. The figure demonstrates the closest
approach to saturation and to quadratic behavior achieved
with the samples used in this research. The magnetoresistance

is henceforth assumed to be prOportional to Bm and the values

in [AV

AV

 

 

O

0

AV = V1(B) - V1(0) = V1(0)AQ/€ vs. B.

AuSn IX

curve a, Fig.

0
0
O
G
O
O
O
0

AuSn
CUPVG

90 G 0 GO 0 O

1 Field in kG

o
18 0
O
o
O
o
o
I
d, Fig. 16

ea coco 0000000

 

 

30___

25‘...-

20...

l5.__

h)___

5...
001;} °
Figure 15.

saturating and

25

quadratic behavior

5f

Graph showing approach of magnetoresistance to

 

 

 

2.0 .. Au Sn I
a, I.3
b, L0
L0—
08 r- °'°'°
§ L
a
\
e _
o.
4
d,a O
0.2—
.I I I
500 I000 5000
BR KILOGAUSS
Figure 16 Log-log plot of Ao/p vs. BR for AuSn I.

 

-58-

of m are determined by log-log plots of Ap/p versus BR.
(-R = resistance ratio - see pg. 2) Ap/p is customarily
plotted as a function of BR so that, assuming Kohler's rule<52)
is valid, samples of different resistance ratios and there-
fore different purities can be compared at the same effective
field. That is, the important quantity with respect to the
high-field galvanomagnetic prOperties is the value of
wbr = uB a BR, and not the value of the field alone.

Fig. 16 is a log;log plot of Ap/p versus BR for AuSn I.
The field direction for the four curves are those marked of
Fig. 14. The numbers at the right of the letters are the
values of the exponent m at the highest field of the curves,
56 kG. The almost complete saturation of Ap/p when B’is
nearly parallel to J (curve d), indicates that the electrons
are largely in the high field region. This follows from

the theory in section 2; the resistivity tensor element 922

is always independent of field ig_the high field limit.

 

Fig. 17 is a graph of the magnetoresistance versus
magnetic field direction for AuSn IX, a sample for which
J was 20 away from the basal plane and 120 away from a (1010}
plane. This sample had a higher resistance ratio (116) than
AuSn I. This is reflected in the higher maximum value of
Ap/p (~4) and in the sharpness of the peaks and dips of the
rotation plot. The field dependence of the magnetoresistance
at the three points marked on Fig. 17 is shown on a log-log
plot in Fig. 18.

-59....

 

4.0—
a
.. AuSn IX
§I§ a: 50 HE
G 9.
2.0...
c
1.0-— Q 0
<"I°2IoTo>
I I I I

 

 

-90 -60 ~30 0 30 60 90
MAGNETIC FIELD DIRECTION

Fngtre 17 Graph of Ap/p vs. magnetic field direction for

AuSn IX.

 

-6 O...

 

2!£)-’

(3.2!r-

 

(DJ

AuSn II b, l.I

 

III 1 11

 

 

Figure 18

'0 BR KILOGAUSS

Log-log plot of Ap/p vs. BR for AuSn IX.

IC)

-61-

Referring to Table II, the summary of galvanomagnetic
properties, it is seen that for a compensated metal (ne = nh),
for a general field direction, Ap/p « B2 (case 11 of the
table). Except for the fairly rare occurrence of cases IV
and V of Table II, Ap/p saturates for a compensated metal
only when there are cpen orbits in one direction and a e:F/2
(case III). Since the magnetoresistance of AuSn never tends
toward saturation except when B,lies in a symmetry plane of

the crystal, we conclude that it is compensated metal. The

 

fact that the maximum value of the exponent m in Fig. 18 at
point a is 1.6 rather than the 2.0 expected from theory pro-
bably means that a small fraction of the electrons are not

yet in the high field region at 56 kG. The fraction must be

a fairly small one or the nonvanishing of Ox in (30) will

y
cause the metal to behave like an uncompensated one. For
example, consider a compensated metal having two groups of
carriers with mobilities differing by an order of magnitude.
Suppose that the electrons and holes had mobility ”e and uh,
reSpectively. Let ”e = 50 uh. Suppose the magnitude of

B has a value such that ueB = 5 so that uhB = 0.1. At this
value of field therefore, the electrons will be marginally

in the high-field region but the holes will be far from it.
In this case, although ne = nh in Oxy in equations (30) and
(31), in effect nh is zero as far as the high-field galvano-
magnetic properties are concerned. In the example cited, one
would expect the galvanomagnetic behavior to be rather more

like an uncompensated metal than a compensated one.

-62-

These kind of arguments explain why, for a transition metal
like Pd which has s and low mobility d electrons, one must
have very pure samples (R ~'2000) and go to fields of order
100 kG to observe high-field galvanomagnetic effects.

Because of the difficulty of preparing very pure samples
of many elemental metals, the galvanomagnetic properties of
some of them have deviated slightly from the ideal m = 2.0
and m = 0.0 behavior expected from the theory. In practice,
the behavior of a compensated metal is recognized by the
conditions Ap/p > 1 and l < m 3,2 for a general field dir-
ection whereas for a compensated metal, Ap/D «'1 and 0 < m < l
for a general field direction.(52) Compensated metals other
than AuSn for which m is appreciably less than 2.0 are Mo,
Re, Pt, Fe, and Pd for which m ~a1.8(53) and Be for which
m ~v1.6.(54)

At point c in Figs. 17 and 18, where B,is in the basal
plane, Ap/p tends toward saturation with m = 0.6. Similar
behavior is observed in Fig. 16 at point 0 when B,is in the
basal plane. This indicates that there are gpen orbits

 

along [0001]; i.e., the saturation is caused by case III
behavior (Table II) with a a:fl/2.

At point b on Fig. 17, there is a relatively sharp dip.
The field at this point is nearly parallel to [0001] and the
exponent m is 1.1. At a similar orientation of B'in Fig. 14,
m = 1.0. We return to this point later.

_53-

A reference to Table II shows that the Hall coefficient
can give useful information about'Ve and Vh, the electron
and hole volumes of the Fermi surface. However, we have
inferred from the magnetoresistance behavior that Vé = Vh;
Table II shows that Vé - Vh cannot be measured directly for
a compensated metal from the Hall effect. The utility of
the Hall effect is therefore somewhat less in compensated
metals than in uncompensated ones. Nevertheless, Fig. 19
shows a plot of the magnitude of the Hall voltage versus
magnetic field direction for AuSn IX. 0° on the plot cor-
responds to B nearly parallel to [0001] and 190° corresponds
to B, in the basal plane (see stereogram in Fig. 17). The
sign of the Hall coefficient is negative. The magnitude of
the Hall voltage at the central peak when B_is parallel to
[0001] is used later in this section. Fig. 19 also shows a
rather broad maximum at i90° where B_passes through the basal
plane.

No attempt has been made to analyse the behavior of the
transverse-even coefficient, RTE‘ For an uncompensated metal,
the transverse-even voltages can be used with the longitudinal
voltages to give the direction of Open orbits directly.(5°)
Essentially, one measures the components of the electric
field along the directions of an Open orbit by measuring the
B2 part of this field on three orthogonal pairs of voltage

probes. For a compensated metal, however, it is difficult to

-51-

.XH case new compoonwe UHOfim oesosmas .m> omupfio> fifimm

 

 

 

ma onsmfim
column-o 0.0-h. o Focmmz
om+ o . on:
o o, AOAV
_ _ _

AM_HH
D.
wo-nvv "mm “H
A
o
AVNHH
e
,o
a
m
human
A

n¥¢

MH cm3<

 

 

-65-

separate the effects of the B2 voltage due to case ll be-
havior from the case III Open orbit B2 voltage (Table II).
The transverse voltages can be used to yield confirmatory
evidence for Open orbits if one has a large amount of data
on crystals of the same purity and different orientations.(57)
AuSn'V, having a resistance ratio of 25, was the lowest
purity sample on which a liquid helium run was done. The
magnitude of the magnetoresistance for this sample was only
about 0.7 at 50 kG. A rotation plot of Ap/p showed very little
anisotropy with the exponent m having the value of about one
everywhere. Undoubtedly, this behavior is caused by an in-
sufficiently high value of th' A very crude estimate of the
value of wbr can be made for a compensated.meta1 using the
(58)

formula from the two-band model with ne = nh,

9%- = ueBuh .

If ”e is assumed to be equal to uh, then if Ap/p 2.: 1, ch a: 1.
Therefore, with this crude model, mbr < l for AuSn V. Actually,
we know that th 2.1 for at least some of the (low mass) elec-
trons in AuSn'V at 50 kG, because dHyA oscillations have been
observed at 60 k0 with samples having a resistance ratio as
low as l3.(59)

Fig. 20 shows the current and field directions for the
two samples AuSn III and AuSn VIII. The magnetoresistance as
a function of magnetic field direction for these two samples

is shown in Fig. 21. The field directions labelled in Fig. 21

-66-

‘1‘

 

EVIII

H\X

:IVIII'” “'1 y (0110)

 

< 1120>
<1010>

Figure 20 Stereogram showing current and field directions
for AuSn III and AuSn VIII.

 

 

 

 

 

AuSn III
39°{l6 from [000:]
3° from {I I20}
I if I
u, {tom} {”20} {IOTOI
p.
E B=SOkG
:3
E AuSnllX
a: -’ 60° from [000:]
a J I 4" mm {IOTO}
(I
‘1
3F.
1 HI l
(0000 {IO 0} (0000

MAGNFTIC FIEL D DIRECTION

(the zero level of magnetoresistance is not shown)

Figure 21 Graph of Ap/p vs. magnetic field direction for
AuSn III and AuSn VIII.

-68-

are planes in which the field lies as it moves along the
great circles of Fig. 20.

The resistance ratio of AuSn III was 42. As in the case
of AuSn V, the anisotropy of the field dependence of AO/p .
was not sufficiently great to relate it to topological pro-
perties of the Fermi surface. The field dependence of the
magnetoresistance was m g 1.4 for B S. 30 kG and m -v-1.1 for
B a-56 kG. AuSn III was the highest purity sample grown
that had an orientation approaching the hexagonal axis. The
rotation plot, Fig. 21(a), roughly shows the six-fold sym-
metry of the crystal about the [0001] axis, but the aniso-
tropy of Ap/p is only about 10% of the total magnetoresistance.

Fig. 21(b) is a rotation plot of the magnetoresistance
for AuSn VIII, a sample having a resistance ratio of 73. pg
is 60° away from [0001] and 4° away from the (0110) plane.
There is a fairly sharp dip in the center of the curve which
corrBSponds to the field lying in the (OIIO) plane. The

1.4

field dependence changes from B at the peaks of Fig. 21(b)

to B1.0

at the dip in the center. This suggests that there
are Open orbits along the [OIIO] direction and hence along all
(1010) directions. The cause of the dip would be the angular
dependence of case III, Table II where a = 86° 3: r/2. A
possible reason why the magnetoresistance does not saturate

more strongly than m = l is that the Open orbits along <10I0>

are caused by a small ratio of D to A, where D is the area of

-59-

the Fermi surface traversed by open orbits and A is the total
area of the Fermi surface. The magnetoresistance for a = m/2
then saturates at a large value A/D only when uOB >> A/D,
where “o is the mobility of the open orbit electrons.(°°) If
this is the correct eXplanation, then the present measurements
are in an intermediate range where uOB ~IA/D and complete
saturation should be achieved by employing higher fields or
purer samples to increase the mobilities.

Another possible explanation is that the Open orbits are
caused by magnetic breakdown.(°1) Then the ratio D/A can be
regarded as a function of B, starting from zero at low fields
and increasing to a critical field Bb’ at which.magnetic break-

down is complete. Bb is given by(62)

MD'beF = (A6 )2 3

where As is the energy gap where the magnetic breakdown occurs,
andlmb = eBb/mcc is the cyclotron frequency in the field Bb of
the carriers of mass mo in the orbits which break down. Ac-
cording to this explanation the present measurements are in the
region where breakdown is only partially complete, i.e., E5 a:

100 kG.

TOpology of the Open Sheet of Fermi Surface of AuSn

We next summarize the results obtained from high-field
galvanomagnetic measurements and use these results to construct

a tOpological model for a multiply-connected sheet of the Fermi

-70-

surface of AuSn.

l. AuSn is a compensated metal: Ve = Vh'
2. There exist Open orbits along [0001].

3. There may be Open orbits along (1010) directions.
4. When §_is parallel to [0001], there is a tendency

for Ap/p to saturate with.m.=:l.0.

We discuss these points in turn. 1. That AuSn is compen-
sated is consistent with the first of the rules for com-
pensation given in section 2. Since s1 = 82 = 2 for AuSn,
compensation would be expected from (44) and the discussion
following it. Further, if we assume valences of one and four
for gold and tin, respectively, then compensation in AuSn is
consistent with (45) since nv = 10. 2. The nature of the
rotation curves, Figs. 14, 17, 19, suggests that the band of
cpen orbits along the hexagonal axis may be fairly wide; the
dips or peaks as B’passes through the basal plane begin when
§,is as much as ~20° away from the basal plane. We are thus
led to pose the following model of Fermi surface for AuSn:
the Open sheet consists of fairly large undulating cylinders
having their axes along [0001]. 3. and 4. As discussed
above, the relatively sharp dip in Fig. 21(b) suggests that
there are Open orbits along <10I0) directions. Suppose that
there were small cylinders whose axes were along <10I0)
directions. These small cylinders explain both point 3. and
also point 4. above. The <10I0) cylinders would make the
[0001] axis a singular field direction and the magnetoresistance

-71-

/ (1120)

, ---t- <10IO>

 

 

 

 

 

 

 

 

 

 

 

Figure 22 TOpological model for the Open sheet of Fermi

surface for AuSn (not to scale).

-72-

would approach saturation because of case V (Table 11) be-
havior. Fig. 22 shows a topological model of the Fermi sur-
face of AuSn which is consistent with the above observations.
Essentially, the model consists of large undulating cylinders
directed along [0001] which are connected at their"bu1ges"

by small cylinders along <1010> directions. It is to be em-
phasized that only the toEology of this figure is significant.
Since‘Ve = Vh, there exists another piece or pieces of Fermi
surface having character Opposite to that of the Open sheet.
Because the normal to most of the surface area of this sheet
lies in the basal plane, one would expect AuSn to have a high
resistivity along the 2 axis and this is indeed found to be
the case; pc/pa 2: 1.5(63) where pc and pa are resistivities
measured along the g’and g'axes, reSpectively.

An estimate of the diameter d of the <10I0> cylinders
can be made as follows. Let B,be along the singular field
axis, [0001]. It is seen from Fig. 22 that for this field
direction there are no open orbits. Appendix D shows that
the sign of the Hall effect at a singular field direction
is given by the character of the simply connected sheet or
sheets of the Fermi surface. Since the Hall coefficient is
negative when B,is parallel to [0001], the Open sheet of
AuSn in Fig. 22 is a hole sheet. Appendix D also shows how
to compute the <10I0) neck diameter d for a model similar to

Fig. 22. The result is that the volume measured by the Hall

-73-
coefficient is

V = A

BZ d ’

where ABZ is the hexagonal cross sectional area of the
Brillouin Zone normal to [0001]. Us1ng the Hall voltage

at the peak of Fig. 19, d is calculated to be O.35(2wb3).
Assuming that the cylinders have circular cross sections,

the area of the cylinders calculation from the above value

of d, Ad, is 0.12 A'2. This area could also be measured
directly by the dHyA effect if the field were along a (1010)
direction. The deA areas have been measured by Jan gt,gl,(°4)
but only for the [0001] and <1120> directions. If we assume
the <10I0> cylinders are sufficiently long, the area of the
neck A can be computed from A<1l20)’ the dHyA area measured when
p is along <1120>. Since [1010] and [1120] are 30° apart,

A = cos30 A<1120) . Using this formula, one of the dHVA

0
periods A<lI§O>, gives A = 0.13 A‘2

which is fairly close
to the area determined from the Hall effect. However, this
does not prove that there are necks like the ones shown in
Fig. 22; the period used to obtain the above area could have
come from a completely different piece of Fermi surface.

One of the interesting tranSport properties of AuSn is
its thermoelectric power, (TEP). Jan and Pearson(°5) have

measured the TEP of single crystals of AuSn as a function of

crystal orientation. They found that there is a large positive

-74-

phonon-drag peak when the sample axis is along (1120) dir-
ection but essentially no phonon-drag peak when the axis is
along the [0001] direction. From this observation and the
anisotrOpy in the resistivity, pc/pa'2:1.5, Jan and Pearson
speculated that the Fermi surface consisted of undulating
cylinders along the g'axis. This would give a large umklapp
contribution to electron-phonon scattering in the basal plane
but very little contribution in the 2 direction. The Fermi
surface of AuSn appears to be relatively complicated. We
have found that there are at least two pieces of Fermi sur-
face and three dHyA periods were observed for each of the
two field directions mentioned earlier. In View of this
fact, the ideas of Jan and Pearson can only be regarded as
highly Speculative; the electron-phonon interaction is very
incompletely understood at present, even for the metals with

(66)

simple, well-known Fermi surfaces.

0n the Electronic Structure of AuSn

As mentioned in the Introduction, it is difficult to
predict the number of electrons in conduction bands in inter-
metallic compounds. We have seen that AuSn is compensated
which is consistent with nV = 10. The single-OPW model of
the Fermi surface with nV = 10 was constructed using the
Harrison method.(°7) Because nV is so large, the model is

quite complicated; there are electrons in zones 3 through 8.

-75-

Further, no meaningful correlation could be made between
the areas of the model and the measured dHyA areas. Since
there is a screw axis and glide planes in the AuSn structure,
the energy bands may stick together on certain points of the
Brillouin zone faces.(°°) This complication may mean that

a double-zone scheme should be used rather than the single»
zone scheme. In order to rigorously determine where the
bands stick together on the B2 faces, it would be necessary
to determine the dimensionality of the group of the wave-
vector £82 at the various BZ points.‘°9) In addition, the
effect of Spin-orbit coupling should be considered to deter-
mine degeneracies.

Since the OPW model, at least in its simplest form and
ignoring the complications mentioned above, does not seem to
agree with experiment, we consider an alternative approach
to the electronic structure of AuSn, that of Pauling.(7°) By
considering the saturation magnetic moment of transition metals
and their alloys as a function of atomic number, Pauling arrives
at a value of 5.56 for the valence of c0pper, silver, and gold,
where valence here means shared electron pairs (covalent bond)
plus metallic orbitals (conduction electrons). The valences
of elements to the right of cOpper are 4.56 for zinc, 3.56
for gallium, 2.56 for germanium, and 1.56 for arsenic. The
same valences are assumed for elements below each of the above
listed ones. In Pauling's theory, for a substance to be a

metal, there must be 0.72 metallic orbitals per atom. The

-76—

metallic orbital permits the unsynchronized resonance of
electron pair bonds from one interatomic position to another
by the jump of an electron from one atom to an adjacent atom
and this leads to stabilization of the metal by resonance
energy, and to the electrical prOperties of metals.(7l) We
simply quote from Pauling's book his explanation of the
structure of AuSn in terms of the valence electrons of gold

and tin:(72)

"Each tin atom*~ is surrounded by six gold atoms at the
corners of a trigonal prism, with the distance Au-Sn =
2.847 A, and each gold atom is surrounded by six tin atoms
at the corners of a flattened octahedron, and also by two
gold atoms, at 2.756 A, in the Opposed directions through
the centers of the two large faces of the octahedron. The
tin atoms are arranged in the positions corresponding to
hexagonal closest packing, but the axial ratio c/a has the
value 1.278 instead of the normal value 1.633. We predict
for gold the valence 5.56, and for tin either the metallic
valence 2.56 or the covalence 4. The Au-Sn distance is
much too small to correspond to valence 2.56 for tin, but
it agrees well with the valence 4. With this valence the
Au-Sn bonds have bond numbers 2/3, [bond number equals the
ratio of valence electrons per atom to number of bonds],
and the interatomic distance predicted with use of the
si ale-bond radii 1.338 A for gold and 1.399 A for tin is
2.8 3 A, in almost exact agreement with the experimental
value 2,847 A. Accordingly the tin atom is quadrivalent,
without a metallic orbital, and its four valence bonds
resonate among the six positions connecting it with the
ligated gold atoms. These bonds use up 4 of the total of
5.56 valences of gold. If the axial ratio of the crystal
had the value 1.633, the remaining 1.56 valences of gold
would not be utilized; however, by compression along the c
axis, keeping the Au-Sn distance constant, the gold atoms
that succeed one another along the c axis can be brought
into a suitably small distance from one another to permit
Au-Au bonds to be formed. The bond number predicted for
these bonds is 0.78, and the Au-Au distance that is pre-
dicted is 2.741 A, in close approximation to the observed
distance 2.756 A. The system of metallic valences and radii
accordingly provides an explanation of the interatomic dis-
tances, including the compression along the hexagonal axis
to give the abnormally small axial ratio."

-77...

If the 1.56 valences per atom of gold which are in the Au-Au
bonds are regarded as weak bonds, in comparison to the Au-Sn
bonds, then one might expect about 3 conduction electrons per
unit cell for AuSn. This is in contradiction to the 10 con-
duction electrons per unit cell used in the single-OPW model
of the Fermi surface. If, in the future, the geometry of

all portions of the Fermi surface of AuSn becometwell-known,
it will be interesting to see if the ideas of Pauling bear

any relation to the actual number of electrons per unit cell
in Conduction bands. Another result; bearing on this question
is the existence of superconductivity in AuSn.(73) The tran-
sition temperature is 1.25°K. According to Matthias,(74) a
necessary condition for the occurrence of superconductivity

in compounds and alloys above 0.1°K appears to be that the
average number of valence electrons per atom cannot be smaller
than two or larger than eight. If, as in Pauling's theory of
AuSn, we regard the electrons in two tin atoms per unit cell
as simply forming localized bonds with some of the valence
electrons of gold, then AuSn would have an average of 1.5
valence electrons per atom. The existence of superconductivity
in AuSn is therefore in disagreement with Pauling's theory.
Again, an accurate determination of the Fermi surface geometry

would settle this question.

6. Conclusions

The results we have described represent the first
observations of high-field galvanomagnetic effects in inter-
metallic compounds. We have shown that these effects can be
observed in these metals with resistance ratios of the order
of 100 and fields of the order of 100 kG. The open orbits
observed in AuSn have been used to construct a tOpological
model for the Open sheet of Fermi surface in this metal.

We have implicitly stated that the interpretation of
high-field galvanomagnetic properties is considerably aided
if there are data on a large number of high purity, accurately
aligned samples available. Therefore, good metallurgical
equipment and techniques for accurately seeding and zone-
refining samples are needed unless, of course, the samples
can be obtained from other sources.

It can be argued that the necessity to use such high
fields is a disadvantage because it is likely that magnetic
breakdown will occur causing open orbits to appear which are
not normally present on the Fermi surface. Hewever, cpen
orbits due to magnetic breakdown can be identified because
samples of differing purity will not obey Kohler's rule if
magnetic breakdown is present.(7°) Again, this would neces-
sitate very accurate control of sample orientation and purity.

It is obvious that an investigation of a given inter-
metallic by both the high-field galvanomagnetic effects and
the deA effect would yield much more information than by

-78-

-79-

either of the techniques alone. As stated on p. l, the
interpretation of dHyA data is very difficult for compli-
cated metals, except if one has a relatively good model to
guide the interpretation.

Finally, we have seen that there have been two main
approaches to understanding the properties of intermetallic
compounds, valence bond theory and band theory. As we have
discussed for the case of AuSn, the valence bond theory is
based on the concepts of bond directivity, resonating bonds,
bond lengths, etc. In band theory, however, crystals tend
to assume a structure such that certain Jones zones are al-
most exactly filled. The reason is that if the Fermi sur-
face based on the free electron model approaches the zone
boundary, then the development of the energy gap at that
face lowers the energy of the occupied states relative to
the unoccupied states and this stabilizes the structure.

The criteria determining the efficacy of these two approaches
to intermetallics would be considerably clarified if the
Fermi surfaces of a number of them were known. As we have
shown for AuSn, high-field galvanomagnetic effects can be
used to study open orbits and to determine the state of com-
pensation in intermetallic compounds. The experimental de-
termination of the state of compensation, in itself, will be
significant in the sense that any model of the Fermi surface

must be consistent with this.

Appendix A
Bridgman Crystal Growing Furnace
Fig. Al shows the Bridgman Furnace used to grow

single crystals and the electronic control circuits used

to stabilize the temperatures to i 1°C.

-80-

-81-

 

 

1;
Ir
I

 

 

t

 

230
News

 

 

 

 

 

 

 

 

/
12,-)/’r

 

 

Recorder

Con‘I’roIkr

 

°\

T crmocoufks

 

iii-Ii

.:\

 

 

 

Lg; ~
N u.“
Dc‘I'QC‘I'or

 

 

 

 

 

 

I I

 

 

 

"E;

 

 

CH:
AMP

 

Figure A1 Schematic diagram of Bridgman Furnace and

electronic control circuits.

 

 

 

Appendix B

Crystallographic Angles for Hexagonal Crystals;
1.10 S c/a S 1.90

The following tables list the angles in degrees between
the various crystallographic planes of hexagonal crystals for

1.10 S c/a S 1.90.

Table BI. Crystallographic angles between planes (hlklll)

and (h2k212) for arbitrary c/a.

(210) (110) (120) (010)
(100) 19.1066 30.0000 40.8934 60.0000
(001) 90.0000 90.0000 90.0000 90.0000

-82-

PLEASE NOTE:

Some tables pages are not

original copy. Type tends

to fade. Filmed as received.
University Microfilms, Inc.

Table BII .

x. '
var“. ‘
. .- .
o'bb AIO
' O‘ C
‘.O‘ A. 'O
1 C
&O&.'_ ‘1.
a. 0"\ g‘
.L»: 1'.
AO.“Y ‘W.
‘ ' ‘
oOl‘a a‘YO
e 1' Q
a...) AIO
s «'1 ‘
«LO‘l Iv.
. . .
$.55: ‘6.
LOL/ 1".
AOh-V A~JO
LOO... A-‘O
'
‘.h.¢—. l—IO
' V
‘ l ‘
OO—Y A“.

o
I

I
a.
(
o

o
I
I
I
o

Osaka Lu.
.I—s.’ ‘U.

1 ’ a
‘Ow—IU AU.
AOVO AV.

T V
*On—r'h Iv.
‘ O 4,
AO‘U Al.
‘ 1 7
‘.U-l LI.
1.-.] L‘.
f,

r- r»
O O

( I

I

to r—-
O O

O
O
I

\
*.
O

A

y .
O
.I
(
F
O

' .
‘O‘YO AU.
10‘14. x...
I ' ‘

‘Ohlv A“.
5. -'~1 .54.
‘.‘.d 5“.
Q I ‘

.0 tv .~_.O
o " .

.O'ln 4,--.
O. f... Lb.
V . o

‘.1-r 3“.
1 ‘ .‘
bOw‘d A.-.

HHF—dh—‘h‘
O
I
-I
J
h.
(
O

O

Crystallographic angles between (001) and

various planes for 1.10 S c/a S 1.90

1 . -
1
v O — ' ‘ V - & V 8.. — V -4 O v
x - v ‘ s -—v ~
~— - V § I . - Q —- a _ .q. . v .v ~— . . u .v K. J . -—--— I v ~ 0 .
a -, . . . . n q ‘ . q . . ,—
c I .- «n I O . - l V .— O A -, A ‘— - O - — '7 fi .4 O -— A ‘- 0... ~.. -- O v
, 1 c -~. . A. — .- ._, ., _ .
v a .3 A l O 4 a- \— -_ ~ . ‘— L. V .1 ~- . - _ ed I»; . J O I v 1 $ .. a- . u—
‘ ‘ .- A q - f. o l 1‘ ‘
~—-- ‘- 5 ._. . v - x - _.- v . - \ y -4 . A _\y . w L ._ - — . \.
'V o i 1
l I . .1 O ' t - a. - O \4 L _ ~ ~ O - . s. A A O E. - 7 _ -.. .
.-- . -~ . g V
.
0 “OJ - .4 O v -— V _. __ O .- - O _- ~_. ~ _. O . - .- . O y
c a . l— ‘ -‘ — 1 q l '1 \
I ‘4 A \_. . g o u -— - T . J .1 V \g 4 . - $ Q ~ g . ' . —- a- - O ‘—
' A .3 s . . ’\ ' ~ ‘ "
._ V- A. - O — ..4 .._. ' -_ 7 O ._ I», y -« ' O V .3 - v V - O -. , .. -- - I Q
.
I
luv A v . —- d- v I - I O . .— - I —’ O h—"- \ I - . ~— v V .1 - ~- . ‘
.. I , '
v ‘— V L ~_/ . r — H ~ n— ’ . "’V r ‘ ~— ‘ . l . V - T -— . -- I A ‘ v .
q , ._- q -. . ' -—-< - u
s, - ‘4 . ,z . A v _ 1. ._ I O I - g \- .. O i _. V ._ O - _ v— .- 7 O L
I s p I ‘
L h— '? A . . ‘— -4 I“ ._ l . A t._.\_, .4 . 1 ‘1 . - ._ . ' EB I 3 ' .
A I r —\ c w s '
I .
./ «V L r . \J L “‘4 O ‘ u .- . h — . 5 _.. — ~— — . - v-— ,1 -- O ..
. .— < \
-.., A-o-‘- ._-ou--vx .. oD-V. ‘.o u..- _ ._.
-y - « . .a I ~ -
I
I z ¢ , . v -y... L- .. . - ‘ 'T V r - O .1 . a I . . — .4 ._. v — _ . V
.. t. s _ Q - . ‘ T .‘
v._ ‘ A a O — I A ' _ —- O S. .- ' - -- O - . l -— ' .4 O .4 - I , - -. . no-
‘I 7» -~ ~ '4. o
L- ‘ .__ a I O — D . — ._ v O x- . —— .- -— —4 O V .4 ' ' 1 ' O A -. v g, - v Q l
. t . A A a. ’ .
" \J ‘ —U . ‘ ~— ‘1 - ‘_ ~— . V v v ‘r -— v . a.“ V'-- . . a4 .— V- T — st . a
a I I ’D q
v I I I... H . - I , v n..~. O s.—.- .. —’ . § ‘7 - O —- I . - g .
- a . . —I 1 q
_, a ., ‘-V . I _ 1 , .._~ . ’T U — ' - -’ O b I! \_/ ¢ 1 l . —4 eye. ~— v . A
.' \ -\ *9 r- x .- v - -
.L V J b-U . ‘— ‘ V d b.“ . a vh—U ~‘ " . - . d -4 I v . VL— O h- — v O -
A ‘ ""V I -) 1 r ‘ g
n.4— bu . I 5 T ._. 4.— ...4 O I -. _ L _. , . ‘ V ‘ v ’ - . b. V I - -. .
. ‘7 . I ' | I . I
— v ‘1’ bu O h -- J. 7 ¢_\./ . I a 4 4 v I . _/ L A l ‘ . ' . -. . I
u ‘ .- \ ‘ c .. ‘ ‘r , . v ~’
I ‘1 Y L. A O \ V" t 1.- I O . U‘r I v a O — ’ bl ' -. O e4 1 l . - V . -
. ' .7 ') -y -
- ~_ ~— A O b ' I a —_ l O t—V'Jh. ‘4 O 1‘... l v ' _. O _. z __ ._. . O L
o q q -- u , — ~ _ ...a ‘ ' _‘ ,
A. -v I O— . . “- I A ‘ L. I O 7 v I v I . ,4 vu o I .v . ‘ UL— ; .a 1' . V
— q ,. ' a ‘ - -‘
~— v ‘7 h a O w I a . _ 4 O u U V I x — O I V .4 a ' v O v ‘ w -4 ,. n O s
. 'V 9 ' ‘ C I I— . ‘ .\ -—v —‘
Saw. --O—/)._.- nO-‘\,--- «'0‘-"'_ °..O---«g ~10:
, — . --e y I , ‘ I a —‘
. -- D — x o . _ v . ._ . o , . - .4 1. o - 1 - . - o a _ A v - o _
I n "~ _,
1' l V L— . O - ~— J - .--._4 O ‘ I v x x O I ' 1 ~ .4 O . _/ I ‘9 ~ b . V
‘ s I r q
a~, ..-_.-,,- .-_..-.i..._ -nag, 0.4-. ,.‘ r.-_ , . -
’ —‘ I -s
wuv e—E-QS l. .--.0 _.- - ~- .1 ‘I~_ t/Q-.'~.. - -.I
. -—.
_. w ; -—-4._. Q -- -2 J .— ...-a O V— - c. V O - ‘ .4» I . O - ' l __ --. .4 . -.
. .
luv b—_-o'~_.1c L_D.--._..-.. - .11... r,.. TIT _--._.
i ' 7 i ‘ ’ , q
4 . I -—‘ _ . ‘— . b. v __ - . - , I ‘ . I V I - ' I . r .' v L " ' . '
. . . . .
o ads—n h-h O ' A v‘— A-. ’ O A \-- A - O 4 -4 ° I - O Q . ~ -- . a
-—u - ‘ .
\4 v i b“... O —« J -. _. g._ . _. _. l v ' V O ‘ _ .. - a O 4 ~— . .. . O \—
' I a v .- a a
_ . "‘ —— s— O - I A - l O V c v :v O __ . - O N. .4 J D O
~ , . . r . .1 ' .
I V v .- s— O o . ' a ._ O - v _ a \— O - ‘ s. .— _. O ‘ .— — - . O v
a " A e
E... ‘1 ~. 1 O —— I _ L— O _ -. \. -\, O l J ‘— ~ -. O - a v . O
- . - .- . . g I _ . .
V v V .__ .‘ O A x. a v v . v V a ~.., . - . .4 - ~— . A ‘u 4 , x.— v O 4

o y.‘

f.

(

’0

b-A

Table BII. (continued)

V v 1 '

4...... $4.44 I -_-.—--._

‘ i

‘O-cvl—u ‘4Ow-_.-v .._,_‘O. .

- . ,.

‘Ovv ‘4014‘,’ 5-».-. ,,

1 , ‘ ..

‘O't ‘4O—JII—J ‘4-~.4O ‘4..—
.

lO‘I—d $4.444- HYOAUV

*1

l O v ‘1 h— V . V I _ V I._ ' . _ . .4
. 'I

O O ‘ d h— V O - .4! 4 V ~_ ’7 . ~_— V

| - . ' u
O O IN! w h— V O h l u ‘1 — O ‘ Q
d O ‘- ‘ h V O ‘4 r ‘— H -— I O - — I
L O ‘1 h- n— v O - .L --— \J — -. O V .. .4
1 I

IL O 4 ~— I— - O .4 g. .4 y ~_ __ . . .4
3 O v ' ._ V O I I .4 _. .. -, . - _ I

* O \— v 5— \,' O 4 I V a “‘ v O 4 v
‘I l ‘
O O ‘1 I — A O \4 4 V . u“ ._ . I .. __
O O u ‘4 _ L O _. v v— - ' O ‘ ' --
' O
O O ‘4 a ‘— -L O ~. 5. V ‘ — .4 O - d .—
O O I V — a O 4 I 4 h x O L - ,
I I ‘ ' .
C O l' O h— L O .4 ‘ v ‘7 —... .4 O - I ..
A O I -— .— . o 4 ._ 4 l. -- 4 o ‘ J 4
, . l »
A O I ‘4 6... 5 O I - . Q ~
' ' I'\ ‘ z.
i O I 1 L— ; O ‘4 - .4 v ._ I. . - v
. I ~, . ‘ -~
O O I .4 §__ O V. U 4 ' ~_. ‘— . _1 v __-
. . l
A O I h-l h...— O ‘ § - d s... -1 . 4 -4 4
-I --
O O I I _— O h..- 4 V s— I . ‘4 _. .._.,
V l
. a
O O I .a 5....— O v L - _, I.— . A . ~4
‘ O l 4 .__—-— O I 4 a 'T - 1' O u _ -
w
I
o O \— V ...~-- O .- -— V —— I O I
O O v & HF- O V .4 — V ~-—- I O I
I
O O \— — —.~—- O I I L. V A . I ¢ __
O O ‘—v V n—‘— O I ‘1‘ - ~ I O ‘ 4/

,4
o
I
I
I
o

V—mt —‘ o —
v I
‘ Ody 1.... . O ‘ .— ‘ _ _.O A ‘v
A O-4v ..__—— O— ' . - i O l __
€
L Owl _.4 O ‘1 - ‘W .. -,. .4 ~ ‘
s
O Ow- g__\4 O ‘4 -— 4 -—-— O i
A OUJ ‘_\.. .vgv. ._¥ . _ ‘
‘ ‘
O O r a Q—VOV4V A— by. I

(3

J

Table BIII.

r— r- r- r— r. r4 r *-

h—Jr—v 5‘

f...

o .. a *.J

V

.u. .o *4

y I

F u

—

V' ,
v a
i

g o
x. -
k. ,

--. 7,.
‘ v
I v

4 d
' v
1 ‘-
f

‘ v
I b
l 0

-. ‘
a A
l b
,

l a
f

I t
4 &

It
I...
la—
‘2
4&-
v
I...
l
c—
l—

I .—
m, \
I b
/
I s—a
~—.
I x;
I 'I
5 .4
»
I w
---
I a
-
I U

f I

Crystallographic angles

various planes for 1.10

1 . . - ‘
‘ .— ‘ —. — . a h
- U — r O ‘ - O ' ._- - . —
, . .- , Q ' .. -. ». . w 0 _ _. ._ —
—- v - . . ‘ - v - - J 5 . I U - 4 t— -
.4 a -. 4 . - .s ' ». a 5 . v ' g ._ .
. _ . tr —
i ‘l ‘4 v . l - q - _. . , - ,
" ! ‘
' V ‘ \r . ‘ -’ ‘ O J- ‘ -‘ . — -
r- 1 — n . 1*
‘. ‘
~ ‘ - V . -- - ‘ ‘ ; . . .— ‘ -.- ‘ . --
_r ‘ u ‘ ‘ f
w l v ‘1 . ‘ ‘ G -— l ‘| . ~— I d ._. v J
\- I ~ V . v a u v I O ‘ _ .. _ -.« V
.
t ~— v L . é -2 .a ~ 4 . . v ‘_ b V - v
z ,.
—- v ~ . o - -- - . o . ‘ - ._ _ ~
'I
' ,
I I V ‘ . — — ‘1 y I . . v 51... _ - ‘
I 4 4
.1 w v . . ' .- g , I . , y. . _ - -
I ' . I
‘4 I v ‘ . - I ~— I v . v , . , - ‘
.
V d w- o ‘ v v -— 4 — . —_ 5 ~— _, .... ‘
.
\.I 'T » _. . - v 4 7 I ‘— . .v 1., .... , \— ._
1. us— . ' .1 - 0 ~ . '7 \- ; ‘ v s
_. , _. .__ . . _, V - . _ . .4 ' v ‘4 -—.
— — v __ . A .... ~— - . —. _. -. .4 ~—
d — ~4 . u v 5 - . u - u— _ Ar
.
a g , ~ ~_ . ' . a l - O b w l v - -
_ .1 ~— v Q ' ’ d. I .. O J l y J ~ -
h. ‘1 .. _. . .. .. V J . ~ . \I I s v --' ‘-
" .
v ‘ v v . I a .. ~- A v . ‘ I 5 I .- ..
‘ l
.._... ‘ *1 ~— . - w 5..., -.._ 1 - . _ 1 v.. "I' M .—
v v v - . ‘ .- ,, . - . ‘1 f V v ‘4 '
u
r V ~— . 5.. _‘ ._ .4 . ~.. . v p -
‘ n
a ,1 ’u— . -' 1 h— I U . v w‘h ‘ —'
. . .
- w ‘4 ’ . - _ . 'v I v . ‘- -_, _/ " .4
' I l
f ‘ v . ‘ . c _ v a , v . . ‘v ' \J v
., - o
“ ' -’ Y . ~» - — 0 '- r g x.
. . .
. .
m . . .-< o _, . . ._ q , _. v _ .. -
.
‘ I (~
*~ v~o‘ »e- --o-.-7 -.-
_ . A ', , x v .
v “‘1‘ v .... . V . t I . . ,, .v . . _. _
-—. 1 .
‘ - . I .. '-
._ .4 ~ ~ . ' v‘ 4. _. : I O Q.— I -r 5 v ~
v .. u w o - ? .4 , I . o ‘1 - _. x w ..
“p ' ‘- 0 ‘ - ‘ v . I o r ' a. ‘ V _.
. b ‘ V' . ‘ —‘\-l v I ; . w h _ \- ‘_ ‘
r‘> ; .
r
‘- v
‘1 L . ‘— — ‘ \_. I ' . V V ’ _ \J v
—. .
I 1 O . ‘, ' r‘ ,A
\J V .
-... L J I , . V ‘ v v ‘. .

between (001) and

S c/a S 1.90

p. '..

.4

Table BIII. (continued

HHF—‘O—‘O—‘P‘

F; *0

F.

r4 b—- o-—- o-

*.L

on 0“ o»- r

...l

4

r« 04 v. r4 o»

’ a

or m U u
cm

L:-

(

(

(Lt

(

p..-

h.

Okf \JC'L‘I

H

IMHO u

L.

p-

'-1

\10 (.1

N

(

\

*4.

y»:

.‘_-.
@114.
"L- #4.
\’.~ ,~
UkJV“?

(\
(
F

(

~J

F.
1 ~— ‘ u
'4 H ‘
‘ U ’ r
~ I l v

’$——l
-v

II-»
-' -l_
.,

A
--V

v

I
4——————.«

- \4 '
— v _, -
’ b h- _

P
V».4‘a_
g4“?

131
-Q;
._.
-,_,
‘V.
&—'u

9

.a‘
w...
-‘.v.4

y- — ‘—
.
A ... --
‘- 7 '
.
— _: ..
‘— v 4

.--O

(i\

A

Appendix C

Extraction of Hall and Transverse Even Voltages from

Transverse'Voltages

Let Vt(I,B) be one of the transverse voltages as
measured, for example, across probes l, 5 in Fig.9 3

I is the sample current and B the magnetic field. Now

Vt(I,B) = v (c1)

H + Ve + V

T

where: VH Hall voltages = transverse voltage changing

sign with I and/or B.

Ve = even voltage changing sign with I but not B.
VT = thermal emf in potential leads.

Ve consists of two parts:

V = 'V

e TE (02)

+ vmis ’

where VTE is the transverse-even voltage which is pro-
portional to B2 and Vmis is the (Ir) drop due to mis-
alignment of the transverse potential probes. Upon re-
versing the direction of the sample current and magnetic

field, we have

V£(I,B) — VH +‘Ve + VTl
Vt(-I,B) = --vH - ve + le

(c3)
Vt(I,-B) = --vH + ve + vT2

Vt("'I,-B): V - V + V

_88-

In writing these equations we have assumed that Vt(I,iB)
and Vt(-I,iB) are measured essentially at the same time

so that the thermal voltage has not changed upon changing
the direction of the current. However, since the direction
of the field is changed with some difficulty in a super-
conducting solenoid, VKiI,B) and V(iI,-B) may be measured
at times differing by several hours; therefore, a dif-
ferent value of the thermal voltage is used for the two

different directions of field. If we define

Avi '5 Vt(I,:i:B) - Vt(-I,:!:B) (05)
then
uvH = AV+ - AV_ (06)
We = AV+ + AV_ (c7)
VTE can be extracted from Vé by determining Ve = Vmis

when B = O. The above equations show that the effect of
thermal emfs goes out if we measure the quantities AVi.
Since the thermal emfs are as much as 0.5uV and the Hall
voltages are only of the order of microvolts, it is very
necessary to measure the transverse voltages as described

above.

Appendix D

The Hall Effect at a Singular Field Direction

Consider a compensated cubic metal having the Fermi

surface shown in Fig. D1. The hole sheet of volume V“ is

h
multiply-connected and the electron sheet of volume v; =‘Vfl

is closed. Let the field be along the z axis. There are no
Open orbits for this direction of field. As in (28) the Hall

effect measures the quantity
v = Ve _ Vh = [[Aeficz) - Ah(kz)]dkz .

Now for ~d/2 s k S d/2, there are only closed electron orbits

Z

 

on the hole sheet of the Fermi surface. Therefore, for this

direction of field

.. ' _ f '-
Vh "' Vh ] A\1nocc(1‘z)dh{z
and

v6 = Vé + [Aocc(1{z)dkz ,

where Aunocc(kz) and Aocc(kz) are the unoccupied and occupied

areas of the hole zone reSpectively, intercepted by the plane

kz = constant. Thus

‘V = 'Ve - Vh = Vé ' Vh +erAocc(kz) + Aunocc(kz)]dkz '
But Vé - VA = O, and

Aocc(kz) + Aunocc(kz) = ABZ(kz) = A = b2‘

-89-

-90-

Hence
V = ijBZ(kz)dkz
2

and for the simple Fermi surface of Fig. Dl, V = Ad = b d.
It is therefore seen that at a singular field direction,
the sign of the Hall coefficient is determined by the char-

acter of the closed sheet or sheets of the Fermi surface.

-91-

2
v
“‘————“ electron
“‘-—————”’ volume‘Vg
V

 

’ A
z
I 9
holes volume V'

J/ h

 

 

 

 

 

 

 

 

 

 

1
___..__/
‘A‘II’V h_' k2 = 0
d
d
b
b

Figure Dl Simple compensated Fermi surface.

ll.

12.

13.
1A.

References

A. B. Pippard, Trans. Roy. Soc. (London) Aggg,

325 (1957).

I. M. Lifshitz, M. Ya Azbel', and M. I. Kaganov,

Zh. Eksperim. i Teor. Fiz. 21, 63 (1956).[English
transl: Soviet Phys. - JETP i, 41 (1957)].

P. Schoenberg, Nature 18 , 171 (1959).

A. V. Gold, Phil. Trans. Roy. Soc. (London) Agil,

85 (1958).

w. A. Harrison, Phys. Rev. IIQ, 1190 (1960).

W. L. Gordon, private communication.

A. Beck, J. - P. Jan, W. B. Pearson, and I. M. Temple-
ton, Phil. Mag. §, 351 (1963).

J. - P. Jan, W. B. Pearson, and M. Springford, Can.

J. Phys. fig, 2357 (1964).

M. Hansen, ed., Constitution g§_Binary_Alloy§_(McGraw-

 

 

Hill Book Company, New Ybrk, 1958), second edition.

W. Hume-Bothery and G. V. Raynor, The Structure of

 

Metals and Alloys (The Institute of Metals, London,
1962), fourth edition.

P. W. Anderson, Concepts in Solids (W. A. Benjamin,

 

Inc., New York, 1963), p. 6.
L. Pauling, The Nature of the Chemical Bond (Cornell

 

 

University Press, Ithaca, New Ybrk, 1960), third
edition, p. 395.
L. Pauling, op, cit., Ch. 11.

We follow the treatment of P. W. Anderson, op, cit.,

pp. 66-71.
-02-

15.
l6.

19.
20.
21.

22.

23.

24.
25.
26.
27.
28.
29.
30.
31.

_93_

P. W. Anderson, 22, £224, p. 68.

J.-P. Jan, W. B. Pearson, M. Springford, Proceedings

of the Ninth International Conference on Low Temper-
ature Physics (to be published).

J.-P. Jan, private communication.

E. Justi and H. Scheffers, Metallwirtschaft 11,

1359 (1938)-

I. M. Lifshitz, M. Ya Azbel', and M. I. Kaganov, 92,935,
R. G. Chambers, Proc. Phys. Soc. London 65A, 458 (1952).
J. M. Ziman, Principles of the Theory 2£_Solids,(Cam-

 

 

bridge University Press, Cambridge 1964) pp. 258-266.
R. G. Chambers, The Fermi Surface (John Wiley, Inc.,

 

1960) p. 100.

A. B. Pippard, in Low Temperature Physics, edited by

 

C. DeWitt, B. Dreyfus, and P. G. DeGennes (Gordon and
Breach, New Yerk, 1962).

A. Baratoff, unpublished.

E. Fawcett, Advan, in Physics 32, 139 (1964).

w. Shockley, Phys. Rev. 19, 191 (1950).

E. Fawcett, op.gi§., p. 159.

E. Fawcett, 92,213,, p. 160.

E. Fawcett, op,gi§,, p. 160.

E. Fawcett, 22,232,, p. 148.

This equation is a generalization for the case of more
than one kind of atom per unit cell, on an equation first
published by E. Fawcett and W. A. Reed, Phys. Rev. 121,
2463 (1963).

32.
33.

34.
35.
36.
37.

58.
39.
40.
41.

12.
43.
14.

45.
46.

47.
48.

49.
50.

-94-

J.-P. Jan, W. B. Pearson, M. Springford, Ref. 16.
W. D. Lawson and S. Nielsen, Preparation gf_Single

 

Crystals, (Butterworths Scientific Publications,

 

London, 1958), p. 62.

M. Hansen, _p_.g_i_§_,_ , p. 233.

M. Hansen, gp.gig; , p. 595.

J.-P. Jan and W. B. Pearson, Phil. Mag. 8, 911 (1963).
J.-P. Jan, W. B. Pearson, A. Kjekshus, S. B. Woods,

Can. J. Phys. 51, 2252 (1963).

J.-P. Jan, W. B. Pearson, and M. Springford, Ref. 8.

E. I. Salkovitz, Trans AIME 1 1, 64 (1951).

D. J. Sellmyer, Trans. AIME 232, 436 (1965).

Taylor Lyman, ed., Metals Handbook (The American Society

 

for Metals, Cleveland, Ohio, 1948), second edition.
Hume-Rothery and G. V. Raynor, 22,313,, p. 193.

M. Hansen, 22,912,, p. 233.

J.-P. Jan, W. B. Pearson, A. KJekshus, S. B. W00ds,
92,313,,

M. Hansen, 22,313., p. 595.

J.-P. Jan, W. B. Pearson, A. Kjekshus, S. B. Woods,
gp.gi§.

J.-P. Jan and W. B. Pearson, Phil. Mag. 8, 279 (1963).
A. C. Thorsen and T. G. Berlincourt, Rev. Sci. Instr. 24,
435 (1963).

V. Rowe, private communication.

P.A. Evetts, Phil. Mag. 19, 339 (1964).

51.

52.
53.
54.
55.

57-

580
59-

60.
61.
62.
63.

64.
65.
66.
67.
68.

-95..

C. S. Barrett, Structure 9;; Metals (McGraw-Hill Book

 

Co., New Yerk, 1952), second edition.

E. Fawcett, QEm£l§°: p. 167.

W. A. Reed and E. Fawcett, Phys. Rev. 226, A422 (1964).
E. Fawcett, 66,623,, p. 177.

N. E. A1ekseevskii,‘V. S. Egarov, Soviet Physics —
JETP 16, 268 (1964).

J. E. Kunzler and J. R. Klauder, Phys. Rev. Letters 6,
179 (1961).

R. W. Stark, unpublished thesis, Case Institute of
Technology, Cleveland Ohio, 1962.

J. M. Ziman, 92-9113." p. 216.

A. Beck, J.-P. Jan, W. B. Pearson, and I. M. Templeton,
Phil. Mag. 6, 551 (1963).

W. A. Reed and E. Fawcett, 2232333

W. A. Reed and E. Fawcett, 66, SEE:

E. J. Blount, Phys. Rev. 226, 1636 (1962).

J.-P. Jan, W. B. Pearson, A. KJekshus, S. B. Woods,
66,623,

J.-P. Jan, W. B. Pearson, and M. Springford, Ref. 8.
J.-P. Jan and W. B. Pearson, 92-2333

P. A. Schroeder, private communication.

W. A. Harrison, 66,623,

'V. Heine, Group_TheoryI;6 Quantum Mechanics (Pergamon

 

 

Press, London, 1960), p. 287.

690

70.

71.
72.
73.

74.

75-

-96-

C. Kittel, Quantum Theory_6§_Solids (John Wiley and

 

Sons, Inc., New York, 1963) p. 215.
L. Pauling, The Nature 6£_the Chemical Bond (Cornell

 

 

University Press, Ithaca, New Yerk, 1960), third
edition, pp. 422,423.

L. Pauling, 22.-2.3.1.9” p. 399.

L. Pauling, 919-2213.” pp. 422,423.

D. C. Hamilton, Ch. J. Raub, B. T. Matthias, E.
Corerzwit and G. W. Hull, Jr., J. Phys. Chem. Solids
29. 665 (1965).

B. T. Matthias, in Progress £2.L°W Temperature Physics,

 

Volume 2 (North-Holland Publishing Co., Amsterdam, 1957)
pp. 138-141.
E. Fawcett, 9p.cit., pp. 167, 189,190.