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ABSTRACT

HIGH-FIELD MAGNETORESISTANCE

OF AuGa, AuPb2 AND AuZPb

By.

Robert Bass

High-field magnetoresistance (HFMR) measurements have
been made on three intermetallic compounds, AuGa, Ausz, and
Ausz, which belong to structural types not previously in-
vestigated by HFMR techniques. The connectivity of the Fermi

surface (FS) of AuPb and AuGa has been determined while the

2
FS topology of Ausz is still unresolved.

All three compounds studied are compensated compounds
in accordance with their even number of electrons per unit
cell as calculated by assuming that Au contributes l electron/
atmm, Ga 3 electrons/atom, and Pb 4 electrons/atom to the con-
<hxmion hand. All three compounds have a FS which supports
Open orbits .

The topology of the AuPb2 FS can be explained by open
Orbit producing cylinders running in the <110> and <101>
directions in k-space. This model of the open FS is consis-
tent with the experimentally determined open orbit direc-
tions, <100>, <110>, <101>, <112> and <211>. Higher order

oPellorbits are also observed. The [001] could not be

 

ill

a.
‘.I
'U

 

 

Robert Bass

investigated for open orbits due to experimental limitations.

The topology of the open AuGa FS consists of a cylinder
running in the [001]. Experimentally open orbits are observed
in the [001] but not in the [100]. The [010] direction could
not be investigated for open orbits.

The topology of the Ausz FS deviates from that expected
of a cubic system since the open orbit directions do not
seem to lie along symmetry directions. More experimental re-
sults are needed before the FS topology can be derived.

The PS topology of AuGa cannot be explained by the
nearly-free-electron (NFE) model constructed by the Harrison
method. The NFE model for AuPb2 constructed for a few magnetic
field directions gives results which are in agreement with
the experimentally determined open orbit directions.

The relatively high degree of crystalline perfection at-
tained in the samples grown, as indicated by residual resistance
ratios ranging from = 300 to z 600, should spur other experi-
menters to investigate the FS of Ausz and AuPb2 by means of

the de Haas van Alphen effect.

HIGH-FIELD MAGNETORESISTANCE

OF AuGa, AuPb2 AND Ausz

By'

Robert Bass

A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Physics

1972

.4

ACKNOWLEDGMENTS

I would like to thank Professor P. A. Schroeder for his
advice, encouragement, and good nature during the course of
this research. His constructive criticism of the first
draft of this thesis is sincerely appreciated. I wish to
thank Mr. Nick Rutter for skillfully machining the sample
holder and Mr. Boyd Shumaker for his assistance in growing
the AuPb2 samples. I am indebted to Dr. Gordon J. Edwards
for the fine job he did in constructing the auxiliary equip-
ment and housing needed to operate the superconducting mag-
net. I also appreciate the guidance and knowledge he unself-
ishly gave during his stay in the laboratory. Last, but cer-
tainly not least, I would like to acknowledge the support
and friendship of all the fine people who made my stay here
a rewarding experience, the men of'the machine and electric
shops, the secretaries, the professors in the solid state
group and my fellow graduate students, especially Mr. Joseph
Pernicone who was able to evoke a smile from me even when
this research was going slowly and there was little joy in my
heart.

The financial support of the National Science Foundation

is gratefully acknowledged.

ii

II.

III.

IV.

VI.

TABLE OF CONTENTS

INTRODUCTION
CRYSTAL GROWTH

THEORY OF HFMR

Orbits

Formal Transport Theory

Compensation of Intermetallic Compounds
The High Field Conductivity Tensor
Open Orbits

Physical Explanation of HFMR

SAMPLE PREPARATION
Crystal Growth
AuGa
NiGe and Others
Ausz and Ausz
RRR Measurements
X-Ray Procedure

EXPERIMENTAL PROCEDURE

Cryostat and Sample Holder
Voltage Measurements

Analyses of Tipping Arrangement

RESULTS
AuGa
AuPb
Aung

CONCLUSIONS

LIST OF REFERENCES

APPENDIX A

The Relaxation Approximation

APPENDIX B

The Superconducting Magnet

iii

Page

146

148

1L

Table

II.

III.
IV.
V.
VI.
VII.

VIII.

IX.

XI.

XII.

LIST OF TABLES

Metallic Intermetallic Compounds
Previously Investigated

Nearly Free Electron Model Applied to
Intermetallic Compounds

Band Structure Calculations

Some CsCl Type Intermetallic Compounds
Some Can Type Intermetallic Compounds
Some CuAlz Type Intermetallic Compounds
Some MnP Type Intermetallic Compounds

Summary of High-Field Galvanomagnetic
Properties of Metals

RRR and Orientation of Samples
AuGa Lattice Constants
FS Parameters for AuGa

Open Orbit Directions in AuPb2

iv

Page

11

11

38
55
77
77

105

Figure
l.
2.

11.

12.

13.

14.
15.
16.

17.

LIST OF FIGURES

Phase diagram of Mg-Ag (From Hansen, Ref. 31)

Phase diagram showing a compound having a
maximum melting temperature at a non-
stoichiometric concentration

Phase diagram with excellent characteristics

Various types of closed and open orbits
(From Fawcett, Ref. 41)

Model of open FS
Phase diagram of Au-Ga (From Hansen, Ref. 31)
Phase diagram of Au-Pb (From Hansen, Ref. 31)

V(B) vs. B for (a) AuPb and (b) AuZPb samples

2
Sample Holder

V(B) vs. B for AuZPbI
(a) Rotation and tipping geometry and
(b) stereographic projection of motion of B

Non-transverse geometry

Comparison of NiAs and AuGa structures. The
lower diagrams are projections taken along

the c axis in NiAs while the upper diagrams
are taken along the a axis. The height of

the atoms above the projection plane is given
by the numbers beside the atoms. Dark spheres
represent Ni(Au) atoms while light spheres
represent As(Ga) atoms.

V(B) vs. w for AuGaI
V(B) vs. B for AuGaI
Ap/po vs. w for AuGaI

Stereographic projection of AuGaI

Page

14

16

18

25
27
43
49
54
60

65

68
70

76
78
79
80
82

Figure

18.

19.
20.

21.
22.

23.
24.
25.
26.
27.
28.

29.

30.

31.

32.

33.
34.

35.
36.
37.

Stereographic projection of AuGaII. X's
designate position of sharp peaks.

Ap/po vs. w for AuGaII

V(B) vs. w for different values of ¢ for
AuGaII

Conventional cell of AuPb2

V(B) vs. w for various values of B for
AuPbZI

Ap/po vs. w for AuPbZI

Ap/po vs. w for AuPb II

2

Ap/po vs. w for AuPbZIV

Ap/po vs. w for AuPbZV

Ap/po vs. w for AuPbZVI

V(B) vs. w for various values of ¢ for
AuPbZI

Stereographic projection showing minima
for AuPbZI

Stereographic projection of HFMR results
for (a) AuszII and (b) AuPbZIV

Stereographic projection of HFMR results
for (a) AuPbZV and (b) AuPbZVI

Stereogram showing (a) open orbit directions
(b) current directions, J, of samples and
(c) planes corresponding to open orbit
directions

Real and reciprocal lattice cells of AuPb2

Brillouin zone of AuPb . Vectors correspond
to directions in k-spage

<110> cylinders generating <100> open orbits
Conventional cell of AuZPb

V(B) vs. w for various values of B for

AuZPbI

vi

Page

82
83

84

92

93
94
95
96
97
98

100

101

102

103

107
109

110
112

115

116

-c 0“"

:o‘“c
a

(-0
(IT

A ,

vll|

l‘
1..

.A

.‘u.

Figure

38.

39.
40.
41.
42.
43.
44.
45.
46.
47.

48.

49.

50.

51.

V(B) vs. w for various values of B for
AuPbII

Ap/po vs. w for AuszI

Ap/pO vs. w for AuZPbIIb

Ap/po vs. w for (a) AuszIII and (b) AuszV
Ap/po vs. w for AuZPbVI

Ap/po vs. w for AuZPbVII

V(B) vs. w for tip angles ¢5|16.S°|
V(B) vs. w for tip angles ¢$|33°|
V(B) vs. w for tip angles ¢$|66°|

V(B) vs. w for various values of ¢ for
(a) AuszIII and (b) AuszIb

Stereographic projections of HFMR results
for (a) AuszIb and (b) AuszIIb

Stereographic projections of HFMR results
for (a) Au2PbVI and (b) AuszVII

V(B) vs. w for Au PbIIb for (a) ¢ = 0°, 11°
and (b) ¢ = 22°, 7.5°

Open orbit directions for four different

cubic FS topologies (From Lifshitz and
Perschanski, Ref. 40)

vii

Page

116
117
118
119
120
120
122
123

124

125

126

127

131

133

INTRODUCTION

The study of the Fermi surface (FS) of metallic inter-
metallic compounds began in 1961 with a paper by Thoren and
Berlincourt on the de Haas-van Alphen effect (deA) in InBin)
Since that time many compounds of various crystal structure
have been investigated and Table I gives a listing of many of
these compounds. The deA effect and high field magnetore-
sistance (HFMR) are the most widely used experiments for study-
ing the F8. The deA effect measures extremal areas of FS
enclosed by the conduction electrons as they undergo periodic
closed trajectories on the FS. If the electrons do not make
closed orbits, that is, the electron does not return to its
initial position on the FS, then the FS supports open orbits
which can be investigated by HFMR. Thus the topology of the
FS is determined by HFMR.

The growth and preparation of samples to be used in deA
and HFMR experiments is of crucial importance. First, single
crystals must be grown and second, the crystals must be of suf—
ficient perfection, as determined by the residual resistance
ratio (RRR). The necessity of using single crystal speci-
mens is obvious since the deA and HFMR effects measure the
anisotrOpy of the FS and depend on the specific direction of
the magnetic field with respect to the crystallographic axes.
A polycrystalline sample would produce an averaging effect

1

Table I. Metallic Intermetallic Compounds Previously Investi-

 

 

 

gated
Crystal
Compound Structure Crystal System Reference
B'CuZn BZ (CsCl) Cubic 2,3
B'AuZn 2
B'Aan 4
B'PdIn 4
AuAl2 Cl (Can) Cubic 5,6
AuGaz 5,6
AuInz 5,6
Aqu2 C2 (FeS , Cubic 7
pyrites

SiPz 8
MgCuz C15 Cubic 9
InBi BlO (PbO) Tetragonal 1
Man2 C14 Hexagonal 10
PtSn B8 (NiAs) Hexagonal 11
AuSn 12,13
PbSb l4
PdTe 14

AuGa B31 (MnP) Orthorhombic 15

 

which could easily obscure the anisotrOpy. The RRR, which is
the ratio of the room temperature resistance divided by the
resistance at 4.2°K, is a measure of the scattering of the
conduction electrons by non-thermal processes at 4.20K. If
the electrons are scattered before they make one complete
orbit on the FS then little information is obtained about

the FS. A high value of the RRR ensures that the electrons
make many orbits if the FS is closed or pass through many
cells if the FS is Open before colliding with scattering
centers.

Samples with a RRR of only 25 can be used for deA mea-
surements(2) while a value of about 100 is generally needed
for HFMR experiments if a field of 100 kilogauss (k6) is
available. The need for more perfect specimens in HFMR ex-
periments is due to the requirement that all the conduction
electrons must be in the high field region, i.e., all the con-
duction electrons must complete many orbits before being scat-
tered while deA measurements can be made on small pieces of
FS which contain a small fraction of the total number of con-
duction electrons.

The Nearly Free Electron model (NFE) is the first rea-
sonable approximation used in interpreting the results of FS
measurements. The success of the Pseudopotential Theory of

W. A. Harrison‘lG)

, and others in explaining the experimental
results of FS studies of the simple metals supports the use

of the NFE model for such metals. The NFE model has also

been used to interpret the results for intermetallic compounds;
such attempts have given mixed results. Table II gives a sum-
mary of the success or failure of the NFE model applied to all
intermetallic compounds studied by both deA and HFMR.

From the theoretical standpoint, the most authoritative
results obtained on the FS come from band structure calcula-
tions. Such calculations have been done for most metallic,
semi-metallic, and semiconducting elements. There have also
been many calculations on semiconducting intermetallic com-
pounds and a few calculations on metallic intermetallic com-
pounds. Table III gives a partial list of the metallic com-
pounds which have been done to date. The theorists are now
able to attack crystal structures with as many as 8 atoms
per primitive cell by first-principles methods such as the APW
calculation done for the B-wolfram structure compound V3Ga(24).
Pseudopotential calculations, using derived values, have been
done on structures with as many as 12 atoms per unit cell
(M92n2)(24). Thus, the more complicated crystal structures
which contain many atoms per unit cell are now theoretically
tractable and the experimentalists should provide the data
necessary to check the theoretical results.

Intermetallic compounds composed of the well understood
metals, semimetals, and semiconductors, have been the first to
be investigated. The elements which form this group are A1,
Mg, Zn, Ga, Cd, In, Sn, Hg, Tl, Pb, Cu, Ag, Au, Ge, Si, Sb,

Bi, Mn, Fe, Co, Ni, Pd, and Pt. The most common structures

Table II. Nearly Free Electron Model Applied to Intermetallic

 

 

 

Compounds
Compound Crystal Structure Successfu1*
AuGaz Cl (Can) Yes
AuSn B8 (NiAs) Yes
B'CuZn B2 (CsCl) Yes
Aqu2 C2 (FeSz) NO

 

*As a first qualitative explanation

 

 

Table III. Band Structure Calculations

 

 

 

Compound Type of Calculation* Reference
B'CuZn K-R, APW, NLPP 17,18,19
B'Aan K-R 17
8 'PdIn LNPP, APW 19,20
B'NiAl APW 21
B'AuZn KKR 21
AuAlz APW 22
AuGa2 APW 22
AuInZ APW 22
AuSn APW 23
Man2 LPP 24
V3X(X=Si, Co, APW 25

Ga, Ge, As)

 

*Abbreviations

APW - Augmented Plane Wave

KKR - Korringa-Kohn-Rostoker
K-R - Kohn-Rostoker

LPP - Local Pseudopotential
NLPP- Non-Local Pseudopotential

formed by these metals will be discussed next.

The BZ(CsCl) structure is a very simple structure con-
taining 2 atoms per unit cell. The space group is cubic
Oi - Pm3m. Table IV is a list of some B2 compounds. Com-
pounds with this structure are attractive for theoretical study
due to the small number of atoms per unit cell. As noted in
Table III, B'CuZn, B'NiAl, B'PdIn, and B'Aan have been in-
vestigated by theoretical methods while only B'CuZn has been
completely investigated experimentally. The HFMR work on
CsCl structure compounds has not been done, with the excep-
tion of B'CuZn, since crystals with sufficiently high values
of RRR are very difficult to grow. The reason for this dif-
ficulty will be discussed in the next chapter.

The B3 (ZnS, zincblende) structure is also cubic with
space group T5 — F43m and with 2 atoms per unit cell. Inter—
metallic compounds like XSb, (X = In, Ga, A1), which have this
structure are semiconductors and have been studied quite
thoroughly both experimentally and theoretically. The PS of
semiconducting compounds consists of small closed regions and
HFMR would not give any new information about the FS.

Intermetallic compounds which form the Cl(CaF2) structure
have been rather extensively studied as can be seen in Tables
11 and III. The C1 structure is cubic with full cubic sym-
metry 0: - Fm3m.and 3 atoms per unit cell. Table V is a list

Of some compounds which have the CaFZ structure. The com-

POunds AuIn2 and MgZPb have been studied by deA but not HFMR

 

'
«a
I.

w.

:

Table IV.

Some CsCl Type Intermetallic Compounds*

 

 

LiAg
Lng
CuPd
CuZn
MgAg

Aan

Ang
AuMg
AuZn
Aqu
MgHg
Mng

NiAl
TlBi
PdIn
NiGa
AgCe

ErIn

Dth
DyAu
HoCu
GdZn
YZn

GdMg

 

*A. E. Dwight‘zs) reports that there are 169 intermetallic
compounds with the CsCl structure, making this structure
the second largest family of compounds.

Table V. Some CaF2 Type Intermetallic Compounds

 

 

AuGa2 MgZSi*
AuAlZ Mnge*
AuIn2 MgZSn*
PtAl2 Mgsz+
PtSn2 PtInZ
ScCuz Ir2P

¥

*semiconducting compound

+semimetallic compound

due to low values of RRR. PtA12 seems to be a good candidate
for study as its phase diagram indicates that the compound
should be growable.

The remaining structures all have at least 4 atoms per
unit cell. Of these, the 88(NiAs) structure with space
group 06h - P63/mmc frequently occurs and is generally formed
by the transition metals Cr, Mn, Fe, Co, and Ni alloyed
with the metalloids 8, Se, Te, As, Sb, Bi and sometimes also
Ge and Sn. One notable exception is AuSn. Pearson's(27)
article on NiAs compounds and related structures is an ex-
cellent reference for these interesting compounds. Four com-
pounds have given deA oscillations: PbSb, PdTe, PtSn, and
AuSn.(14) PtSn and AuSn have been rather thoroughly studied
by deA while there is little information on the FS of PbSb
and PdTe. Only AuSn has been studied by HFMR. Other com—
pounds which seem promising for further study are NiSb, BiPt,
PdSb, and PtSn.

Compounds forming superlattices have been the source of
much experimental and theoretical work. The L12(Cu3Au) struc—

ture with cubic space group O1 - Pm3m and L10(CuAu), tetra-

h

gonal with space group D1 - P4/mmm both have 4 atoms per

4h
unit cell. The F8 of these structures has not been ade-
quately experimentally investigated due to the lack of use-
able crystals, i.e., the RRR has not been higher than about
10.

The Laves phases constitute the largest group of binary

10

compounds. There are three prototype structures based on
magnesium: MgCuz (C15), which is cubic and has space group
0; - Fd3m, Man2(Cl4) which is hexagonal and belongs to space
group Dgh - P63/mmc and MgNi2 (C36) which is hexagonal and
has space group Dgh - P63/mmc. Only the MgCu2 structure con-
tains 6 atoms per unit cell, the Man2 and MgNiz structures
contain 12 atoms per unit cell. According to Nevitt<28),
there are 223 binary Laves phases, 210 have a transition
metal for at least one of the constituents, 152 have the
MgCu2 type structure, 67 the Man2 type structure and only

4 the MgNiZ structures. Only MgCuz and Man2 have been stud-
ied by deA and these compounds would be interesting to study
by HFMR, provided of course that suitable crystals could be
grown.

The C16 (CuAlz) structure has 6 atoms per unit cell and
its Space group is tetragonal D45 I4/mcm. There has been
little experimental work done on these compounds aside from
determining their superconducting properties.(29) Table VI
gives some compounds which have the CuAlz structure. Phase
diagrams of Pde2 and CuAlZ indicate that these compounds
ndght be easy to grow (see Chapter II).

The B31(MnP) structure has 8 atoms per unit cell and is
orthorhombic with space group P-nma. The structure is a
derivative of the B8(NiAs) structure. The deA work of Jan
et al‘ls) on AuGa prompted us to study its HFMR. A list of

compounds which have this structure is given in Table VII.

11

 

 

 

Table VI. Some CuA12 Type Intermetallic Compounds
BFe2 Ger2 AgTh2
Sin2 Rth2 CoZr2
GaZrz Pdpbz CUSDZ
MnSn2 AuPb2 CuTh2
Table VII. Some MnP Type Intermetallic Compounds

 

 

AuGa
PdSi
PtSi
NiGe
IrGe

PtGe

PdGe
PdSn
CrP
MnP
FeP

CoP

CrAs

MnAs

FeAS

CoAs

Rth

WP

 

12

Most of these compounds have suitable phase diagrams, especially

NiGe and PdSi.

I I . CRYSTAL GROWTH

The most important consideration in crystal growth is
the phase diagram. The phase diagram indicates the feasi-
bility of producing single crystals with high value of RRR.
However, phase diagrams given in the literature are not al-
ways correct and errors in the diagrams can cause much dif-
ficulty. It is not possible to tell if the cause of crys-
tal growing failures is due to poor technique or incorrect
phase diagram information.

Figure l is the phase diagram of a typical B2(CsCl)
structure, MgAg. The most important feature of this diagram
is the large solubility of both Ag and Mg in MgAg as indi-
cated by the width of the 8' region. If MgAg is grown from
a melt which is not of exact stoichiometry, then the solid
formed is a solution of either Ag or Mg in MgAg.

The excess Mg or Ag will act as an ”impurity" scatter-
ing center in a crystal of MgAg. Sellmyer‘30) developed an
empirical formula to relate the RRR to the amount of impuri-
ties in a sample. The formula was derived from data on AuSn
but it should provide an order of magnitude estimate for
other systems.

4
RRR~1_%,1515104 (1)

13

14

.mmm .cmmcmm Sonny m¢1m2 mo Emummflp mmmsm .H whamwm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:m
2. 22323.: :3 5.. use: . 2
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n a .e cow
. . .
. .
~ . .
Ca. 3.” S o and - 0°”
ndo odw flow a
w m.\
I x
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3.: s .3 . x
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a nu
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H x .o
6 II 3.5 .. — . .
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III Mm\ v mm ndu
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6
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. q . q . . . q . q a» m L _ 060.
on 3 an ov an 3 cu m. o_. o w v ~

3:33:33 hzuu «mm .5333

15

where I is the amount of impurity in parts per million. If
a sample is made with 49.90 at .% Ag and 50.10 at .% Mg,
then there would be .1 at .% excess Mg and I = 103 which
would make RRR = 10. The importance of getting as close to
exact stoichiometry is evident.

The likelihood of growing B2 type intermetallic com-
pounds with sufficiently high values of RRR is small. First,
if the sample has an initial composition far from exact stoi-
chiometry, say ~5 at .% off, then the crystal grown will
have a composition differing from the stoichiometric composi-
tion and the RRR will probably be low due to the "impurity"
scattering mentioned above. Also, the rather large solid
solubility of the constituents in the stoichiometric com-
pound indicates that even compounds grown from stoichio-
metric concentrations may not be ordered enough to produce
samples with sufficiently high values of RRR. However, in
Spite of these considerations, B'CuZn has been grown with
RR =- 400.”)

The necessity of having correct phase diagram informa—
tion is seen in Figure ‘2, which is a phase diagram of a com-
pound which has a maximum melting point which does not cor-
respond with the stoichiometric compound composition. If a
sample with composition 1 is grown, it would not form the
solid AB upon solidification. If the sample had composition
2, the initially precipitated material would be AB but as the

temperature drOpped the crystal would become richer in B.

16

TEMPERATURE

 

A AB B
ATOMIC % 8

Figure 2. Phase diagram showing a compound having a maximum

melting temperature at a non-stoichiometric con—
centration.

17

Two methods can be used to grow the stoichiometric com-
pound AB. The melt's composition could contain a large
excess of B and a pulling technique such as the Czochralski
method could be used. Alternatively, a non-conservative
method could be used if more precise stoichiometric control
is desired.

The compound whose phase diagram is shown in Figure 3
has the best possibility of yielding single crystal samples
with good RRR. The important feature of this diagram is the
vertical line at composition AB. Such a compound is congru-
ently melting, i.e., the liquid and solid phase are in equil-
ibrium at the stoichiometric composition AB at temperature
Tm. Also, if the width of the line is small, say less than
10-3 at .%, then the RRR will not be limited by the solid
solubility of A or B in AB.

The initial metal which is used must be of high purity
in order to produce crystals with high RRR. Using Equation 1,
it is evident that "59" purity metal, i.e., 99999 at .%
pure, could produce RRR of about 103 if the RRR only depended
on impurity scattering. Thus, to begin with, the material
must be at least "49" purity and usually "59" or "69" mater-
ial is used if it is available. The cost of such high purity
metal is an important boundary condition, especially in the
case of Au, Pt, and Pd.

Usually the sample must be melted and grown in a cru-

cible. Both components of a binary alloy must not react

18

 

TEMPERATURE

 

 

 

 

 

A AB
ATOMIC % B

Figure 3. Phase diagram with excellent characteristics.

2O

(33) Of

not be used since their vapor pressure is too high.
course, a protective atmosphere of argon or some other non-
reactive gas can be used to cut down the loss of material.
However, even one atmosphere of pressure is not sufficient

in some cases and the use of gases could introduce additional
impurities.

The growth of good single crystals, even the pure metals,
is always an art. The problems involved are numerous.

First, the temperature of the sample must be measured and con-
trolled. Parameters such as growth rate and temperature
gradient are instrumental in the process of growing good
single crystals. There have been many books and papers writ-
ten on crystal growth techniques, yet it still remains an

art and each compound requires ingenuity.

The ultimate test of the single crystal is its RRR. As
the crystal is cooled to 4.20K, it may undergo a martensitic
phase transformation as happens in B'AuZn(34). Such a phase
change is thought to occur when the free energy of the crys-
tal can be lowered by assuming a different structure. It is
difficult to predict such a phase change and the direct ob—

servation of the phase change requires a low temperature

x-ray diffraction unit.

 

A: '1.
v‘ 5&6

III. THEORY OF HFMR

Justi and Sheffers(35) in 1936 observed complicated
anisotropy of the HFMR of several metals which was not ex—
plained until the mid-fifties. Fundamental theoretical work

was done by Kohler<36), Chambers(37), Lifshitz, Azbel, and

. 9, 0
Kaganov(38) (LAX) and Lifshitz and Peschanskle(3 4 ).

Fawcett(41) gives a comprehensive review of the theoretical
and experimental work done on pure metals. Since the theo—
retical derivations appear in the references just cited as
well as in other sources, only a brief review of the assump-
tions and derivations will be given.

In the absence of an external electric field the motion
of the conduction electrons in a metallic crystal is deter-
mined by the applied magnetic and the periodic electric field
due to the lattice. The periodic potential determines the
shape of the FS and produces effects such as Bragg reflection.
A geometric description of the motion of an electron in a.
magnetic field can be obtained by assuming the electron is
reflected by the Bragg planes just as in the case of x~rays
being "reflected" by the planes of atoms.

Suppose that an electronic state is characterized by a
wave-number k and a band index n. Also, assume that the

electron can be thought of as a wavepacket centered at a par-

ticular value of k and that the electrons obey Fermi-Dirac

21

22

statistics. These quasi-particles experience a force F

given by

613-:93 (2)

"1|
II

and have a velocity 6 given by

(3)

<l
II
3nd
<h
w
{'1
.5

where E(k) is the energy of the electron.

The quantity hk, based upon reduced wavenumber, is labeled
6, the crystal momentum and plays the role of the momentum of
the electron in a classical sense. This correspondence is
due to the classical mechanical result which states that the
velocity of a particle is equal to the derivative of the Ham-
iltonian with respect to the momentum.

In the case of an external magnetic field F is given by

F=—l§|§x§ (4)
C

where e is the electron charge in e.s.u. and c is the velocity
of light. The wavenumber k and hence the electronic state

undergo equations of motion given by

= _'h%' 6 x B (5)

94%.

For electrons on the FS 5 is normal to the surface and
thus the electron state moves along a tangent to the F8 in a
plane perpendicular to B. The motion of all the electrons on
the FS can thus be geometrically constructed by taking planes

perpendicular to B for various heights along its direction.

'23
The perimeter of each section defines an orbit in wavenumber
space (k space).
To obtain the motion of the electron in position space
where G = g%-, one merely substitutes this value in Equation

(5) to get

%=—|§;g{}x§ (6)

Therefore the motion of the electron in position space pro-
jected onto a plane perpendicular to B is an orbit which has
the same shape as the orbit in k-space but is rotated through
n/2 about the field direction and scaled by a factor of hc/eB.
The velocity of the electron parallel to E varied periodi-
cally as the k vector makes its orbit in the plane 1.B in
k-space. In general the velocity parallel to B does not aver-
age to zero and the net motion in position space is a helix
with axis in the direction parallel to B.

If the FS is closed, the electron wavenumber will exe-
cute a periodic motion in wavenumber space. The frequency of

this motion is called the cyclotron frequency and is given by

_ 2neB 8A

-1
- (———)
c “2c 3E

 

(7)

(L)

where %% is the derivative of the orbit area (in k-space)

with respect to energy with the component of k parallel to 8

being held constant. For a spherical FS,

where m0 is the rest mass of an electron. In the general case

24

an effective mass, called the cyclotron mass, is defined as

_h2 3A

“H"???—
which allows the cyclotron frequency to be written in the

same form as that for free electrons, namely

= ER.
wc ch

The high field region is defined by the condition

where T is the average time between collisions of the elec-
trons with scattering sites such as impurity atoms, lattice
imperfections, or phonons. This condition must be satisfied

for all the conduction electrons.
Orbits

Figure 4 shows some of the possible orbits for a hypo-
thetical sample cubic lattice. This representation of the FS
is called the periodically extended zone scheme. In this scheme
the FS is repeated throughout k-space and the Brillouin zone
boundaries are omitted. Figure 4a shows closed electron and
hole orbits for BII to [100], a high symmetry axis. Figure
4b shows a band of periodic open orbits - orbits in which the
electron moves from zone to zone in the general direction X
and never returns to its original position. Although this
periodic motion in k-space is mathematically equivalent to

repetitions of the same arc in a single zone, the motion of

 

 

' '53.- -5..I

25

 

1o)-PLANE
I ||UOO]-AXIS I IN (0
' R IT
CLOSED ELECTRON AND PER'OD'C OPEN 0 B
HOLE ORBITS
OPE" 0“" EXTENDED ORBIT
\

 

(C) (d)
I NEAR [Teal-Axe I NEAR (010) - PL ANE
APERIODIC OPEN ORBIT EXTENDED ORB”

Figure 4. Various types of closed and open orbits. (From
Fawcett, Ref. 41)

26

the electron in position space in the plane perpendicular to
B is truly unbounded, neglecting collisions. Figure 4c shows
an aperiodic open orbit in the general direction X. Such an
open orbit is produced when B lies near a high symmetry axis
in a non-symmetry plane. Aperiodic Open orbits appear as
two dimensional areas on a stereogram while periodic open
orbits are one—dimensional, being lines on a stereogram.
Figure 4d shows an extended orbit - an orbit which is closed
but not contained within a single Brillouin zone, no matter
where the center of the zone is located.

Topologically open orbits are equivalent to cylinders.
In Figure 5, if B is in the Ksz plane and thus perpendicular
to the cylinder axis, Open orbits are found in the Kx direc-
tion. If B is tipped out of the Ksz plane, extended orbits

form and with further tipping, simple closed orbits result.
Formal Transport Theory

The most convenient method of treating the collective
notion of electrons in metals is the kinetic method. It is
assumed that there exists a distribution function f(p,§,t)
which gives the probability of finding an electron in a state
in six dimensional phase space with momentum between 5 and
5 + dp, position between E and r + d}, and in the time in-
terval between t and t + dt. If a state is occupied f = 1
while f = 0 if the state is unoccupied.

Using the semiclassical results of the previous section,

the trajectory of a particular electron in phase space can be

27

 

DISTANCE BETWEEN
ZONE FAC ES

 

Figure 5. Model of Open FS.

28

traced. If one particular electron is followed, the value Of
f remains constant. However, assuming that the electron can
be scattered by mechanisms which are not the result of ex-
ternal fields, the electron will undergo discontinuous mo-

tion in phase space. The equation of motion is

if.
dt

8

Efii‘ézii
at + at at | (8)

vie.)
fi'Hl

_ 3f
"‘a'E+

8.! H.

Collision

which is the Boltzman transport equation in the notation of

W. A. Harrisonj42)

Now gg-is equal to the applied force while %%-is the

velocity. Equation (8) can be rewritten

3f 3f - 3f — _ 3f
a—E+—§-F+T_~°V-at|cOiiision

Under steady state conditions-%§
3f

metal §¥-= 0. In the presence of an applied electric and

0 and in an isothermal

magnetic field F is given by
F = - Iel V x B - Iel E (9)

Thus the Boltzman equation becomes

(-|e|E:-13—L§xx7) 3L?—

f
c p at (10)

I Collision

Assuming a magnetic field in the z direction and a small
electric field E = Exx + Eyy + Ezz, the time dependence of

the momenta is given by

29

px = -|e| BVY — |e| EX
py = -|e| Bvx - |e| By
p = -|e| E

2

It is convenient to make a change of variables from px, p

I

Y
pz to pz, 6, and “(9) where e is the energy of the electron

and u(6) specifies the time of rotation of the electron

along its trajectory in momentum space.

Me) = $.—
0
where
= c 92. = 2 2 1/2
t TE"? j VJ_ tang, V..L (VX + Vy)
and
T = Zucmo
0 e8

 

For closed orbits To is the period Of the orbit, and for open
orbits To can be defined as the time required for the elec-
trons to travel from one face of the Brillouin zone to the
opposite face along the open orbit direction.

The time dependencies of u(0) and e are given by

-1 _
- f— (1

O V..L

 

C [v E —v E ])

fi(0)
2B x y y x

e = -Iel V . E

The Boltzman equation becomes

30

 

3f 1 c 8f
|e| 35; E2 + T; (1 v 23 (VXEy VyEx)) g:
‘L (11)
_ IeI ; . g i£ = 3f

3: if I Collision

Assuming that the energy acquired by an electron between
collisions is very small compared to the Fermi energy as well as
the thermal energy kT, the Boltzman equation can be linearized
with respect to E and f can be written

f = f0 — |e|to$.fi (12)
where fo is the equilibrium Fermi distribution, and t0 is a relax-
ation time. E is independent of the magnitude of the electric
field, i.e.

T = E(Pzaeru)

The collision term can be written

H

af __£il
(3t)Collision - tO (3E)(at)Collision

9-3)

Substituting for f and (at Collision

in Equation 11 and neg-
lecting terms quadratic in B, one gets the following equation

t 3w. 3f —
O l + v o _ (ii)

i as — at Collision (13)

To Bu

 

LAK(37) solved Equation (13) for ¢i by a power series method.
The quantitative results depend on the form of the collision
operator. However, the field dependence of the magnetore-
sistance and the Hall coefficient for uncompensated metals can
be Obtained without explicit calculation. Table VIII, page 38,

shows the results which are independent of the form of the

31

collision operator.

Once Equation (13) is solved for w and f determined, the

current density J is given by
3 = iii]- ['5de = 3.137 (14)
h

The conductivity tensor 0 is thus given by

-2e2t

Oij

The relaxation approximation is the easiest method for ob-
taining quantitative results for the Boltzman equation. For
such calculations, a detailed knowledge Of the FS is needed.
Since this information is not available on the compounds in-

vestigated in this research and in addition, the relaxation ap-

proximation is of limited usefulness in multi-banded compounds,

it was not attempted. However, a brief summary of the formula

obtained from the relaxation approximation will be found in

the appendix.

32
Compensation of Intermetallic Compounds

Before the form of the conductivity tensor is given it
is necessary to define the state of compensation of a metal.
A metal is said to be compensated if the total number of
electrons per unit cell of the Bravais lattice summed over
the electron sheets equals the total number of holes summed
over the hole sheets. Fawcett(4l) defines the electron/hole
character of a sheet in the following manner: "the sheet has
an electron/hole character if, for a non-symmetry direction
giving rise to no open orbits, every cyclotron orbit is an
electron/hole orbit, with the exception that, when an orbit
is enclosed by one or more orbits on the same sheet of a1-
ternating electron and hole nature, the character of the sheet
is determined by the outermost orbit".

Consider an intermetallic compound with 81 atoms of metal
A and 82 atoms of metal B per unit cell. The capacity of
each Brillouin zone is two electrons per unit cell (real
space), assuming a degenerate non-magnetic metal or com-
pound. If metal A contributes V

1 electrons per atom to the

conduction band while metal B contributes V2 then

51V1 + 82V2 = 2F + (2J - nh) + ne (16)

where F is the number of full Brillouin zones, J is the num-

ber of hole zones, ne the density of electrons per unit cell

and nh the density of holes. Rearranging the above equation

-nA = (ne - nh) = SlV1 + 52V2 - 2(F + J) (17)

33

where nA represents the net number of carriers per unit cell.

If nA = 0, the compound is compensated while if nA # 0, it

is uncompensated. The quantity Vi need not be an integer.
However, for all metals which are nearly free electron like,
Vi is assumed equal to the valence of the metal. From Equa-
tion (17) it follows that an intermetallic compound cannot

be compensated if there are an odd number of conduction elec-
trons per unit cell where as the compound may or may not be
compensated if there are an even number of electrons per unit
cell. However, all non-magnetic metals as well as intermetal-
lic compounds whose galvanomagnetic properties have been

studied in high magnetic fields have been compensated if

they had an even number of electrons per unit cell.
The High Field Conductivity Tensor

In the high-field limit (mcr >> 1), the electrons make
many orbits before being scattered. In calculating the con-
ductivity, the average value of the velocity taken around
the orbit is used. Assuming B is in the Z direction, it can
be demonstrated that the conductivity in the xy plane goes to
zero as B + w for closed orbits. The form of the conduc-
tivity tensor in the case of closed orbits, assuming a12 # 0,

is given by

 

 

I 11 82 12 B 13 B‘
_ _ l L .1.
° ‘ a12 B a22 Bz a23 B (18)
i -a l -a l a i
13 B 23 B 33

34

where the Onsager relationship
oij(B) = 0.. (‘3)

has been used and only the leading term in (%)n has been re-
tained. The aij's are independent Of the magnitude of B.
The resistivity tensor, which is the experimentally

measured quantity, is obtained by inverting the conductivity

tensor

pij = co-factor oji/Iol
Only the leading terms of lowest order in g in each co-factor
of oij and I0] are retained in calculating the pij's.

Case 1: Closed orbits nA # 0: uncompensated

 

 

 

 

B0
b - b
I 11 ean 13 I
p = B0
ean b22 b23 (l9)
bl3 b23 b33

where Q is the volume of a unit cell. The pxy term gives

direct information concerning n The Hall coefficient RH(m)

A.
is defined as

Lim pxy(B)-pxy(—B) Q

RH(m) = B+m 23 = (he—uh)ec (20)

 

 

and is independent Of the direction and magnitude of B. The
other coefficients of the resistivity tensor cannot be eval-
uated without a detailed knowledge of the shape of the FS and

the collision Operator.

35

Case 2: Closed orbits nA = 0: compensated

In a compensated compound, since n = 0, the coefficient

A

1 . . l .
of the fi term in Oxy ls zero and Oxy : gj-whlle the other

terms remain the same. Thus

 

 

I bllB blZB bl3BI
_ 2 2
p - bIZB bZZB b23B (21)
I -bl3B ~b23B b33

For a compensated metal, pxy does not give any direct informa-

tion about the number of carriers.
Open Orbits

Suppose that one band of open orbits is present and that
they run in the general direction of the x axis in k-space.
The open orbit cannot be considered to be either electron
or hole-like and thus a compensated metal no longer behaves
as it does in the absence of open orbits. The Oxy component
of the conductivity tensor will now contain a linear term
in B-l. Since an uncompensated metal already has a term lin-
ear in 8-1, the presence of Open orbits only changes the co-
efficient Of Oxy and thus invalidates Equation (20) for RH(m).
Open orbits also produce constant field-independent terms in
o

o and Ozy' The conductivity tensor becomes

yy' YZ'

36

 

 

a11 a12 a13
I Bz 13 B I
a
_ ___1_2.
° ‘ B a22 a23 (22’
I _ al3 a a I
B 32 33
Case 3: One set of Open orbits.
Inverting Equation (22), p is given by
b 32 b B b B
I ll 12 13 I
p = -blzB b22 b23 (23)
"b13B b32 b33 I

 

 

'Case 4: Two bands of non-intersecting Open orbits in differ-
ent directions.

In this case all components of pij tend to a limit in-
dependent Of the magnitude of B since all elements of oij
tend to limits independent of B.

Case 5: Singular field directions at symmetry axes.

A singular field direction is defined as a crystal axis
of higher than two-fold symmetry which is at the center of a
region of aperiodic open orbits. The several bands of open
orbits required by symmetry intersect to form closed orbits
of opposite character to that of the surface. If B is along
such a singular field direction

recz 6/2
o = —- (V - A(p )dp ) (24)

..'. ,.
L33:

I “-
r

n
n a

C in!
'N

r v‘

Ni

U

P: c
i

4'.
ch
IL)

 

37

where 6 is the minimum thickness along B supporting Open or-
bits Of Opposite character to that of the sheet, and A(pz)

is the area of the section of the Brillouin zone for constant
pz. Since there are no open orbits when B is parallel to a
singular field direction, in both compensated and uncompen-

sated metals the resistivity tensor is given by Equation (19)

 

but the coefficient of the pxy term is changed from - ecn

Q
ec(nArAn)

A
to - where An equals the second term in Equation (24).
The magnetoresistance is defined as

p. . (B)"p. . (0)
A9(B) = 11 ll (25)
pm) piim)

 

In Equation (22), the x-axis was deliberately chosen as the
open orbit direction. Experimentally, one does not know

a priori where the Open orbits lie and thus they will in
general make an angle a with the x axis which is taken as
the sample axis. In such a case the magnetoresistance would

be given by

9335335 : C + ABZCOSZG (26)
0(0)

where a is the angle between the Open orbit direction in
k-space and the current direction J. The subscript trans
means that J_L B. C and A are constants independent of the
magnitude of B.

Table VIII summarizes the important relationships which

have been discussed in this section.

.wommmlx RH coauomuflo NHQNO ammo sue: a mamas cm mcwxme m.d_mcmam men cw me o w

.muflmcmo ucmRNso was: Rom UHONM owuuomam ..m.w +

 

 

38

 

 

med +3... eeeoesesee
.m z m d I 3.352 .m. 2 SEE .>
3&8on 0.5
E2 7m». 1.35333 E2 s 8eo .3
Hcomuooho one
a moo d Sm «m 2 m 2 a «moo «m 2 E ammo .HHH
A: e u .3
pageaomag
.m z m 2 @3583 .m 2 es. ease 3. .=
e I 3 Ge 1 3e
W». s e. emusmsomfiogs
em 2 m G Ammugsasmv em 2 one come? :4 .H
Ema OEMWWME v moss 238 cows RosamoomSOo m O 33m
35-033339 +20% ash a . a 2 use £98 .«O EB .

 

 

 

 

.mamumz mo mmfluuwmoum ofluwcmmeocm>amw pamfimunwflm mo aumeesm .HHH> manna

 

39

Physical Explanation of HFMR

Consider a metallic sample placed in a magnetic field B
which is directed along the Z axis. Suppose the current axis,
taken to be in the x-direction, is J.B. Assuming that the
current Jx is kept constant, the boundary conditions are

Jy=0 and Jz=0. Assuming Ez=0, JX is given by
J=OE+0E (27)
where Ex and By are related by
O=0E+0E (28)

For an uncompensated compound which has a FS supporting

no open orbits, the conductivity tensor elements are given by

_ -2
0xx - allB
_ —2
Oyy — aZZB
C, =1‘fli’ia- =—.
xy QB yx

in the high field limit. Thus solving for Equation (28) for

E
Y

|e|cn

E — ——9a A EXB
Y 22

The transverse Hall field EY grows linearly with B and as
B + w the angle between the current Jx and the total electric

field ET = Ex + E approaches 90°. Thus the resistivity ap-

Y
preaches saturation. In terms of Equation (27)

 

40

a ezczn2
_ 11 A
Jx - ( 2 fl 2 )EX
B 9 a22
all
and as B + m, —§—-+ 0 and pXX which is given by
B
Ex
0xx = 3— (29)
x

goes to a constant independent 0f the field B.

For a compensated compound with no open orbits Oxx and

oyy have the same form but Oxy ls glven by

_ -2
Oxy — alZB

Thus Ey tends to constant independent of B. In fact, since
there are an equal number of electrons and holes, Ey will be

very small if the mobilities of the carriers are similar and

ET = Ex + By W111 be nearly parallel to Jx' Now JX decreases
as B"2 as can be seen from Equation (27)
_ all a21
J — (——— - -————)E
x BZ a B2 x
22
and pxx' the MR, increases indefinitely as Bz.

When open orbits are present, the situation changes
drastically since the open orbit's current conduction is in-
dependent of the magnitude of B in the high field limit. Sup-
pose that there is a band of open orbits running in the x
direction in k-space. In the high field region, these elec-
trons are restricted to motion in the y-z plane in real
space. Since Jy = 0 is still a boundary condition which must

be satisfied, the electric field transverse to Jx' Ey, must

an!
V51
.0

.v.‘
DIV.

Orfiv

.0 J.

(I.

H
Vb

N

(L:

o.
.v
by.

Q
”-57-
"‘18-”

'4
.‘~‘

v.‘
.Q

‘4

am;

H

t“

41

approach zero - if Ey # 0, the current carried by the elec-
trons on the open orbits would have to be annulled by elec-
trons undergoing closed orbits. Such an occurrence is impos-
sible since electrons on closed orbits are very poor conduct-
ors in the high field region. Thus Ey 2 O and the MR goes as
B2 for both uncompensated and compensated materials.

When the open orbit direction in real space is parallel
to the current Jx' the MR saturates. In an uncompensated ma-
terial the transverse field By is not restricted to being zero
since there are no open orbits parallel to it. Thus By 3 B
and the MR saturates as it does when there are no open orbits.
In a compensated compound since the open orbit electrons
dominate the conductivity and they are independent of the mag-
nitude of B, p saturates. In terms of Equation (27) since Oxx iS

xx
independent of B

 

which immediately leads to px going to saturation as B + m.

x
If the current makes an angle a with the open orbit direc—
tion in k-space, then

l 2 2
p ~ B COS 01
XX

and if e is the angle between the open orbit direction in

real space and the current direction, 0 = 90 + a, then

2 . 2
pxx ~ B Sin 0

in the high field limit.

IV . SAMPLE PREPARATION
Crystal Growth

59.92

High purity Cominco Au, either 69's or 59's, and 69's
Alcoa Ga were used to make the AuGa samples. The Au was
etched in warm aqua regia, rinsed in distilled water and ethyl
alcohol, and dried with a heat gun or at room temperature.

The Ga was not etched but it was poured directly into a high
purity alumina crucible from its plastic envelope after melt-
ing had been accomplished with the aid of a heat gun. The
empty crucible had been weighed before the Ga was poured into
it and thus the amount of Ga could be determined. The re-
quired amount of Au was added, usually measured so that the
compound was stoichiometric. A Mettler balance which could
be read to 0.001 grams, was used. The phase diagram of Au-Ga
is shown in Figure 6 and from this figure it is evident that
AuGa is a congruently melting compound which can be grown from
either the slightly rich Ga or Au side.

The alumina crucible was placed in a high purity graphite
crucible. The graphite crucible and all other graphite cru-
cibles and parts used in this research was then outgassed at
about 1000°C and 10-4 mmHg. The graphite served as a susceptor
for induction heating by a Lepel furnace. A protective atmos-

phere of Ar was used to prevent evaporation. After melting

42

1200

1100

1000

900

000

q
O
O

IEIPERITURE,’C
O
O
O

WEIGHT PER CENT GALLIUM
115 210 2

 

l

500

 

1063°

450

 

400

 

 

350

 

300

 

250

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

500
(An)
ora
400 Be//’
300 205°
/ zn°;:
I
200 }
l
#7
l
‘wc :
P
: w 29.7°
0 I
0 20 30 40 50 60 100
A. Atoulc PER CENT GAUJUM Go
Figure 6. Phase diagram of Au—Ga (From Hansen, Ref.

44

the metals and agitating the melt to ensure that it was thor-
oughly mixed, the crucible was cooled to room temperature,
inverted, and placed on top of a split graphite crucible.

The split crucible had three rectangular slots of 1/16 square
inch by 4 inches. The crucibles were then placed in a clean
quartz or Vycor tube which was then evacuated to 10"4 mm Hg
and filled with l/3 atmosphere of Ar. The tube was sealed

off and placed in an induction coil which could be raised
slowly around the sample tube by a speed reducing system of
two sets of worm and worm gears connected to a Heller variable
speed motor. After melting the sample into the lower crucible
the coils were raised at about 1 inch per hour. The tempera-
ture of the graphite was measured by an optical pyrometer.

As the coils were raised the temperature at the bottom of the
graphite crucible decreased since the coupling between the RF
field and the lower part of the crucible decreased. Even-
tually solidication began when the bottom of the sample reached
its freezing point. As the coil continued to be raised, more
of the lower part of the crucible cooled below the freezing
point of the sample thus allowing the crystal to grow from
the bottom up. Since the temperature of the sample could not
be measured directly, an estimate of the actual temperature
was made by placing some aluminum in a graphite crucible of
geometry similar to those used to grow AuGa. Thermo-couples
were placed in the aluminum and measured the temperature of
the sample while the optical pyrometer was used to measure

the temperature of the graphite crucible. The aluminum sample

45

was found to be about 40°C lower in temperature than the
graphite. Thus the temperature of the graphite was initially
kept at least 50°C above the melting point of the sample.

The procedure just described produced two usable samples.
Other attempts, using the same method, produced crystals with
RRR ‘ 30. In an effort to produce large diameter crystals
from which crystals with desired orientations could be cut,

a vitreous graphite crucible with 3/8" i.d. was tried. The
metals were melted in the vitreous graphite crucible and then
the crucible was sealed in a Vycor tube under 1/3 atmosphere
of argon. The sample was then lowered slowly through a re-
sistance furnace which had a sharp temperature gradient.

The slug had a RRR z 250, but was polycrystalline. In an-
other attempt a horizontal Bridgeman method produced samples
with RRR as high as 470 but this sample was not single.

(15)

Jan et al used samples grown by zone refining which had

RRR=28 in their deA measurements.

NiGe and Others

NiGe was the next B31 type compound grown. Since molten
Ni reacts with carbon, the split graphite crucibles could not
be used. The compound was grown in an alumina crucible of
7/16" i.d. which was placed in a graphite susceptor.

The sample was melted by induction heating and then a
temperature gradient was moved along the enclosed sample by
the coil raiser in a manner similar to that used in growing

AuGa. The sample had a RRR = 30. A Vycor tube was then tried

46

as a crucible since a sample with a smaller diameter was de—
sired. The compound was originally melted as in the first
attempt. Then the slug was placed in a Vycor tube which was
large enough at the top to accept the slug but which then
narrowed to about l/8" in diameter. The tube was sealed
under l/3 atmosphere of Ar and the slug melted down into the
bottom part of the crucible. The sample was then melted by
induction heating and the crystal grown as described pre-
viously. The RF field of the induction heater caused stirring
of the melt which was easily observable. The sample pro-
duced in this manner had a RRR = 100 but was polycrystalline.

Due to the success in growing Au Pb and AuPb the work

2 2’
on NiGe was discontinued. It seems that single crystals of
NiGe with sufficient perfection can be grown. Placing the
Vycor crucible inside a graphite susceptor to reduce the
stirring of the melt might prevent the growth of stray crys-
tals and should be tried in further growing attempts.

One other B31 type compound, PdSi, was grown during
this research. A graphite crucible was used to melt the
metals although carbon is known to be soluble in both liquid
Si and Pd. Despite this, a single crystal of PdSi was grown
but no RRR was taken because the crystal cracked while being
spark planed.

Three other compounds, PtSn, CuAlz, and MgAg were grown
during the course of this research. Since the deA work on

PtSn (11) was complete and the results were inconclusive

47

regarding the number of electrons (8 or 10) per unit cell
needed to explain the data, the HFMR might determine which
number was correct. However, the crystals grown had RRR r 20
and were unsuitable for HFMR experiments. Four different
attempts were made at growing PtSn before it was decided to
try other compounds. The high cost of 59's Pt was a major
consideration for termination of the work on PtSn.

CuAlz, which is the prototype for the C16 structures
was also grown. The initial melting and solidification of
the compound caused a vitreous graphite crucible to break.
This breakage was probably due to the difference in thermal
expansion coefficients for Al and C, that of Al being about
three times greater than that of C at 700°C. An alumina
crucible was also tried but the sample could not be removed
after solidification. The alumina crucible also cracked but
was prevented from shattering by the graphite susceptor sur—
rounding it. These problems could be overcome by being more
careful in heating and cooling the melt or using other types
of crucibles. However, since the Au—Pb compounds were being
tried at the same time, the work on CuAl2 was discontinued
when good samples of AuZPb and AuPb2 were grown.

MgAg, a B2(CsCl) structure compound whose phase diagram
is given in Figure l was also grown. Triple distilled Mg
generously provided by Dow Chemical and 59's Ag were used.
The Mg had a RRR of about 300. A graphite crucible was used
to melt the metals and a single crystal of MgAg was grown with

a RRR = 10. Due to the rather poor RRR of the initial Mg,

48

the difficulty of preventing evaporation of the Mg, and the
phase diagram, this compound would be difficult to grow with

better RRR.

Agsz and Ausz

The phase diagram for the Au-Pb system is shown in Fig-

 

ure 7. Both Ausz and AuPb2 are peritectically formed com-
pounds. The solid of composition Ausz is not in equilibrium
with liquid of the same composition as can be seen from the
phase diagram. Therefore, a peritectic compound must be
grown from a non-stoichiometric melt. The region between
45 - 70 at.% Pb will produce cyrstals of Ausz while the re-
gion between 72 - 84 at.% can be used to grow AuPbZ.

The 69's Pb shot obtained from Cominco was etched in a
1:1 mixture of acetic acid and H202. The Pb was then placed
in a vitreous graphite crucible. No rinse was used since
the bright appearance of the etched metal quickly tarnished
if it was placed in water. The Pb was heated to its melting
point under vacuum and the slug produced was weighed. The
desired amount of 59 Cominco Au which had been prepared in
the same manner as for AuGa was placed in the crucible with
the Pb. The mixture was heated under a pressure of 2/3 at—
mosphere of Ar until all the Au had melted. The sample was
cooled and then turned upside down and remelted to ensure
that all the Au had melted.

In making Ausz either a split graphite crucible similar

to those used in making AuGa or a Vycor tube was used as a

49

VEIGHT PER CED" LEAD

10 20 30 40 50 60 70 00 90
J l l L J 1 l

 

1100 a.

1000

 

\

0.1

 

900

\

 

000

*— szb

0.00

 

§

(0.05 \\

 

 

TEIPERAYURE,’C
O
8

(0.04 3
‘V

S
o

 

‘— ‘0 P02

 

 

4w° ‘\5¢3§fl
400 5' **

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2“h\\“\\‘\\ 3270
300
254° A A. -
O vv~vvv
O
200 fooooooLfi =4‘_
I I (~dm

100

o 10 20 30 40 so so 70 so so 100

An Atoulc PER can: LEAD Pb

Figure 7. Phase diagram of Au~Pb (From Hansen, Ref.

31)

50

crucible. The slug of Au2Pb was placed in the split crucible
and the crucible was put into a Vycor tube which was open at
both ends. A brass water-cooled plug was inserted through
the bottom end. This plug was used to make a larger tempera—
ture gradient along the graphite crucible and to lessen the
effect of any radial temperature gradients. A vacuum seal
was made by a rubber "0" ring quick coupling. The other end
of the tube was connected to a vacuum pumping port. The
Vycor tube was evacuated to 10"4 mm Hg and then refilled with
1/3 atmosphere of Ar. The induction coils were raised along
the sample by the coil raiser described previously. Samples
made by this method had a maximum RRR = 170.

Another method of preparation was tried with the hope
of producing cyrstals with higher RRR. Vycor tubing was used
as the crucible and a resistance heater was used instead of
the induction furnace. The sample was sealed in the tube
under 1/3 atmosphere of Ar and then dropped through the fur-
nace which had two heaters separated vertically. The upper
furnace was held at a temperature slightly higher than the
compound's melting point while the lower portion was held at
200°C. The temperature of the upper furnace was regulated to
about 1°C. The actual temperature of the sample was not
measured but the temperature of the upper half of the furnace
was set so that when the sample was completely removed from
the furnace, it solidified within = 15 seconds. Once the
upper furnace temperature was correctly set, the sample was

lowered through the furnace at a rate of about 1/2" per hour.

51

This method produced a sample with RRR = 300.

The Ausz samples were made by three different methods.
All methods of preparation of the initial slug were the same
as those for Ausz. The first method consisted of placing
the slug in a split graphite crucible and then sealing the
crucible in a Vycor tube. No chill plug was used. The RF
induction coils were then raised along the sample. This
method did not produce single crystals of AuPbZ. The poly—
crystals were probably due to the rapid growth rate of the
samples. Although the coils were moved slowly, the tempera-
ture of the graphite remained fairly constant with the temp-
erature gradient being about 5°C/inch. Thus the amount of time
between the solidification of the bottom end of the sample
and the top end was probably much less than the time required
for the coil to traverse the entire sample. The temperature
was measured with an IR Tronics infrared pyrometer which
could measure temperature in the range 200 - 600°C.

The first successful growth of AuPb2 was accomplished
by using a Vycor tube as a crucible and lowering it through
the two heater resistance furnace as described for Ausz.
This method produced a long single crystal with RRR ~ 320.
Inlother successful method consisted of using an optical zone
refiner. (43) The AuPb2 sample was placed in a Vycor boat,
seéiled in a Vycor tube under 1/3 atmosphere of Ar and heated
untLil the entire sample was molten. The temperature was
gralflually reduced in such a way that one end solidified

first and the crystal grew in one direction. This method

52

produced a large single crystal with RRR ~ 300 from which 4

samples were cut.

RRR Measurements

The RRR was determined by measuring the resistance at

room temperature and at liquid He temperature. A Hewlett- ‘

Packard Harrison 6284A D.C. Power Supply was used as a con-

stant current source and a Keithley 147 Nanovolt Null De-

tector was used to measure the voltage. The voltage and cur-

rent leads were soldered to the samples with Rose's Alloy.
The Au Pb and Ausz samples were superconducting at

2
4.2°K. Roberts(29) reports a transition temperature of 4.42°K

for Ausz but does not report that AuZPb is superconducting.
The highest valued RRR sample, AuZPbIIb, does show very small
voltage difference in zero field but this value is 5 times
lower than the value obtained by extrapolating the magnet-

oresistance voltage to zero field.

The Ausz and AuPb2 samples were placed in a dewar which

was situated between the pole faces of a Harvey—Wells elec-

tromagnet. A Bell 660 Model gaussmeter was used to measure

the field. Since many samples were attached to the RRR
Cryostat at the same time, the position of the individual
Saumples in the magnetic field differed from sample to sample
and the field read on the gaussmeter was not necessarily
the; field at the sample. The RRR cryostat consisted of a

cirflcuit board which had four sets of five voltage leads and

four sets of current leads. The board was mounted on the

53

end of an epoxy rod. The positioning of the rod in the dewar
and thus in the magnetic field was not very accurate, causing
some additional error. The current was reversed in order to
cancel out thermal voltages.

The resistance was measured at five or more fields and a
linear extrapolation was made to get the zero field resist-
ance as is shown in Figure 8. In the case of the Ausz and
AuPb2 samples, the RRR was taken of the initial slug as well
as the individual samples cut from the larger slugs. The
RRR increased with time and a similar effect occurred with the
AuGa samples.

The initial RRR of AuGaI was 130 and the RRR measured
six months later was 170. The initial RRR of the AuZPb slug
from which AuszIIb was cut was 240 and the RRR measured 300
nine months later. The increase in RRR is probably due to
annealing of the samples at room temperature.

Table IX lists the samples used, the RRR, and the cur-
rent direction with respect to the major crystallographic
axis. The RRR are accurate to about 10% and the error may
be slightly higher for the Ausz samples due to the effect of

alloying between the sample and the solder. This alloying

effect is discussed in the section on voltage measurement.
X-Ray Procedure

The samples were cut by a spark erosion machine which
used a .002 inch diameter molybdenum wire. The AuGa and

Ausz samples were also spark planed using a Servomet spark

54

 

 

V(pV)
. ./ I
P x/
x/ 11:
x/
3 V/x/// ,‘ x/ZBL'
/x/ ‘ / x /
/" / X Ausz SAMPLES
2 .—
x/ I
/x//
X/P/x
1:?"
l 1 1 1 1 1 1 1 1 1
O l 2 3 4 5 6 7 8 9 I0
8 IN KG
25 .. x
.1-
Ausz SAMPLES x x / .15
/X

20 _. /
x x/“/
V(pV) ‘4
4

 

 

 

// // x
X X / / ‘3
/ / /" lb :1
IO_ i/ /
X /
/ I/ /x
+ x -2
/
54x1 J 1 L l 1 l + 1 1
0 ' 2 3 4 5 6 7 B to sue)

1Figure 8. V(B) vs. B for (a) AuPb2 samples and (b) Ausz samples.

55

Table IX. RRR and Orientation of Samples

 

 

Current Direction*

 

RRR 8 ¢ 1
AuGaI 170 90 0 90
AuGaII 190 90 57 33
AuPbZI 480 86 54 27
AuszII 330 78 42 37
AuszIV 430 57 75 29
AuszV 400 81 77 18
AuPbZVI 600 32 81 61
AuszIb 165 31 60 86
AuZPbIIb 380 31 66 71
AuszIII 140 28 62 9o
Au2PbV 130 31 61 83
AuszVI 175 23 78 71
AuZPbVII 175 14 79 79

 

 

* 6 is the angle between [100] and current direction 3.
6 is angle between [010] and 3.

w is angle between [001] and 3.

56

cutter. The AuPb2 samples could not be planed since even on
the finest setting enough heat was generated to melt the sur-
face layer making X-raying impossible.

Since the wire spark cutter produced uneven cuts, it
was necessary to improve the surface of the AuPb2 samples
before X-ray pictures could be taken. The best method would
have been to chemically etch the surface. However, it is
very difficult to find etchants which will uniformly polish
both constituents of a binary compound (especially Au com-
pounds). There is a genuine need for references on chemi-
cal and electrochemical etchants which could be used for in-
termetallic compounds.

After failing to find an etchant, mechanical polishing
was tried with success. Jeweler's rouge was used with a
solvent like ethyl alcohol. The polish was applied with a
cotton swave. About 10 to 15 minutes of polishing was neces-
sary to produce a suitable surface for X-raying.

A Noreleo X-ray unit was used with a Polaroid camera
and collimator. Initially only a Cu tube was available.
The radiation from this tube was not sufficient at all wave-
lengths of interest and the X-ray pictures did not contain
enough spots for easy identification. After a W tube was ob-
tained, the Laue pictures improved greatly and the identifica-
tion of symmetry axes was possible.

The voltage.setting on the W tube was very important--
especially in the case of the Au-Pb samples. The setting of

20 RV and 20 ma for about 45 minutes produced usable pictures

57

whereas higher KV settings produced fogged pictures which hid
most of the Laue spots, even at shorter exposure times.

In order to check if the samples were single crystals,
the following procedure was followed. One side of the sample
was mechanically polished and then X-rayed at the top and bot-
tom. If the two Laue photographs were identical, the opposite
side was polished and X—rayed in the same manner. If all four
photographs were consistent, then the sample was considered
to be a single crystal.

A Laue X-ray photograph of the current axis of the sam-
ples was difficult to obtain. The small cross—sectional area
of the samples was difficult to polish and thus the surface
could not be prepared well enough for good quality Laue photo-
graphs. The current directions were determined by stereo-
graphic projections of the Laue photographs taken of the
sides of the crystals as described above.

Since the crystal structures studied in this investi-
gation were of a relatively complicated nature, it was neces-
sary to check the crystal orientations with a GE diffracto-
meter. The diffractometer was absolutely necessary in the
case of AuGa where the lattice constant in the 100 direction differs
by only 2% from that in the 001 direction.

In order to determine what the orientation of the two
AuGa samples was, the fact that the 011 reflection is allowed
while the 110 reflection is extinct in first order was used.
The diffractometer was also very useful in confirming the

orientation of the Ausz and AuPb2 samples when a symmetry

58

axis was far from the normal to the polished side of the

crystal.

V . EXPERIMENTAL PROCEDURE
Cryostat and Sample Holder

The D.C. method was used to determine the field depend-
ence of the magnetoresistance as well as its angular de-
pendence. A constant d.c. current of 1 ampere is passed
through the sample and the voltage across the sample is mea-
sured as a function of the magnitude of the magnetic field
or its direction with respect to the crystal axis.

The constant current source was a Hewlett Packard 6263B
DC Power Supply which is stable to 3 ma in the constant cur-
rent mode. The voltage across the sample was measured by a
Keithley 147 Nanovolt Null Detector. The output from the
Keithley 147 was recorded on one axis of a Moseley 2D—2 x-Y
Recorder as a function of the magnetic field or rotation
angle which were plotted on the other axis. A Hewlett-
Packard 7004B x-Y Recorder replaced the Moseley later in the
research.

Figure 9 is a diagram of the sample holder which is a
modification of the one described by D. J. Sellmyer(44).

The crystal could be rotated about two perpendicular axes
thus allowing the magnetic field to be positioned in any
desired direction with respect to the crystal axis. Rota-

tion of the worm gear about the BB' axis is measured by the

59

60

COUNTER-
RING RETAINING SCREW ' WEIGHT

\$ GUIDE
WORM DRIVE

6 EAR

   

SCREW-IN
PIVOTS

SPIRAL DRIVE GEAR

\»

0..

Figure 9. Sample holder.

61

tip angle ¢ while rotation about the AA' axis is measured by
the angle 0. The sample holder was machined from standard
Delrin 500 rod. This material is easily machinable, inex-
pensive and held up well under recycling between room and He
temperatures.

The sample, of approximate dimensions l/l6" x 1/16" x
7/16" was glued to an insert with a mixture of Duco cement
and acetone and the insert was then pressed into the worm
gear. Two screws position the insert and prevent it from
rotating. The worm gear was held in place by a brass screw
and Delrin washer. These precautions are necessary since the
worm gear has a tendency to leave the plane of rotation when
rotated counter-clockwise. The worm was held in place by a
stainless steel retaining ring which was screwed into the
Delrin. The retaining ring was held in place by a small
brass set screw.

The worm had a rectangular protrusion which could be
engaged by a rod with a slotted end when the sample holder
was in the w = 0 position only. Each revolution of the worm
rotated the worm gear by 5.54° (360/65). The rod had to be
withdrawn when a rotation plot was made.

A spiral gear arrangement provided continuous angular
motion for the rotation plots. The gear had a slotted groove
on tap into which a rod with a screwdriver end was inserted.
The rod was coupled by spur gears to a Hurst CA synchronous
motor which was also coupled to a ten—turn Hellipot potenti-

ometer with a linear tolerance of 1.50%. The voltage across

62

the potentiometer was thus directly proportional to the angle
of rotation. Each revolution of the drive rod produced a ro-
tation of 6°. One degree of rotation corresponds to 3.76 mV
on the potentiometer. Due to backlash in the gears, the repro-
ducibility of the angular rotation was about 4°. However, if
data was taken with the tip angle reset each time, the re-
producibility was better than 2°. A complete rotation of
180° was generally done in 5 minutes.

A brass cylinder was used as a counterweight in order
to prevent the lead wires from getting tangled in the gears
as the barrel was rewound. The brass cylinder was restricted
from random motion by a stainless steel tube press fitted

into the top of the sample holder.
Voltage Measurements

Voltage and current leads were soldered to the samples
using Stay Clean Flux and Rose's Alloy solder which is 50% Bi,
25% Pb and 25% Sn and melts at 96°C. The leads were made
from #40 Nyclad insulated copper wire. The leads were then
soldered to pins on a circuit board which was mounted on the
cryostat just above the magnet. During the experiment,
these connections were always immersed in liquid He which pre-
vented thermal voltages due to the soldering. The external
voltageleads to the circuit board were carefully shielded
and brought out to an aluminum mini-box where they were
soldered to a Au plated low thermal switch with Kester

Soldering Flux #1544 and low thermal solder #SW4337. The

63

input cable of the Keithley 147 was soldered to the switch
with the same solder. The current leads from the circuit
board were #24 Poly-thermalized insulated copper wire. The
current leads were soldered to a banana plug inside a mini-
box mounted on top of the cryostat.

The two voltage leads as well as the current leads sol-
dered to the sample were tightly twisted together to minimize
the pick up voltage generated as the sample holder rotated
in the magnetic field. This voltage was measured by turning
off the current and measuring the voltage across the sample
as it rotated in a magnetic field of 98 kG. Four different
samples were measured in this way and the pick up voltage
varied in a monotonic way, the total variation being 1.5 uV
for the worst case. In one case spikes were observed which
apparently were due to non-uniform motion of the spiral gear,
that is, the gear was probably sticking between teeth. In
any case, this voltage did not cause any problems since the
magnetoresistance voltage was generally at least 20 uV and
the pick up voltage varied by only 1/2 uV at the most in an
interval of one revolution of the spiral gear.

The alloying effect of the Rose's Alloy solder used in
connecting the probes was negligible for the AuGa samples.
The solder could easily be removed with a razor blade leaving
no observable alloying effect. However, the solder did alloy
with the AuPb2 and Ausz samples. The AuPb2 samples did not
show any measurable results due to alloying with the solder

but the Ausz samples did.

64

The probe effect manifested itself when the magnetic
field was swept up or down in a field interval containing
14.6 kG. Figure 10 shows the voltage across the sample as
the magnetic field is swept. The abrupt change in voltage,
and thus resistance, which occurs at 14.6 kG was observed
in all AuZPb samples. However, the change in resistance
was much greater in those samples which had the lower RRR.

In order to determine if this effect was caused by the solder,
a sample which had shown the effect was spark planed until

no trace of the solder remained. A pair of phospher—bronze
clips were pressed onto the sample and served as voltage
probes. No abrupt voltage change was observed as the field
was swept from 6 to 20 k6. Thus the abrupt resistance change
is due to alloying between the Ausz samples and the Rose's
Alloy solder.

The alloy formed around the solder blobs is probably a
superconducting alloy which alters the current flow through
the crystal at low fields. Once a high enough field
(>14.6 kG) is reached, the alloy goes normal and its effect
ceases since a four probe system is used and the voltage leads
draw a very small current. The Keithley 147 has an input re-
sistance of 1x106 ohms while the sample resistance is on the

4 ohms at 4.2°K. In the field region used to

order of 2x10"
study the FS of Au2Pb (> 80 kG) the alloying effect should be
oflittle consequence.

In order to eliminate unwanted voltages from the mag-

netoresistance probes which are caused by thermal voltages

65

Vdm

35m

.Hnmcsa mom

‘I

or

.m> Amv>

.OH messes

Qm

 

 

Hffq

W" (6'8 )A '(GIA

 

66

and probe misalignment, the following sets of voltages should

be taken( 30) ,

_ l _ _ -
vMR - I [V(+I,+B)+V(+I, B) V( I,+B) (30)

- V(-II-B)]

The sample holder was designed to rotate a full 360° in order
to allow the field to be positioned in both the +B and -B
directions. The current could also be reversed. The complete

procedure of measuring V was done on AuGa I, AuPb2 VI,

MR

AuZPb VI and Au2Pb 11b. The AuGa and AuPb2 samples showed

no unexpected results and a measurement of V(+I,+B) was taken

to be equal to VMR'

and these will be discussed in a later section.

For AuZPb IIb some anomalies occurred

The field dependence of the HFMR as expressed by the co-

efficient m is defined by

 

Ap/po = aBm (31)
Since Ap/po = V(31BYIO) where V(B)

is the value of the voltage at the field B; the voltage V(B)
was taken at two different fields and m was calculated by

V(B1)-V(0)
V(le-V(0)
B1

109 “8—2-

 

log
(32)

 

The V(B) was read from the x-Y recorder trace. When the
Keithley 147 was on the 30 uV range, the X-Y recorder signal

could be read to ~ .08 uV, on the 100 pV it could be read to

67

~ .25 uV, and on the .3 mV range it could be read to ~ .8 8V.
Thus the error incurred by reading V(B) from the X-Y recorder
trace was less than the fluctuations caused by the rotation
of the sample holder.

Assuming that the magnitude of B is accurage to 15%,
the value of m is probably accurate to ~ 20% when V(B) is be—
tween 15-30 uV, ~ 13% when V(B) is 30-100 uV, and ~ 11%
when V(B) is 100-300 uV. The higher error for the lower vol-
tages is due to the pick up voltage generated as the sample

rotates in the magnetic field.

Analyses of Tipping Arrangement

Figure 11a shows the voltage probe positions and the
direction of the magnetic field with respect to the crystal
as the sample is rotated. The rotation of the crystal clock-
wise about the y" axis (4) can also be represented by a rota-
tion of the magnetic field counter-clockwise about y" with the
crystal remaining fixed. The same interpretation can be ap-
plied to rotations about the x" axis (w). The magnetore-
sistance probes are numbered 1 and 2 and are separated ~l/8"
while the other four probes are used for Hall voltage measure-
ments. Figure llb shows a stereographic projection of the
motion of the magnetic field with respect to the crystal.

The resistivity tensor is given by Equation (21) when

there are no open orbits and is repeated below

68

.m mo cofluoE
mo :0wuomnoum ownmmumomwmum Any one muumEomm madame» one coflumuom

3v

 

 

 

. A3 Ha madman

 

 

 

 

69

 

 

c 2 c 2 c
I bllB bl2B bl3BI
_ c 2 c 2 c
p (closed) — blzB bzzB b23B (21)
I _ C _ C c I
bl3B b23B b33

When there are open orbits in the X2 direction in k-space and
thus in the X1 direction in real space, the resistivity tensor

is given by

0 O 0 o 0

I bllB b123 b13BI
(0 en) = -b° 13° b0 B.L b0 B (33)
p P 12 22 23

 

 

o o o I
Lbl3B l"23B b33

Now the effect on the resistivity tensor of tipping the crys-
tal from the transverse (3.L.B) geometry will be considered.

Figure 12 shows the non-transverse geometry, i.e., B is
not.L 3. The coordinate systems shown are related in the fol-
lowing way. The S(X1,X2,X3) system is the initial coordinate
system. When y and B = 0, the current direction 3 is along
X1. The S(Xi,Xé,X5) coordinate system is obtained from
S(X1,X2,X3) by a rotation of y about the X3 axis and
S"(Xi,X§,X§), which is fixed to the crystal, is obtained from
S'(Xi,Xi,X5) by a rotation of 8 about X'. The Xi axis is the
current direction which makes an angle 8 with the X1 axis and
an angle 8 with the Xi axis.

The rotation matrices for rotations about the x X2, and

1!
X3 axes are

7O

 

 

 

Figure 12. Non-transverse geometry.

71

 

 

 

 

I 1 0 0 I I cos c 0 -sin 6‘
Rx1 = 0 cos s sin e sz = 0 1 0
I 0 -sin 5 cos eI I sin e 0 cos eI

I cos a sin e 0 I

RX3 = -sin t cos a 0

 

 

I o o l I

where e is the angle of rotation taken to be positive in the
counter-clockwise direction.

The voltage is measured in the sample's coordinate sys-
tem, S"(Xi,X3,X§) and can be related to the resistivity ten-
sors given for the S(X1,X2,X3) system, Equations (21) and

(33). The transformation is given by

p" = RXI(B)RX (Y) RX (Y)qu(8)
2 3 Q 3 2

The measured MR is given by the p componentof the resis-

N
11
tivity tensor

n 2 2 2 . .
p11 pllcos 8cos y+p12cos BCOSYSlny+plBCOSBSlnBCOSy

2 O 2 I 2 a o
+ pZICOS 8c05yslny+p22cos BSln y+p23c05851n851ny
O o 0 O 2
+ p3lcosBSInBcosY+p32c05831n831ny+p33s1n B.

If B is tipped infinitesimally away from the X3 axis in
any direction except in the XZX3 plane, then the open orbit
in the x1 direction is missed while y and 8 remain the same
to a first approximation.‘ Thus as B goes from a direction

which produces open orbits to a direction which does not, the

72

MR voltage as measured by p11 will change. The values of
6i1 for both cases, keeping terms in the highest power of B

only are,

2

u _ 2 2 C C . C .2
p11(closed) — B cos B{bllcos y+2b12c05yslny+b2251n y}

n 2 2 . 2
p11(open) = B balcos BSln y

The ratio of the two values of 011 gives,

n O -2
pll(open) biflfln y

~

 

 

~

pil(closed) bc(c03y+siny)2

if the magnitude of the bEj' 3, represented by the symbol be,
are approximately the same. By sweeping B through a region
of open orbits, a dip in the MR is expected if
baasinzy¢bc(cos y + siny)2 or a peak is expected if

2

o . . 2
b‘asln y>bc(cos y + slny) .

We, as well as other investigators(13'45) , have en-
countered both types of behavior. However, when 8 becomes
large, the dip can be washed out due to the dominance of the
longitudinal magnetoresistance which saturates in all cases.

The angles of rotation of the sample holder do not cor-

respond exactly to y or 8. However, with the help of some

geometry the relationship between the angles is seen to be

COSB = (cosz¢coszxp+sin21p)1/2
siny = tan¢sinw
cose = cosBcosK

73

The rotation plots are taken for different values of 6.
The position of the magnetic field corresponding to sharp
minima and maxima are plotted on a stereogram as is the ori-
entation of the crystallographic axes of the sample. Usu-
ally the sample axis (current axis) did not correspond to a
major symmetry axis. The minima corresponding to a partic-
ular open orbit direction are connected and form a plane in
k space. The direction of the normal to this plane is the
open orbit direction in k or reciprocal space. For compen-
sated compounds, several samples with different current
directions are needed to ensure that all open orbit directions
are obtained. If the angle between the open orbit direc-
tion in k-space and the current direction is small the open
orbit can easily be missed.

The orientation of the samples with respect to the mag-
netic field direction could not be directly determined.
Usually the sample was X-rayed while mounted on the worm
gear which was then placed on a goniometer. Thus errors of
~5° in the orientation of the sample are likely. The exact
orientation was determined by fitting the planes formed by
the sharp minima with the approximate orientation obtained

from the Laue X-ray photographs.

VI. RESULTS
AuGa

AuGa has the B31 (MnP) structure which is orthorhombic
with space group ana and 8 atoms per unit cell. The B31
structure is a derivative of the 38(NiAs) structure and Figure
13 shows how the two structures are related. The lattice
constants(15) for AuGa are given in Table X. Note that the
a and c axes differ by only 2%.

Figure 14 shows a plot of V(B) versus w for AuGaI at
fixed magnetic field B = 82 kG. The letters designate
specific field directions. Figure 15 shows curves for V(B)
versus B for the specified values of w. The plots show the
interval between 22 to 82 kG. Note that curve b in Figure 15
is a very slowly increasing function while curves a and c-
are beginning to show a quadratic field dependence at higher
fields. Since the time required to sweep the field from 0 to
82 k6 is about 40 minutes, this method of determining the
field dependence of the HFMR was not used for economic rea-
sons. Also, the positioning of the magnetic field with re-
spect to some particular direction in the crystal was diffi-
cult and could only be determined to ~4°.

Figure 16 shows how the field dependence of the HFMR was
determined. The rotation plots were taken at two different

fields, 99 kc and 88 k6 with tip angle 6 = 0. By assuming

74

Figure 13.

75

Comparison of NiAs and AuGa structures. The
lower diagrams are projections taken along the

c axis in NiAs while the upper diagrams are taken
along the a axis. The height of the atoms abowa
the projection plane is given by the numbers
beside the atoms. Dark spheres represent Ni(Au)
atoms while light spheres represent As(Ga) atoma

76

 

 

~16 o .6 she 2.0

77

 

 

 

Table X. AuGa Lattice Constants (from Jan et al(15))
Room Temperature Liquid He Temperature (4.2°K)
a 6.260 A 6.196 A
b 3.464 3.488
c 6.396 6.354

 

Table XI. FS Parameters for AuGa

 

 

Reciprocal lattice vectors

°-l
a* = 1.014 A
5* = 1.801 1'1
c* = 0.9889 2‘1

Fermi Energy

E _.

Fermi Radius
kf = 1.511 8‘1
Volume Brillouin Zone
°-3
V = 1.806 A
Structure factors of zero value

hOO: h = 2n+l
0k0: k

k=2n+1
002: 1

2n+l
0km: k+2 = 2n+1

hkO: h = 2n+1

 

78

A060 1'

90- ‘ B=82kG

 

 

l I 4

0 90 l8
9 IN DEGREES 0

Figure 14. V(B) vs. 0 for AuGaI.

.Hmosa now m .m> ime> .mH mussan

 

79

 

 

5me
mm
.8 1 o
L
2.318; I\I11111\11 1.
I|I\II‘I I 1 M
a \ . mm
. In
A
1. m.
A
A
o n
o
J on

 

H 00:4

80

0
.mesm new a .m> Q\oa

mummomo z. 9
cm. om

q

oxmonm
oxmmum

 

unnzwadl

 

.mH musmwm

m

AV—

m. 1.0
.‘AV

om

mm“

.on

81

that %£'= aBm, the exponent m was determined at various values
of w. 0The value of m > 1 for all field directions except
when B is in the [001] where %3 saturates. By tipping the
crystal, i.e., ¢ # 0, the posigion of the sharp minimum could
be followed and it occurred when B was in the (001). Figure
17 is a stereographic projection of AuGaI showing the major
crystallographic planes and directions. The data shown in
Figure 16 was taken with the sample rotated by 90° from its
position in Figure 14.

Assuming that Au contributes one electron per atom and
Ga 3 electrons per atom, then AuGa has 16 nearly free elec-
trons per unit cell and AuGa should behave like a compen-
sated compound. This is seen to be the case since m > 1 for
most directions of the magnetic field. The saturation of
the magnetoresistance when B is in the (001) is due to open

orbits in the [001] since %3 2 aBzcosza when open orbits oc-

cur and a, the angle betwee: the current direction [010] and

open orbit direction in k-space, is 90°. The small dip when

E is in (100) will be discussed after data for AuGaII is given.
Figure 19 shows two rotation plots at different fields,

99 kc and 88 k6, for AuGaII which has current direction ~33°

from [001]. The value of m is greater than 1 for all field

directions with maximum value of 1.8. The interesting fea-

tures of this data are the small peak when B is in (001)

and the small dip when B is in (100).

Figure 20 shows rotation plots of AuGaII for various tip

angles with the magnitude of B remaining constant. A

 

 

AuGa I
DID
IOOII
'00 OOI
3.
(100)
J

 

Figure 17. Stereographic projection of AuGaI.
AuGa ]I

OH) ' ~

 

 

 

J

Figure 18. Stereographic projection of AuGaII. X's designate
position of sharp peaks.

3C)

 

 

83

AuGaII

B=99KG

B=89KG

 

m=l.2
l0 —-
5 -
1 I-
1 A 11
O 90 IBO
‘I’ IN DEGREES
Figure 19. Ap/po vs. 0 for AuGaII.

84

AuGaJI

 

 

B=7Z7kG
TIP ANGLE
e
30
6
E; 3
z:
:> O
E
EE -6
t: -9
a: -30
fig .1
"’ KDFA/
§ 1"
O 90 I80

8 IN DEGREES

Figure 20. V(B) vs. 6 for different values of 6 for AuGaII.

85

stereographic projection which shows the positions of the small (x's)
peaks for rotation plots at different tip angles is given in
Figure 18. The peaks occur when B is in the (001) and they
are due to open orbits in the [001]. Referring back to page
72 where a discussion of the tipping arrangement is given,
when hZIFinZY > bc(cosY + siny)2 such a peak can occur.

The slight dip which occurs when B is in (100) for both
AuGaI and II is probably due to a small number of electrons
making open orbits due to magnetic breakdown. In both samples,
the current direction makes an angle of 90° with respect to
the [100] direction and thus the magnetoresistance should
saturate if there are a significant number of open orbits.
Since no saturation occurs, there are not a large number of
open orbits excited. In addition, the dip is not observed
until high fields are reached and the depth increases as the
field increases; both effects suggest magnetic breakdown
leading to a small number of open orbits.

AuGa is thus seen to be a compensated compound whose FS
supports open orbits in the [001]. The open orbits extend
for a complete 180° angular region. Other high symmetry
directions such as [100], [110], and [101] do not support
open orbits and the [010] direction could not be checked for
open orbits because a = 0 for AuGaI and a = 57° for AuGaII.

A sample with current direction of about 90° from [010] would
be needed to check for open orbits in the [010].
A Harrison construction was done to determine if the

NFE model agreed with the observed topology of the FS. A

86

computer programI46) written by J. Longo for Cl(CaF2) struc-
tures was adapted for AuGa. The program plots the intersec-
tion of the Fermi spheres which are centered on reciprocal
lattice points with planes perpendicular to some specified
field direction. By varying the height and direction of the
planes, the FS can be determined.
A NFE model was used with 16 electron per unit cell which

gave a kf of 1.5 110 A’l. The lattice parameters used were

those determined by Jan‘ls)

at liquid He temperature. Planes
perpendicular to [100] and [001] were taken at .1 A-1 in-
tervals. The height of the unit cell in these directions is
about 1.00 A-l. Table XI contains some important FS para-
meters for AuGa.

The F8 in both directions is quite similar since the lat-
tice constants differ by only 2%. The first three bands are
completely occupied in these directions and there is F8 in
the 4th through 12th bands. The 4th, 5th, and 6th bands are
comprised of small hole-like regions while the 7th band has
large hole-like orbits. The 10th and 11th are electron-like
and there is a very small electron-like region for the 12th
band. The 8th and 9th bands are electron sheets and contain
the open orbit regions. For B||[001] and B||[100] open orbits
run in the [100] and [001] respectively for at least half of
the Brillouin zone volume. Thus, the open orbits in the [001]
are consistent with the experimental topology but the [100]

directed open orbits are not.

The Harrison construction just discussed was done assuming

87

a single zone scheme. However, the B31 structure has glide
planes and screw axes which introduce structure factors of
zero value. The structure factor in x—ray diffraction is a
geometric factor which depends on the positions of the atoms
in the unit cell of the crystal. The structure factor S is
given by

S(hk2) = g fjexp[-2ni(ujh+vjk+wj2)] (34)

where fj corresponds to the scattering power of the jth atom
which is located at position [uvw] in the unit cell and hkl
are the Miller indices. The amplitude of the scattered x-rays
is proportional to S*S. Thus, if S = 0, there is no reflec-
tion from the (hkz) plane. This is a very common occurrence
in x-ray diffraction.

The potential energy of an electron in a periodic crys-
tal can be written

V(E) = Z va1E-E)
2

where E is a lattice vector and Va(r-E) is the potential about

an individual atomic site. Since
V(E+E) = V(E)

V(r) can be written as a Fourier series

V(E) = Z VelG°r
G
where
__ _ - -i?§-E -
VG - 9 I V(r)e dr

88

G is a reciprocal lattice vector.

In the case of a lattice with a basis

v 12-1—6) (35)

where b is a vector from the origin of a unit cell to the

atomic positions. The Fourier component V is obtained by

G
inverting Equation (35)

1 — — — -ia.; .-
VG = 5 X Z I Va(r—2—b)e dr
2 b
(36)
= ‘é-E
vG N f Va(r)e drsG
where SG is the structure factor and is given by

-1§.S

SG = X e (37)
6

This structure factor is proportional to the structure factor
for x-ray diffraction and both terms result from the atomic
positions within the unit cell.

As an electron approaches a Bragg plane in the recipro-
cal lattice it will be reflected if there is an energy gap
across the plane. The energy gap is proportional to VG'
G is zero, VG is also zero and the electron doesn't
undergo Bragg reflection. Such a condition leads to multiple

Thus, if S

zone Schemes if the structure factor is equal to zero across
an entire plane comprising a boundary of a Brillouin zone.
The electrons cross the boundary as though it wasn't there

since the energy difference across the plane is zero.

89

Structure factors of zero value do not ensure that the
energy gap across a Bragg plane is zero since spin-orbit
coupling and other relativistic effects can break the de-
generacy in energy, especiably in the case of heavy metals.
The most notable example of this is the work of Falicov and

Cohen‘47)

who were able to explain the magnetoresistance data
taken on Mg and Zn by calculating the energy gap across the
(0001) plane of the Brillouin zone due to spin-orbit coupling.

Advanced group theory can give some information in the
case of spin-orbit coupling. One can in principle determine
whether or not a degeneracy is broken by spin-orbit coupling.
However, the magnitude of the energy difference, if there is
a difference, between bands cannot be calculated by group
theoretical methods. Double groups are used and in the case
of non-symmorphic space groups the application of the theory
is not trivial.

Magnetic breakdown (MB) can cause a Brillouin zone to go
from a single zone at low fields to a double zone at high
fields since the electrons may have a high probability of
crossing the Bragg plane if the gap is small and the magnetic

field is large. The probability of an electron making such

an interband transition is

where Bc is the critical field for magnetic breakdown and is

given by

K ezch
Bc = —EHEE——

f

nan
in;
5U-

p:
«G

90

where K is a numerical factor of order unity, 69 is the band
gap, mH is the cyclotron mass and ef is the Fermi energy. In

a field of B I 100 kG there is appreciable magnetic break-

9

down if Eg ~ 0.1 eV (assuming cf 2 10 eV). Thus MB is a

complicating factor.

In the case of AuGa there are seven wavevectors which
have zero structure factor in the first octant and magni-
tude less than or nearly equal to 2 kf. They are [100], [010],
[001], [110], [012], [300], and [003]. A Harrison construc-
tion was made assuming a double zone scheme in either the
[100] or [010] direction. The open orbits remained for both
the [100] and [001]. Thus a double zone scheme does not ex-
plain the data. It is possible that some combination of zero
value structure factors gives a FS which has the correct
topology but such a combination is very difficult to hypo-
thesize and the Brillouin zone in such a case would be quite
complicated.

To summarize, the FS of AuGa as constructed by the Har-
rison method assuming a NFE model with 16 electrons per unit
cell does not give the correct topology of the experimentally
determined FS. A double zone scheme assuming that the energy
gap across the (010) or (001) is zero, also does not conform
to the topology. Thus, the NFE model does not seem to work.

‘Ian.et al(15) were also not able to fit their deA data to a

NFE model .

91

AuPb2

AuPb has the C16 (CuAlz) structure which is body cen-

2
tered tetragonal with space group Dig —I 4/mmm. The unit
cell contains 6 atoms and Figure 21 shows the conventional
cell of AuPb which contains 12 atoms. The lattice con-

2
(29) at room temperature are a = 7.325 A and c = 5.655 A

stants
which makes c/a = 0.772.

Figure 22 shows how the magnitude and definition of the
transverse magnetoresistance voltage across the sample changes
with the strength of the magnetic field. The upper most curve
was taken with the Keithley 147 on the 10 uV range while the
remaining curves were taken on the 30 “V range. As the field
increases and more of the electrons enter the high field re-
gion, the sharpness of the dips increases as does the height
of the maxima. The signals are much easier to interpret in
the high field (~80 - 100 kG) range and these graphs show the
advantage of using such fields.

Figure 23 to Figure 27 show the field dependence of the
transverse (¢ = 0) MR at various values of w. The quantity m

is calculated from %B-= aBm. The value of m is seen to be

greater than 1 for mgst field directions and these directions
are general directions, i.e., not parallel to a major crys-
tal axis or direction and not in a major crystallographic
plane. Also, the sharp dips, as will be explained later,
occur when B is in a major crystallographic plane. Such be-

havior of the MR means that AuPb is a compensated compound

2

92

 

 

 

 

 

 

 

 

 

 

 

 

1 cell of Ausz.

iona

Convent

Figure 21.

93

0x

0x

0x
0x

.HNQmflfl. HOW m MO m®5dm> wSOHHm> HON

sum

Em

2% F
3:. 2:

H Nam;

mmmmwmo 2.
cm.

a,

}

.T

.m> 33>

.mm musmflm

 

b?

p

 

(8)/\

SUN“ AthHllBHV

 

94

 

 

AuszI
2 5 —
:4
fl1i:6 / W\\
20 — \\
{x 1"“, m= l4
1' I I
k mzlfl}f\\ ‘ We 99KG
’/ I
| 5 J\ //\ fl . . \
[5P I\ )2, \ ‘ \v I \W
“' k \‘ . ‘fl
8 (II/V
I0 I K
m=l0 m=o.7"‘ I "Fl-0 B=89KG
m= -O.9
5 _
I.—
l l
o 90 I80
41 IN DEGREES

figure 23.

Ap/po vs. 0 for AuPbZI.

 

 

IFigure 24.

Ap/po VS.

95

W

l _
90

IP IN DEGREES

for AuPbZII.

 

ISO

96

 

 

 

 

AP
8
5 .-
)—
F'
| _.
l J
0 90 IBO
‘PIN DEGREES

Figure 25. Ap/po vs. w for AuPbZIV.

 

'3

 

 

 

 

 

Figure 26.

S.
Ap/oo V

. 1
m= |.3 m=l.5 m= LB
B‘BBKG m3“
m=l.0
AuszI

i
r—
t I I
0 90 I80

WIN DEGREES

w for AuPbZV.

 

98

    

 

 

%
25 -
ml=l.4 =l.4 m=l.4
In m=l.6 ‘
/\E\ l/‘A ”FHA News“;
2° AN \
(\V m=l.3
I 5— M\ m= I21 8= 3: KG
m=l.0 L2 m=|.2
0
m=|.|
5 .—
| J j I
o 90 |80
w IN DEGREES

Figure 27. Ap/po vs. up for AuPb VI.

2

99

which has a FS which supports open orbits.

In order to determine the directions of the open orbits,
the samples were tipped with respect to their initial trans-
verse positions through various angles (¢) as was explained
previously. A stereographic projection was used to plot the

position of the minima as the sample was tipped and then ro-

tated through 180°.

Figure 28 shows the MR data for various tip angles.

The negative sign for the tip angle ¢ means that the magnetic

 

field moves clockwise with respect to the sample. Figure 29

is a stereographic projection of all the minima observed for
.AuszI. The dash lines show the motion of the magnetic
field with respect to the crystal axis. The small x's are
tflle position of the minima. The appropriate minima are con—
ruected together to form a plane and the direction of the open
cxrbit in k-space is found normal to that plane.

Figures 30 and 31 show the stereographic projections for
tile other four crystals. The dashed lines have been omitted

e)tcept for the ¢ = 0° and 90° lines. The position of the dips

1111 the Figures 23 to 27 can be followed in these stereographic
E>1§ojections with the following changes: for AuszII, AuszIV
arid AuszV start at the right hand side of the figure, 180°
f1:0mBo and go toward Bo along the ¢ = 0 line. For AuszI and
1\11Pb2VI the starting point is 30' which is the usual manner of
Epilotting the data. The reasons for the exception is that the

(iiata shown in Figures 24, 25, 26 was taken with the samples

rt\Qunted 180° from the position used in taking data for the stereo-

Slraphic projections, Figures 28 and 29.

V(B) ARBITRARY UNITS

V(B) ARBITRARY UNITS

Figure 28.

100

 

 

  
    

 

8-99 kG
TIP ANGLE
..L_ o
3pV .35.
T -30-
-24°
0 9b I80
YIN DEGREES
(a)
AUPDZ I
B-99 kG
._L
3!” "Roman
"'1"
-72°
r -a4-
I

 

 

0 910 I80

YIN DEGREES
(b)

V(B) vs. w for various values of ¢ for AuPbZI.

101

 

 

 

 

Figure 29. Stereographic projection showing minima for AuPb I.

2

102

dun u a.
__.-__-;__e_I,

 

      
    
 
   

fimn\

  

'.(IIOI

O
0

._____.

.Vnn

mm

(b)

Figure 30. Stereographic projection of HFMR results for
(a) AuPbZII and (b) AuszIV.

 

103

AUPDZY

 

Figure 31. Stereographic projection of HFMR results for
(a) AuPbZV and (b) AuPbZVI.

 

104

The directions of the open orbits are indicated by $
on the stereograms. The symbol (xxx) indicates a broad min-
imum. Table XII summarizes the directions of the open orbits
found in the five samples. Note that there are many open
orbit directions and that some open orbits are seen in some
samples but not others. The reason for this is the cosza
factor in the HFMR which may obscure open orbits in compen-
sated materials unless a is close to 90°. However, some open
orbits still show up when a is as small as 45°, see column 2
of Table XII.

The major directions supporting open orbits are the
<100>, <101>, <110>, <112> and <211>. The [001] could not
be adequately tested for open orbits since a was generally

too small, the largest value being 60° for AuPb VI. A ster-

2
eographic projection showing the major open orbit directions,
their corresponding planes, and the current directions Of

the samples tested is shown in Figure 32. The <100>, <112>,
<211>, and <101> open orbits occur whenever B is anywhere in
the corresponding planes. The <110> open orbits could not

be entirely tested to see if the open orbits result when B
was at any position in the {110}. There was a dip in the MR
for B in the {110} up until 20° from [001] but experimental
limitation prohibited investigation of the remaining 20°.

The Bravais lattice of AuPb2 is body centered tetragonal

and a primitive cell can be generated by the following vectors,

105

Table XII. Open Orbit Direction in Ausz

 

Angle Between Current and

 

Direction* Open Orbit Direction Sample
[100] 80 I
[I12] 54
[112] 63
[5111 as
[021] 84
[441] 65
[100] 78 II
[011] 87
[2111' 83
[302] 57
[010] 75 IV
[100] 81
[110] 73
[110] 86
[121] 78
I511] 56
[130] 77
[100] 75 v
[110] 77
[011] 73
[ial] 66

[211] 86

Table XII (cont'd.)

106

 

 

 

Angle Between Current and

 

Direction* Open Orbit Direction Sample
[010] 83 VI
[110] 60
[011] 73 E“;
[01I] 59 t
[112] 86 E
[112] 45

 

*Direction in reciprocal or k—space.

107

/ ll20

 

 

’ Oll

 

 

 

‘0 OT

Figure 32. Stereogram showing (a) open orbit directions,
(b) current directions, J, of samples and
(c) planes corresponding to open orbit directions.

108

 

E1 = g (i + j) + g-E
E2 = %'(i + j) - g-k
E3=§({-5)-§IE
The reciprocal lattice vectors are 1
R1 = -2n I; + g)
§2=21r (§i+%)
fi3=§1(-i+5)

The reciprocal lattice is also body centered tetragonal but
the square base of the reciprocal lattice is rotated by 45°
about the c axis; Figure 33 shows the situation.

The Brillouin zone is constructed by taking the recipro-
cal lattice vectors and bisecting them with planes. The faces
of the zone are formed by the planes which are closest to the
origin in the direction under consideration. Figure 34 shows
the Brillouin zone for AuPbZ. Note that the <110> and <101>
in the reciprocal lattice are perpendicular to the zone faces
but the <100> and <001> are not. The maximum height of the
zone in the [001] is .8873:1 as compared to 1.111 A-1 which is
§* and thus reciprocal space is filled by placing the zones
next to each other in the <110> in the KxKy plane but shifted
from each other when stacked in the K2 direction.

In constructing a topological model for the open orbits

using cylinders which have their axes along the open orbit

109

REAL SPACE RECIPROCAL SPACE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r v-
a) "xa.
. o ~e’ \y
i . 3:1 622:”
6:361;
95 7T WT
t a :1- k— 20"'--II_L
Jé

Figure 33. Real and reciprocal lattice cells of AuPbZ.

110

 

Figure 34. Brillouin zone of AuPb . Vectors correspond to
directions in k—space.

111

direction, it is evident that the cylinder axes should be per-
pendicular to the Brillouin zone faces. This ensures that

the open orbits are periodic and continue uninterrupted in-
definitely when viewed in the periodically repeated zone
scheme. The obvious choice of cylinder directions consistent
with the experimentally determined FS topology are the <110>
and <101>. There can be no cylinders in other directions
which are perpendicular to zone faces.

The maximum possible diameter of the cylinder was calcu-
lated to be .97 A-1 for the <110> cylinders. The <100> open
orbits can be generated by the <110> cylinders. If B is
taken to be in the (100), the planes J.B cut the <110> cylin-
der making an angle of 45° with the <110>. If the diameter
of the cylinders is large enough, there will be no gap of
unfilled space as one goes from the intersection of one cylin-
der with the plane J.B to the other intersection and thus a
band of open orbits is generated in the [100], see Figure 35.
The <112> open orbits can be generated by the <101> cylinders.
If the electron first moves along the [101] cylinder and then
changes to a [011] cylinder in the next zone and then back to
the [101] cylinder and so on, the net motion is in the [112].
The <21l> open orbits can be produced by alternate use of
the <110> and <101> cylinders.

A check was made to see if the NFE model gave results
consistent with the experimentally determined FS topology.

The Harrison construction program mentioned previously was

worked out for AuPb2 using the room temperature lattice

 

112

LL .~

                  

     

 

..........H..M.«.......¢....u.
(5' I? r ’9 b..'. b
so... Mourns Moor ..,
floor ”'4' ”04' ”'4.
4.».267493054,
C” 4.20745”
S'hilf'lbfihfflh

  

OOI

 

  

_.

O
0
II Em
DI

OR

0

<110> cylinders generating <100> open orbits.

Figure 35.

113

parameters and assuming 18 electrons per unit cell. 1A single
zone scheme was used although there are glide planes and screw
axes for this structure which mean many zero valued structure
factors.

The results obtained by the construction were consistent
with the experimental data. For B parallel to [110] open in
orbits running in the [110] were found in the tenth band for
z = o to z = .15 A'l. Also, for B 55° from [110] but still

in the (110), open orbits were found in the tenth band run-

 

ning in the [110]. For B in (010) and perpendicular to the
[101], open orbits were found in the ninth band from Z = .3
to z = .45 8.1.

AuPb2 cannot yet be said to be NFE-like since these re-
sults were only for a few field directions. However, the
results are encouraging and once the deA measurements are
obtained, the NFE model can be tested thoroughly.

To summarize: AuPb2 is a compensated compound in accord-
ance with its even number of electrons per unit cell. The F5
is open, supporting open orbits in the <100>, <110>, <101>,
<112>, <211> and other higher order planes. The Brillouin
zone geometry allows cylinders of open orbits in the <110>
and <101> and these cylinders seem capable of explaining the
occurrence of all the major open orbits observed. The NFE
model constructed in a few high symmetry directions is con-

sistent with the <110> and <101> directed open orbits.

114

Ausz

Ausz has the C15 (MgCuz) structure which is face cen-
tered cubic with space group 0; — Fd3m. There are 6 atoms
per unit cell. Figure 36 shows the conventional cell with 24
atoms. The lattice constant is 27.93 A as reported by Hansen.(3l)
There has been some controversy about the stability and crys-

(31)

tal structure of Ausz. Hansen states that data concern-

ing the cold working of Ausz "indicates that Ausz is un-
stable below some unknown temperature." Our Laue x-ray photo-
graphs taken after the crystals have been cycled between room
temperature and 4.2°K are identical. Also, photographs of

the same samples taken many months apart are identical al-
though the surface of the samples tarnish after a couple of
days and must be repolished in order to obtain good photo-
graphs.

Figures 37 and 38 show how the MR signal increases with
the magnitude of the magnetic field B for two of the samples
tested. The top curve in Figure 37 (B = 21.7 kG) was taken
with the Keithley 147 on the 30 uV range while the other data
was taken on the 100 uV range. Figures 39 to 43 show plots
of the HFMR for the remaining samples tested and give the value
of the coefficient m at various points. The data, especially

the data for Au PbIIb which has the highest RRR, indicate that

2
the FS of Ausz supports open orbits.
The fact that the FS supports open orbits is most clear-

ly demonstrated by Figure 40. The field dependence of the

 

115

 

 

Conventional cell of Ausz.

 

 

 

 

Figure 36.

116

AoZPb I

W 2L7 k6
W 44.8%

 

 

(D
E:
Z
3
)-
tfi 70.0kG
a:
t:
92 96.0 k6
'd
E
>'
O 90 |80
9 IN DEGREES
Figure 37. V(B) vs. w for various values of B for AuszI

AuZPb'JI

 

 

:2
2 8L5 RG
2)
).
fi 66JkG
a:
|—
a V
5 3"”
,. ‘T" 47.6 RG
9 V
:>

o 90 ' l80

v IN DEGREES

Figure 38. V(B) vs. w for various values of n for AuszII.

117

 

 

 

 

 

 

 

 

Ausz Ib
2 O '- m=l.0
'1 m=I 2
m=l.2 =99KG
I 5 B=88KG
V /
A P /
T _~. m= 0.8
mJOoG "1-0.9
m= 0.6
5 .-
l ..
l l
O 90 l80

WIN DEGREES

Figure 39. lip/po vs. up for AuszI.

118

 

 

 

 

 

 

 

 

 

 

 

 

AuszIEb
A P m= L4
‘4‘)... ‘ nn=l£1
m=l.3
B=99 KG
/s=a4
nn=<3£5 f
m=0.6
' 1 1
O 9 0 ISO

4' IN DEGREES

Figure 40. Ap/po vs. w for AuszIIb.

 

 

119

 

?
m- 0'9 m= 0.7

 

9 90 IBO
* m DEGREES
(a)
B-99KG

 

 

 

AP "-
l _. AfifEII
— m-LO for all directions
5 _
' _
1 I
O 90 ISO
9 IN DEGREES
(b)

Figure 41. Ap/po vs. w for (a) AuszIII and (b) AuZPbV.

5.

120

    

   

 

 

 

 

 

 

AuszIE
nwL3
2()t: UFK3
— "FL3
- /\ \ B=99KG
l 5 -
I— \\ B=B7KG
[Lg — \I
3 F \V
l O
In=KJ
5 I
I - I l
O 90 I80
4' IN DEGREES
Figure 42. Ap/po vs. w for AuszVI.
AthbIDII
m=I.3 m=|3
20 I' .
: /‘ B=99KG
- tn=L2 \
' / s-G7KG
‘- .
.5; p. \
R \ /
I J ,
46E. i fl/~\ Kkj/
~13: / ' ‘ ,
IO -U U
:\j rn=LO
m=Q8 _
I- m-O.5
5 : m=0.6
I b I L
O 90 ISO
W IN DEGREES

Figure 43. Ap/po vs. w for AuZPbVII.

‘E‘

121

HFMR has values ranging from m = 1.5 to m = 0. Such an oc-
currence can only be explained by the presence of open orbits.
If the compound is compensated, the field dependence of the
HFMR should approach 2 and m will 2 0 only if there are open
orbits present. On the other hand, if Ausz is uncompen-
sated, m should saturate for a general field direction and
approach the value 2 when B is in a direction producing open

orbits. Thus, whatever the state of compensation, Ausz has

.1
I
I.
O

a FS which supports open orbits.

In order to determine the directions of the open orbits,
the crystals were tipped through various angles ¢ and the
positions of the minima plotted on a stereographic projec-

tion. Figures 44 to 46 show the data for the Au PbIIb sample

2
and Figure 47 shows some data for AuszIb and Au PbIII while

2
Figures 48 and 49 show the stereographic projections.

The symbols used in the stereographic projections are
similar to those used for the AuPb2 samples. The X's desig-
nate minima,j§( designates an open orbit direction, and the
planes which connect the minima are drawn with dash lines.
The principle directions and planes are drawn with solid
lines and are taken from the Laue x-ray photographs. There
was no way of aligning the Laue photographs with some charac-
teristic minima as was done for the AuPb2 samples.

Looking at the stereographic projections, it is evident

that the minima lie on planes but the planes do not corres-

pond to the major crystallographic planes, {100}, {110},

122

 

 

 

AuszIIb
8-85 KG
/f\\ l/P\ E%§E
\ fl“ WI
/ fi I 1 I “ 4&5
./ .‘f/é \|‘\\ I W
E 1]) AM, \ \Il-ll'
éhj—nao v: \V/\ 'J/ \j- 5.5
g n—[E‘QH'I‘ I: i’ \ INN T):—
g / "”71". "J/ I/ :4.
I] .‘ o
r—zERo o 1 1
o «r lav +
(a)
A bIIb
33.6
/A\ ’f\ A AEELE
J \ / \ [VV\ / :f'a5
‘ p I l/ I / U
V p/ d m VA I:
r: : ' C / ' T
g 2ER0I6L1/\\\\' \I / \ '
E 'IJ J I ‘V,’ V
E I {I I / \
g ZERD n/- I/ ‘J \j 55
- ZERO 55‘ 1 L
o- «r uGo-,

 

 

(b)
Figure 44. V(B) vs. w for tip angles ¢$|16.5°|.

‘Figure 45.

123

 

W81 ARBITRMY UNITS
2 3
*IE F

AuszIIb

ANGLE

 

n- ZER

U x/
I

V(B) ARBITMRY UNIT 8

 

tZERO 22'
o.

I J
L33. QM“

J
90'
(b)

M‘f

:ZEBQZZ 1 1—
0° 90° IOO' w
(a)
A bIb
3 “6 Tm
¢

v ,1 m
A W

\/ g:

V T

1
mo' v

V(B) vs. w for tip angles ¢5|33°|.

Figure 46.

V(B)

124

AuZPbIL'b
3:87 KG

 

TIP
ANGLE

 

 

 

m
.—
i 55.
D
E \j/\/
4
E
O
5
§ . .'
> t-z R055° W 66'
UV
ZERO 66’ 1 1
0‘ 90° IBO' 1!
(a)
h Au PbIIb
a-a7xe
TIP
/\ J
Wf‘N/VJ / ‘9"
-44'
m: Q9. A
\A/xf/ \U
A .5 -385-

U W _.:
p—ZERO44 A?”

WIARBITRARY UNITS

 

m... L. "$0....
o- 90° '
(b )

vs. w for tip angles ¢$|66°|.

125

 

 

 

 

 

 

AflfiapbjIE
B=IOO kG
TIP A‘GGLE
a.
I? .2 -
'z'
3
> no °
‘3: ~.
E
65 25°
tr
4
a so °
5 .J—
IOpV
T
D 90 ISO
YIN DEGREES
(a)
Ausz Ib
B-IOO kG
m ANGLE
IOZV
m
r.-
z
D
).
D:
<1
G:
.—
_ tr
4
90"” I80
‘1’ IN DEGREES

(13)

Figure 47. V(B) vs. w for various values of ¢ for (a) AuszIII
and (b) AuszIb.

1.—

 

128

<lll>, and <112>. Thus the open orbit directions do not lie
along the symmetry axes in the crystal as would be expected.
Faced with such a development, the positions of the maxima
were plotted since if the compound was uncompensated the open
orbits would be manifested by maxima rather than minima. Two
such attempts, for AuszIIb and AuZPbVII gave worse results
than plotting the minima.

The data obtained by tipping the crystal is thus hard to

. WGTW’

understand. However, the sharpness of the dips observed in 5'
the AuZPbIIb sample strongly indicates that AuZPb is a com-
pensated compound. Sharp dips can only be observed in an
uncompensated compound when B is in a singular field direction.
Such directions are the <100>, <lll>, and possibly <110>
for a cubic structure. The sharp dips are observed in direc-
tions other than those just given and Au2Pb must be compensated.
The stereographic projections do show some consistent
patterns. The direction of the open orbit lies on a plane
which is formed by minima associated with another open orbit.
Most of the open orbit directions lie close to 90° from the
current direction but their directions seem to be random al-
though AuZPbIIb seems to have an open orbit in the <lll>
and <100>.
One systematic trend not shown on the stereographic pro-
jections can be seen from the raw data. Figure 47a which
shows the MR signal for AuszIII at various D, shows the dis-
appearance of a minima as the tip angle increases. Figure 45b

and Figure 46b also show minima which get rather shallow

129

compared to the minima at tip angles on either side. Such
behavior, one or more deep minima which gets very shallow for
certain tip angles but is relatively deep for other tip
angles, has been observed for all of the crystals which have
been studied by the tip method (AuszV was not studied) ex-
cept for AuszII - not to be confused with AuszIIb which is
a different sample. However, the positiOn of this shallow
dip varies from sample to sample.

There are some very puzzling anomalies aside from the
open orbit directions. The HFMR of AuszII is shown in
Figure 38 for ¢ = 0. This sample was also measured for var-
ious other angles of ¢, although the data is not shown. The
data for all tip angles was lacking in the characteristic
four minima pattern of the other samples. This crystal,
AuszII, and AuZPbIIb were cut from the same sample. The
initial sample was approximately 1/8" in diameter and 4" in
length. Both samples were cut with the current axis approx-
imately parallel to the sample axis. The difference between
the current directions of the two samples should not be more
than 5°. However, the HFMR is drastically different. A
small difference in current directions can produce results
which differ markedly (SnI45) is an example), yet it is dif-
ficult to understand why this type of behavior occurs in
Ausz.

Another sample showing anomalous behavior is AuZPbV.

Figure 41b shows the HFMR for ¢ = 0. This data also shows

little anisotropy. The RRR of AuZPbV is not much different

‘n-nmwveaaawmur

 

130

from that of AuszIII, yet AuszIII shows characteristics of
the other samples. No data was taken for AuZPbV at other
values of ¢.

In order to check that the sample holder was operating
properly and that the voltage probes were aligned properly,
rotation plots were taken with w going from 0 to ~270°.

Such plots were taken on all three compounds studied. The

data for the AuGa and AuPb2 (not shown) samples had the fol-

mar-HEN

lowing characteristics. The dips were observed to be sepa-
rated by l80° i 2° as is expected. The magnitude of the sig-
nal sometimes was observed to vary by about 10% for two posi—
tions separated by 180°. This behavior indicates that the
sample holder rotates about its AA' axis (see Figure 9)_L B.
The difference in magnitude of the voltage signal is probably
due to a slight difference in the magnitude of the field at
the two positions or to probe misalignment.

For the AuZPbIIb sample, Figures 50a and 50b show fea-
tures which are unique. For ¢ = 0, the small dip seen in the
first maximum is not repeated when w is 180° away. For
¢ = 11°, the entire appearance of the first peak is quite dif-
ferent from its appearance 180° away. For ¢ = 37.5°, the dips
and peaks are periodic in 180°. For ¢ = 22°, the first dip is
fairly well reproduced 180° away, while the three dips at
~¢ = 115° are slightly distorted at w = 290°. The data for
the sample indicates that Ausz is extraordinarily sensitive

to the position of B, more so than either the AuGa or AuPb2

samples.

V(B) ARBITRARY UNITS

 

0° 45° 90° 135° l80° 225° 270°

VB) ARBITRARY UNITS

 

l

l l _1 l I o 1 o y
0‘ 45° 90° |35° I80" 225 270 9 (b)
Figure 50. V(B) vs. I for AuszIIb for (a) ¢ = 0°, 11°
and (b) ¢ = 22°, 37.5°.

132

The cause of such anomalous behavior can only be of a
speculative nature at this time. The validation of the fol-
lowing explanation requires additional data on samples with
current directions of high symmetry.

1. In Figures 51a to 51d, the stereograms showing open
orbit directions for four different cubic FS topologies are I
given. The darkened regions are field directions giving rise E:
to aperiodic open orbits and the planes connecting these re- 5‘
gions are field directions which produce periodic open orbits. f«
The undarkened areas are thus regions which support no open
orbits, the so-called general field directions. These dia-

(40)

grams are taken from the paper by Lifshitz and Peschanski .
Although J. T. Longo (46) has pointed out that the analyses
underlying these diagrams is not of general validity, the dia-
grams do illustrate points which are of general validity

and these points are the ones which will be discussed.

First, open orbits are produced when E lies in a high
symmetry plane or when B is near a high symmetry axis. How-
ever, if the diameter of the cylinders which serve as the
model for calculating the open orbit directions is varied,
the area of the darkened regions will vary. Figure 51d shows
a diagram where almost 3/4 of the field directions produce
open orbits. By varying the diameters of the cylinders, the
other diagrams could also be made to have greater open orbit
producing regions.

Now, in a compensated material Ap/po ~ B2 cosza where a

 

133

(b)

(a)

 

 

nt cubic FS topologies.

and Peschanski, Ref. 40).

(c)

it directions for four differe
(From Lifshitz

51. Open

Figure

134

is the angle between the open orbit direction in k—space and
the current direction. In a region where there are aperiodic
open orbits, the direction of the open orbit is determined
by the line of intersection of the plane.1.B with the plane
perpendicular to the symmetry direction about which the aper-
iodic region is formed. Since a will not in general be equal
to 90°, the open orbit may be difficult to pick out in a com-
pensated compound since both open and closed orbits will

have a B2 dependence. As the regions supporting open orbits
increase, as in Figure 51d, the difficulty of identifying two
dimensional open orbit areas increases.

It is not generally difficult to identify periodic open
orbit directions, the lines or planes in Figure 50, but as
the two dimensional regions increase in area, the one dimen-
sional regions decrease and extend over a smaller angular
region. If there are no aperiodic open orbit regions, the
periodic open orbits will generally exist for the entire 180°
angular region. However, when aperiodic open orbits exist,
the angular region for periodic open orbits which connect two
aperiodic regions can be as small as 10° as in Figure 50d.
Thus even periodic open orbits can be difficult to identify
when the extent of the aperiodic open orbit region is large.

Thus, a possible explanation of the data on AuZPb is
that the FS of Ausz is such that there are very large areas
of aperiodic open orbits formed which make the identification
of the open orbit directions difficult.

2. A second possibility is the charge wave theory of

 

135

Overhauser which was proposed to explain anomalies in the HFMR

of potassium.(48)

This theory has evoked mixed reactions and
it is still on relatively uncertain ground. Briefly, the
theory assumes that a charge density wave of wave vector

6, (JQI > 2kf) exists and that 5 can be oriented by an ex-
ternal magnetic field. Since |Q| > Zkf, the charge density
periodicity does not change the connectivity of the FS. How—
ever, additional energy gaps appear due to beat or heterodyne
periods arising from the interaction of the charge-density and

lattice periods. These additional energy gaps are produced

primarily at k vectors which satisfy the equation

i=2"§hk2 : Q

where éhkl is a reciprocal lattice vector. Since these addi-
tional energy gaps depend on the orientation of 5, which in
turn depends on the external magnetic field direction, the
trajectories which the electrons follow will not only depend
on the FS but will also depend on the orientation of the mag-
netic field.

The charge density wave theory could thus explain the
non-symmetry direction for the open orbits as well as the
fact that no two experimentally determined open orbit direc-
tions coincided with each other.

3. Magnetic breakdown (MB) can produce many strange ef-
fects. Electrons undergoing closed orbits can form open tra-
jectories because of MB and vice versa. The quantity which

is usually most affected by MB is the field dependence of the

HFMR. No dramatic changes of field dependence have

 

136

been observed, but magnetic breakdown manifests itself in so
many subtle ways that it cannot be ruled out as a contributing
factor to the present problems. The prediction of what ef-
fects MB may produce when the Fermi surface is unknown is
extremely difficult.

4. It is possible that the HFMR depends drastically on
the composition of the samples. Since Ausz is a peritectic
alloy, the composition of the initial samples varied over a
region of ~10 at.%. Also, as the crystals were being grown,
concentration gradients were undoubtedly produced along the
length of the samples. Thus, if the width of the line at
composition Ausz in the phase diagram is of finite size,
say 1 or 2 at.%, and varies with temperature, the concentra-
tions of the various samples used could have been different.
Now, one must show how this difference in composition can
alter the HFMR.

One possible effect is the following. Suppose that the
FS of AuZPb just misses making contact with a Brillouin zone
face. If, by altering the composition of the compound, more
electrons per atom can be introduced, as would happen if Pb
atoms replaced some Au atoms, then the FS could be made to
touch the zone face and thus produce open orbits. Thus, the
FS would be either open or closed depending on the concentra-
tion of the samples. However, such an argument does not ex-
plain the non-symmetry directions for the open orbits since
the Brillouin zone faces are perpendicular to symmetry axes

and thus the open orbit directions should be in symmetry

 

 

137

directions.

5. There is a finite possibility that the crystal struc-
ture of AuZPb is not cubic at 4.2°K. A martensitic transfor-
mation could occur at low temperature and the resulting crys-
tal structure could be a non-cubic structure. The HFMR data
depends on the geometry of the lattice and thus the data
would not show cubic symmetry.

There are two facts which seem to indicate that a mar-
tensitic transformation does not occur. First, the AuZPb
samples have a relatively high RRR indicative of high crystal-
line perfection. A phase change would probably produce poly-
crystals and grain boundaries in the sample and these imper-
fections would lower the RRR appreciably.

Secondly, since the Laue photographs are identical after
recycling between room and liquid He temperature, the phase
change would also have to produce crystals which have the
same orientation as the initial samples. Such occurrences
seem unlikely.

6. There is a possibility that the data is an arti-
fact of the measurement procedure. Since the solder used to
attach the voltage leads does alloy with the sample, the HFMR
data may be an indication of probe effects. The possibility
cannot be entirely ruled out, but the similarities in the data
for different samples would seem to make it an unlikely ex-
planation.

To summarize, Ausz is a compensated compound and its FS

supports open orbits. The direction of the open orbit do not

138

seem to run in symmetry directions which is at variance

from the expected behavior. There are some consistencies in
the data but there are also anomalies. Some possible explana-
tions have been proposed and evaluated, but the verification

of these ideas requires more experimental work.

 

VII. CONCLUSIONS

The F5 topology of three compounds, AuGa, AuPb2 and
AuZPb, has been studied by HFMR. Each compound represents a
crystal structure which has previously not been investigated if‘1
by HFMR. Since most intermetallic compounds have an even ~

number of atoms per unit cell, the compounds will most like-

 

ly be compensated. The three compounds studied in this re—
search were found to be compensated and the experimental pro-
cedures and techniques used in this research should serve as
a guide to further study of compensated compounds.

The F8 topology of AuGa cannot be explained by a NFE
model assuming Au contributes l electron/atom and Ga 3
electrons/atom. There are open orbits in the [001] but no
open orbits in the [100], even though the lattice constant
differs by only 2% for the two directions. The NFE model
gives open orbits in both directions which is at variance
with the experimentally determined directions. A double zone
scheme was also tried but again the NFE model did not give
results consistent with the experimental findings.

The F8 of AuGa is thus not NFE-like. It is possible that
the FS touches the Brillouin zone face in the [001] but not
in the [100]. The necks in the [001] direction could pro-
duce the open orbits Observed while their absence in the [100]

would explain the fact that there are no open orbits in the

139

 

140

[100]. The distortion of the F8 from the NFE sphere would
be similar to the <lll> neck distortion observed in the noble
metals Au, Ag, and Cu. Another explanation for the non-NFE
like behavior is that there is chemical binding effects not
taken into account by the NFE model.

The F5 topology of Ausz has been investigated in some
depth with the [001] being the only major crystallographic
direction not able to be tested for open orbits. Open orbits
were found in the <100>, <110>, <101>, <112>, and <211> in
k-space as well as some other higher order directions.

A model of the Ausz open FS is proposed which consists
of cylinders running in the <110> and <101> in k-space. This
model seems capable of explaining all the major open orbit
directions. The Harrison construction was performed for a few
magnetic field directions; 18 electrons per unit cell were
assumed. The results of the construction are in agreement
with the experimentally observed <110> and <101> open orbits.
Thus the FS of AuPb2 may be explained in terms of the NFE
model, however, more work is needed before the NFE model can
be verified.

The Ausz results are difficult to interpret. The open
orbit directions appear to be along non-symmetry directions
which is different from the expected behavior. Possible ex-
planations for these peculiar results are proposed.

The high degree of crystalline perfection attained in
the Ausz and AuPb2 samples makes them attractive candidates

for study by the deA method. The results of the deA

yw.* - n.
i.

 

141

measurements would detenmine whether AuPb2 is NFE like. Also,
these measurements could clear up the anomalous behavior ob-

served in Ausz.

 

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7.
8.
9.

10.

ll.

12.

13.

14.

15.

16.

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a.
. g. n «.1...
_‘. Ha-

 

S
E
C
I
D
N
E
P
u

APPENDI X A

The Relaxation Approximation

In the relaxation approximation the collision term is
given by

(3:) = - (fvfo)
at I C011 IIE)

 

where 1(2) is the relaxation time which can depend on the
energy and on the wavevector k.

In using this approximation it is assumed that if the
distribution function f differs from the equilibrium Fermi
function f0, then it will decay exponentially in time to £0.

The Boltzman equation can then be solved for J and the
solution is given by

E(u) = ate-GUIu eau'v(u')du'

where

1 1

a = lelBT - wchH

The conductivity tensor becomes

 

2 mn 2n w
Oij “I3 87 I ”CIIO (oviw'kzh’jh‘ I1 )9 dudu }dkz

which is the Shockley tube integral. This integral is appli-
cable in both high and low fields. In high fields where

146

 

147

wct >> 1 and the periodicity of the electron orbits allows
for simplication, the coefficients of the conductivity tensor
can be calculated if the velocity of the electrons in k—space
as well as the variation of the relaxation 1(a) is known at

each point on the electron orbits.

V

 

APPENDIX B

The Superconducting Magnet

A RCA Type SM2830 superconducting magnet which is
capable of reaching 103 kG was used to produce the magnetic
field. The magnet is made from Nb3Sn superconducting rib-
bons and the magnet has a usable bore of slightly greater
than 1.5". The magnet could be swept at a rate of 2A/min up
to 72A and then at a rate of 1A/min up to 89A, the maximum
current rating.

The magnetic field was measured by a magneto-resistive
coil which is permanently wound on the bore in the center of
the magnet. The magnetoresistor was calibrated by Dr. G. J.
Edwards and Dr. P. A. Schroeder using NMR measurements on
H1, F19 and A127. DHvA measurements on potassium provided
field profiles. The position of the maximum magnetic field
was found to be within 0.1" from the geometric center of the
magnet at 98 kG.

All measurements indicate that the field at the center
of the magnet is axial. The symmetry of the field in the
radial direction was also checked by C. K. Chiang and found
to be symmetric to within experimental error. As a final
check, the HFMR cryostat was rotated by 120°, and 240°, about

the axis of the magnet. The HFMR of the AuPbZI sample was

148

mmrmlw

149

taken with w varying from 0° to 200° for all three direc-
tions. The three plots were identical, showing sharp dips at
exactly the same value of w. Thus the magnetic field has

very little asymmetry at the center of the magnet.

I" :2. -II
If

 

 

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