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This is to certify that the

thesis entitled

Magnetothermal Oscillations in

Graphite Intercalation Compowids

presented by

James Eadus Huffman

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in PhYSiCS

Cit/KW

I Messor

 

Date June 7, 1983

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOlD FINES return on or before date due.

 

DATE DUE D

ATE DUE DATE DUE

 

Wow

4

 

 

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x-~ A"???

j

MAGNETOTHERMAL OSCILLATIDNS IN
GRAPHITE INTERCALATION COMPOUNDS

By

James Eadus Huffman

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1983

ABSTRACT

MAGNETOTHERMAL OSCILLATIONS IN
GRAPHITE INTERCALATION COMPOUNDS

by

James Eadus Huffman

The Fermi surface (FS) of SbCls-GIC's for stage 1 to stage 8 has
been studied using the magnetothermal oscillation method. The majority
of the measurements were performed at about 1K and in magnetic fields up
to 5T (SOKG). Frequencies of the deA oscillations were determined by a
fast Fourier transform analysis of the raw data, using an on-line
computer. The results show a monotonic increase in the area of the FS
with decreasing stage index and a weak monotonic increase of the charge
transfer coefficient, f, with increasing stage index. The shape of the
FS, as deduced from the angle dependence of the frequencies, is
consistent with a cylindrical surface or an elongated ellipsoidal one.
For the stage 6 sample we have also determined the effective mass of the
carriers in the plane perpendicular to the c-axis, and find that
m* = 0.3 mo, in agreement with calculated masses based on the
Slonczewski-Neiss-McClure (SNMcC) model of the band structure of
graphite. Our results are in good agreement with a model for the band
structure of GIC's based on the rigid band model (RBM) when zone folding

along the c-axis is employed.

 

DEDICATION
I gratefully dedicate this work to my wife, Ann, and to the rest of

my family.

ii

ACKNOWLEDGMENTS

It is a pleasure to acknowledge the guidance and helpful insight of
Professor F.J. Blatt. I have also benefited from numerous discussions
with Professors S.A. Solin, S.D. Mahanti, and P.A. Schroeder. There are
many others, too numerous to list, to whom I owe thanks, but in
particular D.L. Edmunds who has been instrumental in setting up the
computer interface. Finally, I sincerely thank my typist, 8. Lindsay,
for her remarkable tolerance, humor, and speed.

The financial support of the National Science Foundation is

gratefully acknowledged.

 

TABLE OF CONTENTS

Page
LIST OF TABLES ................................... . ........ ...........vi
LIST OF FIGURES .................................... ........ ....... ..vii
INTRODUCTIONOOOO000.00....000.00.00.00...0.00.0000... 000000000 O 00000 001
CHAPTER ONE DESCRIPTION AND MEASUREMENT OF THE FERMI
SURFACEOOOOOOOO00OOOOOOOOOOOOOOOOOOOOOOO 000000000000000 2
1.1 Introduction ..... ........ ..................... ....2
1.2 The Fermi Surface.... ...... ........... ......... ...2
1.3 Theory of the Experiment.... ................. ....3
1.3.1 History and Physical Origin of
Quantum Oscillations. .... ..... . ........... 3
1.3.2 ThermOdynamic Ana1ys‘iSooo0.09000000000000008
1.3.3 Discussion of the Result..................12
1.4 General Comments.... ........... ..... ............ .15
CHAPTER TWO EXPERIMENTAL METHOD ....... . ......... ....... ......... ..16
2 1 Introduction. ..................... .. ....... ......16
2.2 Experimental Cryostat ...... . ...... .............. .17
203 [dagnetic Fie‘d” 00000 000 00000000000000.0000. 020
2 4 Measuring Small Temperature Oscillations......... 22
2 5 Data Acquisition and Analysis. .... ..... . ..... ...26
CHAPTER THREE PROPERTIES AND SYNTHESIS OF GRAPHITE
INTERCALATION COMPOUNDS........... ......... . .......... 33
3.1 IntrOdUCtiono00000990000000.0000 OOOOOOOOOOOOOOOOO 33
3.2 Review of Graphite Intercalation
Compounds. ........... ........................ 33
3.3 Preparation of SbCl5 Intercalated
HOPG 0.0000000000000000.0.000.000 00000000000000 38
3.4 Characterization....................... ......... .41
CHAPTER FOUR GIC BAND STRUCTURE AND FERMI SURFACE MODELS ....... ....46
4.1 Introduction. . ............. ..... ........ . ..... 46
4. 2 Graphite Band Structure and Its Fermi
surface 00000 00.00.00.000 OOOOOOOOOOOOOOOOOOOOOOOO .46

iv

4.3 GIC Band Structure and Fermi Surface
Models ........ . ................. . ......... ...51
4.3.1 The Rigid Band Model. .................... 51
4. 3. 2 The 20 Band Structure Model. ........... ...55
4.4 Comments ....... ................. ................. 59
CHAPTER FIVE EXPERIMENTAL RESULTS AND ANALYSIS .................... .60
5.1 Introduction............ ........ . ....... . ........ 60
5.2 Fermi Surface Results.. ....................... ...60
5.2.1 HOPG......................................62
5. 2. 2 Stage 1 SbClS.............................63
5.2.3 Stage 2 SbCls............... ...... ........63
5.2.4 Stage 4 SbCl5.............................66
5. 2. 5 Stage 5 SbCl5.............................66
5.2.6 Stage 6 SbCl5..... ....... .................68
5. 2.7 Stage 8 SbCl5..... ...... ... ....... ........71
5.3 General Observations and Oiscussion..............71
5.4 Predictions from the Fermi Surface Models ..... ...74
5.4.1 The Rigid Band Model........ ..... .........75
5.4.2 The 20 Model....... ........... ... ....... ..81
5.5 Conclusions ..... ... ..... ..... ............ ........87
CONCLUSION....... ................................ . ................... 89
LIST OF REFERENCES ........................... . ...... . ................ 90

3.1
4.1
5.1
5.2
5.3

5.4
5.5

LIST OF TABLES
jgggg
Preparation Temperatures and Ic Values for SbCl5 GIC's..........4O
Slonczewski-Neiss-McClure Band Parameters for Graphite..........49

Experimental Results on HOPG and SbCl5 Compounds................61

Sharing of Frequencies in a Mixed Stage 5 and 6 Compound ....... .67
Published Results from deA and MTO Experiments and

the Present Results on SbCls.......... ....... . .................. 73
RBM Predictions and Observed Results ................. ....... ...78

EF as Determined by MTO and RBM, and Values from Other
Experiments .......... 0.0.... ...... 00.000.00.00. 0000000000000000 .82

vi

Figure
1.1

1.2
1.3
1.4
2.1
2.2

2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3

LIST OF FIGURES

53553
Extremal Cross Section of an Elliptical FS Segment ..... ....6
Elliptical FS Segment and Landau Levels..... ............ ...6
Thermodynamic System......................... ....... .......9
MTO and deA Amplitude Dependence on m* ................ ...14
The MTO Cryostat............................ .............. 18
(a) Schematic of the Magnetic Field Sweep
Control System........................ ....... ........21
(b) H versus t for a l/H Sweep........ ........ . .......... 21
Schematic of the A.C. Bridge Circuit ...................... 24
Computer Interface Block Diagram ......................... .27
MTO Data, Stage 4 SbCls ..... ...... .......... ..... ...... ...29
Fourier Transformed Stage 4 MTO Data ....... . ........... ...30
MTO Data Taken in Manual Mode......... ..... ...............32
Schematic of the Hexagonal Form of Graphite (HOPG) ...... ..34
Schematic of Staging................ ..... .................35

3353] P1ane order in SbC‘SOOOOOOOOOOOOOOOOOOO000000000000O37

Reaction Tube for SbCl5 GIC Synthesis.... ...... . ....... ...39
002 X-Ray Diffraction Scan, Stage 1 SbCl5.................42
Stage 1 Mosaic Spread........ ...... . ........... . ....... ...44
Graphite Brillioun Zone and FS............ ..... . ....... ...48
SHMcC 3D Graphite Band Structure .......... ...... ....... ...48
Schematic of RBM FS... ............ .... ......... . .......... 53

vii

5.1
5.2
5.3
5.4
5.5
5.6
5.7

Stage Dependence of SbCl5 GIC FS.... ...................... 64
Rotation Dependence of deA Frequencies .................. .65
MTO Amplitude versus Temperature. ..... . ...... . ............ 69
MTO Amplitude versus 1/H.............. .................... 7O

RBM deA Frequency versus Ef, Stage 2 SbCls...............77
ZD FS Construction, Stage 1 SbCl5.........................83
20 NFE FS Construction, Stage 6 SbCl5.....................86

viii

INTRODUCTION

This dissertation describes a study of the Fermi surface (FS) of a
group of graphite intercalation compounds (GIC's) using the magneto-
thermal oscillation technique. The subject of GIC's spans many
scientific fields. These compounds have been heavily researched in the
last few years and considerable progress has been made toward an
understanding of the nature of these synthetic, laminar materials.
However some topics, one of which being the electronic structure, remain
controversial. In recent years conflicting FS data on GIC's has been
published. As well, fundamentally different models have been proposed
to describe the electronic structure of GIC's. We have made a careful
experimental study of the F5 for one group of these compounds and have
analyzed our results on the basis of two competing theories. We find

one of these is in better agreement with our observations.

«in

L ..M J: m-

.. ”M A‘ ,
,v. ..
444~14.:?1“4<H it.

 

CHAPTER ONE
DEFINITION AND MEASUREMENT OF THE FERMI SURFACE

1.1 Introduction
In this chapter we will briefly discuss the concept of a Fermi
surface and the quantum oscillatory effects used to experimentally

determine its geometrical properties.

1.2 The Fermi Surface

A metal could be defined as "a solid with a Fermi surface."T Thus,
a complete description of the Fermi surface of a solid is essential to
an understanding of its electronic (metallic) properties. In this
section a brief description of the Fermi surface and its utility is
given.

The Fermi surface (FS) is the surface of constant energy in momen—
tum space or 'k-space' within which, at T = OK, all occupied electron
energy levels are contained. Hence, the FS is not a surface in real
space, but in an abstract, reciprocal space. For a free electron gas,
the FS is a sphere whose radius is determined by the number density of
electrons. The occupied electronic states are distributed uniformly
within the sphere. For a material with a crystalline structure and
containing valence or conduction electrons, the shape of the FS is

lnodified due to the periodic lattice potential. The FS is generally
Sshown within the k-space image of the Wigner-Seitz cell of the crystal,
i.e. the first Brillioun zone (see references 1 and 2). Given a full

description of the FS, properties which depend on the electronic or

TA.R. Macintosh.

quasiparticle distribution, such as transport coefficients and equili-
brium and Optical pr0perties can be calculated.2

The Fermi surface for a material can be calculated from a one-
electron energy band structure of the system, as in section 4.3, then
compared to an experimental measurement of the FS. We now turn to the

method by which this surface can be determined experimentally.

1.3 Theory of the Experiment

In section 1.3.1, we present an explanation of the origin of quan—
tum oscillatory effects and conclude the section by giving a formal
expression for the oscillatory free energy of a Bloch electron gas, as
derived by Lifshitz and Kosevitch.3 This formal expression is related
to a measurable quantity through a thermodynamic analysis in section

1.3.2. The final result of this analysis is discussed in section 1.3.3.

1.3.1 History and Physical Origin of Quantum Oscillations

Quantum oscillatory effects were first observed in 1930 by N. de
Haas and P. van Alphen.4 They discovered oscillations in the magnetic
moment of Bismuth as a function of magnetic field intensity. The value
of these observations in providing FS data was not fully realized until

almost 20 years later when Onsager develOped the relation,5

1 _ 21re 1

In this equation, AC%) is the change in l/H through one complete oscil-
lation, and Aext is the extremal cross sectional area of a segment of

the FS perpendicular to the magnetic field direction (see Figure 1.1).

4

The reciprocal of A(%) is called the de Haas-van Alphen frequency, and
is written as F. This remarkably simple expression holds for all quan-
tum oscillatory effects including, among others, the de Haas-van Alphen
effect, magneto-thermal oscillations, and the Shubnikov-de Haas
effect. Basically, the only difference between these phenomena is that
in each case a different measurable quantity exhibits the deA oscilla-
'tions. For the magneto-thermal oscillations technique (MTO), one
detects lattice temperature oscillations; in the Shubnikov-de Haas
technique (SdH) one detects oscillatory behavior in the electrical
conductivity.

We now give a quasi-classical derivation of Equation 1.1. We begin
by assuming that charged particle orbits in a magnetic field are quan-

tized according to the Bohr-Sommerfeld relation,
P . d? = (n + y) Zwfi'; n = integer, y = constant. 1.2

éorbit

The total particle momentum, P, in a magnetic field is given by

P =IHR + qA/c ; H = curl A,’hR = kinetic momentum 1.3
and when used in the above expression gives
5375. dF=——2%¢+q/c<b=-q/c<i> 1.4

where e is the magnetic flux enclosed in the real space orbit.
Combining Equations 1.2 and 1.4 and substituting (-e) for q,

) 2nhc
e

¢n = (n + Y 1.5

which quantizes the flux through the real space orbits. Real space and

reciprocal space (k—space) trajectories are related through a scale

why?» a

factor, 6r/6k = hc/eH, and a rotation of n/Z so that k-space orbits are
also quantized, i.e.,

An = (n + y) Zne/fic H 1.6
where An is the area in k-space enclosed by the orbit with quantum
number n.

Using this we find that consecutive orbits n and n + 1 of equal

area are separated by fields given by

ufiwmfi-Hfiwfigfi 1.7
which is the Onsager relation (Equation 1.1) provided that An = Aext'
The quantized k-space orbits of Equation 1.6 are shown in Figure 1.2 for
a fixed magnetic field value. These orbits are confined to the
cylinders oriented along kz, for H parallel to 2. Also shown in this
figure is a hypothetical, ellipsoidal Fermi surface. Recall that all
quasiparticles must reside within this constant energy surface, and
furthermore, are now required by Bohr-Sommerfeld quantization to be
distributed on the highly degenerate Landau levels within the Fermi
surface, for H r O. From Equation 1.6 we see that as the magnetic field
intensity is increased, the k-space orbits will also increase in area
and move toward the FS. When one of the orbital areas, say An, passes
through the Fermi surface at an extremal FS point, a maximum number of
quasiparticles on that level must simultaneously make transitions to
lower area Landau levels. This requires the free energy of the particle
system as a whole to change abruptly when An = Aext' In addition, these
energy changes are spaced equal increments of l/H apart. Thus, with the
scflastitution of An = Aext in Equation 1.7 we have the Onsager relation

(Equation1.1).

N A1/H
A ext

 

Figure l.l Extremal cross section of an elliptical FS segment.

 

Figure l.2 Elliptical FS segment and Landau levels.

 

A general and formal result for the oscillatory free energy of a

Bloch electron gas, derived by Lifshitz and Kosevitch is,3

2Vk BT(EH)3/2

1.8
32A(E: ,k z) 9?

F2
2

 

 

3k ext

 

where,
V is the volume of the system

T is the temperature in K

 

 

32A(EF,kZ)1’/2 . .
15 the curvature of the Ferm1 surface along the
akz ext field direction, evaluated at the extremum.
mg is the cyclotron effective mass = h2/2r‘%é EF’ ext
w:=—§‘Ii
mcc

y is a phase constant, =¥Q(quadratic diSpersion relation)

9 is the spectroscopic spin splitting factor = 2

 

T is the life time of a state on the extremal cross section of FS

th

2 is the index for the 2 harmonic.

This equation is valid under the following conditions:
*

we h << Ef, kBT << Ef,F/H >> 1.

_ «he
Where F - 2?? Aext‘

Typically, we will be interested only in the fundamental, i.e. t = 1

*
_"£/wcT was originally derived by

6

term of Equation 1.8. The factor e

Dingle and* accounts for collision broadening of the levels.
lngm

The costjfifiii) factor is due to particle spin.7 Note: when ng/m0 =
0

1,3,5..., the amplitude of the fundamental frequency will vanish.

1.3.2 Thermodynamic Analysis

The formal result of Lifshitz and Kosevitch is now used in a
thermodynamic analysis to obtain a working expression for the MTO tech-
nique. Using an outline of the argument from reference 8, we obtain an
expression for the oscillatory lattice temperature of an isolated
metallic solid.

The simplified system to be analysed is shown in Figure 1.3. The
'sample' consists of the solid under study, plus the sensor to detect
the lattice temperature, and any other thermal loading by the sample
support, wiring, and etc. The specific heat of the sample CH includes
all these contributions, as well as the sample's specific heat. We
include in the analysis effects of a thermal link (dQ in Figure 1.3)
between the sample, and thermal reservoir (cryostat). This thermal link
is required to maintain an ambient sample temperature, otherwise dissip-

ation in the MTO sensor and eddy current heating during magnetic field

 

Thermal Reservoir

(cryostat)

TB) CV=03

 

d’o

 

 

 

 

 

Sample System
(sample,eeneor,
mtg. post , etc.)

TSJcl-I

 

 

-Hd|

dQ is positive for heat flow into the sample system

dW' is positive for work done by the system

Figure l.3 Thermodynamic system.

d'w’

 

 

10

changes would cause a temperature increase. The thermal link is also
needed for initial cooling of the sample. The abridged fomn of the
thermodynamic analysis of our ideal system follows. The first law of
thermodynamics,
HQ = dU + aw
becomes 1.9
do = dU + uw' — HdI
for a magnetic system. We consider H and T as independent variables.
dQ is positive for heat flow into the sample, dN' is positive for work
done by the system, and represents Joule and eddy current heating. The
term -HdI is the work done by the magnetic field, and is related to the
oscillatory free energy. Using simple thermodynamic relations, Equation
1.9 can be rewritten as,8
do = CHdT + T .3157! dH + am 1.10
where CH is the heat capacity of the sample, and T = TS is the sample
temperature. The heat flow through the thermal link (dQ/dt) is then due
to dissipative effects, dw', and the oscillatory lattice temperature
from the MTO. If K is the thermal conductivity of thermal link, then
73%: -K(TS - TB) = -K6T. 1.11
Taking the time derivative of Equation 1.10, and using Equation 1.11,

= d(6T) 31 dH dN'
'K‘ST CHT+Ts§T)HHE+dE 1-12

where we have used T3 = constant so that dTS/dt = d(3T)/dt.

11

We now use the result of Lifshitz and Kosevitch for FOSC to calcu-
8F
osc)

late dI/dT)H by using I = l/V —-1T—-Twe obtain,
BIOSC ) = - A£2nF£ sin {ZflFz + ¢ } e-thi/H 1 13
5| H H172 H 2 '

sz/H cothiblT/H) -1
X
sinh bgT/H

 

which we substitute into Equation 1.12. The first order differential
equation 1.12 is difficult to solve as is, but is soluble under the
following justified assumptions:

«ST/TS < 1 so that TS ~ constant = To

we evaluate Equation 1.12 at an average field H0 where H = Ho

+ at

CH is a constant

K is a constant

dw'ldt is constant = P.

Under these assumptions, the working expression for the MTO technique

 

 

 

 

 

is,
-2kBTOHo3/2(e/hc)3/2
ST + GT' = - 1,14
05C 3 2A(EF,k ) 5%
C 2
H 3&2”
2 ext.
ZHF ZrT
. L n -1 Q
i=1 2 3/2
J6 + (rose/ZHTQ) 2

12

 

*
cos(flgmc2) e_b£Ti/H (szo/H9_;o;h;b£To/Ho) - 1)} + P/K + ce-t/rQ
am an to”%
where,
GTosc = oscillatory lattice temperature due to quantum oscilla-
tions.

GT' = the lattice temperature change due to effects of dissipative

heating, eddy current heating, and the thermal link.

_ c . .
F2 - CHE) Aext (EF)t (the Onsager relat1on)

2 'k *
2n kBmCC 5 Inc

Tosc = period of deA osc1llat1ons

To = thermal time constant of the thermal link = CHL/KA
Ti = Dingle temperature

C = constant

1.3.3 Discussion of the Result

In addition to obtaining the extremal FS cross sections from MTO,
additional information can be obtained by analyzing measured amplitudes
using the amplitude factors of Equation 1.14.

The factor
1/2
__71 1.15
1+(Iosc/2nro)

accounts for thermal damping of the MTO amplitude due to the thermal

H ~ t, r c will be a

l ink. For a linear magnetic field sweep, 05

function of magnetic field since the oscillations are periodic not in H,

13

but in l/H. Thus, this damping term will not be a constant, and the
amplitude as a function of magnetic field will be complicated by the
effects of the thermal link, which are not of interest. For l/H sweeps,
(section 2.3) this term is a constant, provided To is independent of H.

The next factor,

x coth x—l _ *
is a function of the cyclotron effective mass through x = 1.47 x

105 (mZ/mo)T/H (G/K), and accounts for thermal broadening of the Landau
levels. Figure 1.4 is a plot of this function for an experimentally
meaningful range of x values. This function has a weak dependence on x,
for x near 1.6. Also shown in this figure is the corresponding ampli-
tude factor for the deA effect, 1/3 sinh x, which in general displays a
stronger dependence on x over a wider range of x values. The effective
mass can be determined by the following method. For a given deA fre-
quency the MTO amplitude versus temperature is measured for a chosen
magnetic field value H0. Under the assumption x > 2.0, (dependent on
mz/mo) for this range of temperatures and Ho, we can approximate Equa-
tion 1.16 by
2(x-1)e"‘ 1.17
with an error < 10%, see Figure 1.4. From this, curves of A versus
temperature can be generated. The measured amplitude variation with
teuperature should display the same temperature dependence as the factor
1.17 only for the proper choice of mg/mo. The temperature dependence of
/\ for various values of mz/mo is shown in Figure 5.3. The cyclotron
<3 ffective mass determined from the MTO experiment

- * 2
15m =‘fi/2118A/3E
c EF

14

I\e .os\ue mocxae.P "x
.1
we so mucmucmamo muzprQE< <>Iu ecu op: e.~ mczmwu

 

 

 

 

 

. x ,
«.0 O.N V.N O.N 0.9 8.? 0. V.

1 No

I V

. /
x :50 //
r v.
p r x 500 x /
I
l I
I
z
z . . c. .
z
I
, a
’ 1 C
’ o
’

x ...... a: a. .

. .
. :6 w

 

15

The factor
-b T./H
e 3‘ 1.18

_ 5 *
where b2 - 1.47 x 10 mc/mO 2 (G/K),

represents effects of collisions which attenuate the amplitude, and was
first included in the deA theory by Dingle.6 The relation between the
lifetime I of the orbit, and Ti, the Dingle temperature, is given by

Ti =‘h/(2nkBr). 1.19

* I I
For a measured value of 111C we may determine Ti for the same orbit from a

plot of ln(6T/H3/2) versus 1/H at constant temperature. At a fixed
temperature Equation 1.14 gives *
3/2 _ 5 mc

ln 6T/H - constant - [1.47 x 10 (fi—i] Ti/H 1.20
0

so that Ti is given by the STOpe Of the above graph.

1.4 General Comments

Given our MTO expression 1.14 and its interpretation in section
1.3.3 we can now obtain the size and topology of the FS, as well as the
effective masses and Dingle temperatures. The deA effect and the
related MTO and SdH effects have been proven to be the most powerful
techniques available for deducing the geometry of the Fermi surface.
Note that the MTO equation applies to an individual Fermi surface seg-
ment, but the experiment will measure AT =7: (6T)i where the index i

identifies the ith segment of the FS. Hence, identification of indivi-

dual FS segments requires Fourier analysis of the MTO signal.

CHAPTER TWO
EXPERIMENTAL METHOD

2.1 Introduction

A simplified version of the expression for magnetothermal oscilla-
tions (MTO) is (see Equation 1.14):

2nF1(6)
6T = z Ai(T,e,H) sin (———7;———) . 2.1
i

To observe MTO the magnetic field H is swept continuously and the
resulting oscillations of the lattice temperature are detected. The
frequency of these oscillations, Fi(e), the de Haas-van Alphen
frequency, is related to an extremal cross section of the ith segment of
the Fermi surface (FS) through the Onsager relation, Equation 1.1. A
complete description of the geometry of the FS can be obtained by
observing the dependence of Fi(e) on e, the angle between the sample c—
axis (for example) and the magnetic field direction. The amplitude Ai
of the ith orbit is a function of the magnitude of the magnetic field,
the sample's perfection, its average temperature, and the angle 6.
Additional amplitude dependence of ST comes from many sources; e.g., the
time rate of change of the magnetic field, the sample size, thermal
links between the sample and its thermal environment, and so on. Addi-
tional FS information may be obtained from the amplitude factor Ai’
which is a function of the effective mass, m:, of carrier 1, and its
Dingle temperature T1. From the temperature dependence of the

amplitudes A1, the effective masses can be found. Following the discus-

16

17

sion in section 1.3.3, the effective mass can be determined by matching

the temperature dependence of A as calculated from Equation 1.17 with
*

the observed dependence by adjusting the value of mC/mo. This procedure

5>2. The

is well justified under the condition, m:/mo (T/H) 1.47 x 10
Dingle temperature Ti represents the effects of collisions of the ith
carrier. To obtain Ti, the field dependence of the amplitude at fixed
temperature is measured. For I/H sweeps (see section 2.3), T1 can be
obtained frmn a plot of £n(6T/H3/2) versus I/H, which should yield a
straight line whose slope is pr0portional to Ti (Equation 1.20).

Before presenting and discussing the experimental cryostat designed
to observe MTO, we review its design requirements in light of the
above: (1) MTO is observable only at low temperatures, (2) the MTO
sensors must have a sensitivity of ~ 10'6K, (3) thermal loading of the
sample by the sensor must be minimal, (4) the sample must be thermally
isolated, (5) sample orientation relative to the field must be adjust-

able, and (6) the ambient temperature must be adjustable.

2.2 Experimental Cryostat

The cryostat and superconducting solenoid are cooled by a liquid
helium bath in a conventional glass dewar. The magnet is hung from a
frame mounted on top of the-dewar. The cryostat is placed inside the
dewar and is also supported by the magnet frame. A schematic of the MTO
cryostat used in this research is shown in Figure 2.1. The sample (not
shown) is attached, using a low temperature vacuum greaseT, to a Vespel*
Inounting post inside the ring gear. A weak thermal link (#46 copper
:Ifipiezon “N grease," Apiezon Product Ltd.

low thermal conductivity, low specific heat resin made by Dupont
Company.

 

to diffusion pump

control rod

t.

18

I I to booeter pump

 

 

-CI

 

 

 

 

 

 

 

 

1K pot

 

 

 

eemple and eeneor
ring gear \

reference eeneor

A

7/

 

 

 

 

 

 

 

 

 

 

 

 

\\

Figure 2.1

79 (germanium)

1":g (carbon-glass)

SOkG SC"

I

/

The MTO Cryostat.

19

wire) between the sample and cryostat compensates for eddy current
heating and provides a means for obtaining ambient equilibrium. The
sample is otherwise thermally isolated once the cryostat has been evacu-
ated to ~ .5 x 10'6 Torr. The ring gear and sample can be rotated
through 90° relative to the vertical magnetic field direction by turning
the control rod shaft. From the known gear ratios, the angular position
can be reproduced to within 1:0.3° using a revolution counter. The
control rod passes through and is thermally lagged at the '1K pot' and
again at the 4K flange, see Figure 2.1. The rod continues up the vacuum
pumping line and terminates above an O-ring seal where the counter is
situated. The 1K pot is a chamber which holds 250 ml of liquid helium,
and is filled via a needle valve. This chamber is in thermal contact
with the sample region and can be cooled to .9K by evaporation of the
helium using booster and mechanical pumps. The sample region tempera-
ture is monitored by a factory-calibrated germanium resistance
thermometerT and a four-wire conductance bridge. A second thermometer,
which has a weak magnetic field dependence,* is used in a resistance
heater feedback 100p which can provide stable temperatures between 10
and .9K. The heater is a 100 Ohm chip resistor which is in contact with
the 1K pot. The 4K flange serves, in part, to cool the electrical
wiring to 4K. The wiring consists of #40 capper (current leads) and #40
manganin (potential leads) wires. The heat loss between the 1K pot and

4K flange is about ZOmw.

ILake Shore Cryogenics GR-200A—100. Factory calibrated .3-6.0 K.
Lake Shore Cryogenics CGR-I-lOOO. Field dependence .5 T/T (%) at 8T.

20

2.3 Magnetic Field

The magnetic field is produced by a 50kG niobium-titanium supercon-
ducting solenoid.T Field homogeneity is g .1% within 1" of the winding
center. The magnetic field to energizing current ratio for this magnet
is 848.9 Gauss per Ampere. This value was confirmed using an NMR
probe,9 and by performing experiments on a test sample (section 5.2.1).

The magnetic field may be swept in a linear mode, H ~ t, or in a
I/H mode, H ~ t'l. With a linear sweep, the resultant MTO will be
aperiodic, as shown in Figure 2.7. Typical linear sweep rates ranged
from 0.5 to 10 amperes/minute depending on sample size, temperature,
angle, etc. A linear ramp mode was used in acquiring some of the data
presented in Chapter 5. The linear ramp was provided by a commercial
sweep controller.* The remainder of the data, however, was obtained
using a l/H drive. This mode of field sweep has several advantages.
The MTO signal is then periodic in real time so that analog filtering
can be performed to enhance the signal to noise ratio on selected

de Haas—van Alphen frequencies. In addition, the thermal damping

factor,
L 1 11/2
1+(T / l

T
osc Q

2.2

is then a constant. In a linear sweep mode, Tosc is a function of time
(or field) so a systematic error in the MTO amplitude dependence on H
complicates the analysis of Dingle temperatures. Finally, eddy current
heating, which is significant at low magnetic fields, is substantially

reduced since the sweep rate is low at small values of H.

IOxford Instruments Company, Ltd., Model K1034.
0.5. Walker Company.

33

21

‘____ A -

 

ramp controllerchrreM Gunplvfi SCM/ "NJ

 

 

 

 

 

Vp Tut)
(a)
30.0'
-4
(b)

8
£5
‘1

1 so “i

T 1* 1 T F l
1 12 £3 4‘ 55

time (arbitrary units)

Figure 2.2 (a) Schematic of the sweep control system.

(b) H vs t for a l/H sweep.

22

Figure 2.2(a) is a schematic of the control system for the super-
conducting magnet. A voltage Vp(t) is generated by the ramp controller
and is input to the programmable current supply so that

I(t) ~ Vpit) 2.3
where I(t) is the current through the superconducting solenoid. In a
superconducitng solenoid I ~ H, so that H(t) ~ Vp(t). To obtain a
linear ramp from this system, Vpit) = at, where a is the sweep rate. If

a l/H sweep is desired, the program voltage should be,

p C - at

where c is a constant and c > at. This function has a time dependence
such that equal increments in l/Vp (i.e. l/H) occur in equal time inter~
vals, as required to give a periodic MTO signal. The commercial linear
ramp controller was modified to produce a program voltage which approxi-
mates Equation 2.4. In Figure 2.2(b) the magnetic field versus time
obtained by using the modified sweep controller is shown. The output
shown in that figure deviates less than 4% from a true l/H sweep over

wide ranges (> ZOkG) of magnetic field.

2.4 Measuring Small Temperature Oscillations

Typical magnitudes of ST in Equation 2.1 are of order 10'4K. The
method used, capable of detecting temperature changes as small as .5 x
10'6K, employs an A.C. bridge circuit and small carbon composition

resistors.

23

Carbon composition resistors have a1 resistance-versus-temperature

relation at low temperatures given by

A/T
= e(A/T + B) or dR/dT “'EETF' ; T in K. 2.5

T

R

For the resistors chosen for this research, A = 5.24K and B = 6.43.T At
ambient temperatures near 1K a temperature change of 10'5K corresponds
to a resistance change of a few Ohms. The carbon resistors also have a
small specific heat; as mentioned earlier, a small heat capacity
prevents damping of the MTO signal by thermal loading. Two resistors,
matched in resistance-versus-temperature characteristics to ~ 1% from
room temperature to 4.2K were used in the differential input bridge
circuit described below.

We wish to detect temperature oscillations 6T about the ambient
sample temperature. To this end we use two resistors; one, Rs, in
thermal contact with the sample; the other, Rr, attached to the sample
region of the cryostat. We detect MTO by measuring the resistance
difference between the two thermometers while sweeping the magnetic
field. From Equation 2.5 the oscillatory resistance difference must be
pr0portional to GT if 5T < T.

Figure 2.3 is a schematic of the bridge circuit. It consists of a
lock-in amplifier, a ratio transformer, and two independent and isolated
current SUpplies. An excitation voltage of frequency w (117.5 Hz) is
supplied to the primary side of the ratio transformer. There are two

isolated secondary windings labeled N and NB. The secondary windings

1'82 Ohm Ohmite "Little Devil" carbon composition resistors lapped to
.2mm thick, weight ~.OI grams.

 

.Swauccu amuwcm .u.< ago co acumeaeom m.N massed

6:13:51.» Eve Q

.2 . ..-...2

A

 

 

24
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weepmwmmc mcwmcmm pcmccso

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25

provide input to the matched and isolated current supply Operational
amplifiers. The secondary winding NB can be stepped from .0001 to
1.2000 times the number of windings on the other secondary, N. Under
conditions of zero magnetic field and thermal equilibrium between the
sample and the cryostat sample area, the currents IS and Ir through the
resistors are adjusted by changing 8 until the voltage drop across both
resistors is the same. The voltage dr0p across each resistor is brought
from the cryostat (two leads each) to a differential input
preamplifier. The combination of the differential input preamplifier
and tuned lock-in amplifier measures the resistance difference between
Rs and Rr- Once Is and Ir. are adjusted as discussed above, the bridge
is in balance and its output is zero. The balance condition may be
written,

I = 81

S

l“
Irzr = 1525

where ZS and Zr are the impedances of the resistor networks. Note that

2.6

in addition to adjusting B it may be necessary to adjust c1, (Figure
2.3), to assure capacitive balance, which is monitored by an
oscillosc0pe. With the conditions of capacitive and resistive balance
met, we obtain from Equations 2.6,

B = Rs/Rr‘ 2.7
Sweeping the magnetic field then results in an oscillatory lock-in
amplifier output due to MTO. The sample temperature, T, may be measured
by replacing Rr with a variable reference resistor (via a front panel
switch) and adjusting the reference resistor and a variable capacitor
until a balance condition is met. To avoid self-heating in the sensors

the currents Ir and IS must be adjusted using the "current sensing

 

26

resistors" shown in Figure 2.3. The power dissipation in the sensors

should be 5 10-14W. Finally, the actual balance condition is
Rs BRr
WW 3-8

thus, wRC must be less than 1. For very low temperatures where RS and

Rr are large, m may have to be small (10-30 Hz).

2.5 Data Acquisition and Analysis

A minicomputerT was interfaced to the MTO setup for data
acquisition and analysis. Figure 2.4 is a block diagram of the
interface. The data consists of a set of magnetic field values and
their corresponding 6R values. The magnetic field value is monitored
by a digital voltmeter which reads the voltage dr0p across a calibrated
reference resistor in series with the superconducting solenoid. The
sample lattice temperature 6T is proportional to the lock-in amplifier
output, 6R. This output is also analog filtered* to enhance the signal
to noise ratio on chosen FS orbits, as well as to assure cut off of
frequencies higher than the maximum frequency of the fast Fourier trans-
form range, which is given later in this section. The outputs from the
analog filter and voltmeter are passed through analog to digital conver-
ters prior to input to the computer.

An MTO scan is performed as follows. The user determines the
de Haas-van Alphen (or fast Fourier transform) frequency range to be

<:overed. This frequency range is from 0 to

EDigital Equipment Corporation PDP 8/e; 32K core, RK05 magnetic disc
rive.

Krohn-Hite active filter model #3750 with high, low, and band-pass
capabilities.

27

.Emgmmwo xoo_m mummcmch gmpzqsou ¢.N mczmwu

 

 

 

 

 

 

 

/

00.2.2 _ 31:4

_
25.050 060...... D
3555 c5200

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_l..._E..E r \ Tunisi— .a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 
 

 

34 + m .
r\ \ A f
E040 I 79 F I _
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=de .50
Tab—gown NOT! V ”GEO...” 0m:— o¢<1 l £9523sz n25“.
4 5502092.: 5;” 5:2

 

 

 

 

28

fl (file length) [G] 2.9

“1' Hf

where Hi and Hf are the beginning and ending magnetic field values, and
the file length is the number of ST and H pairs to be used (typically
512). The computer requests ”1 and Hf for each scan, and from these
values generates a table of magnetic field values (512) which are equal
increments of 1/H apart. During an MTO scan, the current magnetic field
value, read by the voltmeter, is compared with the computer generated
field value table and when the values agree, a reading is made from the
output of the analog filter (GT) and stored in a local computer file.
By taking 6T readings equal increments of l/H apart, the stored data
appears to be periodic, see Equation 2.1. The local file consisting of
(512) GT and H values can be stored in a permanent file on a magnetic
disc, see for example Figure 2.5. This set of periodic data can then be
fast Fourier transformed to obtain Fi in Equation 2.1, as shown in
Figure 2.6. Note that we have essentially converted Equation 2.1 to

sin (2nFix) ~ GT and then Fourier transformed to obtain Fi’ The trans-
formed data (or the raw data) can be displayed on a CRT, plotted on an
interfaced chart recorder, or printed at a terminal in digital form.
The fast Fourier transform program used can be found in references 10
and 11, which also includes a discussion of the derivation, use, and
limitations of the fast Fourier transform technique. As mentioned
earlier, the ST readings are analog filtered, in part, to cut off
frequencies beyond the maximum of the transform range (given by Equation
2.9). A complete discussion of this requirement is found in reference

10. Details concerning the general use of the computer interface can be

29

2965

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19

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.5... >325... (>30
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IA

Ausuowl JOQJHOJ

31

found in a reference manual which accompanies the MTO experimental
apparatus. The use of this method for data acquisition and analysis has
several distinct advantages over earlier, manual methods (see reference
8). First, the stored MTO data can be recovered when desired. Second,
the MTO data can be transformed (~ 70 seconds) while the experiment is
in progress, giving immediate feedback about the FS data accumulated.
Also, the fast Fourier transform program is sensitive to small harmonic
components which might otherwise be missed.

MTO data can be taken in a manual mode as well (Figure 2.7). Here
the horizontal axis corresponds to magnetic field and the vertical axis
is the output of the analog filter. A manual scan is useful for MTO
anplitude determination of the effective mass and Dingle temperature for
each orbit. Prior to computer assisted data acquisition, a manual scan
can be performed to optimize the experimental parameters such as the
magnetic field sweep rate, amplifier gain, analog filter
characteristics, and determination of Hi: and Hf for the desired deA
frequency range. The manual data can also be hand transformed to verify

the frequencies indicated by the fast Fourier transform program.

32

 

 

 

 

 

.wx m.N_ op o .m ammpm
.0002 _m::wz 0:“ :w :0me mums o»: N.N wgzmpd

:—

07

OS

09

CL

CHAPTER THREE
PROPERTIES AND SYNTHESIS OF GRAPHITE INTERCALATION COMPOUNDS

3.1 Introduction

The first section of this chapter will describe briefly the nature
of graphite intercalation compounds. The remainder of the chapter will
detail the preparation and structural characterization of the SbCl5

compounds used in this research.

3.2 Review of Graphite Intercalation Compounds

Graphite intercalation compounds consist. of the host (material,
graphite, and atoms, ions, or molecules called the intercalant. The
graphite host structure used in this research is the hexagonal form
shown in Figure 3.1, with layers of hexagonal rings of carbon atoms
stacked in a sequence ABAB. This material is referred to as highly
oriented pyrolytic graphite, or HOPG. It is synthetically prepared and
consists of layers of crystallites (~ luM), which are aligned along the
c-axis direction, but not in the basal plane. Within each crystallite,
however, there is c-axis and basal plane order.

Through the process of intercalation, whole layers of the interca-
lant material are ingested into the interlayer spaces between the hexa-
gonal carbon planes. In an ideal compound, an ordered structure occurs

along the c-axis and in the intercalant basal plane. The

33

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.1 Schematic of the Hexagonal Form of Graphite (HOPG).
ao=2.45 A, c0=6.70 A

35

.mcmme pcmpmuemucw pcwmogame mmcwfi 005mmu

.mcmme conemu pcmmmgame mm:w_ uw_om .mcwmmum do ovuwewgum N.m oesmwm

 

 

 

 

 

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36

structure along the c-axis is characterized by a staging index n (Figure
3.2). A stage n compound is one in which nearest intercalant layers are
separated by n carbon layers. In a simple compound, the distance along
the c-axis between equivalent layers, IC, is related to the stage index
n by the phenomenological equation,

IC = [(n-1)dc + up] ; n = 1, 2, 3... 3'1
where dc is the distance between carbon layers in HOPG, 3.35 A, and dp
is the thickness of a packet formed by the intercalant and its bounding
carbon layers (Figure 3.2). The graphite and intercalant layers may
also exhibit a stacking sequence in the GIC (reference 12). The in-
plane structure of a GIC may be ordered as well as shown in Figure 3.3
for SbCl5 compounds. From this structure a layer stoichiometry is
determined from the ratio of the number of intercalants in a layer to
the number of carbon atoms in a graphite layer. For SbCl5 compounds,
the layer stoichiometry is C14SbCl5.

Intercalation compounds are also classified by the nature of the
charge exchange between the graphite layers and the intercalant. In
graphite acceptor compounds (GAC's) such as AsF5 or SbCls, the interca-
lant accepts electrons from the graphite layers, leaving them with some
net positive charge while maintaining overall charge neutrality. For
graphite donor compounds the reverse is true, the intercalant donates
electrons to the graphite planes. The amount of charge transfer is
quantified by the charge transfer coefficient, f, defined as the
fraction of intercalant atoms or molecules that are ionized.

Graphite in its pristine form is anisotrOpic in its electrical and
thermal transport pr0perties. As an example, the ratio of the basal

plane to c-axis conductivity, Oa/oc is about 103. In general, this

 

 

37

 

Note:

Figure 3.3 Basal Plane Order in SbClS.

TheJ7 xfT Rl9.lo primitive cell for the intercalant
is shown in heavy lines, the HOPG primitive cell is
shown in dotted lines. A x B Rd’ denotes an inter-
calant superlattice with primitive translations Aa
and Ba where a is the side of the primitive cell

of HOPG. The intercalant superlattice is rotated

9 degrees relitive to the HOPG cell.

38

anisotrOpy is enhanced for acceptor compounds, °a/°c ranging from about
104 to 105. For donor compounds, the anisotrOpy tends to decrease.13
In both the donor and acceptor compounds, the carrier density (a
function of stage) increases from 1018cm'3 for graphite to ~1021cm”3
for a typical GIC. Thus, the'semimetal graphite is altered to a quasi-
metallic state by intercalation. There are several good reviewslz,13 of
GIC's available, and the interested reader is referred to them for

additional background.

3.3 Preparation of Sb015 Intercalated HOPG

The HOPG host material,r is prepared for intercalation by cutting to
a suitable size, typically to 5mm x 10mm x 1mm, using either a diamond
impregnated wire saw or a kerosene immersed spark cutter. Next, the
HOPG is rinsed with alcohol, placed in an RF furnace and vacuum baked at
1000°C for approximately one hour for cleaning and to remove residues
from cutting. The intercalant SbCl5 is a pale yellow liquid at room
temperature and will solidify at 2.8°C. The Sb015 was generally used as
provided.* In two cases a vacuum distillation was performed to further
purify the SbCl5. However, the GIC's produced with the vendor stock
proved to be identical with regard to their physical and electronic
structure, and the distillation procedure was discontinued.

Compounds of stage 2, 4, and 5 were produced locally by the two
zone vapor transport, or Herold method.14 The intercalant and HOPG are
placed in a reaction tube, Figure 3.4, inside a glove bag filled with an

over-pressure of nitrogen gas. The reaction tube is sealed at the valve

TProvided by A.w. Moore, Union Carbide Corporation.
J.T. Baker Co. SbC15 99.6% purity (carcinogenic).

39

O... 2mm Teflon valve.

ground glass joint __,.

K Torch seal

 

 

‘ L HOPG

I

 

250m

9mm Vycor tube

 

 

 

J__

lntercalant

Figure 3.4 Reaction Tube for SbCl5 GIC Synthesis.

 

40

after the HOPG and intercalant are loaded, then removed from the glove
bag. The SbCl5 is frozen by immersing the reaction tube in liquid
nitrogen. The N2 gas is then pumped off, and glass reaction tube is gas
torch sealed (Figure 3.4). The intercalation takes place when the
sealed reaction tube is placed in a two zone furnace. The high tempera—
ture zone of the furnace controls the temperature of the HOPG, T9, and
the lower temperature zone controls the intercalant temperature, TI-
The stage index of the compound depends on the temperatures TI and T9,
with the important parameter being TI - Tg. Table 3.1 lists the

temperatures required for synthesis of each stage.15’16

 

Table 3.1
Preparation Temperatures and IC Values

for SbClS GIC's.15

Stage Index

 

n Tg(C) TI(°C) 91 i2
2 170 115 55 12.72
4 170 70 100 19.42
5 195 75 120 22.77

 

The furnace was made locally, and the two temperature zones can be
"independently" varied between 50 and 950°C t 1°C. The temperatures T9
and TI are monitored by Chromel-Alumel thermocouples attached to the
outside of the reaction tube next to the graphite and intercalant

respectively. Stage equilibrium occurs within about 120 hours.

41

In addition to the compounds produced locally, the stage 6 and 8
compounds studied were produced by R. Clarke, University of Michigan,
using the above technique. The University of Michigan group also
provided the stage 1 compound on which measurements were made. Stage 1
SbCl5 compounds are made by the direct immersion method. The interca-
lant SbCl5 liquid and HOPG are sealed in a reaction tube, with the HOPG
immersed in the liquid. The intercalation reaction proceeds at tempera-

tures near 95°C and takes several days.

3.4 Characterization

A uniformly staged compound with a small (3 3°) c-axis mosaic
spread is required for measurement of the stage dependence of the Fermi
surface. The c-axis mosaic Spread determines the amount of c-axis
misalignment of crystallites which form the compound. A large amount of
c-axis misalignment will cause broadening of the deHaas-van Alphen
frequencies measured for determination of the Fermi surface. The in-
plane intercalant superlattice (if any) should also be known, since it
is of central importance in one of the GIC Fermi surface models (section
4.3.2).

The stage index and stage fidelity were characterized. on (all
samples before and after Fermi surface measurements. The stage index n
is found by nmasuring and c-axis repeat distance, Ic, and then using
Equation 3.1. This repeat distance is obtained from a 002 x-ray

diffraction scan, see Figure 3.5, by using the Bragg relation,

2x = 21c sin e 3.2
2 = the reflection index for line 002
e = the angle for line 002 (E-axis to x-ray beam)

42

.m.0nm . 00.0m .ceom co.00.....o .0.-. .00 m.m 0230..

700503 94.

 

00.3. 0..
4 l.“ 1.1 1.x, A. l.uuunnu.0‘nuuu;.J.nlnungwgazggu
.000. .500. m .000. m .H H. w
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Autumn. flea-x

43

A = x-ray beam wavelength.

From this equation, IC = 51/2 where S is the STOpe from a graph
of 002 versus sin e as measured by the 002 diffraction scan (Figure
3.5). The repeat distances calculated from our characterizations agree
with literature values for this compound, and are given in Table 3.1.
However, for higher stages (n g 6) the 002 peaks become closely spaced
in 20 and the stage characterization by x-ray diffraction becomes more
difficult. The c-axis mosaic Spread is measured by setting the x-ray
diffractometer e and 29 values to a 002 reflection and then rotating the
sample in the x-ray beam (Figure 3.6). Typical mosaic spreads for these
compounds ranged from 3 to .6°. The x—ray diffraction measurements were
performed with a 12 kw rotating anode x-ray source with a four circle
diffractometer and multichannel analyzer. The x-ray beam source target
was Molybdenum, and the beam was monochromated to
the K0 lines,‘X = .7107A. The resolution of this system in 26 is about
.2°.

The stage 1, 6, and 8 samples loaned to us by the University of
Michigan group had been analyzed for their in-plane structure. Their
studies indicate the presence at room temperature of the intercalant
superlattice shown in Figure 3.3. According to their observations,
this fix .7 superlattice persists to liquid helium temperatures, but
below 220K an additional intercalant superlattice appears.17’20 The new
phase was found to be slightly incommensurate with the HOPG lattice,
with a unit cell about/39x J39.” A group at MIT studying SbC15
compounds also observed a structural transition near 220K, but their
measurements indicate a low temperature in-plane phase without long-

range order (glass-like).21 The c-axis stacking sequence for carbon

layers in SbCls GIC's is discussed in reference 17.

.0 m:.. moo 0;. EOE. ...Q

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45

The intercalation chemistry for group V pentahalide-graphite com-
pounds such as SbCl5 GIC's remains controversial. The following
proposed chemistry has received some quantative support from Mossbauer22
and x-ray absorption edge spectrosc0py,23

3 SbCl5 + 2 e'-——+;> 2 SbClG- + SbCl3 . 3.3
This, however, has not been confirmed by other techniques.22 The bulk
charge transfer coefficient, f, depends on the number of bonding inter-
calants, for example the SbCl6_ molecules in Equation 3.3. The true
intercalation chemistry for the SbCl5 GIC's must be determined before
accurate calculations of the charge transfer coefficient can be made.

Surface studies were performed on stages 2, 4, and 5 using Raman
light scattering and infrared reflectivity techniques. These measure-
ments were used to verify the stage index, independent of the bulk
measurements using x-ray diffraction. In all cases, results of the
surface and bulk studies were in accord and both indicate that SbC15

GIC's remain stable for at least 14 months.

CHAPTER FOUR
GIC BAND STRUCTURE AND FERMI SURFACE MODELS

4.1 Introduction

Experimental Fermi surface data on GIC's became available after
1978.24 Since then several theoretical models have been proposed to
describe the observed results. In this chapter those Fermi surface
models apprOpriate to the SbCl5 graphite intercalation compounds studied
will be described.

The two GIC Fermi surface and electronic band structure models
discussed in this chapter are fundamentally different in their nature
and predictions. Prior to describing these models, both of which are
phenomenological, we present the 20 and 30 band structures pr0posed for

graphite upon which the GIC models are based.

4.2 Graphite Band Structure and Fermi Surface

The electronic pr0perties of the GIC host material, graphite, are
reasonably well understood25 and are represented by a phenomenological
model referred to as the Slonczewski, Weiss, and McClure model
(Sl'lMcC).26’27’28 This model has been used successfully to explain a
variety of experimentally measured properties that depend on the elec-
tronic structure near the Fermi energy, Ef.29:30 The SHMcC model has
31

also been confirmed by more saphisticated band structure models,

except for one detail which will not affect our arguments.32

46

47

The 3-dimensional SHMcC model is based on a k . p expansion in the
basal plane and the crystal symmetry appropriate to HOPG.26 The model
describes the k dependence of the conduction n bands arising from the
weakly-coupled carbon pZ atomic wave functions at the Brillioun zone
edges (see Figure 4.1).

The model may be written in terms of a 4 x 4 hamiltonian for the n

bands,
*
E1 0 H13 H13
1:
0 E H -H
.2 22 23 4.1
H13 H23 E3 H33
*
H13 “”23 H33 E3
The H13 are the interaction terms identified with the overlap and trans-

fer integrals between atoms, within the framework of the tight binding
approximation, and the E1 are the band edge energies. The interaction
terms are,
H =2'1/2(-y +vI‘)oei°‘
13 1 o 4
- i
H23=2 /2(YO+Y4I‘) oecll 4,2

33=y3roem

H
where o is the angle between k and the r K direction, and
r = 2 cos n g ; g = szo/"

o =1/2/3 30K.

The SHMcC band parameters (yo-- 75, A) are given in Table 4.1, a0 =
1.4214 is the in-plane nearest neighbor distance, co = 3.35 A is the

nearest neighbor distance along the c axis, and K is the magnitude of

48

 

 

 

 

 

h
(‘ 2r
H H,
H h0(€$
H' . ,
K H . K '
' m H K eleci'rons
K' H H' K.
. K ' H Acdes
H.
H \JH’
hX

Figure 4.l Graphite Brillioun Zone and FS.

E E . E

l 1
\lenl ‘l

 

 

 

 

 

: A-Z);+275

 

0
OX

Figure 4.2 SWMcC 3D Graphite Band Structure.

 

49

 

 

Table 4.1
Slonczewski-weiss-McClure Band Parameters for Graphite.13
Band Approximate
Parameter Value (eV) Description

YO 3.16 Overlap of neighboring atoms
in a single layer plane.

Yl 0.39 Overlap of orbitals associ-
ated with carbon atoms
located one above the other
in adjacent layers.

Y2 -0.020 Interactions between atoms
in next nearest layers and
coupling between n and 0
bands.

y3 0.315 Coupling of the two E bands
by a momentum matrix glement.
7 determines trigonal
w rping.

Y4 0.044 Coupling of E3 bands to E1
and E2 bands.

Y5 0.038 Interactions between second
nearest layer planes.

A -0.008 Difference in crystalline
fields experienced by in-
equivalent carbon sites in
layer planes.

Ef -0.024 Fermi level measured from

H-point extremum.

50

the wave vector in the kx, ky plane measured from the zone edge. The
solutions to 4.1 are

1
E =1/2(E1 + £2) t[1/4(E1- E3)2 + ‘Y. - “102.2 /2

1 4.3
E =1/2u:2 + E3) 1: [1/2(E2 - E3)2 + We + ”142.21 (2
where,

2
E1 = A + yll‘ +1/2'75 I‘

2
E2 = A - 71 I‘ +1/2Y5I‘

2

The dispersion relations along the kZ axis given by equations 4.3
are shown in Fig. 4.2. The majority hole and electron pockets found in
graphite can be seen to arise from the narrow E3 band. Fig. 4.1 shows
the Fermi surface which results from the SHMcC band structure model.
Due to the small extent to which the Fermi surface extends into the kx,
ky plane, the carrier concentration in graphite is quite low, ~ 10"4
carriers/atom, and thus graphite is a classified as a semi-metal. The
SHMcC model as described above is used in a rigid band model to be
presented in section 4.3.1.

Due to the large observed anisotrOpy in graphite's transport
pr0perties, several 20 band structure models have been proposed.33’34
These 20 models for graphite have been used as a basis for a 20 band
structure for GIC's (Section 4.3.2). Theoretical arguments similar to
those used in obtaining the 3D band structure results shown above can be
used to obtain the 20 graphite band structure. The 20 hamiltonian is 2

x 2 with solutions,

_ 3
EC,V-i.2-Y0bk 4.4

51

where c,v refer to the conduction and valence bands, b is the nearest
neighbor in-plane carbon-carbon distance, 1.42 A and v0 is the
resonance integral between nearest neighbor carbon atoms with a value of

~ 2.4ev.35 The momentum vector k is measured from the zone edges H and
H'(see Figure 4.1) of the carbon Brillioun zone. The simple linear

dispersion relation given by Equation 4.4 is not shown.

4.3 GIC Band Structure and Fermi Surface Models

The graphite band structure models above have been used as a basis
for a theoretical description of the electronic structure of GIC's. The
first theory to be considered views GIC's as modified 30 graphite with a
new c-axis periodicity and an increased carrier concentration. The
second theory views GIC's as modified 20 graphite and has a strong

dependence on intercalant in-plane order.

4.3.1 The Rigid Band Models

hi the case of cfilute graphite intercalation compounds one might
expect the GIC electronic band structure to closely resemble that of
graphite. This is the basis of the rigid band model.36 In it we assume
that the GIC band structure is identical to graphite, as represented by
the SHMcC model, and allow the Fermi level to shift, so as to accomodate
the additional electrons (donor GIC's) or holes (acceptor GIC's) from
the intercalant. For typical shifts of the Fermi level, (>275) the new
carrier pockets will extend from g = -.5 to .5, and also extend farther
into the kx, ky plane. For donor compounds the carrier pockets will be
electron-like, and for acceptors only holes are found in the Brillioun

zone. Although this model was originally intended for dilute, i.e.

52

n {'4 compounds,36 it has been used to describe the band structure of
compounds as concentrated as stage 2.37’38 In presenting this model we
shall consider only the case of acceptor compounds, where the Fermi
level shift is towards increasingly negative energy values.

The carrier pockets arise only from the E3 band. By solving
Equations 4.3 we find the following analytic expression for the cross

sectional area of the hole pockets formed along kZ (see Figure 4.3):

4 n 1

E ) =
2’ F 5272
3aovo (1-v)

A (k

 

C
I

= (Y4/Yo) r .

We now have an expression which gives the size and shape of the new GIC
Fermi surface as a function of the Fermi level shift, which is related
to the concentration of intercalant present in the sample.

A significant improvement can be made to the rigid band model by
including the new c axis periodicity due to staging.37 The hole cross
section as a function of k2 and Ef is still given by Equation 4.5, but
it is in a new Brillioun zone. Fermi surface measurement techniques,
such as MTO, measure the extremal cross sectional areas of the carrier
pockets. To predict these cross sections from the rigid band model we
evaluate Ah at points along kz corresponding to the zone center and new

zone edges. Thus, the Fermi surface is cut at points along kZ given by

a =1/2(Ic°/Ic) i i = o, 1, 2, 3, m 4.6

53

 

 

 

 

 

 

 

 

 

 

 

l

l

‘b’ 13.“? “
(c)

 

(d),

 

Figure 4.3 Schematic of RBM FS.

(a) RBM FS cross section.

(b) FS shown folded back into the new Brillioun
zone.

(c) FS segments from (b) including crystal field
effects.

(d) FS segments shown in the extended zone
scheme.

 

54

where I; is 6.70A and Ic is the c-axis repeat distance for the given
stage index. For a fixed stage, we make a total of m cuts along the
Fermi surface, each of which corresponds to an extremal cross section.
The smallest orbit is given by i ==flL A schematic representation of
this method is given in Figure 4.3.38 To apply the rigid band model in
predicting the GIC Fermi surface, the largest Fermi surface orbit is
used as a parameter in the theory. Ef is adjusted by requiring a match
between the models largest cross section at g = 0, the K point, and the
largest Fermi surface orbit. (However, see section 5.3). WiU1 the
Fermi level fixed, the Fermi surface is given by Equations 4.5 and 4.6.

More recent rigid band theories by Dresselhaus et al.39'41 differ
somewhat in the approach taken, and include another aspect of intercala-
tion, the in-plane order. Briefly, these authors replace the
intercalant layer with a vacuum layer and then zone fold along kz, and
also in the kx, ky plane consistent with the intercalant superlattice.
The orbits predicted from this model differ slightly with those from the
model described above. It is important to note, however, that the
effect of the in-plane zone folding on the predicted orbits is very
small and can be ignored in the rigid band approximation.

To summarize, the rigid band model assumes that the band structure
remains graphitic and that changes take place only in the size of the
carrier pockets and in the Brillioun zone along kz. Also, by matching
the largest Fermi surface orbit with that predicted at k2 = 0, the K
point, all other orbits are fixed. Thus, this theory has one 'free'
parameter. From Equation 4.5, we can find the total carrier density by
integrating over kz, which will be used in determining the bulk charge

transfer coefficient. As will be shown in Chapter 5, the effective mass

55

for each orbit can be calculated from curves of E1: versus orbital
area. Finally, we have assumed no distortion of the graphite band
structure due to intercalation. Intuitively this assumption will fail
for compounds with a high concentration of accepted (or donated)
carriers. The limit of validity might be expressed as a maximum Fermi

energy shift, and a likely value is ~ .8 eV. 42

4.3.2 The 20 Band Structure Models

The electronic band structure and Fermi surface models in this
section are two dimensional, and have a strong dependence on the in-
plane ordering of the intercalant. First to be described is a model
pr0posed for dilute compounds, and then models intended for the more
concentrated compounds, i.e. stages 1-4.

The model for high stage compounds (n > 4) consists of the
following features.24’43 AJong the c-axis we find regions of pure 30
graphite, separated by graphite-intercalant-graphite packets. Each of
these components will have a unique electronic structure. From section
4.1 we already understand the 3D graphite electronic band structure and
Fermi surface. The other component is a metallic sandwich formed by the
intercalant and its bounding graphite layers, and has an electronic
structure dependent on the intercalant superlattice and the charge
transfer. The in-plane superlattice of the intercalant will introduce a
new periodic potential in the now highly charged bounding graphite
planes. As a result, a new Brillioun zone is formed in the kx, ky plane
due to the intercalant. This Brillioun zone is used with a nearly-free
electron model in the Harrison method44 to construct the 20 Fermi sur-
face of the metallic sandwich. The Fermi circle is given by the quadra-

tic free-electron dispersion relation, and by,

56

fN _ _ 2
TSBZ'AFs‘" "F 4.7

where fN is the number of "free“ charge carriers per primitive cell, 582
is the area of the new in-plane Brillioun zone, and AFS is the area of
the Fermi surface. The factor of 1/2 accounts for the charge being
distributed equally between the two graphite layers bounding the inter-
calant. An example of a Fermi surface constructed in this manner is
shown in Figure 5.7.

There are several important aspects of this model. First, because
we have used a 20 construction for the metallic sandwich, its Fermi
surface must be cylindrical, that is, independent of kz. Second, as
mentioned above, the dilute GIC in this model is assumed to consist of
two separate components, one being 30 graphite and the other a metallic
sandwich. Thus, we should observe two Fermi surfaces in dilute GIC's,
one due to 3D graphite as described in section 4.2, the other a new 20
Fermi surface arising from the graphite-intercalant-graphite packets.
Finally, this model has no explicit stage dependence except for small
variations in the charge transfer coefficient as a function of stage.
Thus, to first order, the Fermi surfaces in this model are stage
independent, assuming the intercalant in-plane order does not depend on
stage.

The 20 band structure and Fermi surface models for stages 1-4 are
based on the band structure of 20 graphite, as Opposed to assuming the
nearly free-electron band structure used for dilute GIC's.35 The low
stage compounds are viewed as follows. The charge transfer gives rise
to localized carriers in the intercalant layers and delocalized carriers
in the neighboring graphite layers. The localized carriers in the

intercalant layers form a high potential barrier preventing carrier

57

movement along the c-axis. This barrier isolates the highly-charged,
graphite-like layers between intercalant layers which are then modeled
as 20 electronic subsystems. The sample consists of sequences of these
subsystems.

For stage 1 compounds, with one graphite layer between intercalant
layers, the 20 band structure for graphite presented in section 4.2 is
used as is (see Equation 4.4). The Fermi momentum kf is calculated in
this model from

kF = m 4.8
where ri is the number of holes per unit surface of the GIC. With kf
known, Ef may be determined from Equation 4.4. The Fermi surface is
constructed by centering isoenergy circles of radius kf at corners of
the Brillioun zone generated by the intercalant superlattice, see Figure
5.6. The reduced zone scheme is then used to find the k space orbits in
the kx, k.y plane. In predicting orbits from this model, one of the
experimentally observed orbits is matched with an orbit constructed by
varying kf of the circles used in the Brillioun zone. Once kf is found,
other orbits are predicted, and Ef, the carrier density, and the charge
transfer coefficient are fixed by Equations 4.4 and 4.8.

For stage 2 compounds there are two graphite-like layers between
intercalant layers. The graphite layers interact, modifying the single
layer band structure in Equation 4.4. Without presenting details, the
new band structure is

EC1 = -EV1 =1/2[(Y§ + 9v: b2k2)1/2- Y1] 4.9

- __.1 2 2 221/2
E --Ev2 /2[(vl+9vobkl +Y1]

58

where the new parameter Y1 is the resonance integral between nearest
carbon atoms in the two graphite layers, and depends on the charge
transfer coefficient.35 It is assumed35 that the value of 71 in GIC's
is the same as that for pure graphite. In stage 2 compounds two values
are found fer k.» The Fermi surface is constructed by centering two
isoenergy surfaces of radius kfl and kfz at the corners of the inter-
calant Brillioun zone then zone folding in the kx, ky plane. The values
of kfl and kfz are fixed by requiring agreement with two or more of the
Fermi surface cross sections. The Fermi energy, additional predicted
cross sections, and the charge transfer coefficient are then fixed.
Expressions relating kfl and kfz to Ef, carrier densities, and the
charge transfer coefficient can be found in references 35 and 46.

For GIC's of stage 3 and 4, not all of the graphite layers between
intercalant planes are equivalent.45 In a stage 3 compound, the
additional graphite layer is in a new environment, being bounded on
either side by highly-charged graphite layers. This requires a variable
charge density in the graphite packet between the intercalant layers,
and the resultant band structure is complex. Attempts to apply the 20
model to stage 3 and 4 GIC's have not been very successful45 and will
not be used in analysis of our experimental results.

The 20 models presented are summarized by the following. For
dilute compounds, two electronic structures are present; one
corresponding to 30 graphite, the other corresponding to a 20 metallic
sandwich resulting from, and depending on the intercalant
superlattice. For low stage compounds, the Fermi surface is strictly 20

and again dependent on the in-plane intercalant superlattice.

 

59

4.4 Comments

There are several effects not accounted for by the RBM and the 20
model, and also not included in the working expression for MTO, Equation
1.14. These effects are observable in certain materials. The first
effect, called the B-H effect, was first included in the theory of the
deA effect by Shoenberg.47 A quasiparticle sees not just the applied
magnetic field H , but the flux density B. A calculation by Gold48
shows that this effect causes additional, non-harmonic frequencies to
arise. Given fundamental frequencies Fa and Fb, this effect will create
lower amplitude orbits such as 2Fa, 2Fb, Fa + Fb, Fa - Fb, etc. The
second effect which can also couple fundamental frequencies is called
magnetic breakdown (M80) and was first included in the theory of the
deA effect by Falicov and Cohen.49 Magnetic breakdown arises when the
magnetic potential is sufficient to cause interband transitions which
then couple one semi-classical quasiparticle orbit to another, resulting
in sum or difference frequencies, Fa - Fb, Fa + Fb° The above effects
are discussed in reference 9. .A third mechanism by which fundamental
orbits may be coupled is charge density waves (CDW). New orbits can be
obtained by superposing FS "saddle points" through a CDW translation
vector resulting in sums and differences of fundamental FS orbits, see
references43, 50, 51. Finally, crystal field effects which would cause
gaps to appear at zone edges (Figure 4.3(c)) will result in entirely new

cross sections of the FS, see reference 52.

CHAPTER FIVE
EXPERIMENTAL RESULTS AND ANALYSIS

5.1 Introduction

Fermi surface data from our experiments is given in section 5.2.
In 5.2.1, we present measurements on the graphite host material HOPG,
and 5.2.2 through 5.2.7 cover the results for the SbCl5 compounds. A
general discussion of the data follows this presentation. In section
5.4 the two GIC Fermi surface models of Chapter 4 are applied to our

results. We finish the chapter with our conclusions.

5.2 Fermi Surface Results

Before presenting our data, we make some introductory comments on
the nature of our data presentation and the scope of this study.

Results of Fermi surface (FS) studies on graphite acceptor
compounds performed by other groups,"’3:45’52’53 as well as this study,
indicate a large number of deHaas-van Alphen (deA) frequencies present
in these compounds. The largest frequency is typically more than an
order of magnitude higher than the smallest. Usually, the frequency
spectrwn will also contain several closely spaced frequencies (i.e. a
few percent apart). The data set for each sample studied consists of
several dozen frequency scans, each covering a different range of deA

frequencies. This is done to improve the signal-to-noise ratio for

60

61

Table 5.1

Experimental Results on HOPG and SbCl5 Compounds
(Frequencies in MG)

Uncertainties are typically <2%

HOPG STAGE 1 STAGE 2 STAGE 4 ~STAGE 6 ~STAGE 8
(13.7)
.0670 12.3 9.5 7.05 6.5 ?
5.35
.0496 11.4 7.5 6.80 4.46 2.50
.0061* 10.5 6.6 6.60 2.24 2.11
3.2 6.0 6.38 1.50 1.55
3.68 3.20 1.24 1.08
3.10 2.06 1.00 1.00
2.50 1.72 .84 .38
1.54 1.58 .46 .32
.96 1.32 .41 .24
.56 .84 .38 .23
.42 .23 .15
.10
.075

 

*Uncertainty'~13%.

62

selected orbits, or groups of orbits. However, for the sake of brevity,
only one scan which covers the entire frequency range apprOpriate to
that compound will be presented. The cumulative results from all scans
for each compound are listed in Table 5.1.

The primary focus of this research is to characterize the size and
shape of the F3 for each compound of a given stage. However,
the MTO technique can also be used to obtain information on the
effective mass and Dingle temperature for each observed orbit. Recall
from Chapter One that the MTO amplitude dependence on the effective mass
is somewhat weak, and therefore this technique is not well suited for
effective mass measurements. Attempts at determining the effective mass
for these compounds have been somewhat unsatisfactory. This is due
partly to the above reason, ibut primarily due ,to the difficulty in
separating closely-spaced frequencies, which is required for an accurate
measurement of the amplitude of a given deA frequency. The effective
mass must also be known to determine the Dingle temperature (Equation
1.18). The effective mass and Dingle temperature is obtained for the
lowest frequency orbit of a stage 6 compound, in section 5.2.5, as an

example of the method of measurement of these parameters using MTO.

5.2.1 HOPG

FS measurements were performed on the graphite host. material,
HOPG. These studies, performed at about .9K, confirmed the quality of
the host material and also served as a test of the experimental
technique. The deA frequencies observed are listed in Table 5.1, for
the c-axis parallel to the magnetic field direction. These frequencies

are in excellent agreement with the results of Shubnikov-deHaas3O and

 

63

deHaas-van Alphen29 measurements, and also confirm the SWMcC FS model
described in section 4.2. The agreement of the frequencies given by our
transforms and these earlier results also indicate that the cryostat,
magnetic field calibration, and the computerized acquisition and

analysis techniques are satisfactory.

5.2.2 Stage 1 SbCl5

Experimental FS results on a small (5mg) stage 1 sample provided by
R. Clarke, University of Michigan, are shown in Figure 5.1(a). The
small sample size resulted in a small MTO amplitude due to thermal
loading by the support and sensor. Because of this, and the additional
decrease in amplitude with angle (section 5.3), studies of the rotation
dependence of the deA frequencies could not be performed. The four
frequencies obtained for C II H are listed in Table 5.1. Structural
studies by x-ray analysis indicated a small (~ 5%) admixture of stage 2
in this compound. However, none of the frequencies associated with
stage 2 compounds, from section 5.2.3, were observed in this sample.
Note that the typical stage 1 deA frequencies are at least two orders

of magnitudes larger than those found in HOPG.

5.2.3 Stage 2 SbCl5

Three stage 2 compounds were investigated. Two of the compounds
were produced locally, the third was provided by the University of
Michigan group. The deA frequencies obtained on all three samples are
in agreement, and are listed in Table 5.1. A representative scan for

stage 2 compounds appears in Figure 5.1(b).

Fourier Intensity,

64

 

 

 

 

 

 

 

 

stage 1
(a)
5.0 10.0 15.0. F In MG
stage 2
(6)
922W
1 l 1 A I
5.0 10.0 15.0
.2090 5
(c)
A“ M “j A I g
5 0 10.0 15.0
stage 6,
(d)
HUI)
j 7‘ l i
5.0 10.0 15.0

Figure 5.l Stage Dependence of SbCl5 FS.
All data taken at T~.9K with cllH.

65

SN

.m0.0:0:00.. <>Ic .0 mocmccmamo co.umpom m.m 0.:m..

am am

co.vopoe we womemoa

8V
.

0
.

m

0m” 0.

 

 

    

w“. .0039... ...ll..
.wn. 302.0530

 

m. .0200. u 00......
4 GEm.¢ vouaao
Q 6594.. 0.00..

 

 

N.®

m.®

m.m.

6.“.
.3...

66

Studies of the rotation dependent shift of the deA frequencies,
which yield geometrical information, were performed for angles up to 40°
for several orbits. In Figure 5.2 we display, for the sake of clarity,
only a few representative datum points for the .96 MG orbit of the stage
2 compounds. In that figure we also show some of the results for stage
4 and stage 6 compounds. The rotation data obtained for different
orbits within a given stage, and for different stages, all indicate the
same type of angular dependence. Two theoretical curves are also shown
in Figure 5.2. One curve indicates the rotation dependence predicted
for a cylindrical FS, the other for an ellipsoidal FS, see section

5.4.1. We will interpret the rotation data in section 5.4.

5.2.4 Stage 4 SbCl5

The deA frequencies obtained for a locally produced stage 4 com-
pound are given in Table 5.1. Rotation studies were performed to 35°
and are shown in Figure 5.2. Figure 2.6 is one example of the low

frequency scans for stage 4.

5.2.5 Stage 5 SbC15

Two stage 5 compounds produced locally were studied. Characteriza-
tion by x-ray analysis indicated a presence of stage 6 in the first
sample. The frequencies observed in this mixed stage sample are shown
in Table 5.2, along with the frequencies associated with stage 6. We
see that many of the frequencies attributed to pure stage 6 are shared
by this mixed stage sample. This might indicate that stage mixing in
this sample consists of separate macrosc0pic domains of each stage. The

size of these domains has not been ascertained. Typical data obtained

67

Table 5.2

Sharing of Frequencies in a Mixed Stage 5 and 6 Compound

 

Stage 5 & 6 Stage 6
6.92*
6.5 6.5
5.4 5.35
4.46
4.3*
2.25 2 24
2.04*
1.73*
1.50 1.50
1 25 1.24
1.00 1 00
.89*
.84
.46
.41
.38
.23 .23
.14+

 

*Indicates frequencies attributed to stage 5 and confirmed by the
other stage 5 compound.

+This frequency found in stage 8.

68

on this sample is shown in Figure 5.1(c). Rotation studies (not shown)
were also performed on this sample. Analysis of the data yields the
same behavior as seen in Figure 5.2 for the other stage compounds. The
second stage 5 sample cleaved during preparation and consequentJy gave
poor FS data. However, this sample did confirm the stage 5 frequencies

indicated in Table 5.2.

5.2.6 Stage 6 SbCl5

The stage 6 compound studied was produced by the University of
Michigan group. X-ray diffraction characterization indicated stage 6
for this sample, but stage analysis generally becomes more difficult
with increasing stage index (n g 6), as mentioned in section 3.4. In
addition, one is likely to form mixed stage compounds during synthesis
for stages 6 and higher.

The deA frequencies obtained for the nominally stage 6 compound
are listed in Table 5.1, and a typical frequency scan is shown in Figure
5.1(d). The rotation dependence of the lowest frequency orbit, for
angles up to 65°, is shown in Figure 5.2.

Studies of the temperature dependence of the MTO amplitude for
fields between 3.4 and 20 k0 were performed for the lowest frequency
orbit, .23 MG. As described in section 1.3.3, analysis of the tempera-
ture dependence of the MTO amplitude A1 for a given magnetic field value
will give the effective mass. In Figure 5.3, we have shown the measured
MTO amplitude ratio as a function of temperature for Ho = 15kG. Also
shown in that figure are amplitude ratio curves generated from Equation
1.17 for Ho = 15kG, and m:/mo values ranging from .2 to .4. A good fit

to the experimental data is obtained for m:/m0 = .285. Given this value

69

0 0000m .0.0.0 02mm.
.meapmgmaeme m0m20> mu:u..0&< op: m.m 0.30..

x c. ogawoeanUF

N.N N m.« m.a v.« N.«

 

 

 

 

 

o U
E\ E n E
....

 

 

.wa

Ow

‘(,,uI‘I.)V/(,,ul‘ee')v

2.0 "

InlA
( /H3/2 )

1.0 -i

70

 

 

1 1 A

T I ‘77

J ,2 (x10' 4‘."1
1m

Figure 5.4 MTO Amplitude verses l/H.
.23MG orbit, Stage 6

)

71

for the effective mass, Ti: the Dingle temperature, may be obtained from
the field dependence of the MTO amplitude at fixed temperature from a
plot of 2n(Ai/H3/2) vs. l/H, as shown in Figure 5.4. The slope of the
line generated gives the Dingle temperature from Equation 1.20. From
this data the Dingle temperature for the .23 MG orbit is found to be
2.5K.

5.2.7 Stage 8 SbCl5

Results on a nominally stage 8 compound, also produced by the
University of Michigan group, are given in Table 5.1. General features
of the frequency Spectrum are similar to those shown in Figure 5.1(d)
for stage 6, and hence the stage 8 scan is not shown. Considerable time
and effort was Spent searching for low deA frequencies, corresponding

to those of graphite on stages 6 and 8, with null results.

5.3 General Observations and Discussion

The relation between the structure and concentration (staging), and
the resultant Fermi surface cross sections (deA frequencies), is indi-
cated by Figure 5.1 and listed in Table 5.1. One notes a monotonic
decrease in the observed frequencies as the stage index is increased.
The normalized rotation dependence of the deA frequencies is indepen-
dent of stage and is shown in Figure 5.2.

In reviewing the stage dependent deA frequencies observed, there
are several features worthy of note. First, we see strong evidence of
frequency sharing for different high stage compounds such as stages 6
and 8. For high stage compounds it is difficult to produce good stage
fidelity, and from these results it appears that the different stages

within a mixed stage compound are separated.

 

72

Within a given pure stage compound, it appears that certain

frequencies are arithmetic sums of other frequencies appearing in that
compound. For example, in stage 2, the 1.54 MG orbit could be a sum of
the .56 and .96 MG orbits, indicating magnetic breakdown, or other FS
effects discussed in section 4.4. Possible orbit assignments for stage
2 SbC15 are given in Table 5.3. Earlier results by the deA technique
on SbCl5 compounds stages 2-4 also provide evidence of arithmetic sums
of frequencies, although this was not reported in the article.53 Also,
evidence of magnetic breakdown in MTO experiments on Br,46 HN03,43 and
SbCl524 acceptor compounds has been reported. We must also note that
sum or difference frequencies could occur from the non-linear nature of
our apparatus, in partcular, the MTO sensor, see Equation 2.5. These
non-linear effects, however, are too small to be of consequence.
Earlier FS results on stage 2 and 4 SbCl5 compounds using MTO24
(field modulation technique) and for stages 2, 3, and 4, using the deA
technique have been published.53 The frequencies reported for c u H are
listed in Table 5.3, along with our observations. We discuss first the
results of the other MTO experiment.

Batallan et al.24 reported observing stage independent FS results
for these compounds. In reviewing their article, it can be seen that
the deA frequecy resolution of their apparatus is poor and may have
obscured any stage dependent frequency shifts.

Results of the deA studies on stages 2-4 (Table 5.3) generally
show higher frequencies than those observed in this study. This
disagreement between deA frequencies reported by that group and by
others is not limited to these compounds. For example, the results of

Tanuma et al. Show approximately 10-20% higher deA frequencies for K

 

Table 5.3

Published Results from deA and MTO Experiments

Author

Present Results

Tanuma et al.

Batallan et al.

53

24

 

*Note:

drical, FS segments.
elliptical (section 5.4

and the Present Results on SbCl

(13.7

15.

....
Noah-hm"

..me

HNwalOWNKO
0101

F

11

5

Orbit
Assignment*

 

F8+F5

m
\ICD
'11
....

ooh-mow

1..

‘T1'T1+"I'I‘T1'TI'T'I+71
N

I—‘N

HHHNme

The orbits F —F appear to be cylindrical, or nearly cylin-
could be either cylindrical or

HI—ll—‘NwO‘O‘O‘N

l—‘NWU'I

74

compounds in comparison with SdH studies for the same compounds.37’42
Thus, the disagreement between our values for the SbCl5 compounds and
those of the deA studies of Tanuma et al. may be due to a systematic
error in their experiment, e.g. magnetic field calibration. In general,
their results agree reasonably well with ours if we shift their frequen-
cies down by about 10-15%. We also note that while they required very
high magnetic fields (up to 150kG) to observe the highest frequency in
stage 2, we apparently can detect it with only 50kG fields.. In general,
the largest orbits (large deA frequencies) are best observed using
large magnetic fields. This may indicate that our signal-to-noise ratio
is better. From our results, and from the results of the deA studies,
we believe that this highest orbit in stage 2 is due to magnetic break-
down or other orbit coupling effects, and should not be considered as a
fundamental orbit, see section 5.4.1.

We have found eXperimentally that the MTO amplitude is strongly
attenuated as the angle between the crystalline c-axis and the magnetic
field increases. The signal amplitude decreases below the noise level

for angles above 65°. This is related to the amplitude

2) 1/2
ext

of mE/mo which appears in other amplitude factors in that equation.

factor (BZE/akz in Equation 1.14, and to the angle dependence

5.4 Predictions from the Fermi Surface Models

The two FS models we wish to compare with our results are; the
rigid band model from section 4.3.1, and the 20 model of section
4.3.2. Following the application and discussion of each model we con-
clude by comparing these models in terms of their ability to account for

the observed FS results.

 

75

5.4.1 The Rigid Band Model

The rigid band model (RBM) has been used with some success in
predicting and interpreting the GIC electronic pr0perties as measured by
SdH37 and deA38’52:53»55 techniques. The model, as originally
conceived,36 was intended for dilute, i.e. n > 4 compounds. However, it
has also been applied to compounds as concentrated as stage
2.37252’53’55 Comments on the range of validity for the RBM are found
in section 4.3.1. To apply this model, the Fermi level Shift due to
intercalation must be determined. From section 4.3.1 this is
accomplished by lowering (for acceptor compounds) the Fermi energy, Ef,
until the largest predicted deA frequency (Fermi surface cross section
at the K point, k2 = 0) coincides with the largest observed frequency.
We have no guarantee, however, that the largest oberved frequency is, in
fact, the largest frequency orbit present in the sample, (section
5.3). Once Ef is fixed, other deA frequencies are predicted and
compared with the experimental results. This model predicts an
ellipsoidal FS which, after zone folding, consists of nearly cylindrical
segments, and one small elliptical orbit (Figure 4.3). From geometrical
arguments, the deA frequency shift predicted for an ellipsoidal Fermi

surface segment is given by

2
F(o) _ 2 a . 2 92
F107" (cos 0 +.Ez.51n 9) 5,1

where F(o) is the deA frequency for c u H, F(6) is the deA frequency
at angle 6, and a and b are major and minor axes for the small FS
ellipse at the zone center. For a nearly cylindrical FS segment, the

normalized deA frequency Shift should vary according to

76

;4%%-= cos 6 . 5.2

Details of the application of this model are given only for the case of
the stage 2 compounds; the procedure is the same for all other stages.
The variation of the FS cross sections or frequencies given by the RBM
as a function of Ef for stage 2 SbCl5 compounds is shown in Figure
5.5. These curves were generated from Equations 4.5 and 4.6 with appro-
priate values for IC. For stage> 2, the largest. fundamental orbit
observed is 9.5 MG which implies a Fermi energy of -.77eV. Although we
were able to resolve an orbit at 13.7 MG, we believe that this orbit is
not fundamental, and is in fact due to magnetic breakdown or other
effects given in section 5.3, and should not be used in determining
Ef. Note that the Fermi level shift for stage 2 is near the limit of
validity of the rigid band model, see section 4.3.1. Once Ef is fixed,
the predicted deA frequencies are obtained from the curves
labeled 51 - E4 of Figure 5.5. ‘These frequencies, and the frequencies
observed for stage 2 are listed in Table 5.4. The carrier concentration
n(Ef) can be obtained by calculating the volume within the FS. This is
done by integrating Equation 4.5 over the variable kz with Ef fixed.
The carrier concentration calculated for stage 2 is also listed in Table
5.4. From the carrier density, n(Ef), and the number density, N, of
intercalant molecules, the charge transfer coefficient, f, may be deter-
mined from

"(Ef)

f=T. 5.3

 

77

 

.mFUQm N mmmum .mm m0m20> xucmscme. <>zu 2mm m.m 0000..

 

3.. >20... ...—.0“. m”

N... 4.0. 0.0. 0.0. f

4:...— _ ...r If _ .

nil-II l .f
0’. OIUII’- II”,” I];
it)! I’ll I” /
II I I /
0'00 1’ ’2’ l +
II” D]! I, [
II III ”
IPII ’I I I
20.0 «...-on v. .o l
I I
0” I,
It: I:
0’ II
III ’0
of! [I

.... /. «m

or: ’0
a /

 

 

 

O! ......

5.

Nu

Va

SbCl5
Compound
Stage 2

Stage 4

Stage 5

Stage 6

Frequency
Observed
9.
.5
.6
.0
.68
.10
.50
.54

i—‘NOOOOC‘O‘N

HHHNMQGGN

i—IN-b Oi

i—Ii—H—aN-bmoa

78

Table 5.4

RBM Predictions and Observed Results

5

Frequency

Predicted

matched
7.7

4.0

1.0
.20

matched

1.02

matched
6.5

NwU‘l
@0030»)

1

matched
5.66
4.86

1.00

.12

 

Ef(8V) n(Ef)cm

-.77 1.4x1021
-.65 9.2x1020
-.64 9.2x1020
-.58 7.2x1020

-3

 

.65

.75

.69

79

For total ionization of SbCl5 molecules, f would be one. In calculating
N for the SbCl5 compounds, we have used the accepted Cl4nSbCl5 stoichio-
metry, and assumed that all intercalant molecules transfer charge
equally. As was discussed at the end of Chapter 3, this assumption may
not be justified. However, with this assumption the resultant charge
transfer is listed in Table 5.4.

The above procedure was followed for compounds of stage 4, 5, and
6, and the results are given in Table 5.4. Due to difficulties in
confirming the largest orbits for stages 8, the model could not be
applied to that compounds. It is important to note that the predictions
of this model are based on the assumption that we have observed, for
each compound, the largest deA frequency which is used to determine Ef
and upon which all other calculated quantities, n(Ef), and f are based.

The agreement between the observed deA frequencies and those
predicted by the RBM is only qualitative. This is not surprising in
view of the simplicity of this model. From Table 5.4 it is seen that
the RBM also underestimates the number of deA frequencies to be
observed. This is due to at least two causes. First, the RBM as
applied does not include crystal field effects and other possible phen-
omena which would result in several additional orbits, see section
4.4. Effects of the crystal field were estimated by Tanuma et al. in
analyzing their results on AsF5 compounds, see Figure 4.3.52 Second,
some of the experimental frequencies appear to be harmonics of a funda-
mental and others are possibly due to magnetic breakdown or other
effects coupling fundamental orbits, see section 5.3.

Studies of the rotation dependence of the deA ‘frequencies. on

selected orbits in different stage compounds are shown in Figure 5.2.

80

From the RBM we expect the smallest FS segment to be elliptical, as
shown ll) Figure 4.3. The remaining orbits Should display a frequency
shift consistent with the angular variation predicted by Equation 5.2
for cylindrical Fermi surfaces. In Figure 5.2, curves generated from
Equation 5.1 for b/a = 5, and from 5.2 are shown. The higher frequency
orbits shown in that Figure are consistent with a cylindrical Fermi
surface geometry, at least to 45°. The smallest of the orbits for the
stage 6 compound, .23 MG, was observable only to 65°, where the MTO
signal decreased below the noise level. Unfortunately, data to this
angle can be interpreted equally well in terms of an ellipse, or as a
cylindrical FS segment. Thus, on the basis of the data obtained, the FS
consists of cylindrical, or nearly cylindrical, segments aligned along
k2.

We can calculate cyclotron effective masses from the RBM by deter-
mining the sl0pes of the cross section (frequency) curves evaluated at

Ef (e.g. Figure 5.5) and using,

mc = 2fii5E’ E ’ 5'4

A calculation of m: for the .23 MG orbit of the stage 6 compound gives a
RBM value of mZ/mO ~ .35 which compares well with the measured value of
.3 from section 5.2.6.

The F3 data from deA and SdH experiments have also been evaluated
using the RBM.37’38’52’53’55 The SbCl5 compounds stages 2, 3, and 4
were studied using the deA technique. The published results53 of that
study listed only the charge transfer coefficient calculated from the

RBM, which ranged between .43 and .49; in general agreement with our

81

results. Analysis of low stage K and Rb compounds studied by the SdH
method were analyzed using this model.37 The trend we observe in f
increasing for increasing stage (excluding stage 5) was also observed in
their results. This implies that the intercalant molecules are more
fully ionized for more dilute compounds.

The Fermi energy may be inferred from several other experimental
techniques. To date, however, the only technique other than Fermi
surface methods, is the study of infrared and visual Optical
35,56,57,59 are shown

reflectivity. The results of several experiments

in Table 5.5 along with the values calculated from our MTO measurements

using the RBM. The agreement is in general quite good.

5.4.2 The 20 Model

We now apply the 20 F5 model described in section 4.3.2 to our
results. This model has been used in interpreting both Optical35 and
MTO FS data24'43’45»58 on several graphite acceptor compounds. We shall
present the procedure for our stage 1 compound. The method for
constructing the Fermi surfaces for stages 2-4 is Similar, although more
complex. Following discussion of stages 1-4, we interpret our results
on stages 6 and 8 using the dilute limit of the 20 model.

As discussed in section 4.3.2, the intercalant in-plane structure,
which determines the Brillioun zone fer in-plane zone felding, is of
central importance in this model. A discussion of the issue of what
superlattice is appr0priate to the SbCl5 compounds at low (4K) tempera-
tures is found in section 3.4. The stage 1 compound studied was
characterized for in-plane structure by the University of Michigan

group. We base our calculation of the SbC15 GIC Brillioun zone on

 

82

Table 5.5

O

Ef as Determined by MT

Values in (eV)
Source n=1 n=2 n=3 n=4 n=5

Present
Study -.94 -.77 -.65 -.64

Ref. 56
Optical
Refl. -.72 -.62 -.60

Ref. 35
Optical
Refl. -.75

Ref. 59*
Optical
Refl. -1.2 -.94

Ref. 57

from

Thermopower

Data -.79 -.43 -.3

*Used adjusted values for band parameters.

and RBM and Values from Other Experiments

n=6

-.58

 

83

 

a»

Figure 5.6 20 F5 Construction, Stage l SbClS.

84

the /7 x 77 superlattice structure discussed in section 3.4, which gives
the hexagonal zone shown in Figure 5.6. Assuming this intercalant
structure we use the Harrison construction for determining the FS given
the apprOpriate (stage dependent) band structure.

For stage 1, the 20 band structure is given by Equation 4.4. The
Harrison construction is used in Figure 5.6 where the lst Brillioun zone
is shown and the isoenergy circles of radius kf are drawn at the zone
corners as prescribed by this model. The value of kf is adjusted until
the zone-folded Fermi surfaces best agree with our observations. The
best fit found is shown in that Figure where we have matched the 11.4 MG
orbit by adjusting kf to a value of .47A'1. A second orbit is then
predicted at 13 MG. From this value of kf and Equation 4.4, we find Ef

= -2.4eV for stage 1. To determine f we may rewrite Equation 4.8 as
kf=fiffi=/—2%f/2 5.5

where n is the area of the SbCl5 elementary unit cell, ~ 36A2, and f/2
is the number of holes per carbon atom. From the stoichiometry the
charge transfer is given by (f/2) . 14. This gives the unrealistic
figure of f = 8; we would expect the acid molecule to be at most singly
ionized.

We must note that from the model as described, it is not clear what
criterion should be used for choosing which orbit is matched. If
instead we choose to match the 11.4 MG orbit, we obtain kf = .19A'1, Ef
= -.99eV, and f = 3. However, this choice would predict only one orbit.

The same general procedure is used for predicting the FS of stages

2-4, except that the band structure relations are somewhat more

85

complex. Attempts at applying the stage 2 model were not
satisfactory. For stage 2, two isoenergy surfaces, of radii kfl and kfz
are adjusted to match our results. However, the choice of which
observed orbits are matched again appears to be arbitrary in this model
and leads to a variety of possible interpretations.46 The band
structure for both stages 3 and 4 are very complex and were not applied
to our stage 4 compound.

We now turn to the dilute limit of this 20 model and apply it to
the results obtained for our stage 6 and 8 compounds. According to this
model, we should observe two Fermi surfaces, one due to pure graphite,
the other due to a metallic sandwich consisting of the intercalant layer
and its bounding graphite layers. The F8 of the first component,
graphite, is already known. The F5 of the metallic sandwich is con-
structed as follows. From section 4.3.2 we again use the Harrison
construction, but assume a nearly free electron (NFE) band structure.
The Brillioun zone is again the one obtained from the /7 x /7 primitive
cell and is shown in Figure 5.7. In that figure, we have shown the NFE
isoenergy circle centered at the origin, and adjusted the radius, kf, of
this circle to match the largest observed frequency in the stage 6
compound, 6.5 MG. A single FS ellipse results along the zone edges.
This requires that kf = .64A'1. From Equation 4.7 we determine f using
N = 1 and SBZ = 1.085A'2. Again, the value obtained for f is greater
than one (f = 2.4) which again is unrealistic for SbCl5 compounds. We
were unable to confirm the largest orbit in our stage 8 compound, and
hence cannot apply this model. The second component for dilute
compounds is pure 3D graphite. Thus, we expect to also observe the deA

frequencies of graphite for compounds of stage index 4 and higher.

86

 

 

 

Figure 5.7 20 NFE FS Construction, Stage 6 SbCl5.

87

Referring to Table 5.1, no graphite orbits are observed in either the
stage 6 or stage 8 compounds.

The F5 for the low stage compounds, n = 1-4, and for the metallic
sandwich component of the dilute compounds, are cylindrical. Hence, we
expect the deA frequencies for these cases to shift with angle
according to Equation 5.2. This predicted angular dependence is shown
in Figure 5.2, along with some of the rotation data obtained. AS can be
seen, data for all stages is consistent with either an ellipsoidal or a
cylindrical FS model.

From this model it is difficult to obtain a single value of the
Fermi energy which could be compared with the values in Table 5.5. The
values that we obtain from our best fit to our observations imply Fermi

energy shifts much larger than those listed in that table.

5.5 Conclusions

We evaluate the performance of these two Fermi surface models by,
(1) comparing the predicted Fermi surface'cross sections (deA frequen-
cies) and their angular dependence with our experimental values, and by
(2) comparing the resultant Fermi energy and its stage dependence from
these models with the results of other types of experiments.

The rigid band model (RBM) gives qualitative agreement with the
observed data for the appropriate range of stages studied, see Table
5.4. The 20 model and its dilute limit do not fare well for either the
concentrated or dilute compounds. In regard to the geometrical informa-
tion obtained to date, we cannot state whether the FS is elliptical (3D)
or cylindrical (20). Overall, however, the RBM appears to be superior
in predicting the FS. We also recall here that this model has only one

adjustable parameter, whereas in general the 20 models have several.

88

Probably the single most important parameter, and the one which is
most directly measured both by Fermi surface techniques and by other
experimental methods, is the Fermi energy, Ef. The values obtained from
our experiment as interpreted by the rigid band model, and the values
measured by Optical studies, as well as those inferred from other data
are listed in Table 5.5. The values from the RBM are in good agreement
with those listed in that table. Values for Ef obtained from the 20
model are much larger. Again,the RBM appears to be superior to the 20
models.

In summary, we have made a systematic study of the stage dependence
of the F5 for carefully prepared and characterized SbCl5 compounds.
This study has included a wider range of stages than previous FS
measurements for a single compound, extending from the most concentrated
compound to well within the dilute limit. Our results are sample inde-
pendent for samples of the same stage, and we observe a systematic
decrease in the FS cross sections with increasing stage index. Results
of previous deA FS studies on stages 2 and 4 are in general agreement
with our results.

We have interpreted our data using two competing Fermi surface
models. We find that the rigid band model best describes the observed
FS and that parameters such as the Fermi energy as calculated from the

RBM agree well with other experiments.

CONCLUSION

We have designed and used an experimental apparatus for detecting
magnetothermal oscillations in high conductivity single crystals. A
minicomputer has been interfaced to this apparatus, and is used for data
acquisition and analysis. For high quality crystals this system
provides accurate, high resolution Fermi surface (FS) data.
Measurements can be performed on a variety of sample shapes, weights,
and sizes.

This system has been employed to study the FS of Sb015 GIC's,
stages 1-8. The compounds used irl this research have been carefully
characterized. The results obtained exhibit sample independence for
samples of the same stage index. We have also found that with our
system we are able to detect stage admixtures in the high stage
compounds.

The results of our experiments have been analyzed using two
competing theories proposed to describe the electronic structure (FS) of
GIC's. We have found that one of these models, the rigid band model, is
superior in predicting the observed FS results. Model dependent para-
meters, such as the charge transfer coefficient and the Fermi energy,
calculated from our data as interpreted by this model, are in good

agreement with measurements of these parameters by other techniques.

89

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"I11111111111411l“