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STRUCTURAL PROPERTIES
OF
MISMATCHED ALLOYS

By

Normand Mousseau

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1993

 

' ABSTRACT

STRUCTURAL PROPERTIES
OF
MISMATCHED ALLOYS

By

Normand Mousseau

The problem of understanding the local structure of disordered alloys has
been around for a long time. In this thesis, I look more specifically at the ef-
fect of size—mismatch disorder in binary alloys under many forms: metallic and
semiconductor alloys, bulk and surfaces, two and three dimensional systems. I
have studied the limitations of a central-force model (CFM) and an embedded—
atom potential (EAM) in describing the local structure of binary metallic alloys
composed of Ag, Au, Cu, Ni, Pd, or Pt. Although an analytical model developed
using the CFM explains qualitatively well the experimental and numerical results,
in many cases, it is important to add electronic density effects through a more
sophisticated potential like EAM in order to agree quantitatively with experi-
ment. I have also looked at amorphous and crystalline silicon—germanium alloys.
It turns out that the effect of size-mismatch is the same on a crystalline and an
amorphous lattice. In the latter case, it can be seen as a perturbation of the much
larger disorder due to the amorphisation process. However, the analytical predic-

tions differ, for both the crystalline and amorphous alloys, from the experimental

 

results. If one is to believe the data, there is only one possible explanation for
this inconsistency: large amounts of hydrogen are present in the samples used
for the measurements. Since the data analysis of EXAFS results is not always
straightforward, I have proposed some experiments that could shed light on this
problem. One of these experiments would be to look at the (111) surface of a Si-
-Ge alloy with a scanning tunneling microscope. I also present in this thesis the
theoretical predictions for the height distribution at the surface as well as some
more general structural information about the relaxation in the network as one
goes away from the surface. Finally, I have studied the effect of size-mismatch in
a purely two dimensional lattice, looking for mismatch—driven phase transitions.
Although it is possible to map size—mismatch on an effective temperature at low
disorder, I have not been able to find any indication that a hexatic phase exists
in these 2d systems. Since systems were studied with different potentials in very
large unit cells, the conclusion is that the hexatic phase is not universal for two

dimensional networks.

 

 

De choses répandues au hasard,

le plus bel ordre, l’ordre—du—monde.

Héraclite

iv

ACKNOWLEDGEMENTS

I would like to express my deeper gratitude to my thesis advisor Professor
Michael F. Thorpe for his constant support, his physical insight and for sharing
with me his love of physics. During the last three years, he has guided me with
confidence and humour through the meanders of research. His continual search
for simplicity even in complex problems will always serve as a guide to me.

I am also grateful to Professors C. F. Foiles, S. D. Mahanti, R. Stein, D.
Stump and D. Tomanek for serving on my guidance committee. I would like to
thank Mrs. Janet King for making life in this department so much simpler as well
as MM. Ron Winsauer, Seong-Gon Kim and Dr. George Perkins for helping me
with computers.

I wish to thank Dr. Yong Cai and M. Jidong Chen for their numerous
discussions, comments and help at various stages of this work. I am grateful to
Professor R. A. Johnson for discussions related to chapter 3, to Dr. S. de Gironcoli
for sending me his numerical results used in chapter 4, to Professor F. Wooten for
allowing me to use the coordinates of his a—Si model and to Professor J. B. Boyce
and Dr. G. S. Cargill III for useful comments regarding chapter 5.

Finally, I want to acknowledge the moral support given to me by my
friends B. Djordjevié, D. Vasiljevié, F. Liu, C. Overney and J.-H. Pinto as well
as those from Science Theatre, particularly D. Casavant, D. Bercik, D. Kuhl, J.
Kriessler, J. Discenna, J. Conrad, R. Lambert, and S. Snyder. They all made my

life much more enjoyable while I was in East Lansing. .

 

Financial support from the Natural Sciences and Engineering Research
Council of Canada, the Fonds pour la formation de chercheurs et l’aide a la
recherche du Québec and the Center for Fundamental Materials Research of MSU

without which this thesis could not have been, is also greatly acknowledged.

Contents

Abstract ..................................

Acknowledgments 1 ...........................

1 Introduction

1.1 Order and Disorder ..........................
1.2 Alloys and Amorphous Systems ...................

1.2.1 Short—range order .......................

1.2.2 Medium-range order .....................

1.2.3 Long—range order (LRO) ...................
1.3 Understanding alloys .........................
1.4 EXAFS ................................
1 .5 Conclusion ...............................

2 Theory

2.1 Bond mismatch ............................
2.2 Force constant disorder ........................
2.3 Site—mismatch .............................
2.4 a" and a" ...............................
2.5 Semiconductors ............................
2.6 Correlation ..............................

vi

iv

12
19

21
22
29
29
31
34

35

 

CONTENTS

2.7 Conclusion ...............................

3 Metallic Alloys

3.1 Embedded—atom method .......................
3.2 Central force model ..........................
3.3 Computer simulations ........................
3.4 Results .................................
3.5 Conclusion ...............................

4 Crystalline and Amorphous SiGe

4.1 Solution for binary alloys .......................
4.2 Results .................................
4.3 Effect of amorphisation ........................
4.4 Conclusion ...............................

5 About EXAFS Results

5.1 Assumptions ..............................
5.2 Comparison between theory and experiment ............
5.3 Structural model ...........................
5.4 Discussion ...............................
5.5 Conclusion ...............................

6 Surface of semiconductors

6.1 Theory .................................
6.1.1 Bondmismatch........................
6.1.2 Site mismatch .........................

6.2 Topological rigidity parameters ...................

vii

36

39
42
45
47
48
64

67
69
73
78
85

88
89
90
95
105
107

108
109
111
112
113

CONTENTS

6.3 Triangular lattice ...........................
6.4 Silicon-Germanium alloy .......................
6.5 Conclusion ...............................

7 Two-dimensional melting
7.1 Review .................................
7.2 Theory .................................
7.3 Details of simulations .........................
7.4 Results .................................
7.5 Discussion .............. . .................
7.6 Conclusion ...............................

8 Conclusion
A Conjugate—gradient method

B Johnson’s long-range potential

Bibliography ...............................

viii

115
117
128

131
132
139
141
145
155
158

161

165

169

 

List of Figures

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

2.1
2.2
2.3
2.4
2.5
2.6
2.7

3.1
3.2
3.3

Sketch of chemical and structural disorder .............. 3
Characteristics of short- and medium-range order. ........ 5
Nearest-neighbor distance for Ga1-:InxAs .............. 10
X-ray absorption coefficient ...................... 13
Absorptance of Cu at the onset of Cu K—shell ............ 14
Origin of EXAF S ............................ 15
EXAFS oscillations ........................... 17
Fourier transform of the EXAF S signal ................ 17
Perfect triangular network ....................... 22
Triangular network with bond mismatch ............... 23
Variation of the configurational energy. ............... 28
Force constant disorder. ....................... 30
Representation of the topological rigidity parameters. ....... 33
The topological rigidity a” and 2b". . . . . ........... 36
Bond correlations ............................ 37
Nearest—neighbor distances for Au1_,Agx. ............. 49
Same as Figure 3.1, except for Cu1_,‘Aux. ............. 51
Same as Figure 3.1, except for Cu1_dex. ............. 52

ix

 

LIST OF FIGURES

3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11

4.1
4.2
4.3
4.4
4.5
4.6
4.7

5.1
5.2
5.3

5.4
5.5
5.6
5.7

Same as Figure 3.1, except for Pd1-,Ptx. .............
Same as Figure 3.1, except for N i1.,‘Agx ...............
Same as Figure 3.1, except for Ni1_,Aux. .............
Nearest—neighbor distances for alloys relaxed with the EAM potential.
Nearest-neighbor distances for alloys relaxed with the CFM potential.
Same as figure 3.8. ..........................
Bulk and shear modulus for three alloys. ..............

Nearest—neighbor distances in Nio_5Auo.5. ..............

Nearest—neighbor bond lengths for Si1_,Ge, .............
Nearest-neighbor bond lengths distribution for Sio,55Geo,45 .....
Nearest—neighbor bond lengths distribution for SloggGeoJl .....
Next-nearest-neighbor distance in Si1_,Ge,. . i ...........
Strained energy due to length mismatch in a—Si1-zGe,. ......
Nearest-neighbor distance for a—Si1-zGe,. .............

Next—nearest—neighbor distances in a—Si1_,Ge,. ..........

Experimental nearest—neighbor distribution . . . . . . . . . . . . .
Nearest-neighbor length distribution for c—Si1-,Gez: H .......
Variation of the topological rigidity parameter with the mean co-
ordination ................................
Sketch of a cube, tube and plane with open surfaces. .......
Variation of the topological rigidity parameter for open surfaces. .
Sketch of the proposed structural model for SiGe alloys .......
Variation of the bulk modulus B as a function of the mean coordi-

nation ..................................

54
56
59
60
62
63
65

74
76
77
79
80
84
86

94
97

103

104

LIST OF FIGURES

6.1
6.2
6.3

6.4

6.5
6.6
6.7
6.8
6.9

7.1
7.2

7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11

7.12

Triangular network cut along (10) and (11) direction. .......

xi

116

Variation of the topological rigidity parameter as a function of depth118

Topological rigidity parameter as a function of the layer number
for surface (111). ...........................
Topological rigidity parameter as a function of the layer number
for surface (100). ...........................
Height distribution at the surface (111). ..............
Gaussian fit for partial z displacement distribution. ........
Si displacement distribution along 2. ................
Ge displacement distribution along 2 .................

Simulated density plot of the surface (111) of SisoGeso alloy. . . .

Different types defects on the two—dimensional triangular lattice. .
Proposed phase diagram of the two—dimensional melting in the
mismatch-temperature plane. ....................
Pseudo—harmonic potential. .....................
First derivative of the pseudo—harmonic potential ..........
Position-position correlation for a pseudo-harmonic system.
Position-position correlation for a Lennard-Jones system ......
Angular correlations and defects for a pseudo—harmonic system. .
Angular correlations and defects for a Lennard—Jones system.
Configurational energy as a function of effective temperature. . . .
Splay rigidity. .............................
Possible phase transitions for the KTHNY theory in the elastic
constants—temperature space. ....................

Pseudo-harmonic 50—50 alloys with 40% mismatch. ........

121

122
124
125
126
127
129

138

142
144
146
148
149
150
151
152
154

157

LIST OF FIGURES xii

7.13 Lennard-Jones 50—50 alloy with 26% mismatch. .......... 159

List of Tables

3.1
3.2

4.1
4.2

6.1

7.1

Parameters for the Johnson EAM potential. ............ 45
Recapitulation of the results for the CFM and EAM potentials. . 58
Parameters for Kirkwood and Keating potentials. ......... 69
Structural comparison between a—Si and c—Si1_xGe,. ....... 83
Partial displacements at the surface. ................ 123
The three phases of a two—dimensional solid as predicted by the

KTHNY theory with their associated correlation and defects. . . . 135

xiii

 

Chapter 1

Introduction

Until the early sixties, the world as seen by most condensed—matter physicists con-
sisted mainly of perfect crystals. Everything else, doped semiconductors, random
alloys and glasses, was considered dirty and unworthy of research. As mathe-
matical and computational tools developed in the sixties and the seventies, the
importance and interest of these materials began to be appreciated in the physics
community. By their nature, disordered systems are difficult to work with: their
properties can vary widely from sample to sample, they often show more mem-
ory of the preparation method than pure crystals, etc. Moreover, the conceptual
framework used in the study of crystals cannot always be extended directly to dis-
ordered systems. How can we define a defect in amorphous silicon, for example?

In this chapter, I will present some concepts that are useful when working
with disordered systems, with a special emphasis on random alloys and amorphous
materials. I will then examine some of the preceding approaches to understand
random alloy structure and discuss their limitations. Finally, since much of this
new work has been motivated by the development of the extended X—ray absorp-

tion fine structure scattering technique (EXAFS), I will briefly describe the theory

CHAPTER 1. INTRODUCTION 2

behind this experiment as well as its current possibilities and restrictions.

1.1 Order and Disorder

There are many levels of disorder in nature. Doped semiconductors, for example,
are only weakly disordered with typically one impurity per thousand or more
atoms. However, even such small quantities can have a decisive role in determining
the electronic properties of materials. This disorder is very different from that
found in glasses and amorphous solids. It is therefore important to try to classify
these various structures. We can distinguish between two major classes of disorder:
chemical and structural disorder (see Figure 1.1). Each of these two classes can
also be divided according to the length scale characterizing the absence of order.

For chemical disorder, different species of atoms are placed at random
on a crystalline lattice which retains more or less its original symmetries. In
this class are solid solutions (or random alloys), substitutional impurities and
vacancies, the main difference between the first two being the concentration of
the different constituents. Substitutional impurities are typically very dilute and
do not interact with each other; solid solutions are a mixture of two or more atomic
species with comparable concentration. For a long time, physicists believed that
most structural quantities in random alloys could be modeled correctly with a
simple average over properties of the constituent species. However, as it has been
discovered in the last decade, the situation is not so simple.

Glasses and amorphous solids are characterized by structural disorder. In
most cases, the coordination is maintained, but there is no remnant of crystalline
structure beyond the first—neighbor shell; these states are more akin to a liquid

than a crystalline phase. Glasses and amorphous solids can exist in the pure

CHAPTER 1. INTRODUCTION 3

 

(b)

Figure 1.1: (a) Representation of chemical disorder with two different species
distributed randomly on a crystalline network and (b) structurally disordered
lattice.

CHAPTER 1. INTRODUCTION 4

form or alloyed. The question of the glass transition remains one of the most
important unsolved questions in contemporary condensed—matter physics. In spite
of large amounts of work dedicated to this problem over the last two decades, many
fundamental structural and electronic properties are not yet properly understood.
For example, nobody has been able to obtain a satisfying model of the amorphous
silicon structure by computer quenching, raising questions about the time scale
involved in creating amorphous materials and in the validity of the interaction

potentials used for these simulations.

1.2 Alloys and Amorphous Systems

Another important component of classification of these structure is related with
the length scales for which the system is, or is not, ordered. The classification
used here is described in more detail in a review article by S. R. Elliot (1989). It
is usually divided in three length scales: short—, medium— and long-range order,

each associated with different experimental probes.

1.2.1 Short—range order

There seems to be a consensus in the literature about the definition of short—range
order (SRO). The SRO characterizes the purely local structure in a disordered ma-
terial, i.e. the central atom and the first-neighbor shell. It is usually defined by
the first- neighbor distance 1'5,- and by the bond angle 0,5,, as shown in Figure
1.2(a). In a constrained environment, the strain can be minimized by. an equilib-
rium between these two quantities, depending upon the ratio of the angular to

the stretching the force constant. In an amorphous materials (a—Si, for example),

CHAPTER 1. INTRODUCTION 5

   

(a) (b)

Figure 1.2: Characteristics of short- and medium— range order in a diamond
lattice. (a) nearest-neighbor distance r and bonding angle 0; (b) Dihedral angle
between to adjacent tetrahedra.

the average values of r;,- and 9,11; and first-neighbor shell are very similar to the
crystalline one; however, the fluctuations can be relatively large. In an random
semiconductor alloy, although the average values of the bond angle and the first—
neighbor distances are the same as for a pure semiconductor, the n,- distribution
possesses multiple peaks corresponding to preferred values for all the lengths in
the problem.

In alloys, the short—range order is also defined by the correlation between
first neighbors. Usually these correlations can be picked up by EXAF S or other
diffraction experiments if they are strong; however, when the correlation is weak,
varying only by a few percent from the random case, it is very difficult to obtain

reliable experimental data and one must rely on analytical or computational work.

CHAPTER 1. INTRODUCTION 6

1.2.2 Medium—range order

It is possible to subdivide the medium-range order (MRO) in three parts (Elliot,
1989): near, medium and far medium-range order. I will consider here only the
first (N MRO) part, placing the two others with the long—range order.

The near medium—range order is characterized mainly, in semiconductors,
by the relative position of two neighboring tetrahedra, measured by the dihedral
angle. (Figure 1.2(b)). This quantity is very important for amorphous materials
because it is at this level that the differentiation really appears between disordered
structures and crystals. In the case of alloys, the medium-range order is defined
by a correlation between second or third neighbors. This local clustering is almost
impossible to measure experimentally and the effects of such correlation have not
been really studied yet. Generally speaking, because it is too large for scattering
experiments and too small for direct observation, the MRO remains obscure. In
spite of its importance for amorphous and for glasses, very little effort has been put
into studying theoretically or experimentally the impact of varying the correlation
in angle or concentration at this level. With the development of more powerful
computers and more efficient semi- empirical potentials and ab initio methods,

one expects to see major developments this direction over the next few years.

1.2.3 Long—range order (LRO)

The long-range order should be renamed long—range structure since it is difficult
to imagine correlation at this scale; however, we, can here include defects like micro
voids, cracks and structural defects like disclinations. In multilayers, they often
appear in order to release the strain energy due to lattice size mismatch. They

are an indicator of the strain energy as well as the distance over which the lattice

CHAPTER 1. INTRODUCTION 7

perturbation relaxes when submitted to a very local strain, like the introduction
of a larger atom in the lattice.

Although the disordered structures have been divided into many length
scales, it is important to note that there is not always a clear cut distinction be-
tween these different length scales and that they often overlap. However, following
the methods developed for the study of the crystal, most of the theoretical work
in disordered systems has concentrated on the local order, while experiments can
more easily be performed on large samples. In the world of crystals, the fluc-
tuations are small, and average quantities obtained from x-ray diffraction and
other global experiments are representative of their local values; the Bloch the-
orem guarantees a one to one correspondence between the local and the global
properties. But the knowledge of the local situation in disordered materials is
not enough to obtain a global understanding of the structure of the solid and vice
versa. More effort has to be made in trying to link the properties of the disordered
materials with medium and long range order. But for this, one must go beyond

the usual theories of alloys and other disordered systems.

1.3 Understanding alloys ,

Until very recently, the understanding of the electronic properties of alloys was
based on the virtual crystal approximation (VCA). In this theory, all the atoms
are positioned on a crystalline lattice with a lattice parameter following, for most

semiconductor alloys, the Végard’s law (Végard, 1921),

d=(1 —z)dA+1:d3, (1.1)

CHAPTER 1. INTRODUCTION 8

where d is the lattice constant of the alloys A1_,B,. This empirical relation was
proposed in the 19203 to support x—ray diffraction results. Local variations of the
potential are averaged out and this quantity forms the basis for the calculation
of the electronic properties. Since the fluctuations are washed away, the system
recovers its symmetry, and periodicity and crystal states can be assigned. The
VCA constitutes therefore an interpolation between the properties of the indepen-
dent constituents. In spite of these limitations, the predictions of this theory have
had a relative success, with agreement with experiment varying from excellent to
poor. One of the difficulties in judging the quality of this approximation is that
the experimental precision on spectroscopic and other electronic measurements
is not precise enough to constitute an excellent test. Nevertheless, in view of
certain discrepancies, some refinements of the VCA have been developed so as
to include the random fluctuation potential. Since the discrepancies can be due
to many phenomena —random fluctuations, ordering, clustering and impurities,
for example— it becomes important to characterize the effects of each of these
features separately.

The major refinement comes by going one step further, with the coher-
ent potential approximation (CPA). Proposed independently by Soven (1967) and
Taylor (1967) at the end of the sixties, this self—consistent method uses the average
Green’s function instead of the virtual crystal as the zeroth approximation. This
approximation can be seen also as a mean—field approach to the multiscattering
theory. Although there is no formal demonstration of the limits of this approxi-
mation, it usually works very well in the case of small fluctuations but fails when
the fluctuations become very large —in one dimension, for example (Elliott et all,

1974).

ll‘fi

CHAPTER 1. INTRODUCTION 9

At the beginning of the 19805 however, a major blow was given to all the
schemes based on the virtual crystal approximation, shaking most of the concepts
forming the basis of understanding the structure of alloys. In 1983, Mikkelsen and
Boyce published their x-ray absorption fine structure (EXAFS) measurements
indicating that, in the GaInAs random alloys, the Ga—As and In—As nearest—
neighbor distances remained almost constant at all compositions.(see Figure 1.3).
The distribution of nearest—neighbors was therefore not a small deviation from
the Végard’s law, but rather indicated that the lattice itself was very distorted
in order to accommodate the different bond lengths disposed at random through
the lattice. The justification for VCA and CPA were lost in great parts. The first
theoretical explanation came from Zunger and Jaffe (1983) who proposed some
local distortion of the underlying crystal lattice. For large enough mismatch, the
atoms would gain by going locally from the zinc—blende structure to a chalcopyrite
arrangement. They presented a zeroth order theory where the mismatch param-
eter determines how far the chalcopyrite departs from the zinc—blende structure.
Although this model did not include the influence of the concentration on the
position of the center and width of the partial length distributions, it provided a
new view of the alloy problem with two distinct bond lengths averaging out the
Végard’s law.

Along with this important structural discovery was the realization that
the strain, imposed on a binary alloy by fitting on the same lattice atoms of differ-
ent sizes, cannot be neglected when trying to understand the electronic properties
of an alloy. In 1982, Osbourn proposed that strained—layered structures could
display electronic and optical properties not found in unstrained-constituent ma-

terials. Before him, the strain was considered as a necessary evil in the creation

CHAPTER 1. INTRODUCTION 10

 

 

 

 

 

' r T T T I r I T
1
2.60 d
.4
°§ - -
I.“
o » -
3 2.55 - —
I; _ X-RAY . .
5 t .
C VIRTUAL
8 ” CRYSTAL ‘
z - -
0
ii 2.50 - -‘
z. .. ‘ .4
E b- O .4
_ Ga - As d
2.45 4
1 l 1 L 1 l i I 1
0 0.2 0.4 0.6 0.8 1.0
GaAs InAs

COMPOSITION (at in GI1_‘In,‘Asl

Figure 1.3: Partial and total first-neighbor distance for GanInIAs as obtained
experimentally by Mikkelsen and Boyce (1983).

CHAPTER 1. INTRODUCTION 11

of interesting semiconductor heterostructures. But because the strain produced
internally by mixing size—mismatch elements can be much greater than that which
can be obtained externally, effects on the band structure due to this phenomenon
are not negligible. With a mismatch as small as 2%, the band structure will be
modified by more than 100 meV (Pearsall, 1990). Today, a lot of effort is targeted
to study the strain in multilayered compounds, specifically the reaction of the
interface and the maximum thicknesses one can obtain without creating defects
in the layers. In only a few years, this field has become a technological enterprise
where the discoveries are almost directly applied to the development of optical
devices.

More recently at MSU we worked on developing a structural model of al-
loys that would explain the EXAF S results in a consistent theory. This theory will
be presented in more detail in chapter 1 and forms the basis of my thesis. Using
harmonic interaction potentials, an exact expression for all the partial lengths in
the binary, pseudo—binary, ternary'and quaternary compounds as well as for mul-
tilayered materials has been obtained. The limits of the random model discussed
here are obvious. Almost no disordered system presents perfectly random order,
particularly when it comes to the choice of first and second neighbors. Even in
the case of SiGe, which surely can be dubbed as one of the most perfect random
binary structures, electronically speaking, recent measurements have shown that
there is some correlation in molecular-beam—epitaxy grown films (Jesson et al.,
1992) although these correlations do not appear in the usual methods of preparing
bulk alloys. The structural model is nevertheless extremely useful and must be
considered as a valuable step for the understanding of alloys. First, it represents a

huge step forward, compared with the VCA and CPA theories; second, although

CHAPTER 1. INTRODUCTION 12

it is not simple, it is possible to include corrections in this theory to take into
account the effect of correlation particularly if these are weak like they seem to
be in most semiconductor alloys. Furthermore, if the accuracy of the experiment
could be increased enough to present noticeable deviation from this theory, it

would allow for some quantitative measure of the correlation.

1.4 EXAFS

Since extended x—ray absorption fine structure spectroscopy (EXAFS) represents
the major source of experimental results about the local structure of disordered
systems, it is important to understand the underlying theory and limitations of
the technique. Most of the material of this section has been taken from two review
articles: Hayes and Boyce (1982) and Stern (1985). Although the fine structure
at the absorption edge has been known for a long time, it is only about 25 years
ago with the opening of two facilities producing an intense source of x-rays that
the efforts to really understand the phenomenon were made. In this section, I will
present a very condensed theory of EXAFS, followed by a discussion 'of different
approximations and how they limit the information one can get from this method.

In Figure 1.4, one can see the variation of the x-ray absorption coefficient
of copper metal with the energy of the incident photon. The coefficient decreases
more or less monotonically except at some specific energies when a step occurs.
These steps or absorption edges correspond to the binding energies of shells of
electrons in the atom. In the Figure 1.4, two edges are indicated as the L-edge
(n = 2 shell electrons) and K—edge (n = l shell electrons); just after the edge, the
absorption coefficient shows a very rich behavior with peaks and valleys extending

on about 1000 eV past the edge (Figure 1.5). The oscillations happening in the

CHAPTER 1. INTRODUCTION 13

 

 

 

 

 

 

 

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.—.~ LOEDGE
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O
N
3, soooe ‘
.-
Z
‘.‘..‘
9. 60001- -‘
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U
5 4000- d
.:
Q
C
o
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x G A l

0 4000 8000 I2000

ENERGY 0F X-RAY PHOTO! (CV)

Figure 1.4: X—ray absorption coefficient as a function of photon energy (Stern,
1985)

first 20 eV after the edge are classified as near—edge (NEXAF S) region while
the rest of the structure is called the extended x-ray absorption fine structure
(EXAFS) region.

EXAFS is typically understood in terms of single—scattering phenomena.
A photon excites an electron in the core shell of an atom. The photoexcited
electron wave propagates and interferes with scattered waves from the surrounding
atoms (Figure 1.6). The oscillations measured are a signature of these interferences

and hence, of the local atomic structure. We can express this signature as a

CHAPTER 1. INTRODUCTION ,4

 

Absorptance (arbitrary units)

 

 

 

\J

1 l 1 l 1

 

hm IkeVI

Figure 1.5: Absorptance of Cu metal at the onset of Cu K—shell absorption (Stern,
1985 )

perturbation x(w) over a smooth photoexcitation rate 0°(w) that would be seen

if the atom was isolated
0(w) = 0001) [1 + X(w)l, (1-2)

where x contains the EXAFS information. Since x is due to interference between
backscattering waves and ongoing waves around an atom, the main contribution
comes from the product of the photoelectron momentum k and twice the distance
between the central atom and a neighbor, 27'. If we also take into account the

attenuation of the wave and the reflection coefficient of the neighboring atom, we

can express x as
cha(w) = 2: joo drr2p05(r)2Re [ezik'Aag(k, r)] , (1.3)
a o

where
22'1r2m

R2

 

Aas(k,r) 2 -- t§(-—k, k) exp [-A—(Zl?) + 2ina(k)] . (1.4)

CHAPTER 1. INTRODUCTION 15

 

 

 

 

 

 

 

 

 

 

 

Figure 1.6: Origin of EXAF S. (a) A photon is absorbed by an atom, exciting
an electron from the core state j to a continuum state. (b) The electronic wave
propagates outward and interferes with neighboring atoms (c). (Hayes and Boyce,
1982).

The first equation gives the total contribution of all the atomic species ,6 of the
neighbors to a central atom of species a as a function of the energy hw on the
incoming x-ray photon. The integral is the product of the radial distribution
function pap(r) and a complicated energy term A. This last term includes the
scattering matrix t}, the electron mean-free path length A and the phase shift
due to the potential of the excited atom 170,. However, it does not include many—
body correlations (angular dependence, etc.), which makes it simple to use in the
analysis of the experimental results.

From the absorptance of Cu (Figure 1.5), it is possible to extract the

CHAPTER 1. INTRODUCTION 16

quantity ch (Figure 1.7). If only one distance entered in the radial distribution
function (RDF), we would expect to see only one frequency in the EXAFS os-
cillations; however, there are many frequencies on this Figure. Since it is easier
to understand the RDF in real space, we can Fourier transform (FT) the signal

(Equation 1.3):

a 2 2; f0” dr'r"2Paa(r')€aa(T — r'), (1.5)
where
{03(1', 1'") = FT[W(k)Aa,g(k, r”)], (1.6)

and W(k) is a square window function. So o is essentially a linear combination of
{’s, one for each peak in the RDF. However, looking at the Fourier transform of
the Cu K-shell absorption (Figure 1.8), we see that because of the interference,
the resulting curve is not so simple to analyze. There are many complications
that enter in the analysis of the EXAFS oscillations. I will not examine all of
them but only the major ones in order to evaluate the advantages and limitations

of this method.

0 The general treatment of photoabsorption is a one—electron approximation.
It neglects, among other phenomena, the effect on the core of the excitation
of the photoelectron. While the electron is excited, one can expect that the
core will react differently, altering the cross-section and thus the EXAFS.
Although this effect is somewhat taken care of by using the eigenfunctions
of the 2+1 atom (reflecting the presence of one core hole), all the details

have not yet been worked out.

0 A substantial simplification of the equations for EXAF S results from the

neglect of multiple—atom scattering, with the consideration of only short—

CHAPTER 1. INTRODUCTION 17

 

 

 

 

 

 

T

5

.. 0 T

5

x

a:

-1 1
0 5 10 15 20

11 IA“)

Figure 1.7: The EXAFS oscillations pass the K—shell absorption edge in Cu (Hayes
and Boyce, 1982) .

 

 

 

 

 

 

 

Figure 1.8: The real part and the magnitude of the Fourier transform of the
EXAFS signal for Cu (solid and dotted line, respectively). The near—neighbor
distances in Cu are indicated by vertical arrows (Hayes and Boyce, 1982).

CHAPTER 1. INTRODUCTION 18

range order. It is surely one of the main advantage of EXAFS over LEED
where one must solve the full Schrédinger equation. However, comparisons
of the theory with experimental results in Zn chalcogenides has shown that

the inclusion of the multiple—atom scattering is sometimes necessary.

0 In order to compare the theory and the data, it is necessary to establish
two independent variables: the phase—shift k of the outgoing electron wave
due to the excited atom from the calculations and the x—ray photon energy
hw from the experiment. These two variables are related by the final state

electron energy

E = Eaml‘j'hw (1.7)
5ch2
2m

 

= Eo-f-

The correspondence depends on an accurate evaluation of Ea,“ — E0; a task
which is almost impossible in practice due to the simple models used for
the interaction V0. When comparing systems that are very similar, it is
only required that the difference between the assumed E0 and its real value
be the same for all the cases. E0 is then considered as a fitting parameter
for one known system and checked for consistency in the other materials.
However, this approximation creates a problem when one wants to compare

very different materials.

The general method used to interpret EXAFS results is to compare a
model with the experimental data and to extract the structural information from
the fitted parameters. Since the measurement itself is relatively straightforward,
the obvious limitations of EXAFS lie in the model used for fitting the data. Al-

though the present models can work very well for disordered samples that are

CHAPTER 1. INTRODUCTION 19

electronically close to a known structure, its precision is not so good when there is
no known structure close to the one observed. In order to clarify many important
details of the local structure of disordered systems, EXAF S remains an excellent
tool. However, there is still a need for more complete models that would increase
the accuracy of EXAFS to a fraction of a hundredth of A so that it can serve as

an accurate test for the theories solving problems with this precision.

1 .5 Conclusion

In the next chapters, I will present some results that will help in understanding
the specific role of size mismatch in disordered systems: solid solutions on a crys-
talline and amorphous lattice. It is rather surprising that even for simple alloys,
the local structure has not been understood much earlier since all the theoretical
tools were available. After explaining in some details the general theory developed
by Thorpe, Jin and Mahanti (1989), Thorpe and Garboczi (1990) and Cai and
Thorpe (1992a, 1992b), I look at the deviations from this theory in the case of
metallic alloys, where the concept of natural size, introduced by Pauling, does not
apply very well. Then the theory is checked for SiGe crystalline and amorphous
alloys. Although the crystalline alloys are not really a challenge, the study of the
effect of amorphisation versus size—mismatch on the structure gave us a better
understanding of the strength of this simple topological model. However straight-
forward the theoretical results for the SiGe alloys are, they do not agree very well
with a series of EXAFS experiments on both amorphous and crystalline structure.
The following chapter examines the disagreement and looks at whether the theory
or the experiment fails. One method to determine whether the experiment or the

theory is incomplete or wrong in the case of these alloys is to use a different probe

CHAPTER 1. INTRODUCTION 20

to look at the structure. Chapter 6 extends the Cai and Thorpe theory to the
surface of SiGe alloys, making predictions that can be checked against scanning
tunneling microscope measurements that are more direct that EXAFS. A confir-
mation of the theoretical results on atomic positions at the surface would indicate
strongly that the theory is also valid in the bulk. Finally, I examine the effect of
size—mismatch on a two—dimensional lattice. Many claims have been made in the
last decade and a half about the solid—liquid transition in these systems. In the
hope of improving the understanding of this transition, I have performed some
simulations, increasing the disorder with size—mismatch instead of temperature.
Although not a perfect mapping, the size—mismatch technique has the advantage
of being a static study, allowing us to get much closer to the phase transition

without having to care about the divergence in the fluctuations.

Chapter 2

Theory

The development of the general theory of mismatch alloys was performed over
many years by M. F. Thorpe, S. D. Mahanti and their collaborators. They began
to study this problem in the context of intercalation compounds, trying to predict
what kind of deformation the intercalant would suffer under the pressure of the
intercalated layers and its neighbors (Thorpe, Jin and Mahanti, 1989). From one—
dimension, the problem was extended to two and three dimensions for both bond
and site mismatch.

This chapter presents the general theory for the bond-mismatch problem
with only nearest-neighbor interactions since the formalism is simpler. I will then
only state the corresponding results for site—mismatch disorder without proof and
describe briefly the derivation in the case of a two—- and three—body potential, that

I will present in more details in chapter 4 and following.

21

CHAPTER 2. THEORY 22

A A A A A
VAVAVAVAVAVA
AVAVAVAVAVAV
VAVAVAVAVAVA
AVAVAVAVAVAV
VAVAVAVAVAVA
AVAVAVAVAVAV
VAVAVAVAVAVA

V V V V V

Figure 2.1: A triangular network (Thorpe and Garboczi, 1990)

2.1 Bond mismatch

For simplicity, I will consider a two dimensional triangular network, which is
equivalent to a face—centered cubic network in three dimensions (Figure 2.1). The
following derivation is taken from Thorpe and Garboczi (1990). A
On this network, one distributes at random bonds of natural length L9,
and Lg (Figure 2.2) with probability 1 -:1: and :1: respectively. Each of these bonds
is a Hooke’s spring so the total energy is given by
v =% 2 m,- (L,,- — L9,.)2, (2.1)
<u>
where < ij > signifies that the contribution of each bond is counted only once,

and the average length, by

<L>=(1-:c)<LA>+:c<LB>. (2.2)

CHAPTER 2. THEORY

.D. o..-
o ' ' s T
, o ’0

“ '0

   
  
  

        
   

{w o
v, 0
....
\
I
I ‘\
'l \
“ I on... :’..?
coo...- "‘ "O \“ f
\ 0' \ '
I'
‘.. ‘
‘ ... ..-..
D. O ‘
\‘ "l ‘

.. '....‘ a..-

I
0‘
0
0‘
0
I

Figure 2 2 Relaxed triangular network where two sizes of bonds have been (118-
tributed randomly (Thorpe and Garboczi, 1990)

23

CHAPTER 2. THEORY 24

The only requirement for equilibrium is that the force on each bond vanishes. So,

minimizing the energy with respect to 2', we obtain
0 = 2 K..- [(R, — R.) - Liza-,1 . (2.3)
1

with R4,- the unit vector from R.- to R,- . Since this equation is linear, we can
rewrite it as a weighted sum of the average lengths. If we suppose that displace-
ments from the perfect triangular network are small, then all displacements are

along the bond and the previous equation becomes
0 = (1 — r)KA(< LA > -L‘},) +:1:KB(< L3 > —L%). (2.4)

It is equivalent to say that if we draw any line cutting a plane of bonds, the sum

of all tensions must be zero. If we put K A = K B = K,
<L>=(1-;1:)<LA>+<L3>=(1—T)L%+TL%, (2.5)

which is simply the Végard’s law (Végard, 1921), proposed for alloys more than
70 years ago. Note that this relation is exact for any disorder as long as all elastic
constants are the same for all interactions. I will return to the problem where
there is more than one elastic constant in the next section but until then, I keep
this requirement.

The next problem is to find the average partial lengths < LA > and

< L3 > in the alloys. We start by expressing a particular bond length as
L=R,~-R,- =L+ [(u,—u,-)-R,~,—]R,-,~, (2-6)

where u,- is a small diSplacement and R.)- is a unit vector along an undistorted

bond. Replacing this definition in Equation (2.3), it becomes

0 = [(2 [(u, — 11,) . R, — (< L > -L?,.)] as. (2.7)

CHAPTER 2. THEORY 25
This equation can be expressed in terms of the dynamical matrix 0;,-

Dafiug = v9, (2.8)

ij J t
where I follow the Einstein’s summation index convention and

v? = K2 (L?,— < L >) - R°. (2.9)

J

Inverting Equation (2.8),

119' —G?J-fivf (2.10)

with G, the Green’s function for the perfect triangular lattice. Inserting 2.10 and

2.9 in 2.6,

141' < L > +KZ [RJ‘ ° (Gim - Gjm) - le] (L?m- < L >) (2.11)
lm

< L > +2K Z(R,~,- - Gm, - Rim)L?m,
lm
where I used the identity
2: IL, = 0. (2.12)
1
Equation (2.12) gives the actual length of one particular bond as a function of
all the natural lengths. By averaging over all the bonds we obtain an identity.

One can find the average partial lengths using a bond variable 0,-1- equal to plus

(minus) 1 when the bond is of type A (B). The average value of this quantity is
< 013' >= 1 - 21:. (2.13)

Using a projection operator on the bond type A

l + 051'
—— 2.14
4N(1 — x) ( )
and re—expressing the natural length as an operator
0 1 o 0 01m 0 o
le = 5am + LB) 'I‘ ‘2—(LA “ LB), (2-15)

CHAPTER 2. THEORY 26

it becomes
< LA >=< L > +m’1i372,,,m < (1 + a,,-)(1+ 0,...) > (2.16)
X(R,‘j ' Ggm ' fiijLg - 14%).
Defining
(11$): 4N3“ _ I) —————)i§;n < 01J01m> (R1, Ggm le), (2.17)
Equation (2.17) becomes simply
< L, >=< L > +xa‘(1:)(Lf’, — L%) (2.18)
and, similarly,
< L3 >=< L >-(1- 2:)a (:1: )(L0 - L0). (2.19)

Now, let’s take a closer look at a‘(:1:). If the bonds are distributed randomly on

the lattice, the pair—correlation function becomes
< 03501", > — < 0.5 >< 01m >= 4.7:(1 — I)6,‘j'1m. (2.20)
So Equation (2.17) is no longer 1: dependent

a’ = 2K [E,--(G,~,-—G,~,)-R,-,~] (2.21)

= 211/2
2

3,
where i j is a nearest—neighbor bond, d, the dimensionality of the network and
z, the coordination number. This result was first found by Feng, Thorpe and
Garboczi (1985) in the context of rigidity percolation.

It is also possible to obtain the fluctuations of the bond lengths. One can

calculate directly by squaring Equation (2.17). In the random distribution limit,

 

CHAPTER 2. THEORY 27
we find that all the distributions are identical, and have widths

<L‘,’,2>-<L‘,’,>2 = <L‘1’32>-<L‘},>2 (2.22)

= 2(1— :1:)a"(1— a')(Lg — L2,)?

This result is not as surprising as it may appear. It simply represents the local
nature of the relaxation. Since no bond knows the total concentration for A and
B bonds on the lattice, on a local scale, bonds from different species see identical
configurations and hence have similar distributions. Finally, one can obtain the
total strain energy of the bond—mismatched lattice by the back door, using the

Feynman—Hellman theorem (Feynman (1939))

%< €(p) >= ($2). _ (223)

where e is the energy density. From Equation (2.17), we see that only the natural

lengths enter in the average bond energy which we can express as
< 6 >= ADE,“ + 8L5},2 + CLng. (2.24)

The derivative with respect to L9, gives

 

6
if; = 2AL‘3, + CLg. (2.25)
Comparing this result with
66 o
m = K(< LA > —LA)1 (2.26)

and using similar equations for [1%, one finds easily that the average bond energy
is

e = $30 — x)a'(L‘3, — L2,)”. (2.27)
All these results are shown in Figure 2.3 with a comparison between theory and

computer simulations. In all cases, the agreement turns out to be excellent.

CHAPTER 2. THEORY .28

 

 

 

    

 

 

 

 

 

 

 

I 1' I I
1.0 ~ (a) .
<03) .. . f
g 0.8 — .. -- . . 1
I- " .
g 0.4 - . (dA) ..,
u, .
2 0.2 - 1
0 ’ .
L l J I
0’20 I T I r
(b) . «am-<1»:
0.15 ' ’
g 0.10
0.05
0
l I f I
000 b (c) _
g 0.04 - ..
0.02 r- -
o L L L l
0 02 0.4 0.0 0.0 1.0
x

Figure 2.3: Variation of the confi urational energy, the width of the partial len th
distribution and the partial lengt s as a function of concentration :1: in bond a1 oy
141-38,. The d’s are renormalized lengths such that the topological rigidity a" is
given by the difference < d3 > - < dA > (Thorpe and Garboczi, 1990).

CHAPTER 2. THEORY .29
2.2 Force constant disorder

The first condition put on the pair potential was to demand that all the force
constants be the same. This limitation comes from the form of Equation (2.8).
We know how to solve this equation exactly only when the disorder is limited to
the vector v, on the right hand side, not in the dynamical matrix itself. It is,
however, possible to develop an effective—medium theory that will allow a certain
understanding of the role of the force—constant disorder in changing the exact
solutions presented before. Since I will discuss this problem with more details in
chapter 3, I will only add that the main effect of this disorder is to slightly curve
the straight lines of the Z-plot as can be seen in Figure 2.4. In the case of bond
disorder, the effective medium theory does a very outstanding job in predicting
the correct curvature. Also, the curvature is noticeable only when the difference
between the force constants is large (K A = 2K 3 in Figure 2.4). When the two
force constants vary only by 10 or 20 percent, the lines are straight enough that

we do not need to bother with this effect given current experimental accuracy.

2.3 Site—mismatch

For the site—mismatch problem, the solution is almost the same although the
analytical part is a little more difficult. Instead of two partial lengths, there are

three. Defining the average length as
< L >=(1— :1:)2L3M + 2:1:(1— 101233 + 3214(1)”, (2.28)
Chen and Thorpe (1992) have shown that

< L1,. > = < L > +a'22A — m" [L993 - L?“ + (2:1: —1)A] , (2.29)

CHAPTER 2. THEORY 30

 

 

 

 

 

 

 

 

1 1 1 1
1.0 - (a) + t,
00 _ x " x’ _
as °'° “‘9’ . j i i.»
g 06 " f '7’
a 1 -
g 0.4 b a,” 1
g1 0'” (CIA)
( 0.2 1' ’d” + +.-+—+--+-_+-—i
O 1?..- -+ d
1 1 1 1
0-20 m 1 1 1
(b) l’q’To‘sn
0.15 l- ”d (d2) ‘(d)2 \‘ ""
(D d, b\
E 0.10 - ‘.\ -
g ’0’ ‘\
I, 2 q
0.05 - d! . (db-<0.) x -
’I ‘0“0-‘0"°-'fl--.a--n ‘\‘
0 4.0" L 1 m 1 “I":
l r r 1
(0)
0.06 F 0"‘°'~. '1
p’ x
> I” h‘
0.06 '- d \\ -(
g ”l b‘\
I, \
\
0.02 . 1'? b -1
l' \\
I, “\
0 ' L 1 1 1 ‘
0 0.2 04 0.6 08 1 0
X

Figure 2.4: Variation of the confi rational energy, the width of the partial len th
distribution and the partial lengt s as a function of concentration 2: in bond al 0y

241-33, and K A = 2K 3. The 11’s are renormalized lengths (Thorpe and Garboczi,
1990). 9

CHAPTER 2. THEORY 31

< L33 > = < L > +a'(l — T)2A + (1 — 2:)a" [ng — L?” + (22: —1)A],
< LAB > = < L > +a‘a:(1 — 13)A - (:1: —1/2)a”[L%B — L?” + (2.1: -1)A] ,
where
A = L333 + L3,, - 2L3,B (2.30)

and

_ K
— 16Nz(1— :1:)

 

Z < (1+ a,)(1+ 0,)(0, + am) > 11,-,- - Gim - Rm. (2.31)

ijm

The energy density is
E: -§I(.1:(1—.1){(1—a1")[1+6(1+2..~.—1)]2 (2.32)
+2(1- a')5%(1- 17)} (Lisa - L302.

with

A
6 = . 2.33
(L9... — L9...) ( ’

However, the computation of the partial length fluctuations become extremely

 

difficult unless one sets A = 0, which corresponds to defining L23 as the arithmetic

average of the two other lengths. In this case,

< LEM > - < L111 >2 < L233 > - < L33 >2 (2.34)

= <LiB>—<LA5>2
l

= 5:1:(1 — 1:)a"(1 — 11").

Again, and for the reason given before, all the distributions are identical.

2.4 a* and a**

In the two previous section, we have encountered two parameters: a“ and a".

The first one appears in the bond—mismatch problem and the second one in the

CHAPTER 2. THEORY 32

site—mismatch problem —alone if L A3 = (LAA + L3 3) / 2. The knowledge of their
value gives a complete understanding of all quantities presented before and some
more that I will discuss in the next section. In order to understand the meaning
of these parameters, let’s take a look at Equations (2.17) and (2.31). We first see
that there is no disorder included in their definition. Both parameters depend
only on the Green’s function of the perfect crystal. Since their value reflects the
topology of the system, they are called topological constants. Note also that the
summation is only on the nearest—neighbors and so its value reflects the local
connectivity of the lattice.

Even if Equations (2.17) and (2.31) define completely the two topological
parameters, it is useful to describe them in a more physical fashion. As mentioned
previously, a" was previously introduced by Feng et al. (1985) when solving a
topological rigidity problem. In that paper, the authors obtained the topological
parameter by calculating the force needed to pull on a bond with force constant
K but inserted in a lattice. Obviously, this force will be greater than if there was

no lattice. It can be described as

K i
F = —u (2.35)

where u is the displacement from the equilibrium value of the bond. When a‘ =
1, it is as if the network was non—existent or floppy. The other limit, extreme
rigidity, is achieved for a‘ = 0, where only an infinite force can produce a finite
displacement.

Similarly, one can obtain a physical meaning for a". In this case, it
represents the response of the lattice to trying to open up a cage in order to intro-
duce a larger atom. Figure 2.5 gives a physical representation of the topological

parameters. This simple picture works well for bulk alloys. However, it needs

CHAPTER 2. THEORY 33

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b)

Figure 2.5: Physical representation of the topological rigidity parameters. (a) a‘;
(b) a”.

some refinements when one looks at the surface as we will see in chapter 6. From
the definition 2.17 for a‘, for a two—body potential, the more neighbors an atom
has, the more rigid is the lattice locally. This relation also applies for a". For
an unstable lattice, the previous equation is invalid and the topological rigidity
parameters are simply 1.

Finally, in many instances, especially those involving angular terms, it is
simpler to measure directly the topological rigidity parameter from the Z—plot, i.e.
from the average partial lengths as a function of concentration. From Equation
(2.17), one easily sees that by taking the mismatch Lg - L2, as the unit length,
then the distance between the two partial lengths is a', for bond mismatch and

a" for site—mismatch with A = 0.

 

CHAPTER 2. THEORY 34
2.5 Semiconductors

In the case of a semiconductor, it is imperative to include an angular term in the
potential in order to stabilize the lattice. The Kirkwood potential is preferred
over the more usual Keating potential since Kirkwood separates completely the
angular term from the stretching term,
a 2 [3 l
V = 5 2 (L1,- — L9,.) + gL3 Z (coso,,-. + 5). (2.30)
<U> <th>

Here, the two force constants C: and 0 have the same dimension while LC is the
average length in the system. Using again the first order approximation in the
displacement from equilibrium as given by eq. 2.6, it is possible to express the

potential energy in a quadratic form
1
V = §u+Mu + u+<I> + E, (2.37)

where u is the vector displacement from the equilibrium position, M is the connec-
tivity matrix and 1’ contains the internal strain due to disorder. The derivative of
this equation with respect to 11 gives us an equation with the same form as Equa-
tion (2.8) and hence, can be solved exactly. Cai and Thorpe (1992a) obtained a
complete solution for quaternary semiconductors.

In chapter 4, I will present the binary case that is a special case of the
general solution. However, a few details should be pointed out here. First, the
topological parameters are function of the ratio of the angular force constant
to the stretching force constant, fl/a. For a zero angular force, the diamond
lattice is completely floppy and therefore unstable. The typical range of this
ratio, for semiconductors, is between 0.1 and 0.2 with the notable exception of

diamond where the ratio is more like 0.6. Second, the solution found by Cai

CHAPTER 2. THEORY 35

and Thorpe includes the complete first— and second—neighbor distances for all the
partial lengths. For the second-neighbor bond length a new topological parameter

is introduced
.. 8
b (Ii/a) = al/EZ, 0131.13", (2.38)
J

where j’ is a first neighbor of i while 1" is one of its next—nearest neighbors. The
physical significance of this parameter is related to the opening of the shell asso-
ciated with the second neighbors. Although, there is no formal relation between

a“ and b“, it has been found numerically that for most of the range of the values

3/0

fit'

a

b.‘ = .
2

 

(2.39)

Figure 2.6 shows these two topological parameters as a function of the ratio of
force constants. Since the edge of the curve on this figure has a large uncertainty,
one can not rule out that this relation holds everywhere. We have not been able,
however, to provide a formal proof or a physical argument supporting this relation.
Nevertheless, since it works numerically, it is possible to use it in order to simplify

the second-neighbor equations.

2.6 Correlation

It is possible to include correlation effects through the topological constants. Go-
ing back to the definition of, say a‘, the correlation can be included in < 03-1-01", >.
Thorpe and Garboczi (1990) and Chen and Thorpe (1992) studied correlated sys-
tems for bond and site mismatch respectively. This problem is very difficult to
solve analytically in all cases execpt for a handful of special functions.

Using a scheme forcing clustering for the first neighbor interaction, Thorpe

CHAPTER 2. THEORY 36

 

  
 

 

 

 

“ I I T I I T r I I I r I r Id
‘1? C I I 7 .
3 1.0 m
a -1
o I I
o _ .
£1 0.8 r '1
39. P <
5 ' ‘
~ 0.6 P— ” ——
3 C 2b 2
03 P
2 ~4
0 0.4 b
a I-
E2 I' 1 J 1 L I 1 1

0.0 0.2 0.4 0.6

3/0

Figure 2.6: The topological rigidity a“ and 2b”, dash and solid line respectively.
This data is from a numerical calculation (Cai and Thorpe, 1992a).

and

Garboczi(1990) obtained the results shown in Figure 2.7. In spite of
clustering, one could easily fit a straight line through the simulation and conclude
that there is no correlation. Because of this, it would be useful to be able to
introduce correlation only when the experimental results are precise enough to

distinguish between correlated and uncorrelated alloys.

2.7 Conclusion

The general theory of mismatched alloys developed during the last five years
present a very useful scheme in order to understand the meaning and significance of
recent experimental measurements. Although an exact. solution is obtainable only

in certain limits, it is possible to solve it numerically in more complex situations.

 

CHAPTER 2. THEORY

37

 

 

 

 

 

 

 

 

 

I1 I r I
1.0 - (a) .
(do) H"
2 0.8 - ° -
g 0.6 I" 1
11.1 (01)
g 0.4 - <11) -
1.1.1 .
3 0.2 L ’ -
0 /i .
l l I l
0'20 I I I T
(b)
0-15 - (cm-(0)2 a
Q 0.10 - -
3 o o
o o
0.05 )- o o (9121* (d ‘02 o o
o a I l L
I I I r
0.08 "' (C) ‘
> 0.00 _ 2
i 0.04 - -(
0.02 - -
o I L i l
0 02 0.4 0.0 00 1.0
X

Figure 2.7: Variation of the confi urational energy, the width of the partial len th
distribution and the partial lengt s as a function of concentration :1: in bond al oy
A143,. A positive correlation term has been added. The d’s are renormalized

lengths (Thorpe and Garboczi, 1990).

 

CHAPTER 2. THEORY 38

The physical content of the equations is simple and robust since the problem
involves only the topology and connectivity of the pure crystal.

It would be feasible to add complexity to the equations by adding force
constant disorder and correlation but, as shown here, this does not lead to impor-
tant changes in the answers but, especially in the latter case, demands a much

higher knowledge of the local order that we actually have for most alloys at present.

Chapter 3

Metallic Alloys

In the last few years, the problem of length mismatch in alloys has received con-
siderable attention both theoretically and experimentally. As I discussed in the
previous chapter, Thorpe and Garboczi (1990) have recently solved this problem
analytically for alloys with equal harmonic spring interactions between nearest—
neighbor atoms, joined by bonds with different natural lengths. Experimentally,
the development of extended x—ray absorption fine—structure spectroscopy (EX-
AFS) has provided more information concerning the near—neighbor lengths in
alloys. These partial lengths are of primary importance for a proper structural
characterization and in understanding of the deviations from Végard’s law.
From this point of view, the study of fcc binary metallic alloys is very inter-
esting since for these compounds, the deviations from V‘égard’s law are much more
significant than for example in semiconductors, as we will see in the next chapter.
The development of the embedded—atom-method (EAM) potentials (Daw and
Baskes, 1984, and Finnis and Sinclair, 1984) which have given reliable results for
the energies of pure metals, with and without impurities, and binary alloys (Daw

and Baskes, 1984 and Foiles 1985), makes it possible to take into account, in a

39

 

CHAPTER 3. METALLIC ALLOYS 40

simple way, some of the electronic effects. The EAM potentials are, in general,
fast to compute, allowing the use of very large supercells, which is important when
trying to extract statistical information from random alloys. The version chosen
for the present study is the Johnson EAM potential (Johnson, 1988, 1989, 1990),
developed recently and possessing two major advantages over other EAM poten-
tials for this study: 1) it is completely analytic and 2) it requires no additional
parameters for the alloy once the parameters for the pure metals are fixed. The
lack of additional alloy parameters has ultimately proved to be a problem, as there
are no adjustable parameters to fit experiment. In this chapter, I show that the
Johnson EAM potentials are remarkably good in some cases (e.g. NixAu1-x) but
quite unsatisfactory in others (e.g. alloys containing Pt).

Deviations from Végard’s Law for metallic alloys have been known since
the beginning of x—ray measurements more than sixty years ago. Until recently,
however, Végard’s Law was actually nothing more than an ad hoc assumption
(Végard, 1921), but it has been shown that Végard’s Law is to be expected only
in those cases where there is length mismatch accompanied by no changes in the
force constants (Thorpe 1990). As these conditions never occur in reality, the
discussion must always be about the magnitude and sign of the deviations from
Végard’s Law. These deviations are small in semiconductors, but can be much
larger in "metals. As discussed in the introduction, during the fifties, a few models
were proposed to quantify these deviations (Fournet, 1953 and Friedel, 1955),
but they all started with the assumption that the solid solution forms a perfect
network i.e., all the lengths are identical.

In this chapter, I present computer simulation results from the EAM po-

tential. I also present both analytic and computer simulation results from a much

CHAPTER 3. METALLIC ALLOYS 41

simpler spring model. This central force model (CF M) uses only nearest—neighbor
central forces between an atom and its 12 neighbors. It has the virtue that it can
be solved analytically in some cases, and can be used in conjunction with the
EAM results to assess the sensitivity of the results to the local environment. The
CFM is completely independent of the local configuration and incorporates the
transferability of the force constants for a particular type of bond for all environ-
ments. On the other hand the EAM effectively modifies the force constants of a
particular bond in response to the local environment around that bond. This is
accomplished via the embedding function. I shall refer to this effect as electronic
density effects in the rest of this chapter. I show that the EAM potential gives
results in reasonable agreement (better than CFM) with experiment for most of
the binary alloys composed of Ag, Au, Cu, Ni and Pd while alloys containing Pt
all deviate strongly from the EAM results. In this work, I use EAM to refer to
the Johnson EAM, and note that other versions of the EAM for alloys may well
give different results. Surprisingly, CFM gives good agreement with experimental
diffraction data for the N iCu, PdAg and all Pt alloys. These results suppose, as
we shall see in Sec. 3.4, that the electronic density effect is least important for
alloys containing Pt where one can treat the atoms as rigid objects connected by
elastic springs. I also present some analytic and computer results for the bond-
length distributions in alloys. The general features of these distribution are not,
in general, very sensitive to the particular model since they depend on topological
quantities like the number of nearest neighbors and are found to be remarkably
wide, of the order of the length mismatch itSelf.

Another quantity of interest is the variation of the elastic constants with

concentration. I have computed the bulk and shear moduli and found that these

 

CHAPTER 3. METALLIC ALLOYS 42

quantities are also rather insensitive to the model used. In particular, both the
EAM and the CFM show very similar results for the bulk modulus. As there exists
almost no experimental data for the elastic moduli in bulk alloys, the results I
present here must be considered as predictions to be confirmed or otherwise by

future experiments.

3.1 Embedded—atom method

The embedded—atom method was first proposed by Daw and Baskes (1983, 1984)
and was based on the quasiatom or effective medium theory (Stott and Zaremba,
1980). In the quasiatom theory, an impurity interacts with a local and almost

uniform environment. We can describe the energy as

Equas = EZ (ph(R)) (31)

with E2 the quasiatom energy of an impurity with atomic number Z at site R
for a host electronic density p1,. One can of course add an elastic deformation
term Em to take into account the lattice deformation but this technique can
only study single and well-defined impurities, not cracks nor surfaces. Daw and
Baskes extended this formalism to treat all atoms as impurities embedded in the
host consisting of the other atoms. Therefore, using simple energetics, it became
possible to go much beyond the pair potential without a significant increase in the
efforts. At the same time, Finnis and Sinclair (1984) introduced a tight-binding
model that turned out to be completely equivalent mathematically to the Daw
and Baskes model; only the physical interpretation differs. Since these potentials
are empirical, this difference leads to no consequence.

Before discussing further the details of the potential, I want to mention

 

CHAPTER 3. METALLIC ALLOYS 43

a few points. The use of a general EAM potential for alloys commands that the
function describing the atom and its reaction to the local electronic density be the
same for all metallic species involved in the calculation. Recently, Jones (1990) has
shown that in general the results for alloys are less accurate with such potential
rather than with one expressly developed for that particular alloys. However,
when the need comes to compare many different alloys, the EAM remains the
most sophisticated method for the study of large scale statistics. In retrospect,

the EAM potential is based on the two following approximations:

1. The assumption is made that the charge density of each atom is not changed

from the isolated atom density.

2. The background charge density into which the atom is placed is assumed to

be adequately represented by the value at the nucleus of the atom.

However these approximations are not as crude as they may sound since
the potential is largely empirical. Moreover, that the EAM potential has proven
to be very accurate in many different situations.

As I already mentioned, the Johnson (1989) version of the EAM potential
was used for the simulations, and I will summarize it below. In EAM,'the electron-
density functions are only determined to within a scaling factor. For pure metals,
this factor rescales the embedding function. But the situation is very different
in alloys where the relation between the electron density of the two components
strongly affects the mixing energies (Foiles, 1985). The interest in the Johnson
potential is that it is completely analytic and requires no extra parameter for the
alloys. All the parameters are determined using the atomic volume, the cohesive

energy, the bulk modulus, the average shear modulus and the vacancy—formation

 

 

CHAPTER 3. METALLIC ALLOYS 44

energy for the pure metals. The approximation used by Johnson is based on
the preservation of the invariance of the energy under a gauge transformation
involving the embedding function and the pair potential. This choice of constraint
is of course as arbitrary as the arithmetic mean used by Foiles (1985), but has
the advantage of being fairly natural within the EAM formalism. In the notation
used by Johnson (1988), the EAM potential is defined by:

. 1 ' .. ’ .

E: = 2171101“) + 53;) <I>"(r.-,-).p.- = grew. (32)
where E, is the total internal energy, p,- is the total electronic density due to
neighbors j at site i, F ‘(p,-) is the embedding energy for atom i and ¢(r§j) is a
repulsive ion-ion core pair potential. The prime on the summations indicates that

the self terms 1' = j are excluded. The functions used in the potential are defined

as follows: . .
9‘10) = g [f—fiqflr) + (if—30(1)] (3.3)
and
073‘ 0718' 1710"
F‘(p) = —E;'(1—In[(f—) D (g) 45.14%) , (3.4)
where
1‘0) = fiexp [-13:11/11 — 1)] (3.5)
and
¢I(r) = (biexp [—7‘.(r/rf3 — 1)] . (3.6)

The parameters used in this work are the same as those given by Johnson (1989)
and are shown in Table 3.1.
The EAM potential for pure cubic metals is not restricted by the isotropy

relation Cu = 2044 + Cu of the CFM, which is rarely obeyed in metals. However

CHAPTER 3. METALLIC ALLOYS 45

Table 3.1: Parameters for the Johnson EAM potential. fl is the atomic volume

3 . I
(A ), only the ratios of the f, are relevant and the other parameters are dimen-
sionless.

Atom 0 f. <15. 0 fl 7

 

Cu 11.81 0.30 0.59 5.85 5.85 8.00
Ag 17.10 0.17 0.48 5.92 5.96 8.26
Au 16.98 0.23 0.65 6.37 6.67 8.20
Ni 10.90 0.41 0.74 4.98 6.41 8.86
Pd 14.72 0.27 0.65 6.42 5.91 8.23
Pt 15.06 0.38 0.95 6.44 6.69 8.57

the EAM potential does allow only two independent elastic constants, and as a
result imposes the general EAM condition Cu = C44+C12 which is reasonably well
obeyed for many fcc metals (J acobsen, 1988). I have found (numerically) that the
relation Cu = C“ + Cl; still holds for alloys, so that there are only two, instead
of three, independent elastic constants. As the bulk modulus B = [Cu + 2012] / 3
and the Voight average shear constant G = [3044 + (Cu - 012)] /5 are fitted in
the EAM potential for the pure metals, I use these as the two independent elastic

constants for the alloy.

3.2 Central force model

In order to get an idea of the importance of the redistribution of the electronic
charges in the alloy, and to gain some perspective on the EAM model, I have

compared the results of the EAM potential with a simple nearest—neighbor spring

CHAPTER 3. METALLIC ALLOYS 46

model (CFM). The total energy for an A1_,,Bx alloy is given by

l o 2
E = 2 2 1(.-,- (L1,- — L,,-) (3.7)
<i.j>
where the A-A and B—B spring constants and equilibrium lengths are taken from
the parameters for the pure metal as given by Johnson (1989), and the additional

alloy parameters are chosen to be

 

KAB = i (3.8)

and the natural (unstrained) lengths are

1
L248 = ‘2‘ (L314 '1” 14%8) . (3.9)

The angular brackets < > in the summation in Eq. 3.7 indicate that each
nearest-neighbor bond is only counted once. The spring constants are functions

of the bulk modulus B and the nearest—neighbor distance L‘J (Feng et al., 1985)

3L°
K = MB. (3.10)

The mean and variance of the bond length distribution become particu-
larly transparent within the CFM when all the force constants are equal, as the
model can be solved analytically. The mean lengths and their variances can be
expressedas a function of a”, the topological rigidity parameter defined in terms
of the radial force on the 12 nearest neighbors, required to open up a cage. These
results have been described in the previous chapter.

It is also possible to find the effective elastic constant Ke by an effective
medium theory (Thorpe and Wang, 1987) when the force constants vary. It can
be shown within the CFM that the equations for the lengths and spring constants

decouple so that the effective spring constant K,3 is independent of the length

CHAPTER 3. METALLIC ALLOYS 47

mismatch. From effective medium theory, and using the relation (3.8), we have

(1 — your“ — K.) 2(KBBK.)
K.(1- P1) + KAAPI 11’.(l - P1) + Keep:

 

 

:0, (3.11)

where p1 = %(1 + 0“) determines the initial slope for the conductance (Thorpe
and Wang, 1987, and Cai et al., 1990), which obeys the same effective medium
equation as the spring constant. Equation (3.11) is a quadratic for Kc, which
always is sublinear and monotonic in the concentration 2:. As we will see in the
Sec 3.4, these analytical results are very close to the ones obtained with the EAM.
The CFM is a very simple and somewhat crude model, but gives surprisingly good

results for some of the alloys, particularly those containing Pt.

3.3 Computer simulations

The computer simulation results shown in this paper for both the EAM and CF M
have been obtained by statically minimizing the total energy using a conjugate—
gradient program developed by Press at al ( 1986) (see appendix A). I have checked
this algorithm against the simplex algorithm as well as against analytical results
(see next chapter). The conjugate—gradient method is in general faster than the
simplex method and at least as accurate. Moreover, it is in almost perfect agree-
ment with the exacts analytical results presented for c-SiGe alloys. The simula-
tions have been performed with 4000 atoms in an fcc arrangement with a cubic
supercell and periodic-boundary conditions. In the relaxation, all the atoms were
free to move and the volume of the supercell could change while remaining cubic.
The elastic constants are computed by varying the shape and/or size of the unit
cell appropriately and re—relaxing the system. This method leads to three figure

accuracy, which is comparable or better than the available experimental results.

CHAPTER 3. METALLIC ALLOYS 48

It was found that the supercell was large enough so that it was not necessary to

do any ensemble averaging.

3.4 Results

In Figs. 3.1 — 3.6, I compare the EAM and CFM computer simulation results
with diffraction data where available, and with EXAF S data for Ni1.,,Aux in
Figure 3.6. In Figure 3.1 for AuAg, the EAM results agree extraordinarily well
with diffraction data and are clearly superior to the CFM results. Because Au
and Ag atoms have very similar sizes, one Would expect this case to be quite
uninteresting as indeed the results are for the CFM. The experimental data show
a minimum in the mean length at about :1: 9: 0.4. Fournet (1953), using an elastic
sphere approximation, predicted that the deviation form Végard’s law should be
maximum for very small length mismatch but gave no number and was working
in the virtual crystal approximation. As seen in Figure 3.1, the EAM follows
almost exactly the experimental data except at low concentration of Au, where
the difference is only due to the fact that the lattice parameter of pure Ag has not
been taken at the experimental temperature to fit the parameters for the EAM
potential. For this alloy only, the simulation results have been obtained using a
polynomial potential with a cut-off radius after the third shell of neighbors. This
potential is very similar to the one described here, for more details see Appendix B.
I used a longer range interaction for this alloy because Johnson (1990) mentioned
that it is the only one which shows some change in energy with the range of
interaction. All the other alloys studied here are quite stable under such a change.
Note that the condition Cu = C' 12 + C“ is not obeyed in either the pure materials

or the alloy due to the further neighbor interactions. It is worth mentioning

 

. CHAPTER 3. METALLIC ALLOYS 49

 

lilfrrrIT—T’I—YrifrrrrrT—T—I
' Ant-3A8: J
2.90

 

 

 

A 2.05
°:~L

O

O

I:

O

.03

.9.

9 2.90
z

z

2.65

 

 

0.0 0.2 0.4 0.6 0.8 1.0

Concentration 1:

Figure 3.1: The mean A-A, A-B, B-B nearest-neighbor distances and the lattice
parameter are shown for the alloy A1-,Bx. Dashes are for A—A distance; dot—
dashes, for A-B distance; dots, for B—B distance; and the continuous line is for
the mean distance. The symbols are experimental data. The upper panel is for
the embedded atom method (EAM) and the lower panel is for the central force

model (CFM). This figure is for Au1-,Ag,. The diffraction data for Au1_,,Agx

(Karmazin, 1969) is shown as diamonds.

CHAPTER 3. METALLIC ALLOYS 50

that Ackland and Vitek (1990) also see a minimum with their EAM potential
but around :1: z 0.5. The behavior of the Au—Au bond length in this alloy is
quite strange; instead of becoming closer to the length of the Ag—Ag bond, it
decreases by more than 0.07A (3%), close to 10 times the length mismatch! This
behavior can be explained by looking at the parameters for the potential: the
easiest way for the Au atom to satisfy its need for a large electronic density is to
become closer to another Au atom. But, besides the mean length which agrees
with the experiment, it is impossible to tell whether or not this behavior is real.
As this behavior is probably too small to be discerned by EXAF S, only ab initio
calculations, especially of Au—Au pairs in Ag, can give further insight into this
phenomenon.

In Figure 3.2, the results for CuAu alloys show that the EAM and CFM
are about equally good, but do show systematic discrepancies with the diffraction
results. The results for CuPd. in Figure 3.3 give impressive agreement with the
diffraction results for both EAM and CFM. The partial mean lengths are also
similar, suggesting that electronic density effects do not make much difference
here. In contrast the results for PdPt in Figure 3.4, show that the CFM is superior
to the EAM in reproducing the diffraction data. The CFM produces a very narrow
range of mean lengths in this case with a minimal amount of bowing. The results
of EAM and CFM are quite remarkably different in N iAg as shown in Figure 3.5,
giving bowing with opposite signs. There is no experimental data on NiAg as it
phase separates.

In Figure 3.6, I present the results for N iAu alloys, which are particularly
important at the present time, as this is the only fcc metallic alloy for which there

is EXAFS data (Renaud et al., 1988). This alloy was chosen for the first EXAF S

CHAPTER 3. METALLIC ALLOYS 51

 

rTrTIrI—I T—I TTTI I TIT—TrrI—T fiIT

    

Cu,_,Au,

2.8 *-

NN Distance (A)

 

 

 

 

 

 

1...1...11-.L.1...L1....:
0.0 0.2 0.4 0.6 0.8 1.0

Concentration x

Figure 3.2: Same as Figure 3.1, except for Cu1_,Aux. The diffraction data for
Cu1..,Au, is shown as diamonds (Linde, 1932; Nix and Macnair, 1941; Newkirk,
1953; and Lihl et al. 1971).

CHAPTER 3. METALLIC ALLOYS 52

 

h

' Cu1-,Pd,

2.8 TYTWT'rrrIflrTerIIfi‘T'TITj
. i
T

\
\

U
o
I 0
.AL‘. A
V'V VVT‘

NN Distance (A)

 

 

 

 

 

LLLLIIALLlllllLllLLILIH

0.0 0.2 0.4 0.6 0.8 1.0

Concentration x

Figure 3.3: Same as Figure 3.1, except for Cu1_dex. The diffraction data for
Cu1-,Pd, is shown as diamonds (Pearson, 1958).

CHAPTER 3. METALLIC ALLOYS 53

 

‘11Yr1rTT'II‘rrYVITrl—TIYITTfi'
l

NN Distance (A)

 

 

 

CFM 1

 

2.70 .LLL1411...L1“+.1UU _
0.0 0.2 0.4 0.6 0.8 1.0

Concentration it

Figure 3.4: Same as Figure 3.1, except for Pd1_th,. The diffraction data for
Pd1-,Pt, is shown as diamonds (Darby and Miles, 1972).

CHAPTER 3. METALLIC ALLOYS

 

 

 

 

 

 

I
r
i
1
A "
04
v
0
0
:3
d
a 1
U)
0—0
Q -
- - .a
z ,,
z 2.8 *— ...’ I-’ i
:- ..- .I. /q
I / d
/
/
// d
r- .‘I.’ x/ ' -*
2.6 FD...~"./' // .—
U.” /’ J
C,’ CFM
lllLLWLllJLJlLlIJlJlJJ

 

0.0 0.2 0.4 0.6 0.8 1.0

Concentration 1:

Figure 3.5: Same as Figure 3.1, except for Ni1-,Ag,.

 

CHAPTER 3. METALLIC ALLOYS 55

study because the length mismatch between Ni and Au is about 1.5%, which is
around the upper limit for forming solid solutions (Hume-Rothery, 1948). The
EAM results are good for all the partial length distributions and clearly superior
to the CF M. Note that some of the error bars on the EXAF S results are quite
large. Note also that the EAM and CF M give bowing of opposite sign for the
mean length. The difference between these two sets of results, suggests that
electronic density effects are significant in N iAu alloys. The agreement obtained
here between experiment and EAM is good. The large experimental uncertainties
at low Ni concentration cannot discriminate between the EAM and the results with
the Morse potential, used by Renaud et al. (1988), but the Morse potential gives
a crossing of the Ni-Ni and Ni-Au curves which is perhaps less acceptable that
the results presented here. Comparing with earlier EAM simulations by Ackland
and Vitek (1990), using the Finnis—Sinclair model for CuAu and AuAg, and Foiles
(1985) using the Daw-Baskes model for CuN i, we see that the overall behavior of
the Johnson EAM potential produces a smoother variation of the lattice constant
with concentration.

In Figs. 3.7-3.9, I show only the more successful of the EAM and CFM
results for each alloy, as determined by comparison with the diffraction data. In
all nine cases shown, one method was clearly superior to the other. Surprisingly,
when examining results in Figs. 3.1-3.9, the CF M is superior to the EAM in at
least as many cases as the EAM is superior to the CFM. Table 3.2 also shows
that the there is no clear preference for one approach over the other, which was of
considerable surprise to us. With alloys containing platinum, the Johnson EAM
potential does not give the right curvature for the mean length. Johnson already

showed that his potential is weaker for Pt. It seems that the crude electronic

CHAPTER 3. METALLIC ALLOYS 56

 

}rTTTlFrIIII—TrT—TITUFIIVII

 

2.8

 

NN Distance (A)

 

 

2.8

 

CFM ‘

ILALLJJLLRILLILLJIIIIIL

0.0 0.2 0.4 0.6 0.8 1.0

 

Concentration 1:

Figure 3.6: Same as Figure 3.1, except for Ni1_,‘Aux. The diffraction data for
Nil..,,Aux is shown as stars (Renaud et al., 1988). The EXAFS data, with error
bars, is also taken from Renaud at al.. The diamonds, squares and small circles
refer to the N i—Ni, Ni—Au and Au—Au nearest-neighbor distances respectively.

CHAPTER 3. METALLIC ALLOYS 57

model used in the embedded—atom method is not sufficient for platinum. But
surprisingly, the CFM (with no adjustable parameters) is in very good agreement
with the mean experimental nearest-neighbor distance for all but one alloy for
which the embedded—atom potential fails (see Table 3.2). In Figs. 8 and 9, the
simulations using the CFM and the diffraction results for the PtAg, PtAu, CuPt
and N iPt are shown. There are small discrepancies for the N iPt alloy, although the
curvature is negative, the same as in the experimental results. For PdPt, shown
in Figure 3.4, the CFM does not give better agreement with experiment than
the EAM potential, both having the wrong sign for the deviation from Végard’s
law. Figure 3.8 shows CFM simulations for NiCu and PdAg, which, as one can
see, agree with the experimental data. In the next chapters, we will see that
semiconductors can be described analytically by a simple harmonic spring model
(with angular forces) because they approximately obey Pauling’s rule of additivity
of atomic radii, as expressed in Eq. (8). The results of simulations would suggest
that platinum alloys also respect the Pauling rule with no or little anharmonicity
in the potential.

A selection of results for the elastic constants is shown in Figure 3.9;
experimental elastic data for alloys are virtually non—existent. But the smooth,
rather uninteresting, behavior in every case might lead us to suppose that they
are not too far from the real behavior, especially as the CFM and EAM give very
similar results. The relation Cu = C“ + Cu holds for every concentration x
within the limits of the precision on the numerical calculations using EAM for the
alloys. In Figure 3.10, I present the bulk modulus and the Voight average shear
modulus, computed using the EAM. The results for the pure metals are exact as

they were used as input in determining the parameters in the EAM. Computing

CHAPTER 3. METALLIC ALLOYS 58

Table 3.2: Alloys for which the EAM or the CFM gives the lattice parameter in
agreement with experimental diffraction results (shown by a bullet). The crosses
indicate disagreement with experimental results, and the blanks indicate that no
experimental data was available. The third columns refers to the Figure number(s)
in this paper where the results are shown.

Alloy EAM Central Force Figure

 

AuAg o x 1, 10
CuAg o x 7
PdAg x o 8
CuAu o o 2
NiAu o x 6, 10, 11
PdAu o x 7
NiCu x o 8
CuPd o o 3
NiPd o x 7
PtAg x o 8, 10
PtAu x o 9
CuPt x o 9
NiPt x o 9
NiAg 5

PdPt x x 4

CHAPTER 3. METALLIC ALLOYS 59

 

firTfYfi—fTTY—fVIYf—r—Tlfijr’

Cu1-3Agx . . . fl .

     

2.8 _

NN Distance (A)

 

 

   

LLL1A+LALAJLL1LLAALLAA

0.0 0.2 0.4 0.6 0.8 1.0

Concentration x

Figure 3.7: Showing the EAM results for Cu1-,Ag,, Pd1_xAu,‘ and Ni1_,‘de in
the three panels. In all cases, I show only the EAM results which were clearly
better than the CFM results when compared with diffraction data (Pearson, 1958;
Nagakura et al., 1966; Mealand and Flanagan, 1964; and Bidwell, 1964).

CHAPTER 3. METALLIC ALLOYS 60

 

 

 

2.9,L
[
2.8
e
o 2.55
o .
G
d
«on!
.9. -
O
z 2.50 CFM
2 .. :it‘ul‘ «mun

 

Ll
f‘ril'YYTITrVT—IT'fifiT

2.9— ' Pt,_,Ag,

 

 

 

‘

ALAILLQIlLHLlLULLLAA

0.0 0.2 0.4 0.6 0.8 1.0

 

Concentration it

Figure 3.8: Showing the CFM results for Pd1_,‘Agx, Ni1_,‘Cux and Pt1_,,Agx in
the three panels. In all cases I show only the CFM results which were clearly
better than the EAM results when compared with diffraction data (Coles, 1956 ;
Owen and Pickup, (1934); Lihl et al., 1971; Johansson and Linde, 1930, Novikava
and Rudnitskii, 1957, Klement and Lui, 1963; and Ebet et al., 1983).

CHAPTER 3. METALLIC ALLOYS 61

the bulk modulus with both the spring model and the EAM potential, I find that
its behavior is not very sensitive to the model used, and well described by the
simple effective medium theory as given by (16). This effective medium approach
must be applied separately to the bulk and shear moduli by fitting the end points
via K AA and K33. Note that the effective medium theory is designed to get the
correct initial slope for small 1: and 1 — a: (Thorpe and Wang, 1987). I find also
that the behavior of the elastic constant is very different from that determined
by Ackland and Vitek (1990). The calculation shows no change of curvature with
the concentration in any of the alloys studied in this paper, as they observed for
CuAu. Some experiments would be useful here.

Figure 3.11 shows the bond—length distribution for the CFM with all
the force constants equal and for the EAM potential. The CFM with equal spring
constants can be solved exactly to give the result 2.22 for the widths, in agreement
with the simulation results shown in Figure 3.11. The widths for the three peaks
for the CFM in Figure 11 are all equal and indeed the shapes are all similar apart
from a vertical scale factor. The CFM results are rather different from the EAM
results, but have the important common feature that the widths are comparable
to the peak separation. The full width at half the maximum, compared to the

peak separation < L33 > - < L M > is given by

 

(1 - a"')

a.’

2\/;r(l - x)ln2 (3.12)

where I have used the relations given in chapter 2 (Eqs. 2.29 and 2.34). The
lattice enters through the topological rigidity a". At a: = § the ratio (3.12) is
much larger for metals, 2 1.48, than for semiconductors, 9: 0.18, where a similar
relation exists, so that the length distributions are wide in the metal alloys studied

here as seen in Figure 11. In semiconductors the component peaks are more

CHAPTER 3. METALLIC ALLOYS 62

 

 

 

 

 

riflftvfifTfirerrrTrrr'

ALL 1 LI LLLLLLILALL

02 28'- r'JFT f'firjl'rjfrrrr1

o ' .4

o ’ '-.’-'

a b ’/.o'..". l
o ,a’ .

a I...” ;, 4

0‘ '° 1

a F 1'. 'O.

z 2.8'*.- CFM "

z , 4

35 Ni Pt ’

l-x x ‘1:

I." "d

I o‘,’- A

’0’”. ..-"°.. if

206 ’ .’...,o.'... q;

’ .0 4

"o..-'. l,

. - CFM ,

ALLLIL..AL.LL.LLL..I...L

 

0.0 0.2 0.4 0.6 0.8 1.0

Concentration 1:

Figure 3.9: Showing the CFM results for Pt1-,Au,, Cu1-,‘Pt, and Nil..,,Ptat in the
three panels. In all cases I show only the CFM results which were clearly better
than the EAM results when compared with diffraction data (Pearson, 1958).

CHAPTER 3. METALLIC ALLOYS 63

 

 

 

 

 

* *rl glrrrr' r1 ”7
’ ul—xAgx i
1.0:- a
. l
D «1‘
0.5:- s
A I- :
s —_—_—____w
04: 'nYlLLL.1...TI .114 ‘
rfT, .4. "'r' -1
>
2:: 1.0]? '1
:1 t ‘
“g 05 K 4
a o : \\\ 1[
.2 u \§~——-—1
a L . 1 -..1... L .. L- .
g .rrrr--.-,fi IL..-J....
—n ; t1
“3 1.5
1.0
0.5
0.0

 

 

 

 

0.0 0.2 0.4 0.6 0.81.0

Concentration 1:

Figure 3.10: The bulk modulus (solid line) and the Voight shear modulus (dashed
line) for three different alloys obtained from the computer simulation with the
EAM potential as described 1n the text.

CHAPTER 3. METALLIC ALLOYS 64

separated. These results make it very hard to justify using any virtual crystal
type of approximation for the electronic properties of alloys, even at the crudest

level.

3.5 Conclusion

This work gives no clear result in determining as to whether the central force
model (CFM) or the embedded—atom method (EAM) is superior. For platinum
alloys and a few others, the CFM is clearly superior, but in all other cases the EAM
is better. This may be related to the amount of electronic density effects in the
alloy. The Johnson RAM is very successful in N iAu alloys; the only case in which
EXAFS data is available. The surprising contraction of the Au-Au bond in Ag
rich AuAg alloys within the EAM, points out that more ab initio calculations are
needed for both single and pair defects in metals. Armed with this information, it
will be possible to construct EAM potentials that will interpolate over the whole
concentration range. In the absence of such calculations, the present results can be
taken as a guide, but with a good deal of skepticism. The CFM explains why the
widths of the length distributions are so wide, when compared to semiconductor
alloys. This is because the fcc metal lattices have a topological rigidity parameter

a " z 0.24 as compared with semiconductors where a" 2 0.8. The width of
these distributions means that the virtual crystal approximation is particularly
inappropriate in metallic alloys.

The effect of variation of electronic density is important in many cases and
is taken into account by the embedded-atom-method potential and is sufficient

in many cases to give the correct variation of the lattice parameter with the

concentration for most alloys, not containing Pt. The elastic constants show

CHAPTER 3. METALLIC ALLOYS 65

 

trTrrfTrIIt‘rWrrti‘rI

1000

1000

Probability (arbitrary units)

 

 

 

 

2J4 2&6 213 (10

Bond length (A)

Figure 3.11: The bond length distributions for N i—N i, N i—Au and Au—Au nearest—
neighbor distances in an N io,5Auo,5 alloy, computed using both the CFM (with no
variation in the force constants) and the EAM.

CHAPTER 3. METALLIC ALLOYS 66

monotonlc behavior with the concentration, and seem to be insensitive to the

model used.

Chapter 4

Crystalline and Amorphous SiGe

The SiGe alloys represent an important opportunity for the micro-electronic in-
dustry because of their opto—electronic properties when deposited as a multilayer
alternating with Si. To fully understand their properties, it is necessary to obtain
a good knowledge of the effect of length mismatch, which is about 4%, on the
strain. Recently, a few experimental and simulational papers have been published
on this topic (de Gironcoli, et al., 1991; Matsuura et al., 1991; Weidmann and
Newman, 1992; Incoccia et al. 1985; and Nishino et al., 1988). In this chapter, I
examine the mismatch problem in bulk semiconductor binary alloys, which are a
special case of the general theory for quaternaries, presented in Cai and Thorpe
(1992a, 1992b). I will concentrate on studying the effect of length mismatch for
the SiGe alloys and show that it is possible to understand most of the important
structural features with the help of simple analytic results.

Amorphisation is also a very important phenomenon in semiconductor
technology, as amorphous materials often share similar electronic properties with
crystals while being much easier to grow (Stutzmann et al., 1989; and Mackenzie

et al., 1985). Very little work has been done to date to study the effect of amor-

67

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 68

phisation on the length mismatch problem. Incoccia et al. (1985) and, a few years
later, N ishino et al. (1988) have measured the Ge—Ge and Si-Ge nearest-neighbor
distances in hydrogenated amorphous silicon (a-SiGe) but I am not aware of any
computational or analytical work. Schemes for constructing a perfect tetrahe-
dral amorphous structure (i.e. every site is fourfold coordinated) with periodic
boundary conditions were introduced a few years ago (Wooten et al, 1985; and
Wooten and Weaire, 1987). When studying amorphous systems, analytical results
for structural properties are usually difficult to obtain since one cannot generally
use expressions valid in the pure crystal limit. However, the situation in a—SiGe
is made tractable because the length mismatch in SiGe is small and therefore
the strain induced by the alloying can be treated as a perturbation on the ideal
amorphous network. Such behavior allows us to use the theory developed for the
crystalline case without any major modifications. Végard’s law is again predicted
and the topological rigidity parameter a" is unchanged by the amorphisation such
that the mean partial length equations are still valid and shown to be independent
of the amorphisation. For the next-nearest— neighbor distances, one must include
some effects due to the amorphisation and we will see that all the length distribu-
tions are completely dominated by the amorphisation, which leads to much larger
angular distortions between the tetrahedral bonds, than does the alloying.

In section 4.1, I describe the theory for a binary alloy of the form A143,,
for Si1-,Ge,. with results for the nearest and next—nearest neighbors. In section
4.2, I compare the analytical results with computer simulations using the Kirk-
wood model and with some ab initio results recently obtained by de Gironcoli et
al. (1991). In section 4.3, I examine the effect of amorphisation on the strain

energy by substituting Ce for Si atoms in a a-Si supercell constructed by Wooten

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SICE 69

Table 4.1: The force constants a and B in N/m for the Keating and Kirkwood
models obtained from the elastic modulus Cu and the bulk modulus B = §(Cu +

2012) (Bublik et al., 1974).

Kirkwood Keating
(1 [3 a 5
Si 53.09 13.81 48.49 13.81

 

Ge 42.28 11.30 38.51 11.31

(1991) using the Wooten, Winer and Weaire algorithm (1985). I compare the
results with EXAF S experiments performed on hydrogenated amorphous silicon—

germanium alloys by Incoccia et al. (1985) and by Nishino et al. (1988).

4.1 Solution for binary alloys

As discussed in chapter 2, analytical results have been obtained using the Kirk-
wood potential,

V: g 2 (ng-L?j)2+-§LZ Z (C036;jk+§)2, (4.1)

<ij> <ijk>

where a and 19 are taken to be the same for all bonds. This is a reasonable assump-
tion in the case of semiconductor alloys since the topological rigidity constants
are not very sensitive to changes in the elastic constants and, as can be seen in
Table 4.1, differ only by about 20% between pure Si and pure Ge (Bublik et al.,
1974). As there is no evidence for any appreciable bowing in the partial lengths,
there is no reason to go beyond this assumption for SiGe alloys. The mean length
L, is introduced in (4.1) so that the ratio 18/ a is dimensionless. For the potential

(4.1), the mean length follows Végard’s law.

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 70

The natural (unstrained) lengths Lag,- and Lche are obtained from the

pure crystals while the mixed length Lgiac is taken to be the arithmetic average

I F"

LgiGe = ([12:15; + LgeGe)' (4.2)

Q

N

Although inappropriate for most metal alloys, the additivity of the atomic radii
(Pauling, 1967) has been found to hold generally for semiconductors. Martins and
Zunger (1986) obtained a value very close to sum of the radii of Si and Ge in a
first-principles self-consistent calculation of an ordered zincblende SiGe structure.

The solution for binary alloys can be obtained directly, as mentioned
earlier, as a special case of the general solution for quaternaries presented in Cai
and Thorpe (1992a) where A = C = Si and B = D = Ge in A1_,BxCl_,,Dy. The

mean lengths are then given by

L8 = (1 _ x)LgiSi + ngeGe

< L513; > = Le - 30" (143.0. " Lgisz')
< LGeGe > = < L315i > +0" (LgeGe "’ 1431's.“)
1
< Lsgae > = §(< LGeGe > + < Lsgsg >) . . (4.3)

The topologiCal rigidity parameter, a“, is defined by a lattice integral and is
related to the ratio of force constants fl/a by an interpolation formula (Cai and
Thorpe, 1992b):

u _ 1+ 1.249(fl/a)
- 1+ 3.600(fl/a) + l.171(fl/a)2'

 

(4.4)

that is valid for any reasonable value of fl/a.
The second moments of the three length distribution functions about their

centers are given by

< LSiSi > - < Lsgsg >2 = < [420803 > - < LGch >2

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 71

= < 142510: > - < Lsgac >2
1 .. 2
= 5'30 — 2:)a1 (LgeGe — [43:31) (4-5)
where a'i" is defined as

a(W1!)
8y '

where y = fl/a The third moments can also be computed; they are all identical for

 

a," = (4.6)
the three distributions and vanish at :1: = 0.50 (Cai et al., 1990). Using supercells
of up to 125 000 sites relaxed with a Kirkwood potential, Weidmann and Newman
(1992) confirmed numerically that the bond length distributions are identical for
Si- Si, Si-Ge and Ge—Ge at all concentrations. 4
The strain energy per atom can also be written as a function of the topo-

logical rigidity parameter
a .. .
6 = 555(1 _ I)(l _ 0' )(LgeGe _ LgiSi)2' (47)

From the general theory for quaternaries, one can obtain all the next—

nearest—neighbor distances,

< L >332“ = L2“

(LgeGe—Lg'iSi) \/§ ..(2$-1+€2) f..( €1+€3
+ 2 3a 2 + 8b 2x+1— 2

 

where L2“ = ML: is the mean next nearest neighbor distance and e is +1 for
Si, and —1 for Ge. It easy is to see that all the distances are, contrary to the
results obtained for pseudobinary alloys (Cai and Thorpe, 1992b), a translation
of the same straight line. Here, as the two sublattices contain the same atoms,
one cannot distinguish between them and the slope for the six curves is the same.

There are two groups of curves, corresponding to the type of atom sitting at

 

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 72

the center of the triplet. Using results from Cai and Thorpe (1992a), the mean

next-nearest neighbor distances can be written explicitly as

L0 _ O... '
< LSiSiSi > = LGn _( Gch Lbzbz) fian- $+ \/§bu 223]
2 l. 3 8

L0 -111. _ '
< LSiSiGe > = Lretnn _ ( GcGe 5131) fian- I + \/§bn (21. _ 1)]
2 L 3 8
L0 - L0,. .
< LGeSiGe > : Licmn _ ( GeGe .5151) \/§aat 1' + \fibaa 2(3 _ 1)
2 3 8
and

L0 — L0... .
< LSiGeSi > = L211". _ ( GeGe ‘ $1.51) flaw-(x _ 1) + \/§bxu 22
2 3 8
L0 — L°.. . 8
< [15‘0ch > = LGn _ ( GeGe 2 .5351) [\/;a“| (I _ I) + fibu- (227 __ 1)]

L0 —L°- -
< LGeGeGe > = L:nn_( Gch2 515:) [flu-111(3— I)+\/§b”2(I—I)] .

(4.9)

 

 

 

 

 

 

 

The topological rigidity parameter b" is also defined by a lattice integral and
describes the opening of the second-neighbor shell when applying a force on the
first neighbors. As explained in chapter 2, it is found numerically that b" = a" / 2,
with the Kirkwood potential (Equation 4.1) (for all values of fl/a).

Therefore all the partial nearest and next-nearest-neighbor lengths de-
pend on only a single parameter, a“. This parameter can be obtained from the

Z plot via
a" _ < 1’0ch > — < LSiSz’ >

 

LgeGe — ngsg (410)

where the partial lengths are taken at the same concentration. One can then

predict the width of the nearest—neighbor bond length distribution (4.5) and all

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 73

the next-nearest-neighbor distances using (4.9). The slopes of the set of parallel

lines for the mean next—nearest—neighbor lengths (4.9) is given by (since L2“ is

8 0 0 11 m-
\/;(LGeGe - LSiSi) [1 - 16a ] (4'11)

and the distance between the curves in the same group is ‘/3/ 128 a" ( 2;ch — Lgai),

also :1: dependent)

while the distance between the two groups is ‘/2/ 3 a" (Law, — 14955,).

4.2 Results

To ascertain the quality of this theory, as applied to binary semiconductor alloys,
I performed simulations using the Kirkwood model from which the analytical
model has been developed. I statically relax unit cells of 8(12)2 = 13824 atoms
with periodic boundary conditions using the conjugate gradient method.

From Figure 4.1, we can see that the analytical results and the simulations
performed using the Kirkwood potential correspond closely. This agreement was
expected from previous studies of similar systems which showed that the approach
used holds to a very good accuracy up to a mismatch much greater than the one
studied here (Thorpe and Garboczi, 1990; Cai et al, 1990). Also shown in this
figure are results obtained recently by de Gironcoli, Giannozzi and Baroni (1991)
who have computed the structural (and thermodynamic) properties of a relaxed
random solid solution of Si1_,Ge,, by expanding the pseudopotential to second
order about the virtual crystal. In order to compare our respective simulations, I
adjusted a” = 0.707 from their results, which corresponds to fixing the ratio fl/a
at 0.20 using Equation (4.4). By so choosing a", one determines the mean bond

lengths for Si-Si, GLLGe and Si—Ge in Figure 4.1.

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 74

 

 

 

_IIII[fITrTrr11IrITrIIIII
A
°<2
v
4:
.0-3
Q0
:3
0.)
F!
‘U
1::
O
m

llllllllLllLlLlllllille-

 

 

0.0 0.2 I 0.4 0.6 0.8 1.0
SI x Ge

Figure 4.1: Mean nearest-neighbor bond lengths for Si1-3Ge,. The solid line is
the analytic theory compared to simulations for the Kirkwood model shown as
the solid symbols. The open symbols are from de Gironcoli et al. (1991).

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 75

It is also possible to compare the nearest—neighbor distribution predicted
by the theory with the data computed by de Gironcoli and collaborators as shown
in Figure 4.2(a). Around a: = 0.50, the third moment of the distribution is
small and the distribution is well described by a gaussian with the width given
by Equation (4.5). De Gironcoli’s length distributions are wider than predicted
by this model. This difference lies in the fact that they obtained their results
with constant chemical potential instead of constant concentration, broadening
the distribution. In Figure 4.2(b), the same peaks are shown but they have been
moved to place their center at the origin. The heights have been rescaled but not
the widths. As predicted by the theory, all the peaks are identical. In Figure 4.3,
we see the distribution for x = 0.11 with the gaussian second moment predicted by
the theory. The small asymmetry is explainable by higher order moments. Note
that again the three peaks scale perfectly as expected. it is particularly surprising
that this scaling still holds in the dilute limit of small x or (1 - :r).

The length for the Si-Ge bond lies within the experimental error limits
of a recent EXAFS experiment performed on silicon—rich SiGe/Si(100) films by
Matsuura, Tonnerre and Cargill III (1991) where they obtain (for low. concentra-
tion of Ge), 2.375 :1: 0.02A. This error is much too big to eliminate any model of
Si1-,Ge,. To distinguish between models, an experiment would have to have an
error of no more than :t0.002A.

This theory gives the exact Végard’s law while we know from X—ray
diffraction that there is a downward bowing with a maximum deviation of about
4% (Dismukes et al., 1964). It would be possible to implement this bowing by
changing the length of the heterobond (see Appendix A). The causes of this small

effect are subtle and would require much more advanced simulations to understand

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 76

 

u:
—1

 

III‘I’IIIU‘IUITITIIIIj

 

 

lllllllllllllllllll

2.35

 

TFI1II’IIrVJTTIV'IIUTII
.‘

(b)

Length Distribution

5, ’1.

" 7+1] LlLLlll

-0.02 -0.01 0.00 0.01 0.02

Length (A)

Figure 4.2: (a) The nearest-neighbor length distribution for Si1-,Ge, at :r =
0.45. The solid lines are gaussians using the correct weights, centers (3) and
widths. The solid symbols are from simulations using the Kirkwood potential.
The open symbols are from the simulation by de Gironcoli et al., with squares:
Si-Si; triangles: Si—Ge; diamonds: Ge—Ge. (b) The three peaks have been moved
to have their maxima at the same position and their height has been rescaled.
Symbols have the same meaning has in (a). The solid line is a gaussian obtained
with the parameters from the analytic theory (5).

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 77

 

 

 

:I I I I [T I I I I I I I I 1
— ——l
P , -
E (3) ° 5103903041 1
_ U . —-‘
P d
: o ‘ I
:- ‘. '1
r- . o d
: a ° 2 i
L— . ° 0 ‘1
P . o ' 1
t. . 1113.313;me .1

N
(a)
O
N
0
C3

240 245

 

lrIIIlTIIIIIIIIIIIIIl
I

(13}

Length Distribution

LlllllL-

-0.02 -0.01 0.00 0.01 0.02

Length (A)

Figure 4.3: (a) The nearest-neighbor length distribution for Sil-¢Gez at :1: =
0.11. The solid lines are gaussians using the correct weights, centers (3) and
widths. The solid symbols are from simulations using the Kirkwood potential.
The open symbols are from the simulation by de Gironcoli et al., with squares:
Si—Si; triangles: Si—Ge; diamonds: Ge—Ge. (b) The three peaks have been moved
to have their maxima at the same position and their height has been rescaled.
Symbols have the same meaning has in (a). The solid line is a gaussian obtained
with the parameters from the analytic theory (5).

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 78

them. In their calculations, de Gironcoli et al. (1991) obtained a 4% upward bow-
ing from their ab initio calculations.

In Figure 4.4, I show the nextwnearest—neighbor distance for all the possi-
ble triplets where the solid lines are the theory, obtained from Equation (4.8), and
the solid symbols are from the simulation. As noted in section 4.1, all the curves
have the same slope and divide in two groups. I have not found any experimen—
tal data on these quantities nor any other calculations. But as calculations and
experiments performed on Ca, I nlqu are in very good agreement with this the-
ory (Thorpe and Cai, 1992b), one expects the same in the binary semiconductor

alloys.

4.3 Effect of amorphisation

To understand the effect of amorphisation on the length-mismatch problem, one
notes that the strain energy introduced by the length mismatch between Si and
Ge is much smaller than the strain energy due to the amorphisation. One would
therefore expect the two phenomena to be largely independent of each other since
the tetrahedral topology is conserved. The effects of each kind of disorder can
than be simply added, at least to a first order approximation. we. can see that
there is no correlation between) the two effects by computing the strain energy
due to the length mismatch in the amorphous network. Figure 4.5 compares
this energy with the one predicted by Equation (4.7). I subtracted the strain
energy due to amorphisation, leaving only the term due to the length-mismatch
distortions. The latter is about two orders of magnitude lower that the strain
energy due to amorphisation.

It is useful to compare the magnitude of the disorder induced by the

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 79

4.00

3.95

Length (A)

3.90

3.85 .

 

IIIIIIIIITIIIIFIIIIIIrI

' .GeGeGe
' j ‘ SiGeGe
' —. SiGeSi

  

_ GeSiGe
_ SiSiGe

 

SiSiSi

 

lllllLllLl+lllllllllllll

 

 

0.0 0.2 0.4 0.6 0.8 1.0
Si x Ge

Figure 4.4: The next-nearest—neighbor distance in Si1_,Ge,. Solid lines are the
analytical results using the theory and the symbols are from computer simulations
using the Kirkwood potential. .

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 80

 

JTIIIIIIIIIIIIIIITFI

1.5

1.0

0.5

IIIIIIIIIIIIIII

 

Energy ( meV/atom )

 

 

00 llllllllllllllllll

'0.00 0.25 0.50 0.75 1.00
SI ' x Ge

Figure 4.5: The strained energy due to length mismatch in a—Si1-3Gex. The
large) component due to amorphisation has been subtracted. The symbols are
m computer simulation using the Kirkwood potential and the line 13 the analyt1c

result.

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 81

length-mismatch and the amorphisation. The comparison is made for a--Si1.,,GeJc
and the numbers are obtained from the relaxation of a giant supercell containing
eight times the 4096 atom amorphous supercell recently constructed by Wooten
using the Wooten, Winer and Weaire algorithm (1985; and Wooten and Weaire,
1987). I used a giant supercell to increase the probabilities of finding specified
triplets at low a: and (1 — x). This structure is perfectly tetravalent and can be
considered as a perfect amorphous structure, containing no dangling bonds. The
cell has been relaxed with the Kirkwood potential using the parameters described
in the previous section. Because the length mismatch in SiGe is small, about 4%,
and the network is relatively floppy, the values of the structural quantities in crys-
talline SiGe random alloy do not have a large variance. On the other hand, the
amorphisation entails considerable disorder leading to wide distributions. For ex-
ample, in Table 4.2 we see that the deviation of the bond length Si—Si in crystalline
Sio,5Geo,5 is 0.2% while it is ten times larger (2.0%) in the pure a—Si. Similarly,
the standard deviation for the bond angle is 11° in the amorphous network and
only 0.77" in crystalline Sio_5Geo,5.

What effects should the amorphisation process have on the theory pre-
sented in section 4.1? Because the equations for the partial nearest—neighbor
distances are linear, depending only on the average lengths, they remain valid
here and the mean nearest-neighbor distance in a-Si is identical to the crystalline
one. The only question to settle is whether or not a“ will remain the same. From
the definition, the topological rigidity constant represents the force needed to open
up a cage in the equilibrium structure. In the harmonic approximation which con-
stitutes the basis for this theory, a" should remain essentially unchanged as long

as the coordination stays fourfold. Indeed, Figure 4.6 shows that the theoretical

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 82

predictions are well supported by the simulation.

The next—nearest—neighbor distances must be examined carefully since
they involve second-order equations. In the pure length-mismatch problem, one
can neglect the deviations from the perfect tetrahedral angle and obtain a very
good agreement with simulations and experiments. In the amorphous material,
the angular deviations are important and one cannot neglect them.

For a triplet i j k in pure a—Si, the next—nearest—neighbor distance is given

by
$"=Iwnn¢A+ msmaB mAm

where ng and ij are nearest—neighbor distances and the angles %— (13,1 and §—¢3
are the angles between the nearest—neighbor vectors and the next nearest bond
ik.

Taking the average over all the triplets and neglecting correlations between

angle and bond length, we obtain
< L’""‘ >= 2Lg,5, < sin¢ > (4.13)

since in a tetravalent amorphous system to first order the mean length is the ideal
length in this problem. Using the fact that, in this system, the distribution of
tetrahedral angles is almost gaussian (Mousseau and Lewis, 1990), we need only

the first two moments:

<0>
2

 

<¢>= (4M)

where 0 is the bond angle. If the angular distortions are large enough, we cannot
approximate < sin¢ > by sin < 05 >, and we must include higher order terms.

Expanding around < 05 >, we obtain

< sin¢ > = sin(< 03 > +605)

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 83

Table 4.2: Values of various quantities in the a—Si network and in the crystalline
binary sem1conductor Sio,5Ge0_5.

Slo,5Geo.5 a-Si

 

Ar5,5, (A) 0.005 0.050
< 0 > (deg) 110.05 109.17
A6 (deg) 0.77 11.

= sin < 03 > cos(6¢) + cos < 05 > sin(605). (4.15)

Averaging over all angles, the second term on the right—side vanishes and we can

expand the cosine:

< sin 05 > = sin < 05 >< cos(6¢) >
= sin < 05 > e’i<(5‘f’)2>
= 0.9954 sin < 45 >

= 0.9935 sin 050 (4.16)

taking the mean angle and the deviation for a—Si values of a-Si given in Table 4.2.
2450 is the tetrahedral angle in a perfect diamond lattice. Using (13), we obtain
< L""" >= 3.807 instead of me = 3.833 given by \/8/—3 Lg,5,-. This change can
seem insignificant but it is easily perceived in computer simulations as shown in
Figure 4.6, which should be compared with the crystalline result in Figure 4.1.
When the length mismatch is superposed on the amorphisation, Equation
(4.9) remains valid except that the average next-nearest-neighbor distance is now
given by Equation (4.13) where the bond length Lia-S,- is replaced by L2" obtained

from Végard’s law,

L2“ = 0.9935 (/8/3 ((1 — mugs, + xL%.c.) , (4.17)

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 84

 

TIIIIIIIIIIIIIIrIIIrTITI

Amorphous

 

lllllllllllllLLllllllllL

0.0 0.2 0.4 0.6 0.8 1.0
SI x Ge

Figure 4.6: The nearest—neighbor distance for a—Sil_,Ge,. The symbols are from
simulations using the Kirkwood potential and the lines are the analytic theory.

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 85

leading essentially to a downward displacement of the six next—nearest—neighbor
curves for a small mismatch. We can also neglect the corrections in the second
term on the right-hand side of (4.8) since the are already very small compared
with the average next-nearest-neighbor distance. Figure 4.7 shows the simulation
and theoretical results. There is some noise in the computer data, due to the wide
distribution but the agreement with the theory is excellent.

These results contradict EXAFS experiments performed on a—Sil_,Ge,,:H
by Incoccia et al. (1985) and repeated using a different analysis method but with
the same results by Nishino et al. (1988). Both groups observed that the Ge—Ge
and Si-Ge bond lengths remained unchanged as the composition 2: changes. Such
a behavior is highly surprising and not believable. In order to follow Végard’s law,
the Si-Si nearest-neighbor distance would also have to remain constant, indicating
a topological parameter a“ = 1, meaning a completely floppy network. The only
way to bring such freedom to the amorphous network is via the introduction of
large amounts of hydrogen. By saturating dangling bonds, the hydrogen atoms
decrease the effective coordination of the network. The next chapter will examine
this problem in more detail; besides, more recent experiments have shown that
the uncertainties in the measurement are much too wide presently to challenge

this theory (Matsuura et al., 1991)

4.4 Conclusion

The simple analytical model presented here gives results which are in general
agreement with ab initio calculations. For the random crystalline solid solution,
the calculations by de Gironcoli et al. (1991) give even the wrong curvature for

the mean length emphasizing that it would require a more SOphisticated theory

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 86

 

 

 

 

3.95 ~Amorphous ‘ GeGeGe
" SiGeGe
°$
01
g 3.90 (GeSiGe
g SiSiGe
.21. SiSiSi
“O
§
2 3.85
3.80 '
0.0 0.2 07.4 0.6 0.8 1.0
Si x Ge ‘

Figure 4.7: The mean next—nearest—neighbor distances in a—Si1-,,Ge,. The solid
lines are the analytic results and the symbols are from simulations using the
Kirkwood potential.

CHAPTER 4. CRYSTALLINE AND AMORPHOUS SIGE 87

than is presently available to obtain the small deviations away from Végard’s law.

This theory provides a simple understanding of the mean partial nearest
and next nearest—neighbor lengths, in the absence of bowing. To go beyond this
and to be able to predict the bowing with any confidence is a formidable task. This
theory predicts that the width of the distributions of nearest—neighbor distances
will be all identical; a prediction which is robust when compared with parameter—
free simulations and I also show that all the next-nearest-neighbor distances have
the same slope. There are very few experimental results on these alloys; the
calculations and theory are consistent with (Matsuura et al., 1991).

Finally, contrary to experimental results obtained a few years ago (Incoc-
cia et al, 1985; Nishino et al., 1988), I predict that the amorphisation will have
no effect on the length mismatch problem, when all the atoms are tetravalently
bonded. The strain energy produced by the amorphisation is larger by two or-
ders of magnitude, compared to the strain energy produced by the alloying. The
length mismatch disorder can be treated as a perturbation on top of the disorder
already present in the amorphous network. This leads to very similar results for
the mean lengths to those found in the corresponding crystalline alloy. The widths
of the length distributions, for nearest neighbors, is dominated by the distortions

produced by the amorphisation.

Chapter 5

About EXAFS Results

With the development of EXAF S techniques, there has been a renewed interest on
the part of experimentalists to examine the local structure of disordered materials.
Some of the most exciting work in the past decade has been on crystalline semi-
conductor alloys (Balzarotti, 1987; Podgorny et al. , 1985); both III—V and II-VI
compounds. Many different materials have been examined at various composi-
tions and a very consistent picture emerges. The mean bond length of chemically
specific pairs, for example Ga—As bonds in Gazlnlqu, is composition depen-
dent but much less strongly so than the overall mean bond length. The overall
mean bond length, or unit cell size, can be obtained independently from X—ray
diffraction experiments, and is always found to be close to linear (Végard’s law,
1921) in agreement with the result expected if a weighted average of the different
bond lengths is found using the EXAFS results. The weaker composition depen-
dence of the chemically specific mean bond lengths is due to the soft nature of
the zincblende lattice.

The situation is very different for binary semiconductor alloys made from

the group IV elements Si, Ge and Sn. Recent experimental EXAF S results involv-

88

CHAPTER 5. ABOUT EXAFS RESULTS 89

ing the alloy Si1-,Gex have shown that the Ge—Ge and Ge—Si mean bonds lengths
are independent of the composition :13, for both crystalline and amorphous alloys.
Similar composition independent bond lengths results have also been reported for
SiC and GeSn alloys, but these systems are more complex because of the likeli-
hood of some local graphitic bonding of the C in SiC, and the possibility of local
metallic bonding associated with the Sn in GeSn. This chapter will therefore try

to understand the divergence between experiment and theory for Si—Ge alloys.

5.1 Assumptions

In the previous chapter, I have extended the theoretical description of the local
structure of crystalline, and with minor changes, amorphous binaries to group
IV alloys. Recent EXAFS measurements on a-SiGe and e—SiGe compounds (Ka-
jiyama et al., 1992; Incoccia et al., 1985; Nishino et al., 1988) have shown quite
severe disagreement with both CT theory and simulation predictions. The goal of
this paper is to examine these discrepancies between experimental EXAFS work
on one side and theoretical and simulation work on the other, as well as to propose
a new structural model that may resolve these discrepancies.

In order to solve the set of equations for the partial lengths, using a

Kirkwood potential (see chapters 2 and 4), I have used two major assumptions
that are worth repeating here:
(A) The elastic constants are the same for all interactions. The topological rigidity
parameter a“ is not very sensitive to changes in the local force constants, and
the elastic constants for the pure Si and Ge differ by about 20%, which will only
produce a hardly perceptible bowing of the Z-plots to be discussed later.

(B) The length Lgiac is the arithmetic average of the two other partial lengths.

CHAPTER 5. ABOUT EXAFS RESULTS 90

The additivity of atomic radii is found to hold generally in the case of semicon-

ductors.

5.2 Comparison between theory and experiment

The theory predicts an average length following Végard’s law, and the three paral-
lel partial lengths varying linearly with the concentration 2:. As I have commented
earlier, similar straight line behavior is predicted and has been seen experimen-
tally in ternary III—V and II—VI compounds (Balzarotti, 1987). For the binaries,
there exists a few experimental papers on SiGe alloys, using either X—ray mea-
surements or EXAF S on the Ge atoms. Kajiyama et al. (1992) measured the
mean Ge—Ge and Ge—Si bond lengths in bulk c—SiGe alloys; Incoccia et al. (1985)
and Nishino et al. (1988), in bulk a—SiGe alloys; and Matsuura, Tonnerre and
Cargill (1991) measured these lengths in SiGe/Si(100) films. There has also been
some work on a—GeSn and a—SiC alloys; both by the same group (Pascarelli et al.,
1992a and 1992b). However, since Sn and C can have variable bonding (Sn can
have local metallic bonding and C can have local graphitic bonding), conclusions
in these cases are more difficult to draw and will not be directly addressed here.
Moreover, because the CT theory applies only to the bulk, I will concentrate on
the three first papers (Kajiyama et al., 1992; Incoccia et al., 1985; and Nishino
et al., 1988). Nevertheless note that all these experiments cited above on SiGe,
SiC and GeSn alloys are consistent with the result that there is no composition
dependence of the chemically specific bond lengths, for both crystalline and amor-
phous samples. In the three papers on SiGe, both the Ge—Ge and the Ge—Si mean
nearest-neighbor bond lengths have been measured using Ge EXAFS, but not the

Si—Si bond length. All of these experiments agree on the fact that the measured

CHAPTER 5. ABOUT EXAFS RESULTS 91

mean bond lengths do not vary with composition. The exact bond length varies
only slightly and within a 0.02A error bar from one paper to the other, but the
conclusions are the same: the Ge—Ge and Si—Ge nearest—neighbor distances are
independent of concentration. Since the maximum variation from the length pre-
dicted by the theory is 0.026A, the predictions of this theory are at the limit of
the error bars.

Looking at Equation (4.3), we see that the only way to obtain flat curves
for Ge—Ge and Ge—Si bond lengths is for the network to be floppy, i.e. to have
a" very close to 1. Even relaxing the condition where l have used the same force
constants for the three different kinds of pair interactions, one finds in the low

concentration (single defect) limit (Thorpe, Jin and Mahanti, 1989).

(1 - a“)asaoe
(1 - a")as.'sa' + 0"01516e

 

< LSiG’e >= LgiSi + 3 ] (LgiGe - [43151) 1 (5-1)

where I have introduced different central force constants 05,-5.- and 05503 for the
Si—Si and Si—Ge bonds. Different angular force constants can be incorporated
by adjusting a”. I have also not assumed that Lgice = (Lg-S,- + Lgch)/2. So,
independently of the values of the force constants, we can only obtain a flat
EXAF S curve (i.e. independent of composition 1:) with a“ = 1. The first way
to achieve this result is to affect the value of a“ by varying the ratio fl/a. From
Equation (2.31), we have the behavior of a" as a function of the ratio of the
angular force 3 to the bond stretching force a from CT. To obtain a" = 1,
we need a ratio fl/a that is essentially zero, and certainly much smaller than
the usual range 0.1 — 0.2. In order to explain their results, Kajiyama et al.
(1992) suggested that the angular term 3 is anomalously small compared to the
stretching term a but for this explanation to be valid would require a ratio fl/a

of less than 0.05 which would lead to a much weakened shear modulus. Direct

CHAPTER 5. ABOUT EXAFS RESULTS 92

measurements of the elastic constants in SiGe solid solutions grown by evaporation
in the absence of hydrogen show no such weakening (Bublik et al., 1974). It
would be interesting to repeat these experiments on hydrogenated samples. Also
indirect measurements like Raman spectra can be well reproduced theoretically
using the same force constants for all bonds and taking into account only the
mass differences (Yndurain, 1978). The Raman peak at about 470 c111‘1 in pure
silicon can be rescaled using only the appropriate reduced mass to the Ge peak at
280 cm‘1 and to the new Si-Ge peak centered at about 380 cm“, all to within 5
percent, confirming that the force constants are close for the three kinds of bonds.
The samples used for the Raman scattering were polycrystalline and should not
contain hydrogen. It is therefore very unlikely that any appreciable change in
the ratio fl/a occurs, and I dismiss this possibility. One does however expect
a softening of the elastic constants in hydrogenated samples —not due to the
weakened angular forces— but due to the mechanical degradation of the network
caused by the presence of large amounts of hydrogen.

In their paper on c—SiGe alloys, Kajiyama et al. (1992) measured the
lattice constant as a function of the composition :1: using X- ray diffraction. It
is worth mentioning that previous measurements showed a small deviation from
Végard’s law of less that 4% (Dismukes, Ekstrom and Paff, 1964) compared with

the much larger 16% presented in Kajiyama’s paper. Using the (exact) sum rule
< L >= (1 - 1:)2 < L555; > +2:r(1 — I) < Lsgac > +132 < LGeGe >, (5.2)

and the values for < LSi-Gc > and < Lake: > obtained by EXAFS, it is possible
to predict the expected value of < L555.- >. If Végard’s law is exactly obeyed, then

(5.2) gives < L355,- > = Lag,- which is also concentration independent. However

CHAPTER 5. ABOUT EXAFS RESULTS 93

the measured X—ray diffraction data (Kajiyama et al. 1992) can be fit with
< L >=(1- $)Lg15i + ILche ‘— 017(1— 13), (5.3)

where 0 is the bowing parameter. From (5.2)

0 1'
< LSiSi >= LSISI' "" 6'1:

(5.4)
showing that the composition dependence of < L35; > will become large as the Ge
composition (1 — 2:) increases, as shown in Figure 5.1. I emphasize that the result
(5.4) is exact and independent of any model. It would be useful to have EXAF S
experiments on the Si available so that this quantity would be available to provide
an important internal consistency check. It would be quite unusual if the length
of the Si-Si bond decreased upon the introduction of the larger Ge. Theoretically
this behavior has been predicted in some metallic alloys (e.g. AuAg) where it is
attributed to the effects of variation of electronic charge density. However, we
cannot use the same argument here since Si and Ge are in the same column of the
periodic table and charge transfer effects are negligible. It is difficult to justify
the Ge—Ge and Si-Ge bond lengths remaining so constant compared with such
a marked change in Si—Si bond length, although such an occurrence would not
violate any geometrical constraints.

Another way to increase floppiness in the alloy, would be to decrease the
average coordination number of the atoms. This is a likely explanation since in
all three experiments (Kajiyama et al., 1992; Incoccia et al., 1985; Nishino et al.,
1988), the sample preparation took place in the presence of hydrogen. Hydrogen
is a network terminator, and attaches itself to one (or more) of the four bonds

formed by the Si and Ge ions. As a consequence its main effect on the network is

to decrease the effective coordination and hence the rigidity of the network.

CHAPTER 5. ABOUT EXAFS RESULTS 94

 

 

 

 

2.5 r I I I r I 1 1 v T I I I . I r I 1 1 r r l r r I I Id
_ . - ............ ' .............. . . . . .’. ’. .
l- “ ’ ’ " ’ «I
02 2,4— . .. ...“ ...... ’ 1.’..'¢’..’... .. .......... . ........... -.':
V - ’ ’ a
a: " .— " ’ ’ ’ 'i
g . ............................................................... u
on b -
c - d
.9. o
2.3 *"" o .—
13 _ .
I:
O L «1
m .1
2.2 — ...-1
)- o q
I m 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 . I 1 1 1 1 l.4
0.0 0.2 0.4 0.0 0.8 1.0
Si Concentration 11 Ge

Figure 5.1: A Z-plot from the Kajiyama et al. (1992) experiments. The Ge-
Ge and Si-Ge bond lengths measured by EXAFS are shown by full squares and
the average length obtained by X-ray diffraction, by crosses and the dashed line
is Végard’s law. The Si-Si bond length (empty circles) was deduced using the
definition of the average len ths. The solid line is the quadratic relation given in

the text and using 0 = 0.06 . The horizontal dotted lines are guides to the eye.

CHAPTER 5. ABOUT EXA FS RESULTS 95
5.3 Structural model

I have examined the effect of the inclusion of hydrogen on the topological rigidity
of the network. Hydrogen is a terminator that makes a single bond to either
Si or Ge, both of which remain four—fold coordinated. Hydrogen that goes in
interstitially or as molecular hydrogen is irrelevant to the mechanical properties
of the network and will be ignored here. The method used here gives an estimate
for the variation of a" with the degree of hydrogenation. The numerical method
used is the same as the one presented in the previous chapter where the Kirkwood
potential is used and the static relaxation is accomplished using a conjugate—
gradient algorithm. I have introduced hydrogen into the network by cutting a
bond and inserting two hydrogens. This is an unphysical procedure as the Si-
H and Ge—H bonds lengths are too large (about 1.5A) compared with 2.35 and
2.43A for Si—Si and Ge—Ge bonds respectively, for this to occur. Nevertheless,
since I consider the hydrogenated bond as absent, this procedure does encompass
the main effects of hydrogen which is to act as a terminator and hence lower the
mean coordination of the network (He and Thorpe, 1985). In the simulations, the
bonds are removed randomly , but never attach more than two hydrogens to a
Si or Ge atom. This procedure can be accomplished with equal computational
facility for both the crystalline and amorphous networks.

It is possible to obtain the limits of a" analytically, i.e. the fully coordis
nated and the floppy limit when the rigidity vanishes. In between these two limits,
one has to resort to computer simulations. The four—fold coordination limit of a"
is found from Equation (4.4) to be 0.707 for a ratio fl/a = 0.20. The floppy limit
is obtained by looking at the rigidity percolation transition, using a mean field

approach. This has been used elsewhere (Thorpe, 1983) and has been shown to be

CHAPTER 5. ABOUT EXAFS RESULTS 96

very accurate. The network becomes floppy when F, the number of zero frequency
modes is greater than zero (Thorpe, 1983). The number of zero frequency modes
is equal to the number of degrees of freedom minus the number of constraints.
Because I don’t allow more than two hydrogens per Si or Ge, the whole network
remains fully connected. Counting 4/ 2 central force and 5 angular constraints per

Si and/or Ge, and 1/2 central force constraint for each H in Sil_,,Ge,:Hy, one gets

F = 3N(1+y)—N[2+5]—Ny[%]
= N(gy-4). (5.5)

For F =0,

cmoo

y, = = 1.6 (5,6)

which corresponds to a yp / (1 + yp) = 8/13 = 62% atomic fraction of hydrogen.
The essential idea is that each hydrogen atom terminates a bond and
has no other interactions. Since hydrogen does not participate in connecting the
network together, being a terminal bond, one can consider it as being absent
for most purposes. The simulation procedure consists then in removing bonds
at random in fiSiGe or a—SiGe solid solutions to obtain the topological rigidity
parameter a" associated with a certain concentration x and y. This is done by
computing the various mean distances, as in Figure 5.2 and fitting the resulting
straight line Z-plot found with the single unknown parameter a". This is a
somewhat indirect procedure, but appropriate in the present context. Indeed it is
probably the best way to obtain the topological rigidity parameter in amorphous
networks. The quantity of hydrogen needed to take the topological rigidity a" of
the lattice to unity (i.e. the floppy limit) would be extremely large (62% atomic
fraction) and it is difficult to imagine such amounts of hydrogen distributed evenly

throughout the sample.

CHAPTER 5. ABOUT EXAFS RESULTS 97

 

 

 

 

20450 bl U j I I I I I I I T VT I I I I I I I l U U r I
:.—-.—-r-t-O"."."r-.-.- ‘Ge-Ge
2.425 F- d.
a? . .
v )- d
‘5 2.400 :— — 81-61:
g C .- — O - " T . — . - ..
-o C. _‘
c: 2.375 _ .
o . .
a: L . s1-s1
2.350— --.._.-—.--O-"."'.".".—_:
l l l l I I l l l l L I. j l I l l l l I I I I I r
0.0 0.2 0.4 0.0 0.8 1.0
5‘ Concentration x Ge

Figure 5.2: A Z-plot for ¢:—Si1.xGe,:Hy from a computer relaxation of a crystalline
alloy. The solid lines correspond to a topological rigidity parameter a" = 0.707
and the dashed lines are a fit to the simulation results (shown by solid circles)
where y = 1.0 and a" = 0.923. The fraction of hydrogen is determined by
assigning two hydrogens to each removed bond.

CHAPTER 5. ABOUT EXAFS RESULTS 98

The bulk calculations for Figure 5.3 have been made for randomly re-
moved bonds in crystalline as well as amorphous cells of 215 = 32,768 atoms
with periodic boundary conditions, constructed by Wooten (1991) following the
Wooten, Winer and Weaire algorithm (1985). The Wooten sample contained
2” = 4,096 atoms but was replicated 8 times to form a larger cube. I then dec-
orated the full lattice randomly with Si and Ge atoms following the procedures
described in chapter 3. In the previous chapter I have shown, using computer
simulations, that since the connectivity is the same in the crystalline and perfect
amorphous Si structure, the topological rigidity a" is identical in both systems.
Moreover, as the strain due to alloying is about two orders of magnitude smaller
than the strain due to amorphisation, one can treat the former as a perturbation
and use the CT theory developed for crystalline alloys. The results for all the
partial lengths in both the amorphous and crystalline structures are always the
expected set of three parallel straight lines that can be fit with the single parame
ter a”. From computer simulations, 1 find that the topological rigidity parameter

a” is linear in the mean coordination < r > of the network and so can be written
*. 5 it I
a = §[(4- < r >) + (10 (< r > —2.4)] (5.7)

where < 1' >= 4 — y is the mean coordination of the network '(ignoring the
hydrogen), and the topological rigidity parameter a” = a5‘ = 0.707 when no
hydrogen is present.

The precise way in which the hydrogen is distributed in the network is ir-
relevant as far as the topological rigidity is concerned, as long as it is homogeneous
on some reasonable length scale. It is therefore possible to think of a physically
more plausible model that could result in the same amount of floppiness in the

network. I have studied the topological rigidity for finite pieces of crystalline

1.0 l r I l __

0.9 — —‘

oi . j
_ O Amorphous (bulk) ‘

_ D Crystalline (bulk) .

_ I Cube .

_ 0 Tube _,

0.7 __ O Plane _

b l l l l I l l l l J L l l l I l l I L I L J

CHAPTER 5. ABOUT EXAFS RESULTS 99

 

    

 

 

 

2.0 2.5 3.0 3.5 4.0
Mean coordination <r>

Figure 5.3: The variation of the topolo ical rigidity parameter a" with the mean
coordination < r > in SiGe alloys. T e open symbols are for simulations with
periodic boundary conditions (bulk) and the solid symbols for (clusters) with free
boundary conditions. The solid line is a linear interpolation between the floppy
limit a“ = 1 and the a" = 0.707 for the perfect network, given in the text.

CHAPTER 5. ABOUT EXAFS RESULTS 100

571-306, alloys of different sizes and shapes with free surfaces. The three shapes
I used were cubic (I x I x I), tubular (I x I x L) and planar (l x L x L), where I is
a variable and L was fixed at 54A which one can regard as infinite (Figure 5.4).
There is no hydrogen implanted in these pieces. A physical realization of this
situation would be that all the dangling bonds at the surface of these pieces are
saturated with hydrogen. Again the topological rigidity parameter is extracted
from the computer simulations by fitting the three parallel lines in the Z—plot
like that in Figure 5.2 with the single unknown parameter a". We do indeed
always obtain simple straight line Z—plots from the simulations on cubes, tubes
and planes. The results in Figure 5.3 shows that the geometry of the defects
is almost irrelevant to the topological rigidity a"; the mean coordination of the
system uniquely determines its topological rigidity.

For these structures shown in Figure (3), the open surfaces are saturated
with hydrogen. One is therefore free to choose any geometry, for a given concentra-
tion of dangling bonds, without affecting the mechanical rigidity of the network.
A probable structural model would be one with a high density of thin planar
cracks with a typical size of the two regions between cracks being about 10A as
deduced from Figure 5.5. A sketch of this model is presented in Figure 5.6. These
cracks will be filled with hydrogen saturating the dangling bonds of the surfaces.
Such a structure would be almost completely free of strain due to the alloying,
resulting in an almost flat Z—plot, and would also respect the random proportion
of Si-Ge to Cue-Ge bonds found experimentally from EXAFS (Kajiyama et al.,
1992; Incoccia et al., 1985; Nishino et al., 1988.

From this model, it is also possible to find the softening of the elastic

constants. This is done by applying an external strain to the model and then

CHAPTER 5. ABOUT EXAFS RESULTS 101

[XIX]
[XIXL I

[XL xL

Figure 5. 4. A sketch of the cube, tube and plane, where the long dimension

L= 54A and t e short dimension can be varied. The results obtained were not
sensitive to the ength L which can be regarded as infinite.

CHAPTER 5. ABOUT EXAF S RESULTS 102

 

 

 

 

T l I T I I I I I V I l I V :d
1.0 - T I
.. . I ..
I ' - .
O . _
0.9 r.— . ' .
O - O O .1
O l' .
fl . . . .
. e .
- I Cube ‘
i Me .
0.7 -- 0 Plane Bulk -
l l l l l l _1 l l l l L l l I I l d
0 10 30

. 20
Length 1 (A)

Figure 5.5: The variation of the topological rigidity parameter a" with the length
scale I for the cubes, tubes and planes shown 1n Figure 5.4. Here the large dimen-

sion L = 54A can be regarded as infinite.

CHAPTER 5. ABOUT EXAFS RESULTS 103

 

Figure 5.6: A sketch of the proposed structural model for SiGe alloys. The dark
region is network and the open cracks are saturated with hydrogen to tie off the
dangling bonds.

CHAPTER 5. ABOUT EXAFS RESULTS 104

 

1.2 1+ I r r r I I I U l I U I T l' I I Y T

1.0

l

0.8
0.8
0.4

0.2

Bulk modulus (normalized units)

IIIIUIIIIUIIUITIIUIIIII’TrUII

llllllLLllllLllelll

 

 

 

 

1*
o

2.0 2.5 3.0 3.5
lean coordination <r>

Figure 5.7: The variation of the bulk modulus B as a function of the mean coor-
dination < r > in c—SiGezH. The solid symbols are from a simulation on a sample
with planar cracks like that shown in Figure 5.6 and the solid line is Equation
(5.8).

again minimizing the energy as before. The difference in the relaxed energy leads
to the elastic moduli (He and Thorpe, 1985). Figure 5.7 shows how the bulk
modulus varies with the effective coordination < r >, in a bulk c—Si where planar
cracks where introduced. Theses cracks were created the following way: in a
crystalline cell of 215 = 32,768 Si atoms, the position and direction of a plane
were chosen at random; all the bonds that crossed the plane in a square area of
side 2 unit cells around the origin of the plane were removed.

I can compare these results with an expression found by He and Thorpe

llL

th

CHAPTER 5. ABOUT EXAFS RESULTS 105

(1985) for removing bonds at random. This expression is valid around the rigidity

threshold, but is not supposed to be critical behavior,
8 = 0.328 (< r > —2.4)‘°5 (5.8)

in units where the bulk modulus is 1 at full coordination (< r >= 4). Within the
numerical accuracy, we see that the distribution of the hydrogen does not effect
the bulk modulus either. We see once again that the mean coordination is much
more important that the detailed geometry in determining the mechanical / elastic
properties of the system. For hydrogenated samples near the rigidity threshold,
which are required to explain the EXAFS results, the elastic constants should be
an order of magnitude lower than in SiGe alloys that do not contain hydrogen.
Sample annealing, allowing for the removal of cracks and hydrogen, would increase
the rigidity of the lattice, and hence produce more composition dependence of the

EXAFS results and larger elastic moduli.

5.4 Discussion

How does one account for these fundamental disagreements between theory and
experiment in crystalline and amorphous alloys? It is possible that the angular
forces fl become anomalously small in the alloy: much smaller than in pure Si or
Ge. This could be due to some complicated charge transfer effects of the kind
that are known to occur in metals. However no such effects have been seen in any
III—IV or II-VI compounds and so this possibility has to be regarded as extremely
unlikely. It is possible that the presence of hydrogen might substantially weaken
the angular force B, but no evidence of such a phenomenon has been seen in

hydrogenated amorphous silicon. For example Maley and Lannin (1987) showed

, -‘l

CHAPTER 5. ABOUT EXAFS RESULTS 106

with Raman scattering that the position of the peaks remains the same in a—Si
with or without hydrogen.

A second possibility is that the EXAFS data has been incorrectly in-
terpreted. While some theory is necessary to extract the number and distance
associated with the various nearest neighbor pairs, the EXAFS technique has
been very successful in other semiconducting alloys for nearest neighbor distances
and it is difficult to see why it would fail here. However, although these results
are all consistent with a flat curve, we must not forget that the error bars on
the measurements are around 0.02A. It is true that the size mismatch between
Si and Ge is only 5% but this is comparable with the 8% bond length mismatch
in for Ga—As and ln—As bonds GaInAs for example. One should also note that
all the experiments cited agree that SiGe forms a good random solid solution, as
evidenced by the relative weights in the EXAF S components. It is true that if the
Si and Ge were phase separated, this would account for the observed composition
independence of the Ge—Ge and Si-Ge bond lengths. However this would lead to
very little weight in the Si-Ge component, as these bonds would only occur along
the interfaces between the two phases.

Finally we have the possibility that there is a very large, between 45%
and 65% atomic fraction of hydrogen. Although this is a huge amount, we can
find proportions of hydrogen in a—Si and technological a-Si that contain typically
between 10 and 30 atomic %H (Pankove and Johnson, 1991). Such material would
be mechanically weakened and show very soft elastic behavior. when compared
with crystalline Si and Ge (see Figure 5.7). ,I have been unable to find any mea-
surements of the elastic constants for hydrogenated material. Maley and Lannin

(1987) also pointed to a marked sharpening of the TO peak associated with hy-

 

CHAPTER 5. ABOUT EXAFS RESULTS 107

drogen and due the improvement of the local structure in a—SiGezH alloys with
up to 40 atomic %H. Also the same results are obtained experimentally for both
crystalline and amorphous SiGe alloys and it is even harder to envisage how so
much hydrogen could be incorporated uniformly into the crystalline alloys. How-
ever, the presence of planar cracks on a very short length scale can apply to both
crystalline and amorphous alloys and is compatible with the preparation method

used to obtain these samples.

5.5 Conclusion

EXAF S is a very powerful technique that has opened up a new field of mea-
surements i.e. the local structure of disordered systems. However, because this
technique is an indirect one, one has to remain very careful in interpreting the
results. The curious results for SiGe alloys open up many questions which will
require more experiments to sort out. In particular, I would suggest EXAF S

measurements on SiGe samples not containing hydrogen.

Chapter 6

Surface of semiconductors

The development of tools to observe directly the surface with an accuracy of
about 0.01 A vertically has opened a whole new field of observation which is very
important in order to sustain the rapid technological inventions in the field of
semiconductors. The surface effects, for example, will become noticeable when
trying to construct thinner and thinner samples. The study of surface can also
provide, indirectly, some information on the nature of the bulk disorder which
would complement results obtained from other experimental methods like EXAF S.

In this chapter, I propose to expand the theory used in the preceding
chapters to cover the surface problem. I will study the surface relaxation of 2D
triangular lattices as a test case, as well as surfaces of zincblende binary alloys. I
will show that the successful theory developed previously by Thorpe, Garboczi and
Cai (Thorpe and Garboczi, 1990; Cai and Thorpe, 1992a and 1992b) is portable
to this problem and is very helpful in putting together a clear image of the re-
sults. This theory predicted correctly distortions of the lattice for the ternary
semiconductors although there are still conflicts between it and experimental re-

sults regarding the binary semiconductor SiGe. I am, however, confident that the

108

CHAPTER 6. SURFACE OF SEMICONDUCTORS 109

theory will prove to be accurate also in the latter case. In section 6.1, I present
the theory adapted to the surface problem and in section 6.2, I discuss different
methods to obtain the topological rigidity parameters in lattices. I then present,
in section 6.3, the triangular lattice results and in section 6.4, the diamond lattice,

SiGe crystalline alloy with (100) and (111) surfaces.

6.1 Theory

The formalism developed and presented in chapter 2 was applied to crystalline
and amorphous bulk alloys. However, this fermalism can be extended to solve
the problem of surface relaxation in semiconductor alloys. I will repeat a few step
from chapter 2 in order to show that the existence of a surface does not preclude
the application of this theory and that the formalism is general enough to include
the loss of symmetry due to the surface.

I can start directly from the demand that fully relaxed structure must

have no net force,
2 K,,-(L.-,- - LEE-m..- = 0- ‘ (6.1)
Therefore, if one draws a plane across layers I , along any direction, the total force

must also vanish:

(l - :r)2KA,4(< LAA >1 _L31A) +2.2(1- x)KA3(< LAB >1 -1813) (6.2)

+$2K33(< L33 >1 -L%B) = 0. (6.3)
If the elastic constant K M = [(A3 = K 33 = K then, using the general definition

< L >1=(1- :1?)2 < L,“ >1+2x(1— x) < LAB >1+:1:2 < L33 >1, (6.4)

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 110

with I being a layer index, one obtains
< L >,= (1 — 3ft)” + 2::(1— x)L§’,B + $2L%B. (6.5)

On any layer, as well as between them, Végard’s law is therefore obeyed. I can
now address the question of partial length average. Assuming small displacements

in the solid, one gets, as in chapter 2,

L11 Rn‘ - Rj (6'6)

= 141+ [(u, - U1) ' Rileij

where L1 =< L >1 R?)- and RE),- is the unit vector of a perfect lattice and u.- is the

displacements from this lattice. Using the previous equation in the force equation

0 = z: L:+1(u.-- u.) ~11?) — Lam..- (6.7)

one can write

Du = v, (6.8)

where D is the dynamical matrix of the system or the inverse Green function

u = —Gv (6.9)
and
v = :(L1— Lang), 7 (6.10)
so that i
L.) =< L >1+K?:R?j-(G.-m - GJ-m) '- R?m(L?m— < L >1). (6.11)

This equation is exactly the same as the one presented in chapter 2 except that

the value of < L >1 can vary from layer to layer. In a homogeneous solid, we can

CHAPTER 6. SURFACE, OF SEMICONDUCTORS 111

drop the layer index. In the case of a bulk solution, one can easily obtain G’gm
and from there solve exactly for the partial lengths. By the surface, however, each
layer will present a different average and the case is not so simple. For example,

close to the surface, the relation
2: Ru = 0 (6.12)

no longer holds and i and j indices are not symmetric anymore.

6.1.1 Bond mismatch

The previous arguments are identical for both bond and site mismatch; only during
the averaging does the nature of the disorder, and hence, the correlations, appear.
And one would not expect, a priori, to obtain the same form at the surface as in
the bulk. Nevertheless, by doing the complete algebra one finds that the bond
mismatch problem leads to the same solution as previously found regarding the

average lengths

< L; >1 = < L >1-:ra,'(L%— Lg) . (6.13)

< LB >1 = < L >1 +(l — x)af(L% - Lg)
where the topological rigidity parameter a; now varies through the layers
K
a] = N 2R3- -(G.'1 - 2011' + ij) ' 1191-. (6.14)
‘1'

In this equation, the summation over i and j is restricted over their respective
layers defined by I. With the loss of symmetry across the surface, the Green’s
function cannot be obtained through the reciprocal space. In the real space, the
equations are non-linear and direct calculation can become very cumbersome. I

use a’ from the definition of Equation (6.14).

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 112

6.1.2 Site mismatch

For the site—mismatch problem, the situation is similar to the bond problem. After

averaging over the different partial lengths, one obtains

< LM >1=< L >1 +ra,“(L?M — L233) (6.15)
and
< L33 >1=< L >1 -(1— x)a,"(L?,A — L333) (6.16)
where
519': 11,-,( G1,, — ,m)-R1m(61k + 61... + 8,1 + 6,-m). (6.17)

Nijkm
In the bulk, the symmetries are recovered and the equation becomes identical to
the one described in chapter 2.

In site—mismatch alloys, another quantity becomes interesting at the sur-
face: the average displacement u1 from a reference position. In the bulk, for
symmetry reasons, this quantity vanishes everywhere. At the surface, however,
the symmetry is broken along z and should reflect in this quantity. Averaging a,-

over the sites A of a particular layer, from Equation (6.9),

=—KZG:;"R§’ (L9, —< L >1) (6.18)
J'kfi
we obtain
2 KxUJ 22 22 2
< “A >1=—2N“; Lola);(Gfi -G:j) Rij’ (6.19)
and
K 1-

< 11; >1= -(—2N—fl(LgB — LL) 2 (fo — 05;) 125,. (6.20)

*5

Because of symmetry,

< u; >1= < ui’, >1= < 113 >1=< uB >1= 0. (6.21)

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 113

However, the interesting quantity is the difference between the two displacements

z I{ ”T ‘2 z
'.7

i.e., it is independent of the concentration and varies only with the layer and the

size mismatch. If one also adds the difference in radius, defining the maximum

height at the surface, the total distance between the two peaks in an alloy is

 

K L0 — 0

Aug -.- 1+ N: (fo — 05;) 12:, ( BB 2 L“). (6.23)
11

One can easily verify that the average over all the species indeed vanishes.

Moreover, in the bulk, the partial average displacements < uA >1 and < 113 >1

vanish due to the symmetries of the Green’s function and the lattice.

6.2 Topological rigidity parameters

There are many methods to obtain the topological rigidity parameters of a system.
One would be to solve directly Equations (6.14) and (6.17), using the Green’s
functions of the system. For a bulk crystal, this method can be useful, providing
for exact analytical result; however, for non—symmetric systems, like surfaces,
clusters or glasses, obtaining the Green’s functions can require huge efforts.
Another method is to use Equations (6.14) or (6.16). If one replaces a

bond of length Le by one of length L’e in a perfect crystal, the energy changes by
AB = K(L — LL)? - K(L — Le)2. (6.24)
The force of the bond on the lattice is

F = —— = K(L; — L). (6.25)

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 114

But the force can also be expressed by
K
= ;(L — L'c) (6.26)

and so

AL = L — L; = 111*(L'c — L.) (6.27)

which is simply Equation (6.14). It is therefore possible to obtain the topological
rigidity parameters by applying forces on atoms of a pure system or by alloying
a lattice and relaxing the whole network. In most cases, the latter method is
used because it allows a better comparison with experiment. However, for the
local calculation of the topological rigidity constants (i.e., for a kink, an edge or a
cluster), it is often simpler to use forces. For a‘, a force must be applied at both
ends of a band while it must be applied on all neighbors of a particular atom,
radially, to obtain a" for each bond around the central atom.

I have verified numerically the equivalence between using alloying or local
forces to obtain the topological rigidity. In two dimensions, the local force method
in unstable under relaxation close to the surface. As discussed in the next section,
there is nothing surprising about this since there are problems with fluctuations in
two dimensions. I have also verified this equivalence for a" and a" on a diamond
structure and it works perfectly within the numerical error. It is therefore possible
to extend the concepts and formalism developed in the previous section to almost
any configuration.

Although the application of the force is direct for a bond case, it becomes
trickier when trying to obtain a“ at the surface. In the bulk, one can pull on
all bonds symmetrically thus involving only the pair term. At the surface, the
symmetry is broken and if one pulls only on the neighbors, a net force would

result, moving the system as a whole. A solution can be to add a counter force on

CHAPTER 6. SURFACE OF SEMICONDUCTORS 115

the central atom to compensate and keep the total force zero. Since full symmetry
is lost, there could also be a contribution of the three—body term to the energy.
Happily, this term vanishes to first order with a Kirkwood potential and the

relation between the force and alloying described here is recovered.

6.3 Triangular lattice

The first system looked at is the triangular lattice. Although it does not exist in
nature, it provides a useful test for the theory both because of its simplicity and
of its two dimensionality which allows a simple graphical representation. For this
lattice, I have used the harmonic potential described in chapter 2.

I have done the simulation using slabs with periodic boundary conditions
along x and surfaces perpendicular to y. The relaxation was performed using a
conjugate-gradient method presented in Numerical Recipes (see Appendix I). In
the triangular network, there are two important directions, (10) and (01). Figure
6.1 (a) and (b) shows these two surfaces in a disordered state. The (10) surface
has a low coordination number, three nearest-neighbors, while the sub-layer has
five. This very low coordination leads to an unstable surface. On the other hand,
the (11) surface, which is closer the (111) surface of semiconductors and of most
interest, is more stable due to its higher coordination: the (11) surface has a
fourfold coordination and all the sub-layers are perfectly coordinated.

In this chapter, I have used the experimental definition of the topological
rigidity

a._<LA>1-<’LB >1
1" 0 O
104-LB

 

(6.28)

This definition has the merit of being simple to implement and remains valid

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 116

        
   
  
     
 
 

(11)

   

 

111111111111111111111111 11111 W111111111111111111111111111111111111111111111 1
“"1111 111111111111111111111111111111111111MIAIIIIIIIIIII“WM111111111111111111111
1111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111IMill11111111111111!(11111111111111!)
1111111111111111111111111111111111111111111111111111111111111111”!1111111111111111111111111111
1111111111111111111111111111)!1111111111111111]MW111111111111111111111111(11111111111111
1111111111111111111111111111111111W111111111111111111111'111'

 
 

        

       
     

 

(10)

   
     
 

     

     

111111111

11111) 1111 ’111’

1111 111111111

1111111111111
1111111111

1111
111111

1

 

1
1
11

 

1111111
111111

 

1

 

/\/"\A

\1’\ (“N

R

w
\/W

1111111

 

  
   

 

 

 

 

 

 

 

 

 

 

 

 

</\[‘\/’\]

 

 

 

 

 

 

 

  

 

1 1‘ WV!

Figure 6.1: Triangular network cut along (a) (10) and (b) (11) direction. Mis-
matched bonds have been distributed at random with a. 50—50 % probability.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 117

in almost any geometry. Figure 6.2 shows the value of a; as a function of the
layer index I in a 100 x 100 slab, averaged over 100 configurations. As expected,
the surface presents a much floppier relaxation than the bulk; from a‘ = 2/ 3, in
the bulk, it reaches 0.82 for the first interlayer and 0.80 for the top intralayer
along the (11) surface. So, we obtain an enhancement of the topological rigidity
of about 25%. The difference in topological rigidity between the first inter and
intralayer can be understood in the following manner. Inside the top layer, bonds
are confined along 2: because of the periodic boundary conditions. Note that the
problem would be the same for an infinite slab. However, interlayer bonds have
an important component along y, which is almost free of topological constraint
close to the surface. It is therefore easier for a bond in the first interlayer to keep
its natural length than for an intralayer bond.

Besides the topological constant, it is important to look at the fluctuations
in height at the surface. Since we are interested in probing the surface itself,
such a quantity should provide us with useful information. However, the position
fluctuations diverge in two dimension as was first demonstrated by Landau (1937)
and Peierls (1934) in the 19303. We can see the effect of fluctuations in the large
averaging needed to obtain an accurate value for the topological rigidity. This
peculiarity brings many new questions to the surface of a 2D configuration. I will
not elaborate more on this problem in this chapter, delaying the discussion until

the next chapter which is strictly concerned with 2D phenomena.

6.4 Silicon-Germanium alloy

I have also studied the surface relaxation for SiGe alloys using the Kirkwood

potential described in chapter 2. As shown previously, the use of different elastic

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 118

 

 

1: T T r T I I I I I I I I I I fi I I I I I I L
)- -1
0.9 — AzoBao 0 Inter—layer -‘
1 0 Intra—layer 1
P d
orsL— -—
L .1
0.7 -- _
- 1
0.6 _ _.1
1. .1
.- 1 l l l l I l 1 1 l l l l 1 l I l l l l I l-l

 

 

 

o 50 100 150 , zoo
Layer number

Figure 6.2: Variation of the topological rigidity parameter a‘ as a function of
depth in a triangular slab cut along (11) direction. The results are an average
over 100 configurations of a 200 x 200 sites cell with a 2% mismatch.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 119

constants for the three different interactions (Si-Si, Ge—Ge and Si-Ge) leads to
almost identical results since they differ by only 20%. To remain closer to the
analytical calculations, the same elastic constants are used for all interactions.

I used periodic boundary conditions along :1: and y, leaving the two sur-
faces perpendicular to z. The two simplest directions are the (100) and (111)
surfaces. As for (10) in the triangular network, the atoms at the surface (100)
have few neighbors (only two), they are relatively unstable and hence, reconstruct
easily. One method to inhibit surface reconstruction is to satisfy the surface dan-
gling bonds with hydrogen. With only two bonds attached to the bulk, the surface
Si atoms are etched away during this process. The atoms at the surface (111),
on the other hand, have three neighbors and form a relatively stable structure
(Schulze, 1983; Ancilotto, 1992; Itaya, 1992; and Steinmetz, 1990). The hydro-
genation process leaves a perfect surface with no reconstruction and should allow
measurement that can compare with the theory and simulations presented here .
All the results can be understood in terms of the formalism developed in section
6.1

The relaxation process is much faster in 3D than in 2D. Without aver-
aging over many configurations, the a“ value of the bulk shows only very small
oscillations. Because of that, the averaging has been made over 20 configurations
and the céll contained 31,250 atoms (25 x 25 x 25 unit cells). The relaxation has
been done using a conjugate-gradient method.

As predicted by the theory, the average bond length on each layer < L >1
remains constant throughout the slab < L >1E< L >. Because the subsurface
atom is tied to the bulk by only one bond, we expect, and indeed get, a” to be

very floppy at the surface and even more in the subsurface layer. Figure 6.3 shows

CHAPTER 6. SURFACE OF SEMICONDUCTORS 120

the variation of a“ as one moves closer to the surface. For the first two interlayers,
a“ is about 17% larger than in the bulk. The surface effect decreases very fast,
the third interlayer already gives a a" value about 1% only larger than the bulk
value; even at a few layers under the surface, it becomes difficult to distinguish the
environment from that of a deep bulk. Figure 6.4 presents the topological rigidity
as a function of the distance from a (100) surface. Since the coordination at the
surface is lower than for (111), the variation of a“ is greater by about 27%. The
same results are obtained for a“ using the local forces method instead of alloying,
as discussed in section 6.2.

Table 6.1 shows the value of Au along different directions and for two
concentrations 15—85 and 50-50. As predicted by Equation (6.22), the partial
displacement vanishs in the bulk and are non-zero only for the layers close to the
surface, along the z direction. As predicted from the symmetry of the system,
Au vanishes along a: and y. The difference between the average height of each
species is independent of the concentration which is very similar to the difference
< L33 >1 - < L,” >1 is also concentration independent.

The height at the surface is described by Equation (6.23) where the radius
of species is also included. From Table 6.1, and considering a mismatch of 0.09A
for SiGe, the separation between the centers of height distribution at the surface
should be" around 0.085A. However, it would be possible to determine locally
the species only if the partial distributions were completely separated. Figure 6.5
shows the density distribution at the surface; it also presents the partial distri-
butions. As one can see, these partial distributions are gaussian like (Figure 6.6)
where the width of the distribution follows more or less a". The overlap between

the peaks is too large to permit an atom by atom distinction between species at

 

61*dt

CHAPTER 6. SURFACE OF SEMICONDUCTORS 121

 

 

 

 

0.90 I I I I I I I I I I I I 7 I I I T I 1
_ 1 1 r I 1 I I 1 d
P 1
CL85r—' __
C 1 1- :
01K)_- .—
" 1
(175r—- _d
b 4
1 :
I 1
(170"—' __
I 1
0.65 l l l l 1 l l l l l l J l L 1 l l 1 l l I l l l l I l

 

 

 

O 10 20 30 40 50
Layer number

Figure 6.3: Topolo 'cal rigidity parameter a" as a function of the layer number
from a surface (111 in SiGe alloy, averaged over 5 configurations.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 122

 

0.95 I I I I I 7 I If I VI I I I I I T I I T I
I l I l I

IIII

—1
—1
q
.4

0.90

  

0.85

0.80

838*

0.75

 

0.70

IIITTIIIITIIIIIIIIIIIIIII

llJlIlLilIllLlIllll

 

 

l l L I 1 I 1 l l l I l l l l I l l l l I l l l l I
o 10 20 30 4o 50

Layer number

I

 

0.65

Figure 6.4: Topological rigidity parameter a“ as a function of the layer number
from a surface (100) in SiGe alloy.

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 123

Table 6.1: Partial displacements and their difference perpendicular (z) and parallel
(:r and y) to the surface at various depths for two concentrations of SiGe alloys.

Slsoceso Sl15G€35

 

Surface
<u,> -0.0010 -0.0014

< u, > 0.0017 -0.0011
1
< u. > 00434 0.0446

 

 

< uz > -0.0011 -0.0018

 

the surface.

Although the floppiness is isotropic in the bulk, it is not at the surface, as
we could expect (Figure 6.7 and 6.8). On the top layer, the width of displacement
along 2:, y, and z is almost the same, with 2 being only slightly larger. However,
on the sub-surface layer, the difference is very important. The values of the width
of distribution along a: and y is bulk—like while along 2, it retains almost the
surface value. The effect of the layer directly over it does not seem to have any
noticeable effect on the z relaxation. It is easy to understand this phenomenon
directly from the geometry of the layers. The bond between the second and third
layers is initially only in the z direction, because there is not much push from the
top layers, it relaxes almost fully by disturbing the second layer which is almost

free, not the third one which has all the bulk behind it.

As seen in the theory section, all the quantities described previously are

 

2 Distribution

CHAPTER 6. SURFACE OF SEMICONDUCTORS 124

 

 

 

 

.- I I I I I I I I I I I I I I Iirfi I T T I I I I I fiI .1
25 '— —1
2° F SlozoGeoso ‘1
1. I
1' I \ ‘
15 "" {/i '—
C ' I
" _I- .1
lo 1— d
1- " d
b ’ d
5 L 1' .2
I ‘ " \ -' I
.A .

C , , \ .

0 an! -' L 1 I T P - 1
-O. 10 -0.05 0.00 0.05 0.10

Distance (Angstroems)

Figure 6.5: Height distribution at the surface (111) of SlgoGego (solid line). The
dashed and dot—dashed lines are the Si and Ge height distributions respectively.
The curves are smoothened results from a 35, 152 atom cell.

 

2 Distribution

CHAPTER 6. SURFACE OF SEMICONDUCTORS 125

 

IIITTIIIIIIIIIIIrTIIIIIIIII
D d

25
20 Sl0.20Geoso

15

10

1114I1L11I1L11I11LIIA‘111l

IIIIIITf‘I’IIII1IIIIIIIIII

 

 

 

l

 

-0.05 0.00 0.05 0.10
Distance (Angstroems)

-O.10

Figure 6.6: Gaussian fit (solid line) for partial z displacement distribution in
SigoGego (a). The dashed and dot—dashed lines are the smoothed Si and Ge
distributions respectively, the solid lines are two gaussians with identical width.

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 126

Z Distribution (Ge only)

125

100

75

50

25

 

TIIIIIIIIIIIIIIIIIIIIIII

- .1
-0.10

 

 

 

 

I I I I I I I I I I I I I I I I I I I I I I I I I I I 1
Sl0.20G9000 .1 .1
I. I :
\Vl' \ l :
1 i. 1 I '3
l . ‘ .
'1“! 1 z
I 1 -—
I '1 ‘ I
I l 1 2‘
I I .II 1 _1
J ' 1. 1 I
I v - 1 ..
' 1.‘ ‘- .'. ‘
/ ‘. \.\.. _
I 71.1 ‘I I_ I
’I ." ‘. 1 1
1 I’ll. 1 1 L l 1 r 1 I‘I.’\l A l 1'1 1 d
-0.05 0.00 0.05 0.10

Distance (Angstrom)

Figure 6.7: Si displacement distribution along 2 at the surface (111), the sub—
surface and the center of a slab of SlgoGeso (solid, dash and dot-dash lines). The
curves are averaged over 20 configurations from a 16, 000 atom cell.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 127

X Distribution (Si only)

 

 

b I I I I I I I I I I I I I I I I I I I I I I I I I FI ‘
150 T SiozoGeoso 1 ‘i
_ .‘ii'. i

1- " -l

- l1 .

l

I— ‘ I .1
100— 9 1 —
- l ' .1

' I: (‘1 ‘

.. I - 1 .1

u— 5' I -I

.'Ii '.

50 — l-l ‘. l ""
- AI .

_ Ii ‘41 .

- ,1! , k .

.l |‘\
01 ..1.. Our....1..,.\h 1-.|...l

 

 

-O.10 -0.05 0.00 0.05 0.10
Distance (Angstrom)

-.

Figure 6.8: Ge displacement distribution along :2: at the surface (111), the sub—
surface and the center of a slab of SlzoGego (solid, dash and dot-dash lines). The
curves are averaged over 20 configurations from a. 16, 000 atom cell.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 128

function of the mismatch and the concentration of species. These relations have
been spelled out in section 6.1 and provide and easy check of the simulation. I
have therefore performed relaxation at two different concentrations (1: = 0.85 and
0.50 in Si1-,,.Ge,) and verified that all quantities behaved as they should. From
the results presented in this section, the called z—plot is indeed respected for any
layer; the topological rigidity parameter a)“, although a function of layer number,

remains unchanged when varying the concentration of species.

6.5 Conclusion

Although reconstruction is often present at the surface, it is interesting to be able
to develop a phenomenological theory that would allow some understanding of the
relaxation phenomenon at the surface. I have presented such a theory here and
applied it the 2D triangular lattice and the SiGe (111) and (100) surfaces. These
predictions could be verified (1) by measuring the distances between the centers
of atoms at the surface or (2) directly, using some kind of scanning microscope
(see Figure 6.9). Recent developments in scanning force microscopy give some
hope to see a precision along the vertical displacement good enough to see the
distributions described here (Meyer and Heinzelmann, 1992; Meyer, 1993).

The study of surfaces is also important because in emphasizes the effects of
mismatch. All the distributions are enhanced and therefore easier to see, the strain
is reduced at the cost of increasing the distortions in the crystalline lattice. As we
could see, the concept of topological rigidity allows us to describe quantitatively
the floppiness of the network as we approach the surface. In both the 2D and
3D cases, the effects of surface on the relaxation die out very fast in the bulk,

after only a couple layers. I have also shown that it is possible to calculate the

 

CHAPTER 6. SURFACE OF SEMICONDUCTORS 129

 

Figure 6.9: Simulated density plot of the surface (111) of SisoGe5o alloy. The
darker peaks are Si atoms and the lighter, Ge.

CHAPTER 6. SURFACE OF SEMICONDUCTORS 130

topological constant locally simply by applying a force on a few atoms. This
method is much simpler that trying to obtain a‘ or a" directly from the Green’s
functions, particularly in non—symmetric configurations.

Although a“ and the displacement distribution are not directly related,
the same theory applies at the surface and in the volume. If the surface results are
in agreement with the theory then one can hope that the theory is also valid in
the bulk. Therefore, a measurement of the surface relaxation can give important

clues about the volume strain in the limits of this theory.

 

Chapter 7

Two-dimensional melting

The concept of bond and site mismatch can also be useful in two dimensions
where the stability of phases is very sensitive to disorder. Unlike surfaces, which
are quasi two— dimensional systems, these systems are restricted to motion in the
plane only, leading to unique behavior. This peculiar behavior of phases in two-
dimensional systems has been predicted a long time ago, independently, by Peierls
(1934) and Landau (1937) who showed that a long—range crystal cannot persist
at any finite temperature. More recently, Mermin (1966) was able to show that
only the position-position correlation would decay at any finite temperature while
the angular correlation would remain long—range. A few years later, Kosterlitz
and Thoul_ess (1973) introduced a theory of melting particular to two-dimensional
systems that would be driven by defects (vortices in superconductors, point defects
in crystals).' This theory was refined by Young (1979) and Nelson and Halperin
(1978) in the case of elasticity. The KTHN Y theory of two-dimensional melting
predicted a third phase, included between the solid and liquid phases, that was
called hexatic. This phase has been the subject of a large number of studies since

1979, but, to this date, it has not been found in any perfectly two—dimensional

131

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 132

system.

After reviewing the KTHN Y theory of melting, I will present an applica-
tion of the size—mismatch theory to the two-dimensional case. I will then describe
the details of the simulation and present the results. Almost all the theoreti-
cal and experimental study of the two—dimensional melting has concentrated on
temperature as the driving force. However, the temperature is fairly difficult to
control with a great accuracy close to a phase transition. It is possible to define in
size—mismatch systems, a quantity that is strictly equivalent for the description of
effects of disorder, to the temperature. By using size mismatch, one can therefore
drive the phase transition from a different perspective, offering complementary
results to the temperature driven transition. Also, since with size mismatch a
simple static relaxation is enough to reach equilibrium, it is possible to use much
a larger system size which is always needed when one wants approach as close as

possible a phase transition.

7 .1 Review

The proof of the crystalline instability in two dimensions is very simple and can
be obtained by various ways. I will be using here the Landau theory of phase
transitiorLand the general discussion he gives in his book on statistical physics
(Landau and Lifshitz, 1980).

Defining u(z, y) as the displacement from equilibrium r(z, y), the proba-

bility of any thermal fluctuation is given by

w o: exp(—AF/kBT), (7.1)

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 133
where
AF = [(F — Pym (7.2)

is the deviation of the total free energy from its average value and F is the free
energy per unit of area. Taking the harmonic approximation for interactions in

the solid, the free energy becomes

AF:

wlllv—

V2 uikulkAtlmnkm kn, (7.3)
ijk

where /\ is a real tensor, and is a function of the direction of the vector k and ng
is the first derivative with respect to 1:,- of the momentum space representation of

the displacement
u = Zuke‘l”. (7.4)
I:
To obtain the mean square value of the displacement, we use the usual

statistical technique

2mm} uikulk C’AFM/kfl
Z{U(I’)} c-AFTUUICBT

 

< u,ku1k >= (7.5)

where the sum is over all possible displacement fields u(r). Solving for this equa-

tion, we obtain

< u u > (z) A (7 6)
1k ‘1‘ A km kn 3
and I
_ < uikulk. >= 0 (7.7)
for k’ 75 —k.

The mean values are found by summing over the k’s. The mean square

displacement vector is therefore given by, changing the discrete sum into an inte-

gral:

 

< |u|2 > = T f A” (n) (d: , (7.8)

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 134

T 2w l/ddk
z (2702/0 A”(¢)d¢/o T'

If one performs the integral over Is, one finds that it diverges logarith-

 

mically as k -> 0 or, in the real space, as the distance goes to infinity. The
divergence of the mean square displacement implies that, on average, the atoms
on a heated lattice find themselves at an infinite distance (logarithmically speak-
ing) from the point they were at, at zero temperature. For any fluctuation greater
than W 3 a, the ideal lattice spacing, it becomes impossible to assign a
posteriori a crystal point to an atom, and therefore the long—range periodicity is
lost. Thus, in two dimensions, thermal fluctuations always destroy the long—range
periodicity of a lattice, for any non-zero temperature.

The lack of long—range periodicity does not preclude all long—range order.
As noted by Landau in 1937, “[i]f the body is isotropic, then p = const; however,
from p = const it does not follow that the body should necessarily be isotropic”
(from Brock, 1992). For example, the angle between the local crystallographic

axes and the axes of the ideal lattice (Mermin, 1968)
1 1
0(1),!” = 5(63111, - agar), (7'9)

where u(x,y) represents the displacement field vector introduces in its Fourier

transform a factor I: more than in the displacement field itself:

1 .
0(z, y) = 5 2(Ik,u,,k — ikyu,,k)e‘k'“. (7.10)
I:

Therefore, the thermal average becomes

:r
(4r)2

 

21v . l/d
/. f.(¢)f.(¢)A..((n))d¢/o kdk. (7.11)

<02 >=

with f,(¢) = cos(¢) and fy(¢) = sin(¢). In this thermal average of the angular

fluctuations, all divergence is lost and the quantity < 02 > is well defined. Because

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 135

Table 7.1: The three phases of a two—dimensional solid as predicted by the
KTHNY theory with their associated correlation and defects.

Solid Hexatic Liquid

 

 

Ca( R) Algebraic Exponential Exponential

decay decay decay

 

C 9(00) Constant Algebraic Exponential

decay decay

 

Defects Pairs of Pairs of Disclinations

dislocations disclinations

 

 

 

 

all the fluctuations do not diverge at d -—1 0, this phase is called quasi—long-range
order.

A few years later, Kosterlitz and Thouless (1973) proposed a model for
melting in two dimensions. This model was applied in multiple contexts from
superconductor-normal state to solid-liquid transition. In the latter case, it pre-
dicted that the melting would be driven by the separation of dilute dislocation
pairs. Later and independently, Nelson and Halperin (1979) and Young (1979) dis-
covered that another transition could be predicted from the Kosterlitz—Thouless
theory if one would let the dislocations break into pairs of disclinations. The new
phase was introduced between the solid and liquid phase and was called the hex-
atic phase in reference to liquid crystals where this type of oriented phase happens
frequently. Table 7.1 presents the different phases as well as the kind of defect

and correlation associated with them.

In two dimensions the translational order Cg(R), defined as

00(3) =< PG(R)PG(0) > (7-12)

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 136

with

pea?) = e‘G'lR+“<R”. (7.13)

can be rewritten as

00(12): R-"Gm (7.14)

for T < TM, using the logarithmically divergent < 712 > from Equation (7.9).

And 170(T) is function of the Lamé elastic constants of the solid (p and A):

3713 + An
4WR(2#R + AR).

 

7200") = kBTlGl2 (7-15)

So the translational order decays algebraicallyto zero at large distance in the solid
phase. The only defects present are isolated pairs of dislocations with opposite
Burger’s vector (see Figure 7.1). These defects affect the lattice only very locally;
they do not perturb the general properties of the solid and so they only renormalize
the Lamé constants. The deformation due to these pairs decays as R” and they
can therefore be seen a the elastic equivalent of electric dipoles.

As one raises the temperature to T > Tm, the pairs of dislocations begin
to unbind, creating a new phase similar to what is found in liquid crystals. Above
Tm, the finite number of free dislocation leads to an exponential decay of the

translational order,

CAR) ~ eXp(-R§+(T)) (7-16)

where the correlation length 5+ diverges as T approaches Tm from above

5+ ... 617%.? (7.17)
with
17 = 0.36963... (7.18)

 

 

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 137

However, if one looks at the orientational order, one finds that the correlation

decays only algebraically with distance
< ¢‘(R)1b(0) >~ Rm”) (7.19)

where

17(3) = e6“) (7.20)

and 0 is given by Equation (7.10). The 6 appears because of the local coordination
on a triangular lattice; for a square lattice, it would be 4. Although in this phase
the translational order is completely lost, there is still quasi-long range order for
the orientation of the solid. Since there is still some angular correlation in this
phase region, Nelson and Halperin (1979) dubbed it hexatic, in reference to the
typical names given for phases in liquid crystals.

Until now, the disclinations, a missing or surplus row of atoms on a lattice,
have not been considered since they are very unfavorable in view of their high
energy cost (see Figure 7.1). As free dislocations appear on the lattice, they offer
a screening for the disclinations, producing a weak logarithmic binding. Since
dislocations can be seen as a bound disclination pair, it becomes possible to apply
again the KT theory to study the unbinding transition of disclinations. Above
the second transition temperature T1, both the translational and the orientational
order decay exponentially, signature of a liquid phase.

After the KTHNY theory of two-dimensional melting was proposed, a
large amount of theoretical and experimental studies was performed in order to
find the hexatic phase. However, the search forsignatures of two different second-
order phase transition has been unsuccessful until now. For a more detailed review
of the work done in this field, the article by Strandburg (1988) remains appropriate

since few major articles have appeared since then. On the experimental side, the

CHAPTER 7. TWO-DIMENSIONAL MELTING 138

 

(bl

 

Figure 7.1: Different types defects on the two-dimensional triangular lattice. (a)
dislocation (pair of disclinations); (b) disclinations (Brock et al., 1989). The vector
labeled bin (a) indicates the Burger’s vector associated with the dislocation.

CHAPTER 7. TWO-DIMENSIONAL MELTING 139

hexatic phase has been found only in liquid crystal where one can argue about
the dimensionality of even mono—layered films; it is difficult to consider these long
molecules as a perfect two-dimensional system. There are interactions along these
molecules that can distribute the strain along the third dimension also. Some
work has also been performed with magnetic bubbles and colloidal suspensions
(see Murray, 1992, for a review of this work) but the phase transition happens at
a much higher density of defects than predicted by the KTHN Y theory. From the
theoretical side, the efforts have gone on two fronts. The first one, analytical, was
the development of first-order transition theory for the two-dimensional melting,
analogous to what is found in three dimensions. One of the major problems
with the KTHNY theory is that the density of defects must be low enough so
that onelcan neglect the interactions between them. In most simulations, the
number of defects present at the phase transition is large enough so that it is not
clear if this approximation misses important physical interactions. Chui (1983)
and Kleinert (1983) have proposed another mechanism for melting that do not
support the presence of an hexatic phase. However, there are also problems with
the different approximations present in the later theories. The second theoretical
efl'ort was directed to trying to simulate directly the two—dimensional melting but
this project turned out to be much more difficult that predicted. Many groups
have tried' to obtain a phase diagram for two—dimensional melting using different

configurations and potentials but nobody has yet seen the hexatic phase.

7 .2 Theory

The problems with computer simulations are numerous: finite-size, sensitivity to

boundary conditions and difficulty in handling the long—range fluctuations close

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 140

to the phase transition. One of the solutions is to use size mismatch to drive
the transition. Thorpe and Cai (1991) have shown that it is possible to map the
mismatch to the temperature in two-dimensional harmonic solids. The efl'ective

temperature was found to be, for bond mismatch,
kBTD = K1:(1-— x)(L% — L9,)2, (7.21)

and

1 .
ICBTD = 51(230 — (CHI/$3 — [491,02 (7.22)

for site mismatch, when L913 = l/2(L?M + L998). In the rest of this chapter, I put
k3 = 1 and K = 1.

The relation 7.22 is found by comparing the equations for the correlation
as a function of size mismatch with the one found in the KTHNY theory. In the

case of translational order, the position—position correlation is

 

(70(3) = 91);? (7.23)
where
‘2 o 0 2
n= 377.?“ -..) [cuss—L10] . (724)

Which is equal to Equation (7.15) for kBT described by Equation (7.22) and the
Lamé constant A = p = x/3K/4.

Si—milarly, by using similar derivations as those used in Thorpe and Cai
(1989) for the bond mismatch, one finds that the angular correlation in the site

mismatch case are

09(00) = 5‘“, ' (7.25)

where

 

o _ o 2
W6 = 2.466x(1 — :13) [(LB: L :AA)] . (7.26)

CHAPTER 7. TWO-DIMENSIONAL MELTING 141

These results are valid only in the solid phase and are not directly exten-
sible to predict a phase transition since the derivation assumes that the network is
perfectly coordinated. The inclusion of defects would call for an effective—medium
theory similar to the KTHNY derivation and therefore lead to identical results.
However, there are fundamental differences between temperature and mismatch
disorder in the localization of the second compared with the former. Nevertheless,
following the exact results described earlier, we can propose a phase diagram in
the temperature-mismatch plane (see Figure 7.2). It would he therefore possible
to pass continuously from the pure mismatch disorder to the pure temperature
disorder. There is of course a difference between these two disorders: the equi-
librium phase. In mismatch alloys, the random configuration is never the lowest
energy one. Even if the potential favors equally phase separation and ordering,
the equilibrium phase would be a mixture of phase separated and ordered regions.
In spite of this difference, such alloys exist in real life because the time scale for
reaching the lowest state is too long, this is why the analogy proposed here can

be regarded as exact.

7 .3 Details of simulations

For simulations, it is important to have a finite-range potential, to allow for the
formation of defects, i.e. variable number of neighbors, which is not included
in the harmonic theory deve10ped in the preceding section. However, I had to
keep a quadratic term around the minimum of the potential so that it would be

possible to compare the results in the solid phase with the analytical theory. I

 

 

' CHAPTER 7. TWO-DIMENSIONAL MELTING 142

 

 

Mismatch

 

 

 

 

 

——

Figure 7.2: Proposed phase diagram of the two—dimensional melting in the
mismatch-temperature plane.

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 143

had therefore constructed a pseudo-harmonic potential composed of four parts:

12
(f) + 61 for r < r1 (7.27)
V(r) = 11’2(r — reg)? + £2 for r1 < r < r2
11'3(r - r3)? for r < r3 (7.28)

0 for r > r3,

where K2, r1, req, 7'; and r3 are variables and

 

K, = 19M (7.29)
7‘2 _' 7‘3
62 = [1,203 — reg)? — 113(7) — 1‘3)2
r 6
0' = [l\2rf3(rcq — rl)]1/ (7.30)
6
12
61 = [(203 - 7‘2)2 - £2 — (1)
1'1 -

The shape of this potential is shown in Figure 7.3 and its derivative, in
Figure 7.4. Typically, I chose the same depth for all interactions, insuring that
there would be no tendencies to cluster or phase separate. Also, the AB length is

always taken to be
1
2

I have also performed simulations using a truncated Lennard-Jones po-

L913 = (L313 + Lisa) ‘ (7-31)

tential (truncated between the 3rd and 4th neighbor) in order to verify which

results are general and which ones depend on the nature of the potential. The

functional form of the potential is

a 12 a 6
V(1) _ e K?) (7) J , (7.32)
where I defined the 6 related to the partial lengths AA, AB and BB such that

the depth of the potential is constant:

__ 07213
6A3 — 6AA '7- (733)
0AA

 

V(R)

CHAPTER 7. TWO-DIMENSIONAL MELTING

144

 

 

 

 

Figure 7.3: Pseudo—harmonic potential.

 

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 145

and

2
a

688 = 6,1,4 +38- (7.34)
”M

By using the geometric average for the AB ideal bond length,

(7‘3” + T033) < 7‘23 (7-35)

NIH

0 _
rAB"

with

7.0 7.0
7“ =fi—M (7.3.6)
”‘3 1/r9é. +293.

where rfw is the mixed length needed for no favoring clustering or phase separa-
tion. In the present situation, clustering would be favored at T # 0.

The minimization procedure was completed using the conjugate gradient
algorithm described in Appendix A. The simulation presented in this chapter were
performed using large cells of 10,000 sites with randomly distributed species A and
B. In most cases, the concentration was 50-50 or 25—75.

Finally, it is no longer possible to obtain the neighbor simply from a list,
as was done in the previous chapters. I have used the standard Voronoi algorithm
to find, uniquely, the first neighbor shell. Although not perfect for very distorted
lattices, it is the most effective and stable method for defining clearly the local

environment.

7 .4 Results

The first step was to verify the application of the harmonic theory to the pseudo-
harmonic potential in the solid phase region and to verify the validity of the
effective temperature as the main disorder parameter. Figure 7.5 also presents

results at different concentrations but with the same TD. Figure 7.5 also shows

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING

146

 

F(R)

 

 

.—

Figure 7.4: First derivative of the pseudo—harmonic potential.

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 147

that the pseudo— harmonic potential agrees well with the theory until the first
defects develop. Again, the results are in excellent agreement with the theory.
This is not the case with the Lennard-Jones potential which is highly anharmonic
even at very low mismatch although the general behavior is consistent. As the
effective temperature increases, the Lennard—Jones system departs strongly from
the harmonic theory well before the first structural defect appears (see Figure
7.6). The abrupt change of slope at large R in Figures 7.5 and 7.6 is only due to
the finite size of the lattice.

I have also tested the validity of the parameter TD after the creation
of defects. For a harmonic potential, this parameter is strictly valid; however,
nothing guaranties that the relation holds when the first defects appear.

As the number of defects created increases, a phase transition occurs and
the nature of the correlation functions changes. The translational order decays
exponentially while the angular correlation decays algebraically. One can see the
latter results in Figure 7.7 which shows the value of angular correlation coefficient
at R -» 00 as well as the number of defects as a function of TD. It indicates
that around TD = Tm = 0.005 a transition occurs. However, this transition is
not the one predicted by the KTHNY theory. For TD < Tm, we find essentially
no defects in the solid: in a 100 x 100 lattice, they begin to appear only as we
reach Tm.” Contrary to the prediction of the KTHNY theory, there are no pairs
of Burger’s vectors in the lattice at low mismatch. For To > Tm, we assist at
the creation of a large number of defects. However, almost none of them unbinds
into pairs of dislocations. Only by going to very large mismatch do we begin to
see the unbinding predicted by Kosterlitz and Thouless; however, the density of

defects becomes large enough such that the mean-field non-interactive KTHNY

CHAPTER 7. TWO-DIMENSIONAL MELTING 148

 

1.00 ‘ jlflnl l '—

 

0.95

c.<R)

0.90

 

FIFIITIIIIIIII

0.85

 

 

1....—
N
U1
...—a
O
N
C

Figure 7.5; Variation of the position-position correlation as a function of distance
at two effective temperatures. The solid curves are from the theory, the open
squares are for a 50-50 concentration using the seudo—harmonic potential on
a 100 x 100 site lattice; the crosses are for a per ectly harmonic potential on a
200 x 200 site lattice; and the closed circles are for a 25-75 concentration with
pseudo-harmonic potential on a 100 x 100 lattice.

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 149

 

1.0 .. .

 

 

 

Figure 7.6: Variation of the position- osition correlation as a function of distance
at two effective temperatures using t e truncated Lennard-Jones potential. The
solid curves are from the theory and correspond to the effective temperature at
which the systems where relaxed. The full symbols are for a 50-50 concentration
on a 80 x 80 site lattice.

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 150

 

   

spagaq 10’ 0110;)

 

 

1.0, I ' ' 1.0
C
0.8:— \ '1 0.8
p \O a
’8‘ 0.65- \ -3 0.6
‘3 : ‘o . ' :
. \ u
U 0.4:- . —; 0.4
: h 9 :
0.2} , x -i 0.2
sow'lwdfll'blu 0.0

 

0.00 0.01 0.02 0.03 0.04
Effective Temperature TD

Figure 7.7: Angular correlations (circles) and number of defects (squares) as a

function of TD using the pseudo—harmonic potential. The solid line is from the
theory, the dash line is a guide to the eye, the open symbols are from a 50 - 50
alloy while the closed ones are from a 25 - 75 alloy.

theory can hardly be thought to apply. Here again, the lattice relaxed using
a truncated Lennard-Jones potential departs from the harmonic theory for the
angular correlation at very low effective temperature and it becomes therefore
difficult to point to a phase transition.

When looking at the total configurational energy for the pseudo—harmonic
potential as function of temperature (Figure 7.9), we find an inflexion point at
To = Tm, indicating a second-order phase transition at this point. Even by

going to very large temperature, up to 4Tm, when it is almost impossible to find

 

CHAPTER 7. TWO-DIMENSIONAL MELTING

1.0

0.8

0.6

Ca(°°)

0.4

0.2

00

0.00 0.01 0.02 0.03 0.04

 

   

IIIIIIIIIIIITIIII—IIIIII
O

lllllllJllLLJlllllll

 

 

l l l l 1 l l l l 1 14 I l J l 1

1

Effective Temperature TD

1.0

0.8

0.6

0.4

0.2

0.0

151

spajaq JO ouog)

Figure 7.8: Angular correlation (circles) and number of defects (squares) as a
function of TD using the truncated Lennard-Jones potential. The solid line is
from the theory and the symbols are from a 50 - 50 alloy.

‘1

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 152

 

“180000 IIIIi IIIfIIIIIIrITr‘IIIIfi‘I’IITr

lLLl

- 182500

- 185000

Energy
0

- 187500

9

O

- 190000

 

 

[Mr1111ti‘lrfr11tillrrfti
" L l
llllllLLll 1111 11111

-192500 Ill; lilflllllllfllLJlJlLllLLl

0.000 0.005 0.010 0.015 0.020 0.025 0.030
To

 

Figure 7.9: Configurational energy as a function of effective temperature To for
a lattice relaxed using the pseudo-harmonic potential.

a relatively stable minimum, there is no sign of a second phase transition as
predicted by the KTHNY theory. Since the depth is not the same for the three
interactions with the truncated Lennard-Jones potential, the total configurational
energy does not indicate so clearly the presence of a phase transition. These
effects are convoluted with the difference in the potential depths between the
species. However, it is clear, from directly looking at the cell, that there is no
phase transition at a low concentration of defects corresponding to the KTHN Y
solid-hexatic transition.

Another characterization can be performed by measuring the elastic con-

, “h

CHAPTER 7. TWO-DIMENSIONAL MELTING 153

stants of the system. For a triangular network with nearest-neighbor interactions,
only one elastic constant defines the full macroscopic elasticity. However, as the
number of defects increases, the lattice loses its symmetry and the need for more
elastic constants appears. Since the shear modulus vanishes in the liquid phase
while the bulk modulus remains finite, the choice of these two elastic constants
seems a good one: monitoring the variation of the elastic constants with effec-
tive temperature should provide a reliable test of phase transition. Since the
shear modulus would also vanish in the hexatic phase, another elastic constant
is needed. The most obvious one is the splay rigidity (Figure 7.10) which, in the
solid phase is simply

S = C44 = §Cu. (7.37)

(Clio

As we go to the hexatic phase, the orientational order is still present so it costs
energy to apply a splay deformation to the system. In the liquid, any deformation
that preserves the volume costs no energy so the splay rigidity should vanish at the
hexatic-liquid transition. As a signature of the behavior of the elastic constants
for the different phase of the KTHN Y theory of melting is presented in Figure
7.11. Note that the measurement of the bulk modulus does not really bring any
information on the status of the system but does provide a useful scale against
which to check the two other elastic constants.

It. turns out that the measurement of these quantities is not as simple as it
may seem. As soon as the defects appear, the lattice becomes extremely unstable
against even very small deformations. Some defects appear and then disappear,
creating jumps in the total energy and us preventing from obtaining a significant
value for the elastic constants. Although straightforward and easy to interpret in

theory, the actual computation of elastic constants turns out to be a real problem

CHAPTER 7. TWO-DIMENSIONAL MELTING 154

 

 

Figure 7.10: Splay rigidit . The arrow indicates the direction of the strain field
applied to measure the sp ay rigidity.

CHAPTER 7. TWO-DIMENSIONAL MELTING 155

 

Solid Hexatic ; Liquid

 

 

Elastic constants

 

 

 

 

Figure 7.11: Possible phase transitions for the KTHNY theory in the elastic
constants-temperature space.

and more work is needed to find ways to stabilize the system enough to be able to
obtain reliable results. The problem does not seem to be the choice of potential
since these difficulties are found using a Lennard-Jones potential. It comes more

probably from the strain on the lattice at these high TD.

7 .5 DiscussiOn

From all the different tests and simulations performed on lattices statically re-

laxed using a pseudo—harmonic or Lennard—Jones potential, one can draw two

 

 

CHAPTER 7. TWO-DIMENSIONAL MELTING 156

conclusions: (1) The harmonic theory extends very well to the pseudo—harmonic
potential for T < Tm and only at very low mismatch for the Lennard—Jones
potential. (2) There is no sign of a hexatic phase or any phase transition 2‘). la
Kosterlitz-Thouless. The first point emphasizes the validity of the TD parameter
to study and understand the properties of the lattice until the first defects appear.
This one—to—one correspondence between size—mismatch and temperature disor-
der allows one to compare two very different methods of driving a phase transition.
The temperature disorder creates a globally equilibrated system while the size—
mismatch disorder is only locally equilibrated. Waiting long enough, at a non zero
temperature, we would see the atoms segregating in ordered or phase-separated
alloys. However, as with glasses, on normal time scales, the two dimensional size
mismatched alloys can be considered as equilibrated. As the mismatch increases
and defects are created, the effective temperature is no longer a meaningful quan-
tity. Once defects appear, the local stress as well as the number of contact points
between the two species of the alloy will determine the overall properties. For
example, by looking at Figure 7.12, which indicates the defects and the bond type
on a lattice with 40 % mismatch, one sees that there is a tendency for the pairs of
dislocation to form in a diamond configuration with alternate species on the cor-
ners. It is therefore natural that To defined previously no longer gives an accurate
picture ofthe situation. In other terms, TD is valid only when harmonic terms
dominate. The creation of defects, being non-harmonic, opens other avenues for
decreasing the stress that are not accounted for in the previous theory.

The second point is very important in the search for the elusive hexatic
phase. It shows that static systems will not organize themselves in a hexatic

phase. Any local strain (due to the local and frozen nature of mismatched alloys

 

157

CHAPTER 7. TWO-DIMENSIONAL MELTING

.nflhfifirorfionfimimwhvuhfik6......
2.....saauntaaae

......an 4 4 1 «>4... a... :34
”a...afianfi».$.§asses...
‘h...¢bfi.‘§§‘b~’4><>40xD§4.00W£V"‘>"§‘
bfiflflu REV wqfihv >4>4>fq>4>4>§1>1m
<><><>4uwav:bafihioedwhhafiv.
Fi’flb‘fi‘» rawfirabfifigsol
.."5‘4DAVMPQEV4>§1myflbfib§0hzufi
......
4 p
esteemefiaewmmacaw»...
I 4 4 .. '
Wo>o>0bweofingumamwpavardih?
easesufisreuw? ...«04»<>§«. .«
Aug-5.! AvasédV A. \ {iv p4>4>1sllxea
“V‘COQQVAHV‘V‘VdafllV ... .
pmwpahcsmwpavahrfcba . a a « <><>4><s< 4

B bonds; a open square indicates an atom

11 atom with five—fold coordination.

Figure 7.12: Section of a 50—50 alloy with 40 % mismatch relaxed using the
pseudo—harmonic potential. The solid line represents A — A bonds, the dash one,

A — B bonds, and the dotted line, B —
its 7 first neighbors and a full circle, a

CHAPTER 7. TWO-DIMENSIONAL MELTING 158

by opposition with the non-local nature of temperature disordered systems) will
preclude the separation of Burger’s vectors. Only when a very high density of
defects screens considerably the interaction do some dislocation pairs unbind.
In the case of pseudo—harmonic potential, one can almost fill up the lattice with
defects before single disclinations appear. This effect could be due to the choice of
the potential parameters, too deep or too shallow. However, it would not explain
the second— order phase transition indicated by the configurational energy when
the first defects appear, at Tm = 0.0005. This transition is totally different from
the expected KTHN Y one. The new phase seems extremely unstable and efforts
to calculate the elastic constants have shown that a very small perturbation can
create or destroy hundred of defects. One could believe that the problem lies
in the potential. However, although many results differ between the pseudo—
harmonic and the Lennard—Jones potential, there are two constants: (1) Even
at very high mismatch, single disclinations do not appear; and (2) the density
of defects at which some unbinding happens turns out to be very high and a
theory ignoring interactions between pairs of dislocations cannot be valid. Figure
7.13 represents a lattice relaxed with Lennard-Jones potential, even at very large

defect concentration, one finds only very few single disclinations.

*7 .6 Conclusion

Although the search for hexatic phase comes in empty—handed once again, the
study of mismatch—driven phase transitions gives us a better understanding of
the limits of the harmonic approximation and of the KTHNY theory. Because
static relaxation is more straightforward than Monte—Carlo or molecular dynam-

ics, it is possible to look at much bigger samples. The correspondence between

 

 

159

CHAPTER 7. TWO-DIMENSIONAL MELTING

...... u. u
. .. ... .. .... .
. . . . ... . .. .
. . .. . . . .
. . . . . .
. . .. .-
.. ..
....
. . .. .
.
. . . . .
o. .-

.. .. .....flfia

anew/sf

z.

._ .. . . .. ... is

hr

.....K1
1\.. \ \

mu

.....
0 .........

...x. .Q.

       

  

./ ./.
. . .a...a..... .‘IH... . ..H...I
...a....flmu...w.w. ....

.01..

.6 ix swims?

the

gfibors

Figure 7.13: Section of a 50—50 alloy with 26 % mismatch relaxed usin
Lennard—Jones potential. A open square indicates an atom with 7 first nei

and a full circle, an atom with five—fold coordination.

CHAPTER 7. TWO-DIMENSIONAL MELTING ' 160

temperature and mismatch is exact at low TD, allowing a meaningful comparison
between both. However, this relation breaks down somewhat when the first de-
fects appear. Also, lattices become unstable at large mismatch and create chains
of pairs of dislocations that eventually fill up almost all the lattice without evi-
dence of a second phase transition. Since the details of the phase transition in the
system presented here differ dramatically from what is predicted by the KTHN Y
theory, it is possible to conclude that the KTHN Y transition is not always favored

in 2D and, at least, does not reflect the behavior of mismatched systems.

 

Chapter 8

Conclusion

The knowledge of the structure of a solid is primordial to a satisfying under-
standing of its electronic properties. Too often, physicists make very accurate
calculations based on a poor model. In this thesis, I have tried to improve the
understanding of the structural properties of relatively weakly disordered systems:
binary solid solutions. Even with this low disorder, the structure turns out to be
very different from what was predicted using perturbation theory; instead of a
large distribution around the linear combination of the properties of each species,
we find multiple peaks centered close to the individual natural lengths.

In the case of metallic alloys, the simple Hooke’s spring approach pursued
here does_ not always give precise results; it is often necessary to account for
the changes in the electronic environment that affect the quality of the chemical
bonds. Nevertheless, the qualitative predictions of the harmonic theory remain
valid: small rigidity parameter, three—peak distributions, etc. It is to be noted
also that the more complex embedded—atom potential fails as often as the spring
model to replicate the average length of the alloys.

Using the Cai and Thorpe (1992a) theory for semiconductor alloys, 1

161

 

 

CHAPTER 8. CONCLUSION 162

have calculated the structural properties one would expect for crystalline and
amorphous SiGe solid solutions. I have shown that the effect of size mismatch on
amorphous semiconductor is roughly the same as that predicted for a crystalline
one. In the former case, it is possible to treat size mismatch as a perturbation
compared to the large disorder introduced in the process of amorphisation. How-
ever, it turns out that the predictions for crystalline and amorphous SiGe alloys
are in disagreement with recent experimental results. Because the topological
rigidity parameter, which determines all the properties of the disordered network,
depends only on the connectivity of the perfect lattice, it is difficult to believe that
the predictions are wrong qualitatively. Some more experimental work remains to
be done on this problem in order to obtain a definitive answer.

I have also extended the formalism to the surface. Although it is not
possible to obtain analytically the value for a", the general ideas developed pre-
viously still hold. At the surface, the difference between the average partial lengths
is larger than in the bulk, due to the lower coordination. Having predicted the
height distribution as it should be seen from the surface, we can only wait to
see how this prediction will compare with experiment. A consistent result at the
surface would probably indicate that the bulk predictions are equally valid. It is
obvious that in the treatment of the surface problem, I have not considered any
reconstruction. First, due to the generality of the principle, one would expect to
obtain a qualitatively similar result. Second, by covering the surface (111) with
hydrogen or by heating it up to 600 C, it is possible to deconstruct completely
the surface and to obtain a structure similar to what I have studied here. I have
also showed the equivalence between the topological rigidity calculated globally

and locally. This equivalence will be very helpful in understanding the rigidity of

 

 

CHAPTER 8. CONCLUSION 163

non-symmetric systems.

Finally, I have attempted to look at the solid-hexatic mismatch-driven
phase transition. At low mismatch, a perfect correspondence between temperature
and mismatch can be found that will predict the appropriate structural correla-
tions. However, I could not find any trace of the hexatic phase or Kosterlitz-
Thouless type of phase transition. The problem of hexatic phase has been around
for almost 15 years but it remains highly controversial. However, we have seen
that it is not universal in the size-mismatch problem. Moreover, at low mismatch
(strictly equivalent to low temperature), there is no sign of the bound dislocations
predicted by the KTHN Y theory, even in systems much larger than studied previ-
ously. As the first defects appear, a transition seems to occur but with the density
of defects increasing very rapidly there is not even the appearance of phase tran-
sition or global unbinding of defects to be seen. With the creation of defects, the
correspondence between mismatch and temperature is weakened leaving a direct
extrapolation of the results between both kind of disorder somewhat less direct.
It remains, nevertheless, that the results speak not for a KTHNY-type phase
transition but for a more conventional one that remains to be fully characterized.

A lot of questions have been raised in this thesis. I believe that they
will be answered in a near future. Many of the answers lie on the experimental
side and one can only hope for the development of better analysis for EXAFS
measurements. This way, it would be possible to settle the questions about the
predictions for SiGe random alloys. It would also complement happily the series of
numerical results obtained for the metallic alloys particularly regarding the strange
behavior observed in the AuAg alloy. Some more theoretical work is still needed

to understand why the EAM and CFM potentials seem to be so complementary

 

CHAPTER 8. CONCLUSION 164

in their predictions. The study of surfaces is still in its infancy and so I am
confident that we will soon have some experimental results to compare with the
many predictions presented here. Some research would be useful to characterize
quantitatively the effect of size—mismatch on reconstruction. It may be possible to
prevent reconstruction by using large enough impurities at the surface. Although
2D melting has been heavily studied during the last decade, I believe we have
here a solid proof that the hexatic phase is not universal. However, more work is
needed in order to develop efficient tools to characterize the phase transition. The
choice of elastic constants seems very appropriate but many problems remain to
be solved before one can use them, in a computer simulation, as a good indicator
of the structural phases. I have no doubts that many of these questions will be

addressed in the next few years.

Appendix A

Conjugate—gradient method

The problem of finding the global minimum of a multidimensional surface is gen-
erally impossible to solve. However, there exists multiple techniques that allow
one to find a local minimum. In the case discussed in the main text, with relatively
weak disorder and fixed coordination, we can be almost certain that the minimum
found by a straightforward and linear method will lead to the global minimum. In
view of this insurance, the conjugate—gradient method (CGM) was used in every
case. The following discussion is based on the one given in Numerical Recipes
(Press et al., 1986).

The first step in the minimization is to convert the multidimensional prob-
lem to a set of line minimizations, i.e. that we try to find a scalar A that will
minimize the energy E (x + Ay). The problem is now to choose appropriately the
vectors x and y. The CGM is based on the fact that the gradient of a function

miminized along a vector x will be perpendicular to it at the minimum.

To see how this method works, let’s start by a Taylor expansion around

165

 

APPENDIX A. CONJUGATE—GRADIENT METHOD 166
some point x0:

E(x): E()x0 )+Z:—' :3le 33+" 22??2 (If-:17; fllrox$1+ " (AJ)

By keeping only up to the quadratic term, we get
1
E(x)2c+b-x+§x-A-x. (A.2)
Using this latter equation to obtain the gradient of the total energy,
VE=A-x+b. (A.3)

By posing VE = 0, one finds a set of linear‘equations to solve by inverting the
matrix A. However, this method would require the knowledge and the storage of
a matrix N x N. The CGM demands only the total energy and its gradient, an
order N in information.

So, let’s take two sets of vectors g,- and h,- with go = ho and a recursion
defined as
gm = g. - AIA ° hi (AA)

11i+1 = Si+1+7£hi1

where orthogonalisation for g and conjugation for h are required:

51.1.1 ° 3; = 11.4.1 ' A ° I1; = 0. (A.5)
This latter requirement defines the two scalars

A,- = 3%in (A.6)

and
“=8i+1'A'hi
7' h,-A-h,~.

 

(A.7)

 

APPENDIX A. CONJUGATE—GRADIENT METHOD 167

Moreover, these relations also imply (as can be proven by induction) that for all

 

i 961'
gg°gj=hg'A'hj=0. (A.8)
It is also possible to rewrite equations A.6 and A.7 as
_ = Si+1 'Si+1 = (gi-H " 8i) '8i+1 (A 9)
g.- - g.- g. - g.- ’
and
g.- ' hi
A,- = —. .
h.‘ . A . h.‘ (A 10)

Although the last two terms in equation A9 are equal when the function is exactly
quadratic, most of the time, the total energy used is not and therefore the orthog-
onalisation is not perfect. The first term on the right is named after Fletcher and
Reeves while the second one is called the Polak-Ribiere method. I have used the
latter.

The interest of the CGM is that it does not require the knowledge of the

Hessian matrix A. Suppose that one has
g,- = —VE(x,-) = —A - x,- -+- b. (A.ll)

one also sets g,-+1 = —VE(x,+1) where X1.“ is the new minimum of the total
energy obtained by minimizing the energy along a vector h,-.

Since

5+1 = A ' (X; '1' Ah.) + b (A.12)

= Si - Ahi.
at the minimum, obtained by varying A,

ll; ° VE = -h.' ° 31.1.1 = 0. (A.l3)

 

APPENDIX A. CONJUGATE-GRADIENT METHOD 168

Using this condition and equation A.10, A13 is the same as A5. It is therefore
possible to find the minimum of the total energy without ever evaluating the

Hessian matrix A.

... .‘

 

Appendix B

Johnson’s long—range potential

As I mentioned in chapter 3, the gold—silver alloy was treated a bit differently
from the other alloys. Instead of using a first-neighbor interaction, I have used a
long-range one, going up to the thirteenth neighbor, developed also by Johnson
(1990). It is essentially an extension of the embedded—atom potential presented
in chapter 3.

The basic equations are the same as the one given in chapter 3. The

energy is a sum of a repulsive pair potential and an attractive electron-density

term
E: Z: 45(7‘0') +ZF(’°‘)’ (B-1)
- P.- = Effie). (3-2)
#1
with
1/3 a: 1 16 1
W) = E.{1+..[(%e) ’1l}6a[(p)/_] (3.3)

_¢ (3.)“ _ 1(a) ,
Pe ,6 Pe
The parameter 0 depends on the core energy EC, the bulk modulus B, and the

169

APPENDIX B. JOHNSON’S LONG—RANGE POTENTIAL 170

atomic volume (I, a = 1/9128/ EC. The exponents fl and 7 are given by

 

fl:

 

J 159013.135,
(Elf; + EC)El1/,

 

1590(13,’ + E.)
= . 13.5
7 \J E(quc ( )

where G is the shear modulus.

 

Moreover, in this potential the electron—density function and the pair
potential are given by

N) = f. [1]” (3.6)

and
rle

0(7‘) = .1. H (8.7)

where the parameters fe and 03., are scaled a weighted sum of the atoms in the

first n neighbor shells:

 

Ec _ 71 Ni

fe — 5309 SH " E; k?) (8'8)
Be It Ni

3. _ 23:, S, _ g kg, (13.9)

where N,- is the number of atoms in the ith shell and k,- is the distance a the ith

shell to the central atom in units of r1, r,- = 16,-r1.

 

 

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