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\frontmatter
\title{Raman Spectra \& Growth of SiGeC }
\author{\includegraphics[scale=.5,draft=false]{shield.eps} \\ \\ A thesis submitted for the degree of\\
          Doctor of Philosophy of the University of London\\ \\ by \\ \\ 
         Simon Scarle\\  \\ Department of Physics\\ King's College London}
\date{\today}
\maketitle
%this chapter* puts the number right at bottom of page
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To

me Mam 'n' Dad

\noindent{for}

unerring support, money, beer, laughter \& embarrassment.

\chapter{Why?}
\begin{quote}`` Sometimes you get the feeling the gods are trying to tell you  that a particular career is not for 
you. Unfortunately, when Simon Scarle was at school he had a couple of chemistry teachers who didn't seem
to have got the message. One of the teachers received a clear signal when he lost the tip of his finger
while trying to pick up a piece of sodium. Oblivious to the gods' warning, he went on to set fire to himself 
in front of the class, by siting in a pool of acid that had been spilt on a bench. Insensible as ever, he didn't
realise he had a problem until one of his pupils pointed out that smoke was coming from his trousers.

However, he did know that some chemicals were safer in a fume cupboard and one day placed an open bottle
of bromine in one. Unfortunately, the fume cupboard vented straight on to the roof and three dead pigeons
plunged past the window during the lesson.

Despite the obvious dangers of chemistry, Mr Scarle decided to opt for the subject at A level, encouraged
by the fact that the A level teacher had previously been a research chemist. However, he began to suspect 
that this might not guarantee his safety when she stood in front of her new class and recalled some highlights
of her earlier career at a chemicals factory. She described how a report on a new compound simply stated: 
``Compound catastrophically decomposes above approximately  $50\:^\circ{\rm C}$''-- in other words it had
exploded, destroying her lab in the process.

On another occasion she told her eager pupils how she had managed to launch a poisonous gas cloud over
the factory. Mr Scarle decided that the gods might be trying to tell him something too. He is about to embark
on a PhD in physics. ''
\end{quote}

\begin{flushright}extract from Feedback, New Scientist 3 August 1996, issue No2041\end{flushright}

\chapter{Abstract}
\begin{quote}
`` I checked it very thoroughly, and that quite definitely is the answer. I think the problem, to be quite honest with you, is that you've never actually known what the question is.''
\end{quote} 

\begin{flushright}Deep Thought \end{flushright}
\setlength{\parskip}{.5cm}

This  work contains a study of the variations in the local vibrational modes within $\rm{Si}_{1-x-y} \rm{Ge}_x \rm{C}_y$ alloys brought about by changes in the nearby environment. This data was then used to produce Raman spectra.

This work also contains  a Monte-Carlo simulation of the growth of binary and tertiary alloys of silicon, germanium and carbon. 

Through out this a valence force potential has been used. The parameters of which were found by fitting to phonon frequencies of pure materials at symmetry points of reciprocal space, atomic positions taken from a separate local density approximation calculations for given alloy structures, and local vibrational mode data for single substitutional atoms in otherwise locally pure materials.

\setlength{\parskip}{0cm}
\chapter{Acknowledgements}
\begin{quote}
`` If there's anything more important than my ego around, I want it caught and shot now."
\end{quote}

\begin{flushright}Zaphod Beeblebrox \end{flushright}

I would like to thank the following in various orders :

\begin{description}
\item[Elaine Lewis] For loving this daft waistcoat obsessed loony
\item[Karl \& Wahid] For having the peace, tranquility and solitude of their office ruined by me
\item[Simon Owen] For helping with my bouts of home sickness, with large quantities of beer
\item[The Jubilee-uns] Manchester wouldn't have quite been the same without you all
\item[Ted Lowther] For the local density approximation data
\item[Dr A Mainwood, Dr C Creffield \& Prof S Sarkar] My three supervisors in descending order of gratitude
\item[The Hitch-Hiker's Guide to the Galaxy] For the quotes
\item[EPSRC] For funding this research
\item[\LaTeX2e] With which this whole thesis was typeset
\end{description}

\& finally thanks to me, myself and I, without whom this would not have been possible.

\newpage
\addcontentsline{toc}{chapter}{Contents}
\tableofcontents
\newpage
\newpage
\addcontentsline{toc}{chapter}{List of Figures}
\listoffigures

\mainmatter
\setlength{\parskip}{.5cm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Introduction}

\begin{quote}
`` QUOTE GOES HERE ''
\end{quote}

\begin{flushright}Who Said That? \end{flushright} 

\section{The Elements in Question}

Silicon chips have become a basic fact of life. Each year, these chips are expected to get faster. But as the circuitry is scaled down to achieve these greater speeds, it becomes clear that the needed characteristics eventually break down when you try to reach beyond a certain length scale. Instead of making things go faster by making them smaller, another tactic would be to alter the properties of the material in the device to gain this additional performance.

In the last few years, various alloys of the group IV elements (C, Si, Ge and Sn) have been investigated for use in  hetrojunction devices compatible with current Si integrated circuits. There has been especial interest in SiGe, indeed some high speed SiGe chips have already gone into production by IBM.

\begin{figure}
\begin{center}
\includegraphics[scale=1,draft=false]{sigwafer.eps}
\end{center}
\caption[SiGe wafer]{A SiGe wafer \copyright IBM}
\end{figure}

The existence of germanium had been predicted in 1871 by Russian chemist Dmitry Mendeleyev, who called the hypothetical element ekasilicon. But it was fifteen years before it was discovered by Clemens Winkler, while he was analysing the mineral argyrodite at the Freiberg School of Mining. Shortly after 1945 germanium's properties as a semiconductor were found, and germanium remained the major semiconductor building material until the early 1960s, when silicon replaced it. It therefore seems ironic that the use of SiGe could improve on the useful properties of silicon, but virtually every major telecommunications company is now working on SiGe technology. 

There has also been a lot of interest in SiGe containing Carbon, this is because assuming Vegard's Law [check] a ratio of about 9:1 Germanium to Carbon should lattice match SiGeC to Silicon, getting around critical thickness limitations inherent in SiGe technology. Including carbon in SiGe, means that we can use both alloy concentration and strain as variables for altering the properties of the material. The problem being that carbon is relatively insoluble in both Silicon and Germanium, whilst SiGe is a completely miscible system. Further more formation of various kinds of SiC are thermodynamically favoured at high growth temperatures. 

Carbon is also a technologically important impurity in silicon due to it being found at concentrations as
high as $10^{18} \rm{cm}^{-3}$ in Czochralski grown samples, some of the defects thus formed
have been said to exhibit metastability \cite{song} \cite{zhan}.

\section{Structure of this Thesis}

[Do ref's here automatically not put in]

Chapter \ref{raman} contains work to produce the Raman Spectra for various alloys of SiGeC, taking into account how local environment affects local vibrational modes [LVM]. In Chapter \ref{grochap} there are further results from simulating the growth of these alloys.

Whilst the Appendix tells of how the parameters of the potential were arrived at.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The Raman Spectrum} \label{raman}

\begin{quote}
`` WHAT SAID''
\end{quote}

\begin{flushright}Who Said That? \end{flushright}

\section{Background Theory}

\section{Calculating a Phonon Dispersion Curve}

[Description of Phonon Dispersion Curve]

To calculate the phonon dispersion curve for a given lattice \cite{ziman}, we must first define the 
lattice displacements: $\mathbf{u}_{s \mathbf{l}}$ is the vector giving the displacement
of the $s$th atom, in the $\mathbf{l}$th unit cell from its equilibrium position. Then if it's
mass is $M_s$, the kinetic energy of the whole system is given by
\begin{equation} \label{ek}
E_k=\sum_{s \mathbf{l}} \frac{1}{2} M_s | \dot{u}_{s \mathbf{l}} |^2 .
\end{equation}

We are assuming here a lattice with a basis, the label $s$ runs through the one or more atoms in the 
unit cell, and the $\mathbf{l}$ vectors form a regular array which positions the unit cells.

If we take the potential energy of the crystal to be of the form
\begin{equation}
E_p= V(\mathbf{u}_{s \mathbf{l}}),
\end{equation}
so that the energy of the whole crystal depends at any one moment on the displacements, and assume
that $V$ is at a minimum when all $\mathbf{u}_{s \mathbf{l}}$ are zero. If we then produce a multi-variable
Maclaurian's series for $V$, using the Cartesian components $u^j_{s \mathbf{l}}$ of  $\mathbf{u}_{s \mathbf{l}}$, ($j=x,y,z$), to second order we get
\begin{equation} \label{v}
V=V_0 + \sum_{s \mathbf{l} j} u^j_{s \mathbf{l}} \left [ \frac{\partial V}{\partial u^j_{s \mathbf{l}}}  \right ]_0
+ \frac{1}{2} \sum_{ss', \mathbf{l} \mathbf{l}', jj'} u^j_{s \mathbf{l}} u^{j'}_{s' \mathbf{l}'}
\left [ \frac{\partial^2 V}{\partial u^j_{s \mathbf{l}} \partial  u^{j'}_{s' \mathbf{l}'} }  \right ]_0.
\end{equation}

We can ignore the constant term, $V_0$, and the coefficients of the linear terms must be zero as we
are near equilibrium. This leaves us with the quadratic term.

From equations (\ref{ek}) and (\ref{v}) and using the techniques of Lagrangian classical mechanics, we get the
equations of motion for the displacement components
\begin{equation}
M_s  \ddot{u}^j_{s \mathbf{l}} = - \sum_{s' \mathbf{l}' j'} \left [ \frac{\partial^2 V}
{\partial u^j_{s \mathbf{l}} \partial  u^{j'}_{s' \mathbf{l}'} }  \right ]_0 u^{j'}_{s' \mathbf{l}'}.
\end{equation}

This represents a huge quantity of coupled equations, namely three for every atom in the
crystal, but we can simplify things by writing this as a Cartesian tensor, thus
\begin{equation}\label{coup}
M_s \ddot{\mathbf{u}}_{s \mathbf{l}}=- \sum_{s' \mathbf{l}'} \mathbf{G}_{s \mathbf{l}; s' \mathbf{l}'} \cdot
\mathbf{u}_{s' \mathbf{l}'},
\end{equation}
where,
\begin{equation}
G^{j j'}_{s \mathbf{l}; s' \mathbf{l}'}=\left [ \frac{\partial^2 V}{\partial u^j_{s \mathbf{l}} \partial 
u^{j'}_{s' \mathbf{l}'} }  \right ]_0 .
\end{equation}

The terms in the summation can be seen as the force acting on the $s$th atom in the $\mathbf{l}$th cell due
to the displacement $\mathbf{u}_{s' \mathbf{l}'}$ of the atom on the $s'$th site of the $\mathbf{l}'$th cell. As
we are dealing with a lattice with a basis, what ever the definition of the forces $\mathbf{G}$ can only depend
on relative positions. That is 
\begin{equation}
\mathbf{G}_{s \mathbf{l}, s' \mathbf{l}} = \mathbf{G}_{s s'}(\mathbf{h}), \; \; \rm{where} \; \; \mathbf{h}=
\mathbf{l}'- \mathbf{l}.
\end{equation}

So the coupled equations (\ref{coup}) become these translationally invariant  ones
\begin{equation} \label{coup2}
M_s \ddot{\mathbf{u}}_{s \mathbf{l}} = - \sum_{s' \mathbf{h}} \mathbf{G}_{ss'}(\mathbf{h}) \cdot
\mathbf{u}_{s,\mathbf{l}+\mathbf{h}}.
\end{equation}
as is necessary by Bloch's theorem. [CHECK THIS] We now assume there is a solution to these equations such
that $\mathbf{u}_{s \mathbf{l}}$ is defined with respect to time for all $\mathbf{l}$, so that there is
a wave-vector $\mathbf{q}$
\begin{equation} \label{bloc}
\mathbf{u}_{s \mathbf{l}}(t)=e^{i \mathbf{q}\cdot\mathbf{l}}\mathbf{u}_{s \mathbf{0}}(t),
\end{equation}
here $\mathbf{u}_{s \mathbf{0}}(t)$ is the displacement of the $s$th atom in the unit cell that has been taken
to be the origin of the lattice vectors $\mathbf{l}$. This states that the $s$th atom of ever unit cell is moving in the
same direction and with the same amplitude, but that the phases varies as we go from cell to cell.

For each solution of the equations of motion there exists a value of $\mathbf{q}$ such that (\ref{bloc})
is satisfied. To find which solution applies to which wave-vector, substitute (\ref{bloc}) in (\ref{coup2}) to
get 
\begin{equation}
M_s \ddot{\mathbf{u}}_{s,\mathbf{0}} e^{i \mathbf{q} \cdot \mathbf{l}} = - \sum_{s' \mathbf{h}}
\mathbf{G}_{ss'} \cdot \mathbf{u}_{s',\mathbf{0}}  e^{i \mathbf{q} \cdot \mathbf{h}} e^{i \mathbf{q} \cdot \mathbf{l}}
\end{equation}
The factors of exp $[i \mathbf{q} \cdot \mathbf{l}]$ can be cancelled, and our origin is arbitrary so we will drop the
$\mathbf{0}$ as well, 
\begin{eqnarray}
M_s \ddot{\mathbf{u}}_s & = & - \sum_{s'} \left [ \sum_{\mathbf{h}} \mathbf{G}_{ss'} (\mathbf{h}) 
e^{i \mathbf{q} \cdot \mathbf{h}}\right ] \cdot \mathbf{u}_{s'} \nonumber\\
& = & - \sum_{s'} \mathbf{G}_{ss'}(\mathbf{q}) \cdot \mathbf{u}_{s'}
\end{eqnarray}
where,
\begin{equation}
\mathbf{G}_{ss'}(\mathbf{q})=\sum_{\mathbf{h}} \mathbf{G}_{ss'} (\mathbf{h})e^{i \mathbf{q} \cdot \mathbf{h}} .
\end{equation}

To produce the phonon dispersion curve we simply follow ordinary vibration theory and solve the following
eigenvalue equation,
\begin{equation}
\sum_{s' j'} \left [ G^{jj'}_{ss'}(\mathbf{q})- \upsilon^2 M_s \delta_{ss'} \delta_{jj'} \right ] u^{j'}_{s'}=0
\end{equation}
varying $\mathbf{q}$ so that we trace out lines within the Brillouin zone. Solving this
eigenvalue system is equivalent to finding the vibrational modes of the lattice.

\section{Calculating The Raman Spectra}

\section{Results}

We cannot talk about how well this work produces the local vibrational modes of single substitutional atoms, nor the Raman frequencies of the various pure materials. This would be a circular argument, as these values were used in the parameter fitting used. [Appendix]

If SiGe does cluster, we expect the Carbon in SiGeC to mostly be in the Silicon. This is because the solubility of Carbon in Germanium is only $10^8-10^{10} \rm{cm}^{-3}$ \cite{scace} whilst Carbon in Silicon is $3.5 \times 10^{17}  \rm{cm}^{-3}$.\cite{bean}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Growth Simulation} \label{grochap}

\begin{quote}
``Humans \& computers share in the stimulating exchange of  \\ ARRRRGGGGGHHHH! ''
\end{quote}

\begin{flushright}Eddie \end{flushright}

\section{Background Theory}

Homoepitaxial growth is well described with the use of lattice models. There are three basic
varieties of lattice models; lattice gas, QQQQQQQ, and solid-on-solid. In this order they each 
represent a greater level of approximation than the last.

\begin{figure}
\begin{center}
\includegraphics[scale=1,draft=false]{max.eps}
\caption{Lattice gas model}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=1,draft=false]{mid.eps}
\caption{QQQQQQQQQ}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\includegraphics[scale=1,draft=false]{min.eps}
\caption{Solid-on-solid model}
\end{center}
\end{figure}

In the lattice gas simulation we allow some atoms to have no nearest neighbours, and these
are taken to be in a gaseous state. Whilst those with neighbours are taken to be in the solid substrate. In solid-on-solid we only allow the solid state, but we do not allow overhangs or vacancies. The lattice is basically represented by a 2-D array of heights. In QQQQQQQQ we have no gaseous state but we do have overhangs and vacancies. In all cases Monte Carlo methods are used to model deposition, evaporation and hopping of atoms between sites.

An intriguing connection can be made between lattice simulations of material growth, and
an extension of cellular automata. This extension is the so called probabilistic cellular
automata, were a certain amount of die throwing is allowed in deciding where new cells
are to be born and old ones die on the grid. Clearly the birth of a cell can be directly equated
with the absorption of an atom, and the death of a cell equated with an atom's desorption,
whilst diffusion on the surface is a combined birth/death event. In this way the two simulations
become identical.

\section{Simulation Technique}

For simulations of hetroepitaxial growth, the way that the atoms are confined to lattice sites
means that the use of lattice models can become less appropriate. Especially in cases where there
is a high lattice mismatch. Of course the ideal model would be one where atoms are allowed to occupy
continuous space. Molecular dynamics [MD] methods do this but at a great computational cost.
The problem being that time scales of MD (order of the period of atomic vibrations) and growth (order of 1
monolayer $\rm{s}^{-1}$) are many orders of magnitude different.

There are Monte Carlo simulations in which atoms follow paths in continuous space which also
suffer from this problem but to a lesser amount. The atoms usually follow highly convoluted orbits
about their equilibrium positions, and only once in a while make an \textsl{interesting} move to a
new equilibrium. It is of course these \textsl{interesting} moves that are more significant

Silicon, germanium and carbon atoms hold a special relationship with each other, they are isovalent and can all form the same crystal structure, namely the diamond lattice. It has been well established that isovalent impurities prefer to dissolve substitutionally in silicon, so the technique used in this work is a more precise variation on the QQQQQ type of lattice simulation, and is loosely based on work by J. Kew \textsl{et al.} \cite{kew}. Each atom is directly linked to one lattice site, but we allow deviation from the perfect lattice positions. We are assuming that overall the lattice will always have a diamond structure, but that the strains and stresses caused by lattice mismatch will produce some form of distortion. Distortion but no lattice defects, however, steps [and screw dislocations CHECK] can be added by use of distorted boundary conditions.

I will also deal mostly with lower temperatures, as it has been observed that SiC systems have higher substitutional to interstitial carbon ratios when growth temperature is reduced \cite{osten}.

The simulation starts with a substrate of (111) layers with continuous boundary conditions and continues until a set number of atoms have been added/removed. We have three basic processes to do this: absorption, desorption and diffusion on the surface.

The position where adatoms stick and diffusing ones move to, is taken to be a local potential
minimum. The energy from the local valence potential is calculated for the atom in question, and its
nearest neighbours. The possible position of the incoming atom is then altered to minimise the value, and then this is used as part of the Boltzmann factor in the Monte Carlo attempt at the process.

[use of binding energy]

Any empty site that has four occupied neighbouring sites is assumed to be inaccessible to adatoms, 
thus allowing for vacancies in the material. Similarly an atom with four occupied neighbouring sites
is assumed to locked in and unable to diffuse or be desorbed.

[energies]

[Monte-Carlo system used]

The equilibrium concentration of carbon atoms in silicon is expected to be,
$5 \times 10^{22} \exp (-D/kT) \rm{cm}^{-3}$, where $5 \times 10^{22}\rm{cm}^{-3}$ is the atomic
density of pure silicon, and $D=1.98 \rm{eV}$ is the energy of substitution per atom \cite{dal pino}.

\section{Results}

Under simulation of homoepitaxial growth a perfect crystal is grown. The minimum of the valence potential is the diamond structure, and with no lattice mismatch nothing else can form.

Interfaces undergo kinetic roughening whenever the adatoms are deposited at a faster rate than their diffusion
on the surface. This leads to atoms not having time to reach their equilibrium positions. Kinetic roughening is
therefore more likely at low temperature due to the activated nature of diffusion. Atom mobility varies directly and greatly with temperature.

Growing kinetically rough surfaces have often been taken as examples of self-affine objects, which posses the behaviour that on \textsl{short} length scales they appear rough, while on measurements at \textsl{larger} scales find the surface flat. This implies anisotropic scale invariance, whereby distances measured normal to the substrate scale as $h \mapsto f^\alpha h$, where $\alpha < 1 $, while lengths measured in the plane are rescaled as $L \mapsto f L$.

From experiment \cite{lanczycki}, the growth of Silicon on a Silicon (111) substrate between the temperatures of $280 -410\;^\circ \rm{C}$, has been observed to proceed mostly by layer-by-layer island growth. Except small portions of the surface are covered in relatively tall pyramid shaped structures, each surrounded by a bare zone suggestive of atoms being drawn onto the feature. As temperature decreases, the number of these increases whilst their size decreases. But even with these lumps the surfaces atoms are still mostly in the 3 most probable exposed layers.

As temperature increases feature size increase and the number of layers exposed at surface goes down.

In \cite{lanczycki}, they reject that the Ehrlich-Schwoebel barrier effect [explain] is responsible for the bald areas around the features. As there is no evidence of the usual `` Wedding cake'' shape, where new islands form on the backs of old ones as atoms cannot cross the lower island boundary, also if this were the case all step structures would be effected and the otherwise flat growth over the rest of the surface is not consistent with this happening.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix

\chapter{The Potential}

\begin{quote}
``Quote''
\end{quote}

\begin{flushright}Who Said That?\end{flushright}

\section{Functional Form}

For all of this work one potential has been used, the so called Valence Force Potential.
This potential assumes that we are close to an ideal diamond lattice structure, as that structure
forms the zero of it's energy scale. Any divergence from this is an increase in energy.

[Potential diagrams][functional form]

[Talk about $\theta$ and $r$'s, and how dividing up parameters.]

A large parameter set has been taken, each of the above terms has one parameter value for
each arrangement of atoms that can be involved, taking symmetries into account.

[IE]

\section{Fitting the Parameters}

The parameters for the pure materials were produced by fitting the following set of equations [ref].
These equations relate the symmetry points of the phonon dispersion curves to the parameters of
the Valence Force Potential, all in the case of a pure material. The starting values for this search were taken from
[REF].

These equations were also used to check that the program for calculating the phonon dispersion
curve was functioning correctly.

EQUATIONS HERE

The above fitting procedure was done using a least squares NAG minimisation routine to the following
experimental data , silicon \cite{dolling}, germanium \cite{brockhouse}, and carbon \cite{warren}.
\begin{center}
\begin{tabular}{|c||c|c|c|}  \hline
symmetry & \multicolumn{3}{c|}{Phonon Frequency $\rm{cm}^{-1}$} \\ \cline{2-4}
point  & Si & Ge & C \\ \hline \hline
$\Gamma_{2\, 5'}$ & $520.5$ & $300 \pm 10 $ & $1332.5 \pm 0.5$ \\ \hline 
$X_1$ & $411$ & $275 \pm 13$ & $1184 \pm 21$ \\
$X_4$ & $463$ & $230 \pm 10$ & $1072 \pm 26$ \\
$X_3$ & $150$ & $81 \pm 5$ & $807 \pm 3$ \\ \hline
$L_3$ & $114$ & $65 \pm 3$ & $552 \pm 16$ \\
$L_1$ & $420$ & $215 \pm 10$ & $1035 \pm 32$ \\
${L'}_3$ & $489$ & $246 \pm 10$ & $1210 \pm 37$ \\
${L'}_2$ & $378$ & $280 \pm 10$ & $1242 \pm 37$ \\ \hline
\multicolumn{4}{c}{Experimental Data from above references}  
\end{tabular}
\end{center}

These use BLANK techniques to measure phonon frequencies.

[Description of high symmetry points] 

The binary material parameters were found by relaxing a given lattice structure, this structure
had been designed by a simple genetic algorithm so as to contain at least one of each these
mini-structures. This was used in a minimisation process using the relaxed
positions from the Local Density Approximation [LDA] data to fit all parameters, except the 
$f^4_{\theta \theta}$ and $f^5_{\theta \theta}$ parameters.

The LDA considers the overall density of the electrons, not the actual wave functions. It is a reduction from the great computation expense of trying to solve Schr{\"o}dinger's equation on the whole ensemble of atoms and their occupying electrons.

Then the mixed material $f_r$ parameter was also used to match the local vibration mode (LVM) frequency of a single of the lighter atom in a background of the heavier, in a separate minimising process. These two stages were repeated until the minimisation failed to reduce further.

The starting values for the parameters were taken as atom pair-wise averages of the pure material parameters. The $f^4_{\theta \theta}$ and $f^5_{\theta \theta}$ parameters were not fitted any further
as in the case of the pure materials these were usually a couple of orders of magnitude smaller than
the others, and they also involve neighbours at a greater distance.

[Description of LVM]

\begin{center}
$\rm{C}^{12}_s$ in Si $605\rm{cm}^{-1}$ \cite{newman} \\
$\rm{C}^{12}_s$ in Ge $531\rm{cm}^{-1}$ \cite{hoffman} \\
$\rm{C}^{13}_s$ in Ge $512\rm{cm}^{-1}$ \cite{hoffman} \\
$\rm{Si}_s$ in Ge $386\rm{cm}^{-1}$ \cite{yang}
\end{center}

LDA data was produced using the same basic structure but with each possible pair of the 3 atoms,
and this fitting procedure was carried out for each pair.

The tertiary material parameters where then found with a fit to a further set of LDA
data for a given structure with all the tertiary mini-structures,
which was also designed using a genetic algorithm.

\section{Parameter Values}

The following are some of the actual parameter values arrived at and used in this work. Namely, these are the parameters for the pure materials and the mixed material $f_r$ parameters. All are in units of eV\AA$^{-2}$.

\subsection{Silicon}
\begin{center}
\parbox[t]{4cm}{
\begin{eqnarray*}
f_r & = & 9.198 \\
f_\theta & = & 0.1955 \\
f_{rr} & = & 0.1692 \\
f_{r \theta} & = & 0.3075 \\
f_{\theta \theta} & = & 0.02130 \\
\end{eqnarray*}}
\parbox[t]{4cm}{
\begin{eqnarray*}
f^\ast_{\theta \theta} & = & 0.2451 \\
f^4_{\theta \theta} & = & 0.02582 \\
f^5_{\theta \theta} & = & 0.03746 \\
f^7_{\theta \theta} & = & -0.02897 \\
\end{eqnarray*}}
\end{center}

\subsection{Germanium}
\begin{center}
\parbox[t]{4cm}{
\begin{eqnarray*}
f_r & = & 7.862 \\
f_\theta & = & 0.1436 \\
f_{rr} & = & 0.1837 \\
f_{r \theta} & = & 0.2208 \\
f_{\theta \theta} & = & -0.009779 \\
\end{eqnarray*}}
\parbox[t]{4cm}{
\begin{eqnarray*}
f^\ast_{\theta \theta} & = & 0.1146 \\
f^4_{\theta \theta} & = & 0.01428 \\
f^5_{\theta \theta} & = & 0.03237 \\
f^7_{\theta \theta} & = & -0.05558
\end{eqnarray*}}
\end{center}

\subsection{Carbon}
\begin{center}
\parbox[t]{4cm}{
\begin{eqnarray*}
f_r & = & 23.88 \\
f_\theta & = & 2.420 \\
f_{rr} & = & 0.4141 \\
f_{r \theta} & = & 1.994 \\
f_{\theta \theta} & = & 0.1189  \\
\end{eqnarray*}}
\parbox[t]{4cm}{
\begin{eqnarray*}
f^\ast_{\theta \theta} & = & 0.6581 \\
f^4_{\theta \theta} & = & -0.01829 \\
f^5_{\theta \theta} & = & 0.04826 \\
f^7_{\theta \theta} & = & -0.02647
\end{eqnarray*}}
\end{center}

\subsection{$f_r$}
\begin{center}
\begin{math}
\hfill f_r( \rm{Si-C} )=7.999 \hfill  \mathnormal f_r( \rm{Ge-C} )=7.999 \hfill \mathnormal f_r( \rm{Ge-Si} )=7.999 \hfill
\end{math}
\end{center}

\newpage
\addcontentsline{toc}{chapter}{Bibliography}
\begin{thebibliography}{99}
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