\chapter{Molecular dynamics simulations}

The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures.
Both, quantum-mechanical and classical potential molecular dynamics simulations are performed.
While quantum-mechanical calculations are restricted to a few hundreds of atoms only small volumes composed of three unit cells in each direction and small carbon concentrations are simulated using the VASP code.
Thus, investigations are restricted to the diffusion process of single carbon interstitials and the agglomeration of a few dumbbell interstitials in silicon.
Using classical potentials volume sizes up to 31 unit cells in each direction and high carbon concentrations are realizable.
Simulations targeting the formation of silicon carbide precipitates are, thus, attempted in classical potential calculations only.

\section{Ab initio MD simulations}

Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.

\section{Classical potential MD simulations}

\subsection{Initial simulations}

In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized.
Periodic boundary conditions in each direction are applied.
The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved by ion beam synthesis (IBS).
After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted.
The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by
\begin{equation}
 N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3}
                  =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3
\label{eq:md:spheric_prec}
\end{equation}
with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC.
A total amount of 6000 carbon atoms corresponds to a radius of approximately 3 nm, which is discovered to be the minimal size for precipitates in IBS experiments.
In separated simulations these 6000 carbon atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell.
For reasons of simplification these regions are rectangularly shaped.
$V_1$ is chosen to be the total simulation volume.
$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate.
$V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si.
For rectangularly shaped precipitates with side length $L$ the amount of carbon atoms in 3C-SiC and silicon atoms in c-Si is given by
\begin{equation}
 N_{\text{Carbon}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
\label{eq:md:quadratic_prec}
\end{equation}
and 
\begin{equation}
 N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
\label{eq:md:quadratic_prec2}
\end{equation}
Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by equations \eqref{eq:md:quadratic_prec} and \eqref{eq:md:quadratic_prec2} and the resulting carbon concentrations inside these volumes.
\begin{table}
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
 & $V_1$ & $V_2$ & $V_3$ \\
\hline
Side length [\AA] & 168.3 & 50.0 & 49.0 \\
Carbon concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
\hline
\hline
\end{tabular}
\end{center}
\caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 carbon atoms.}
\label{table:md:ins_vols}
\end{table}
The insertion is realized in a way to keep the system temperature constant.
In each of 600 insertion steps 10 carbon atoms are inserted at random positions within the respective region, which involves an increase in kinetic energy.
Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy.
Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures.
Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is  $20\, ^{\circ}\mathrm{C}$ is reached.
Figure \ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
\begin{figure}
\begin{center}
\begin{pspicture}(0,0)(15,17)

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 \rput[lt](3.2,16.8){\color{gray}INITIALIZIATION}
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                                    unit cells of c-Si}}}
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                                    $p_{\text{s}}=0\text{ bar}$}}}
 \rput(7,14){\rnode{12}{\psframebox{Thermal initialization}}}
 \rput(7,13){\rnode{11}{\psframebox{Continue for 100 fs}}}
 \rput(7,12){\rnode{10}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
                                                    \pm1\,^{\circ}\mathrm{C}$}}}
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 \rput(7,9){\rnode{8}{\psframebox{Continue for 100 fs}}}
 \rput(7,8){\rnode{7}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
                                                   \pm1\,^{\circ}\mathrm{C}$}}}
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 \ncline[]{->}{6}{5}
 \trput*{\scriptsize true}

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 \rput[lt](3.2,4.8){\color{gray}COOLING DOWN}
 \rput(3,4.8){\pnode{CD}}
 \rput(7,4){\rnode{4}{\psframebox{$T_{\text{s}}=T_{\text{s}}-
                                                1\,^{\circ}\mathrm{C}$}}}
 \rput(7,3){\rnode{3}{\psframebox{Continue for 100 fs}}}
 \rput(7,2){\rnode{2}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
                                                  \pm1\,^{\circ}\mathrm{C}$}}}
 \rput(7,1){\rnode{1}{\psframebox{$T_{\text{s}}=20\,^{\circ}\mathrm{C}$}}} 
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\end{center}
\caption{Flowchart of the simulation sequence used in the molecular dnymaics simulations aiming to reproduce the precipitation process.}
\label{fig:md:prec_fc}
\end{figure}

The radial distribution function $g(r)$ for Si-C and C-C distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
\begin{figure}[!ht]
\begin{center}
 \includegraphics[width=8cm]{pc_si-c_c-c_thesis.ps}
\end{center}
\caption{Radial distribution function of the Si-C and C-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$.}
\label{fig:md:pc_si-si_c-c}
\end{figure}


\subsection{Increased temperature simulations}

\subsection{Simulations close to the silicon melting point}

