\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.
Since 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}

No pressure control, since VASP does not support this feature in MD mode.
The time step is set to one fs.
Explain some more parameters that differ from the latter calculations ...

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}

In contrast to the quantum-mechanical MD simulations the developed classical potential MD code is able to do constant pressure simulations using the Berendsen barostat.
The system pressure is set to zero pressure.
Due to promising advantages over the Tersoff potential the bond order potential of Erhard and Albe is used.
A time step of one fs is set.

\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.
In IBS experiments the smallest precipitates observed have radii starting from 2 nm up to 4 nm.
For the initial simulations a total amount of 6000 carbon atoms corresponding to a radius of approximately 3.1 nm is chosen.
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.
The two latter insertion volumes are considered since no diffusion of carbon atoms is expected within the simulated period of time at prevalent temperatures.
{\color{red}Todo: Refere to diffusion simulations and Mattoni paper.}
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
 \text{ and} \quad
 N_{\text{Silicon}}^{\text{c-Si}} =8 \left( \frac{L}{a_{\text{Si}}}\right)^3 \text{ .}
\label{eq:md:n_prec}
\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:n_prec} and the resulting carbon concentrations inside these volumes.
Looking at the carbon concentrations simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of carbon.
\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}[!ht]
\begin{center}
\begin{pspicture}(0,0)(15,17)

 \psframe*[linecolor=hb](3,11.5)(11,17)
 \rput[lt](3.2,16.8){\color{gray}INITIALIZIATION}
 \rput(7,16){\rnode{14}{\psframebox{Create $31\times 31\times 31$
                                    unit cells of c-Si}}}
 \rput(7,15){\rnode{13}{\psframebox{$T_{\text{s}}=450\,^{\circ}\mathrm{C}$,
                                    $p_{\text{s}}=0\text{ bar}$}}}
 \rput(7,14){\rnode{12}{\psframebox{Thermal initialization}}}
 \rput(7,13){\rnode{11}{\psframebox{Continue for 100 fs}}}
 \rput(7,12){\rnode{10}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
                                                    \pm1\,^{\circ}\mathrm{C}$}}}
 \ncline[]{->}{14}{13}
 \ncline[]{->}{13}{12}
 \ncline[]{->}{12}{11}
 \ncline[]{->}{11}{10}
 \ncbar[angle=0]{->}{10}{11}
 \psset{fillcolor=hb}
 \nbput*{\scriptsize false}
 
 \psframe*[linecolor=lbb](3,6.5)(11,11)
 \rput[lt](3.2,10.8){\color{gray}CARBON INSERTION}
 \rput(3,10.8){\pnode{CI}}
 \rput(7,10){\rnode{9}{\psframebox{Insertion of 10 carbon aoms}}}
 \rput(7,9){\rnode{8}{\psframebox{Continue for 100 fs}}}
 \rput(7,8){\rnode{7}{\psframebox{$T_{\text{avg}}=T_{\text{s}}
                                                   \pm1\,^{\circ}\mathrm{C}$}}}
 \rput(7,7){\rnode{6}{\psframebox{$N_{\text{Carbon}}=6000$}}}
 \ncline[]{->}{9}{8}
 \ncline[]{->}{8}{7}
 \ncline[]{->}{7}{6}
 \trput*{\scriptsize true}
 \ncbar[angle=180]{->}{7}{8}
 \psset{fillcolor=lbb}
 \naput*{\scriptsize false}
 \ncbar[angle=0]{->}{6}{9}
 \nbput*{\scriptsize false}
 \ncbar[angle=180]{->}{10}{CI}
 \psset{fillcolor=white}
 \nbput*{\scriptsize true}
  
 \rput(7,5.75){\rnode{5}{\psframebox{Continue for 100 ps}}}
 \ncline[]{->}{6}{5}
 \trput*{\scriptsize true}

 \psframe*[linecolor=lachs](3,0.5)(11,5)
 \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{$T_{\text{s}}=20\,^{\circ}\mathrm{C}$}}} 
 \ncline[]{->}{4}{3}
 \ncline[]{->}{3}{2}
 \ncline[]{->}{2}{1}
 \trput*{\scriptsize true}
 \ncbar[angle=0]{->}{2}{3}
 \psset{fillcolor=lachs}
 \nbput*{\scriptsize false}
 \ncbar[angle=180,arm=1.5]{->}{1}{4}
 \naput*{\scriptsize false}
 \ncbar[angle=180]{->}{5}{CD}
 \trput*{\scriptsize false}

 \rput(7,-0.25){\rnode{0}{\psframebox{End of simulation}}}
 \ncline[]{->}{1}{0}
 \trput*{\scriptsize true}
\end{pspicture}
\end{center}
\caption[Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in molecular dnymaics simulations aiming to reproduce the precipitation process. $T_{\text{s}}$ and $p_{\text{s}}$ are the preset values for the system temperature and pressure. $T_{\text{avg}}$ is the averaged actual system temperature.}
\label{fig:md:prec_fc}
\end{figure}

The radial distribution function $g(r)$ for C-C and Si-Si distances is shown in figure \ref{fig:md:pc_si-si_c-c}.
\begin{figure}[!ht]
\begin{center}
 \includegraphics[width=12cm]{sic_prec_450_si-si_c-c.ps}
\end{center}
\caption[Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature.]{Radial distribution function of the C-C and Si-Si distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature. The bright blue graph shows the Si-Si radial distribution for pure c-Si. The insets show magnified regions of the respective type of bond.}
\label{fig:md:pc_si-si_c-c}
\end{figure}
\begin{figure}[!ht]
\begin{center}
 \includegraphics[width=12cm]{sic_prec_450_energy.ps}
\end{center}
\caption[Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes.]{Total energy per atom as a function of time for the whole simulation sequence and for all three types of insertion volumes. Arrows mark the end of carbon insertion and the start of the cooling process respectively.}
\label{fig:md:energy_450}
\end{figure}
It is easily and instantly visible that there is no significant difference among the two simulations of high carbon concentration.
The first C-C peak appears at about 0.15 nm, which is compareable to the nearest neighbour distance of graphite or diamond.
The number of C-C bonds is much smaller for $V_1$ than for $V_2$ and $V_3$ since carbon atoms are spread over the total simulation volume.
These carbon atoms are assumed to form strong bonds.
This is supported by figure \ref{fig:md:energy_450} displaying the total energy of all three simulations during the whole simulation sequence.
A huge decrease of the total energy during carbon insertion is observed for the simulations with high carbon concentration in contrast to the $V_1$ simulation, which shows a slight increase.
The difference in energy $\Delta$ growing within the carbon insertion process up to a value of roughly 0.06 eV per atom persists unchanged until the end of the simulation.
Here is the problem.
The excess amount of next neighboured strongly bounded C-C bonds in the high concentration simulations make these configurations energetically more favorable compared to the low concentration configuration.
However, in the same way a lot of energy is needed to break these bonds to get out of the local energy minimum advancing towards the global minimum configuration.
Thus, this transformation is very unlikely to happen.
This is in accordance with the constant total energy observed in the continuation step of 100 ps inbetween the end of carbon insertion and the cooling process.
Obviously no energetically favorable relaxation is taking place at a system temperature of $450\,^{\circ}\mathrm{C}$.

The C-C peak at about 0.31 nm perfectly matches the nearest neighbour distance of two carbon atoms in the 3C-SiC lattice.
As can be seen from the inset this peak is also observed for the $V_1$ simulation.
In 3C-SiC the same distance is also expected for nearest neighbour silicon atoms.
The bottom of figure \ref{fig:md:pc_si-si_c-c} shows the radial distribution of Si-Si bonds together with a reference graph for pure c-Si.
Indeed non-zero $g(r)$ values around 0.31 nm are observed while the amount of Si pairs at regular c-Si distances of 0.24 nm and 0.38 nm decreases.
However, no clear peak is observed but the interval of enhanced $g(r)$ values corresponds to the width of the C-C $g(r)$ peak.
In addition the abrupt increase of Si pairs at 0.29 nm can be attributed to the Si-Si cut-off radius of 0.296 nm as used in the present bond order potential.
The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
Without the abrubt increase a maximum around 0.31 nm gets even more conceivable.
For low concentrations of carbon, that is the $V_1$ simulation and early stages of the $V_2$ and $V_3$ simulations, analyses of configurations in which Si-Si distances around 0.3 nm appear and which are identifiable in regions of high disorder, which especially applies for the high concentration simulations, identify the \hkl<1 0 0> C-Si dumbbell to be responsible for stretching the Si-Si next neighbour distance.
This excellently agrees with the calculated value $r(13)$ in table \ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \hkl<1 0 0> C-Si dumbbell configuration.

\begin{figure}[!ht]
\begin{center}
 \includegraphics[width=12cm]{sic_prec_450_si-c.ps}
\end{center}
\caption{Radial distribution function of the Si-C distances for 6000 carbon atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of $450\,^{\circ}\mathrm{C}$ and cooled down to room temperature together with Si-C bonds resulting in a C-Si \hkl<1 0 0> dumbbell configuration.}
\label{fig:md:pc_si-c}
\end{figure}
Figure \ref{fig:md:pc_si-c} displays the Si-C radial distribution function for all three insertion volumes together with the Si-C bonds as observed in a C-Si \hkl<1 0 0> dumbbell configuration.
The first peak observed for all insertion volumes is at approximately 0.186 nm.
This corresponds quite well to the expected next neighbour distance of 0.189 nm for Si and C atoms in 3C-SiC.
By comparing the resulting Si-C bonds of a C-Si \hkl<1 0 0> dumbbell with the C-Si distances of the low concentration simulation it is evident that the resulting structure of the $V_1$ simulation is dominated by this type of defects.
This is not surpsisingly, since the \hkl<1 0 0> dumbbell is found to be the ground-state defect of a C interstitial in c-Si and for the low concentration simulations a carbon interstitial is expected in every fifth silicon unit-cell ...

\subsection{Increased temperature simulations}

It is not only the C-C bonds which seem to be unbreakable.
Also the C-Si pairs, as observed in the low concentration simulations, are stuck.
This can be seen from the horizontal progress of the total energy graph in the continue-step.
Higher time periods or alternatively higher temperatures to speed up the simulation are needed.
{\color{red}Todo: Read again about the accelerated dynamics methods and maybe explain a bit more here.}

\subsection{Simulations at temperatures exceeding the silicon melting point}

LL Cool J is hot as hell!

