\chapter{Silicon carbide precipitation simulations}
\label{chapter:md}

The molecular dynamics (MD) technique is used to gain insight into the behavior of C existing in different concentrations in c-Si on the microscopic level at finite temperatures.
The simulations are restricted to classical potential simulations utilizing the analytical EA bond order potential as described in section~\ref{subsection:interact_pot}.
Parameters are chosen according to the discussion in section~\ref{section:classpotmd}.

At the beginning, simulations are performed, which try to mimic the conditions during IBS.
Results reveal limitations of the employed potential and MD in general.
With reference to the results of the last chapter, a workaround is discussed.
The approach is followed and, finally, results gained by the MD simulations are interpreted drawing special attention to the established controversy concerning precipitation of SiC in Si.

\section{Simulations at temperatures used in IBS}
\label{section:initial_sims}

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 in IBS.
After equilibration of the kinetic and potential energy, C atoms are consecutively inserted.
The number of C atoms $N_{\text{C}}$ necessary to form a spherical precipitate with radius $r$ is given by
\begin{equation}
 N_{\text{C}}=\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 \unit[2]{nm} up to \unit[4]{nm}.
For the initial simulations, a total amount of 6000 C atoms corresponding to a radius of approximately \unit[3.1]{nm} is chosen.
In separated simulations, the 6000 C 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 amount of Si atoms necessary to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower Si density of 3C-SiC compared to c-Si.
The two latter insertion volumes are considered since no diffusion of C atoms is expected within the simulated period of time at prevalent temperatures.
This is due to the overestimated activation energy for the diffusion of a \ci \hkl<1 0 0> DB, as pointed out in section~\ref{subsection:defects:mig_classical}.
For rectangularly shaped precipitates with side length $L$ the amount of C atoms in 3C-SiC and Si atoms in c-Si is given by
\begin{equation}
 N_{\text{C}}^{\text{3C-SiC}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3
 \text{ and} \quad
 N_{\text{Si}}^{\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 C concentrations inside these volumes.
Looking at the C concentrations, simulations can be distinguished in simulations occupying low ($V_1$) and high ($V_2$, $V_3$) concentrations of C.
\begin{table}[tp]
\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 \\
C concentration [$\frac{1}{\text{c-Si unit cell}}$] & 0.20 & 7.68 & 8.16\\
\hline
\hline
\end{tabular}
\end{center}
\caption{Side lengths of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incorporation of 6000 C 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 C atoms are inserted at random positions within the respective region, which involves an increase in potential energy that is partially transformed into kinetic energy.
Thus, the simulation is continued without adding more C atoms until the system temperature is equal to the chosen temperature again.
This is realized by the thermostat, which decouples excessive energy.
Every inserted C atom must exhibit a distance greater or equal to \unit[1.5]{\AA} to neighbored atoms to prevent the occurrence of too high forces.
Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e.\ \unit[20]{$^{\circ}$C}, is reached.
Fig.~\ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
\begin{figure}[tp]
\begin{center}
\begin{pspicture}(0,0)(15,17)

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                                    unit cells of c-Si}}}
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                                    $p_{\text{s}}=0\text{ bar}$}}}
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                                                    \pm1\,^{\circ}\mathrm{C}$}}}
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                                                1\,^{\circ}\mathrm{C}$}}}
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\end{center}
\caption[Flowchart of the simulation sequence used in MD simulations aiming to reproduce the precipitation process.]{Flowchart of the simulation sequence used in MD 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 Fig.~\ref{fig:md:pc_si-si_c-c}.
\begin{figure}[tp]
\begin{center}
 \includegraphics[width=0.7\textwidth]{sic_prec_450_si-si_c-c.ps}
\end{center}
\caption[Radial distribution function of the C-C and Si-Si distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of {\unit[450]{$^{\circ}$C}} and cooled down to room temperature.]{Radial distribution function of the C-C and Si-Si distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of \unit[450]{$^{\circ}$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}[tp]
\begin{center}
 \includegraphics[width=0.7\textwidth]{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 C 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 C concentration.
Thus, in the following, the focus can indeed be directed to low ($V_1$) and high ($V_2$, $V_3$) C concentration simulations.
The first C-C peak appears at about \unit[0.15]{nm}, which is comparable to the nearest neighbor 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 C atoms are spread over the total simulation volume.
On average, there are only 0.2 C atoms per Si unit cell.
These C atoms are assumed to form strong bonds.
This is supported by Fig.~\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 C insertion is observed for the simulations with high C concentration in contrast to the $V_1$ simulation, which shows a slight increase.
The difference in energy $\Delta$ growing within the C insertion process up to a value of roughly \unit[0.06]{eV} per atom persists unchanged until the end of the simulation.
The vast amount of strongly bonded 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, such conformational changes are very unlikely to happen.
This is in accordance with the constant total energy observed in the continuation step of \unit[100]{ps} in between the end of C insertion and the cooling process.
Obviously, no energetically favorable relaxation is taking place at a system temperature of \unit[450]{$^{\circ}$C}.

The C-C peak at about \unit[0.31]{nm} perfectly matches the nearest neighbor distance of two C atoms in the 3C-SiC lattice.
As can be seen from the inset, this peak is also observed for the $V_1$ simulation.
Investigating the corresponding coordinates of the atoms, it turns out that concatenated and differently oriented \ci{} \hkl<1 0 0> DB interstitials constitute configurations yielding separations of C atoms by this distance.
In 3C-SiC, the same distance is also expected for nearest neighbor Si atoms.
The bottom of Fig.~\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 \unit[0.31]{nm} are observed while the amount of Si pairs at regular c-Si distances of \unit[0.24]{nm} and \unit[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 \unit[0.29]{nm} can be attributed to the Si-Si cut-off radius of \unit[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 abrupt increase, a maximum around \unit[0.31]{nm} gets even more conceivable.
Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbor distance for low C concentrations, i.e.\ for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
This excellently agrees with the calculated value $r(13)$ in Table~\ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \ci \hkl<1 0 0> DB configuration.

\begin{figure}[tp]
\begin{center}
 \includegraphics[width=0.7\textwidth]{sic_prec_450_si-c.ps}
\end{center}
\caption[Radial distribution function of the Si-C distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of {\unit[450]{$^{\circ}$C}} and cooled down to room temperature.]{Radial distribution function of the Si-C distances for 6000 C atoms inserted into the three different volumes $V_1$, $V_2$ and $V_3$ at a temperature of \unit[450]{$^{\circ}$C} and cooled down to room temperature. Additionally the resulting Si-C distances of a \ci{} \hkl<1 0 0> DB configuration are given.}
\label{fig:md:pc_si-c}
\end{figure}
Fig.~\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 \ci{} \hkl<1 0 0> DB configuration.
The first peak observed for all insertion volumes is at approximately \unit[0.186]{nm}.
This corresponds quite well to the expected next neighbor distance of \unit[0.189]{nm} for Si and C atoms in 3C-SiC.
By comparing the resulting Si-C bonds of a \ci{} \hkl<1 0 0> DB with the C-Si distances of the low concentration simulation, it is evident that the resulting structure of the $V_1$ simulation is clearly dominated by this type of defect.
This is not surprising, since the \ci{} \hkl<1 0 0> DB is found to be the ground-state defect of a C interstitial in c-Si and, for the low concentration simulations, a C interstitial is expected in every fifth Si unit cell only, thus, excluding defect superposition phenomena.
The peak distance at \unit[0.186]{nm} and the bump at \unit[0.175]{nm} corresponds to the distance $r(3C)$ and $r(1C)$ as listed in Table~\ref{tab:defects:100db_cmp} and visualized in Fig.~\ref{fig:defects:100db_cmp}.
In addition, it can be easily identified that the \ci{} \hkl<1 0 0> DB configuration contributes to the peaks at about \unit[0.335]{nm}, \unit[0.386]{nm}, \unit[0.434]{nm}, \unit[0.469]{nm} and \unit[0.546]{nm} observed in the $V_1$ simulation.
Not only the peak locations but also the peak widths and heights become comprehensible.
The distinct peak at \unit[0.26]{nm}, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.

For high C concentrations, i.e.\ the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
The consequential superposition of these defects and the high amounts of damage generate new displacement arrangements for the C-C as well as for the Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
Short range order indeed is observed, i.e.\ the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
This indicates the formation of an amorphous SiC-like phase.
In fact the resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential~\cite{gao02}.

In both cases, i.e.\ low and high C concentrations, the formation of 3C-SiC fails to appear.
With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
However, sufficient defect agglomeration is not observed.
For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.

\section{Limitations of conventional MD and short range potentials}
\label{section:md:limit}

Results of the last section indicate possible limitations of the MD method regarding the task addressed in this study.
Low C concentration simulations do not reproduce the agglomeration of C$_{\text{i}}$ \hkl<1 0 0> DBs.
High concentration simulations result in the formation of an amorphous SiC-like phase, which is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
Keeping in mind the results 
On closer inspection, however, two reasons for describing this obstacle become evident, which are discussed in the following.

The first reason is a general problem of MD simulations in conjunction with limitations in computer power, which results in a slow and restricted propagation in phase space.
In molecular systems, characteristic motions take place over a wide range of time scales.
Vibrations of the covalent bond take place on the order of \unit[10$^{-14}$]{s}, of which the thermodynamic and kinetic properties are well described by MD simulations.
To avoid discretization errors, the integration time step needs to be chosen smaller than the fastest vibrational frequency in the system.
On the other hand, infrequent processes, such as conformational changes, reorganization processes during film growth, defect diffusion and phase transitions are processes undergoing long-term evolution in the range of microseconds.
This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e.\ the system temperature.
Thus, the average time of a transition from one potential basin to another corresponds to a great deal of vibrational periods, which in turn determine the integration time step.
Hence, time scales covering the necessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
New methods have been developed to bypass the time scale problem.
The most famous approaches are hyperdynamics (HMD)~\cite{voter97,voter97_2}, parallel replica dynamics~\cite{voter98}, temperature accelerated dynamics (TAD)~\cite{sorensen2000} and self-guided dynamics (SGMD)~\cite{wu99}, which accelerate phase space propagation while retaining proper thermodynamic sampling.

In addition to the time scale limitation, problems attributed to the short range potential exist.
The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms~\cite{tang95,mattoni2007}.
This is supported by the overestimated activation energies necessary for C diffusion as investigated in section~\ref{subsection:defects:mig_classical}.
Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
The bonds of these C-Si pairs are also affected by the cut-off artifact preventing C diffusion and agglomeration of the DBs.
This can be seen from the almost horizontal progress of the total energy graph in the continuation step of Fig.~\ref{fig:md:energy_450}, even for the low concentration simulation.
These unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
While cohesive and formational energies are often well described, these effects increase for non-equilibrium structures and dynamics.
However, since valuable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature~\cite{mattoni2007}.
Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC~\cite{tang95} but might be rather ineffective for the challenge inherent to this study.

To conclude the obstacle needed to get passed is twofold.
The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
This obstacle could be referred to as {\em potential enhanced slow phase space propagation}.
Due to this, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.

Instead, the approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock~\cite{sorensen2000}.
The TAD corrections are not applied in coming up simulations.
This is justified by two reasons.
First of all, a compensation of the overestimated bond strengths due to the short range potential is expected.
Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS~\cite{nejim95,lindner01}.
It is therefore expected that the kinetics affecting the 3C-SiC precipitation are not much different at higher temperatures aside from the fact that it is occurring much more faster.
Moreover, the interest of this study is focused on structural evolution of a system far from equilibrium instead of equilibrium properties which rely upon proper phase space sampling.
On the other hand, during implantation, the actual temperature inside the implantation volume is definitely higher than the experimentally determined temperature tapped from the surface of the sample.

\section{Increased temperature simulations}
\label{section:md:inct}

Due to the limitations of short range potentials and conventional MD as discussed above, elevated temperatures are used in the following.
Increased temperatures are expected to compensate the overestimated diffusion barriers.
These are overestimated by a factor of 2.4 to 3.5.
Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460--2260]{$^{\circ}$C}.
Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K})~\cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
The simulation sequence and other parameters except for the system temperature remain unchanged as in section~\ref{section:initial_sims}.
Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.

A simple quality value $Q$ is introduced, which helps to estimate the progress of structural evolution.
In bulk 3C-SiC every C atom has four next neighbored Si atoms and every Si atom four next neighbored C atoms.
The quality could be determined by counting the amount of atoms, which form bonds to four atoms of the other species.
However, the aim of the simulation is to reproduce the formation of a 3C-SiC precipitate embedded in c-Si.
The amount of Si atoms and, thus, the amount of Si atoms remaining in the c-Si diamond lattice is much higher than the amount of inserted C atoms.
Thus, counting the atoms, which exhibit proper coordination, is limited to the C atoms.
The quality value is defined to be
\begin{equation}
Q = \frac{\text{Amount of C atoms with 4 next neighbored Si atoms}}
         {\text{Total amount of C atoms}} \text{ .}
\label{eq:md:qdef}
\end{equation}
By this, bulk 3C-SiC will still result in $Q=1$ and precipitates will also reach values close to one.
However, since the quality value does not account for bond lengths, bond angles, crystallinity or the stacking sequence, high values of $Q$ not necessarily correspond to structures close to 3C-SiC.
Structures that look promising due to high quality values need to be further investigated by other means.

\subsection{Low C concentration simulations}

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{tot_pc_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{tot_ba.ps}
\end{center}
\caption[Si-C radial distribution and evolution of quality $Q$ for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and evolution of quality $Q$ according to equation~\ref{eq:md:qdef} for the low concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}. The gray line shows resulting Si-C bonds in a configuration of \cs{} in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of C insertion and the start of the cooling down step. A fit function according to equation \eqref{eq:md:fit} shows the estimated evolution of quality in the absence of the cooling down sequence.}
\label{fig:md:tot_si-c_q}
\end{figure}
Fig.~\ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding evolution of quality $Q$ as defined above for the low concentration simulation.
The first noticeable and promising change in the Si-C radial distribution is the successive decline of the artificial peak at the Si-C cut-off distance with increasing temperature up to the point of disappearance at temperatures above \degc{1650}.
Obviously, sufficient kinetic energy is provided to affected atoms that are enabled to escape the cut-off region.
Additionally, a more important structural change is observed, which is illustrated in the two shaded areas in Fig.~\ref{fig:md:tot_si-c_q}.
%
In the gray shaded region a decrease of the peak at \unit[0.186]{nm} and of the bump at \distn{0.175} accompanied by an increase of the peak at \distn{0.197} with increasing temperature is visible.
Similarly, the peaks at \distn{0.335} and \distn{0.386} shrink in contrast to a new peak forming at \distn{0.372} as can be seen in the yellow shaded region.
Obviously, the structure obtained from the \degc{450} simulations, which is dominated by the existence of \ci{} \hkl<1 0 0> DBs, transforms into a different structure with increasing simulation temperature.
Investigations of the atomic data reveal \cs{} to be responsible for the new Si-C bonds.
The peak at \distn{0.197} corresponds to the distance of a \cs{} atom and its next neighbored Si atoms.
The one at \distn{0.372} constitutes the distance of a \cs{} atom to the second next Si neighbor along a \hkl<1 1 0> direction.
Comparing the radial distribution for the Si-C bonds at \degc{2050} to the resulting Si-C bonds in a configuration of a \cs{} atom in c-Si excludes all possibility of doubt.
The resulting bonds perfectly match and, thus, explain the peaks observed for the increased temperature simulations.
To conclude, by increasing the simulation temperature, the structure characterized by the \ci{} \hkl<1 0 0> DB structure transforms into a structure dominated by \cs{}.

This is likewise reflected in the quality values obtained for different temperatures.
While simulations at \degc{450} exhibit \perc{10} of fourfold coordinated C, simulations at \degc{2050} exceed the \perc{80} range.
Since \cs{} has four nearest neighbored Si atoms and is the preferential type of defect in elevated temperature simulations, the increase of the quality values become evident.
The quality values at a fixed temperature increase with simulation time.
After the end of the insertion sequence marked by the first arrow, the quality is increasing and a saturation behavior, yet before the cooling process starts, can be expected.
The evolution of the quality value of the simulation at \degc{2050} inside the range, in which the simulation is continued at constant temperature for \unit[100]{fs}, is well approximated by the simple fit function
\begin{equation}
f(t)=a-\frac{b}{t} \text{ ,}
\label{eq:md:fit}
\end{equation}
which results in a saturation value of \perc{93}.
Obviously, the decrease in temperature accelerates the saturation and inhibits further formation of \cs{}.
\label{subsubsection:md:ep}
Conclusions drawn from investigations of the quality evolution correlate well with findings of the radial distribution results.

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{tot_pc2_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{tot_pc3_thesis.ps}
\end{center}
\caption[C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures.]{C-C and Si-Si radial distribution for the low concentration simulations at different elevated temperatures. All structures are cooled down to $20\,^{\circ}\mathrm{C}$. Arrows with dashed lines mark C-C distances of \hkl<1 0 0> DB combinations and those with solid lines mark C-C distances of combinations of substitutional C. The dashed line corresponds to the distance of a substitutional C with a next neighbored \hkl<1 0 0> DB.}
\label{fig:md:tot_c-c_si-si}
\end{figure}
The formation of \cs{} also affects the Si-Si radial distribution displayed in the lower part of Fig.~\ref{fig:md:tot_c-c_si-si}.
Investigating the atomic structure indeed shows that the peak arising at \distn{0.325} with increasing temperature is due to two Si atoms that form direct bonds to the \cs{} atom.
The peak corresponds to the distance of next neighbored Si atoms along the \hkl<1 1 0> bond chain with C$_{\text{s}}$ in between.
Since the expected distance of these Si pairs in 3C-SiC is \distn{0.308}, the existing SiC structures embedded in the c-Si host are stretched.

In the upper part of Fig.~\ref{fig:md:tot_c-c_si-si} the C-C radial distribution is shown.
The total amount of C-C bonds with a distance inside the displayed separation range does not change significantly.
Thus, even for elevated temperatures, agglomeration of C atoms necessary to form a SiC precipitate does not take place within the simulated time scale.
However, with increasing temperature, a decrease of the amount of next neighbored C pairs can be observed.
This is a promising result gained by the high-temperature simulations since the breaking of these diamond and graphite like bonds is mandatory for the formation of 3C-SiC.
Obviously, acceleration of the dynamics occurred by supplying additional kinetic energy.
A slight shift towards higher distances can be observed for the maximum located shortly above \distn{0.3}.
Arrows with dashed lines mark C-C distances resulting from \ci{} \hkl<1 0 0> DB combinations while arrows with solid lines mark distances arising from combinations of \cs.
The continuous dashed line corresponds to the distance of \cs{} and a next neighbored \ci{} \hkl<1 0 0> DB.
%
Obviously, the shift of the peak is caused by the advancing transformation of the C$_{\text{i}}$ DB into the C$_{\text{s}}$ defect.
Next to combinations of two \cs{} atoms or \ci{} \hkl<1 0 0> DBs, combinations of \ci{} \hkl<1 0 0> DBs with a \cs{} atom arise.
In addition, structures form that result in distances residing in between the ones obtained from combinations of mixed defect types and the ones obtained by \cs{} configurations, as can be seen by quite high $g(r)$ values in between the continuous dashed line and the first arrow with a solid line.
For the most part, these structures can be identified as configurations of \cs{} with either another C atom that basically occupies a Si lattice site but is displaced by a \si{} atom residing in the very next surrounding or a C atom that nearly occupies a Si lattice site forming a defect other than the \hkl<1 0 0>-type with the Si atom.
Again, this is a quite promising result since the C atoms are taking the appropriate coordination as expected in 3C-SiC.
%However, this is contrary to the initial precipitation model proposed in section~\ref{section:assumed_prec}, which assumes that the transformation into 3C-SiC takes place in a very last step once enough C-Si DBs agglomerated.

To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
The \ci{} \hkl<1 0 0> DB dominated structure turns into a structure characterized by the occurrence of an increasing amount of \cs{} with respect to temperature.
Clearly, the high-temperature results indicate the precipitation mechanism involving an increased participation of \cs.
Although diamond and graphite like bonds are reduced, no agglomeration of C is observed within the simulated time.
Isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignment, are formed.
By agglomeration of \cs{} the interfacial energy could be overcome and a transition from a coherent and stretched SiC structure into an incoherent and partially strain-compensated SiC precipitate could occur.
Indeed, \si in the near surrounding is observed, which may initially compensate tensile strain in the stretched SiC structure or rearrange the \cs{} sublattice and finally serve as supply for additional C to form further SiC or compensate strain at the interface of the incoherent SiC precipitate and the Si host.

\subsection{High C concentration simulations}

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{12_pc_thesis.ps}\\
\includegraphics[width=0.7\textwidth]{12_pc_c_thesis.ps}
\end{center}
\caption[Si-C and C-C radial distribution for the high concentration simulations at different elevated temperatures.]{Si-C (top) and C-C (bottom) radial distribution for the high concentration simulations at different elevated temperatures. All structures are cooled down to \degc{20}.}
\label{fig:md:12_pc}
\end{figure}
Fig.~\ref{fig:md:12_pc} displays the radial distribution for Si-C and C-C pairs obtained from high C concentration simulations at different elevated temperatures.
Again, in both cases, the cut-off artifact decreases with increasing temperature.
Peaks that already exist for the low temperature simulations get slightly more distinct for elevated temperatures.
This is also true for peaks located past distances of next neighbors indicating an increase in the long range order.
However, this change is rather small and no significant structural change is observable.
Due to the continuity of high amounts of damage, atomic configurations remain hard to identify even for the highest temperature.
Other than in the low concentration simulation, analyzed defect structures are no longer necessarily aligned to the primarily existing but successively disappearing c-Si host matrix inhibiting or at least hampering their identification and classification.
As for low temperatures, order in the short range exists decreasing with increasing separation.
The increase of the amount of Si-C pairs at \distn{0.186} could be positively interpreted since this type of bond also exists in 3C-SiC.
On the other hand, the amount of next neighbored C atoms with a distance of approximately \distn{0.15}, which is the distance of C in graphite or diamond, is likewise increased.
Thus, higher temperatures seem to additionally enhance a conflictive process, i.e.\ the formation of C agglomerates, obviously inconsistent with the desired process of 3C-SiC formation.
This is supported by the C-C peak at \distn{0.252}, which corresponds to the second next neighbor distance in the diamond structure of elemental C.
Investigating the atomic data indeed reveals two C atoms, which are bound to and interconnected by a third C atom, to be responsible for this distance.
The C-C peak at about \distn{0.31}, which is slightly shifted to higher distances (\distn{0.317}) with increasing temperature still corresponds quite well to the next neighbor distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
The Si-C peak at \distn{0.282}, which is pronounced with increasing temperature, is constructed out of a Si atom and a C atom, which are both bound to another central C atom.
This is similar for the Si-C peak at approximately \distn{0.35}.
In this case, the Si and the C atom are bound to a central Si atom.

To summarize, the amorphous phase remains.
Huge amounts of damage hamper identification.
The alignment of the investigated structures to the c-Si host is lost in many cases, which suggests the necessity of much more time for structural evolution to maintain the topotactic orientation of the precipitate.
Though, sharper peaks in the radial distributions at distances expected for a-SiC are observed indicating a slight acceleration of the dynamics due to elevated temperatures.

\subsection{Conclusions concerning the usage of increased temperatures}

Regarding the outcome of both, high and low C concentration simulations at increased temperatures, encouraging conclusions can be drawn.
With the disappearance of the peaks at the respective cut-off radii, one limitation of the short range potential seems to be accomplished.
In addition, sharper peaks in the radial distribution functions lead to the assumption of expeditious structural formation.
The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low C concentration simulations.

The question remains, whether these states are only occupied due to the additional supply of kinetic energy and, thus, have to be considered unnatural for temperatures applied in IBS or whether the increase in temperature indeed enables infrequent transitions to occur faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
As already pointed out in section~\ref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02}, IBS is a non-equilibrium process, which might result in the formation of the thermodynamically less stable \cs{} and \si{} configuration.
Indeed, 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and low temperatures to stabilize the 3C polytype.
In IBS, highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile \ci{} atoms.
This is in fact found to be favorable in the absence of the \si{}, which turned out to have a low interaction capture radius with the \cs{} atom and very likely prevents the recombination into a thermodynamically stable \ci{} DB for appropriate separations of the defect pair.
Results gained in this chapter show preferential occupation of Si lattice sites by \cs{} enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.

Moreover, the cut-off effect as detailed in section~\ref{section:md:limit} is particularly significant for configurations that are deviated to a great extent from their equilibrium structures.
Thus, for instance, it is not surprising that short range potentials show overestimated melting temperatures while properties of structures that are only slightly deviated from equilibrium are well described.
Due to this, increased temperatures are considered exceptionally necessary for modeling non-equilibrium processes and structures such as IBS and 3C-SiC.

Thus, it is concluded that increased temperatures are not exclusively useful to accelerate the dynamics approximatively describing the structural evolution.
Moreover, it can be considered a necessary condition to deviate the system out of equilibrium enabling the formation of 3C-SiC, which is obviously realized by a successive agglomeration of \cs{}.

\ifnum1=0

\section{Long time scale simulations at maximum temperature}

As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
Furthermore, these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.

A maximum temperature to avoid melting is determined in section~\ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
This value is almost equal to the temperature of $2050\,^{\circ}\mathrm{C}$ already used in former simulations.
Since the maximum temperature is reached the approach is reduced to the application of longer time scales.
This is considered useful since the estimated evolution of quality in the absence of the cooling down sequence in figure~\ref{fig:md:tot_si-c_q} predicts an increase in quality and, thus, structural evolution is likely to occur if the simulation is proceeded at maximum temperature.

Next to the employment of longer time scales and a maximum temperature a few more changes are applied.
In the following simulations the system volume, the amount of C atoms inserted and the shape of the insertion volume are modified from the values used in first MD simulations.
To speed up the simulation the initial simulation volume is reduced to 21 Si unit cells in each direction and 5500 inserted C atoms in either the whole volume or in a sphere with a radius of 3 nm corresponding to the size of a precipitate consisting of 5500 C atoms.
The \unit[100]{ps} sequence after C insertion intended for structural evolution is exchanged by a \unit[10]{ns} sequence, which is hoped to result in the occurrence of infrequent processes and a subsequent phase transition.
The return to lower temperatures is considered separately.

\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{c_in_si_95_v1_si-c.ps}\\
\includegraphics[width=0.7\textwidth]{c_in_si_95_v1_c-c.ps}
\end{center}
\caption[Si-C and C-C radial distribution for low concentration simulations at {\unit[95]{\%}} of the potential's Si melting point at different points in time of the simulation.]{Si-C (top) and C-C (bottom) radial distribution for low concentration simulations at \unit[95]{\%} of the potential's Si melting point at different points in time of the simulation.}
\label{fig:md:95_long_time_v1}
\end{figure}
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{c_in_si_95_v2.ps}
\end{center}
\caption{Si-C and C-C radial distribution for high concentration simulations at \unit[95]{\%} of the potential's Si melting point at different points in time of the simulation.}
\label{fig:md:95_long_time_v2}
\end{figure}

Fig.~\ref{fig:md:95_long_time_v1} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a low C concentration simulation.
Differences are observed for both types of atom pairs indeed indicating proceeding structural changes even well beyond \unit[100]{ps} of simulation time.
Peaks attributed to the existence of substitutional C increase and become more distinct.
This finding complies with the predicted increase of quality evolution as explained earlier.
More and more C forms tetrahedral bonds to four Si neighbors occupying vacant Si sites.
However, no increase of the amount of total C-C pairs within the observed region can be identified.
C, whether substitutional or as a DB does not agglomerate within the simulated period of time visible by the unchanging area beneath the graphs.

Fig.~\ref{fig:md:95_long_time_v2} shows the evolution in time of the radial distribution for Si-C and C-C pairs for a high C concentration simulation.
There are only small changes identifiable.
A slight increase of the Si-C peak at approximately 0.36 nm attributed to the distance of substitutional C and the next but one Si atom along \hkl<1 1 0> is observed.
In the same time the C-C peak at approximately 0.32 nm corresponding to the distance of two C atoms interconnected by a Si atom along \hkl<1 1 0> slightly decreases.
Obviously, the system prefers a slight increase of isolated substitutional C at the expense of incoherent C-Si-C precipitate configurations, which at a first glance actually appear as promising configurations in the precipitation event.
On second thoughts however, this process of splitting a C atom out of this structure is considered necessary in order to allow for the rearrangement of C atoms on substitutional lattice sites on the one hand and for C diffusion otherwise, which is needed to end up in a structure, in which one of the two fcc sublattices is composed out of C only.

For both, high and low concentration simulations the radial distribution converges as can be seen by the nearly identical graphs of the two most advanced configurations.
Changes exist ... bridge to results after cooling down to 20 degree C.

 
\fi

\section{Conclusions concerning the SiC conversion mechanism}

MD simulations at temperatures used in IBS result in structures that are dominated by the C$_{\text{i}}$ \hkl<1 0 0> DB and its combinations if C is inserted into the total volume.
Incorporation into volumes $V_2$ and $V_3$ leads to an amorphous SiC-like structure within the respective volume.
To compensate overestimated diffusion barriers, simulations at accordingly increased temperatures are performed.
No significant change is observed for high C concentrations.
The amorphous phase is maintained.
Due to the incorporation of a huge amount of C into a small volume within a short period of time, damage is produced, which obviously decelerates structural evolution.
For the low C concentrations, time scales are still too low to observe C agglomeration sufficient for SiC precipitation, which is attributed to the slow phase space propagation inherent to MD in general.
However, a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure is observed.
The amount of substitutionally occupied C atoms increases with increasing temperature.
Isolated structures of stretched SiC adjusted to the c-Si host with respect to the lattice constant and alignment are formed.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.

Results of the MD simulations at different temperatures and C concentrations can be correlated to experimental findings.
IBS studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates~\cite{kimura82,eichhorn02}.
In particular, the restructuring of strong C-C bonds is affected~\cite{deguchi92}.
These bonds preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements.
This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations.
The insertion of high amounts of C into a small volume within a short period of time results in essentially no time for the system to rearrange.
% rt implantation + annealing
Furthermore, C implanted at room temperature was found to be able to migrate towards the surface and form C-rich clusters in contrast to implantations at elevated temperatures, which form stable epitaxially aligned 3C-SiC precipitates~\cite{serre95}.
In simulation, low temperatures result in configurations of highly mobile \ci{} \hkl<1 0 0> DBs whereas elevated temperatures show configurations of \cs{}, which constitute an extremely stable configuration that is unlikely to migrate.
%
% added
This likewise agrees to results of IBS experiments utilizing implantation temperatures of \degc{550}, which require annealing temperatures as high as \degc{1405} for C segregation due to the stability of \cs{}~\cite{reeson87}.
%
Indeed, in the optimized recipe to form 3C-SiC by IBS~\cite{lindner99}, elevated temperatures are used to improve the epitaxial orientation together with a low temperature implant to destroy stable SiC nanocrystals at the interface, which are unable to migrate during thermal annealing resulting in a rough surface.
Furthermore, the improvement of the epitaxial orientation of the precipitate with increasing temperature in experiment perfectly conforms to the increasing occurrence of \cs{} in simulation.
%
% todo add sync here starting from strane93, werner96 ...
Moreover, implantations of an understoichiometric dose into preamorphized Si followed by an annealing step at \degc{700} result in Si$_{1-x}$C$_x$ layers with C residing on substitutional Si lattice sites~\cite{strane93}.
For implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing below and above \degc{700}, the formation of small C$_{\text{i}}$ agglomerates and SiC precipitates was observed respectively~\cite{werner96}.
However, increased implantation temperatures were found to be more efficient than postannealing methods resulting in the formation of SiC precipitates for implantations at \unit[450]{$^{\circ}$C}~\cite{lindner99,lindner01}.
%
Thus, at elevated temperatures, implanted C is expected to occupy substitutionally usual Si lattice sites right from the start.
These findings, which are outlined in more detail within the comprehensive description in chapter~\ref{chapter:summary}, are in perfect agreement with previous results of the quantum-mechanical investigations.

Thus, elevated temperatures are considered to constitute a necessary condition to deviate the system from equilibrium, as it is the case in IBS.
It is concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.~\cite{nejim95}.
This agrees well with previous results of the {\em ab initio} study on defects in C implanted Si, which show C$_{\text{s}}$ to occur in all probability.
However, agglomeration and rearrangement is enabled by mobile C$_{\text{i}}$, which has to be present at the same time and is formed by recombination of C$_{\text{s}}$ and Si$_{\text{i}}$.
Indeed, \si{} is observed in the direct surrounding of the stretched SiC structures.
Next to the rearrangement, \si{} is required as a supply for additional C atoms to form further SiC and to compensate strain, either within the coherent and stretched SiC structure as well as at the interface of the incoherent SiC precipitate and the Si host.
%
In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$~\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.

