-\chapter{Molecular dynamics simulations}
+\chapter{Silicon carbide precipitation simulations}
+\label{chapter:md}
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.
Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed.
+{\color{red}Todo: ... in progress ...}
+
\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.
+Due to promising advantages over the Tersoff potential the bond order potential of Erhart and Albe is used.
A time step of one fs is set.
\subsection{Simulations at temperatures used in ion beam synthesis}
+\label{subsection: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.
$V_2$ 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.}
+This is due to the overestimated activation energies for carbon diffusion as pointed out in section \ref{subsection:defects:mig_classical}.
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
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.
+Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high forces to occur.
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]
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, such conformational chamges are very unlikely to happen.
+Thus, such conformational changes are 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.
+Investigating the corresponding coordinates of the atoms it turns out that concatenated and differently oriented C-Si \hkl<1 0 0> dumbbell interstitials constitute configurations yielding separations of C atoms by this distance.
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.
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}.
\subsection{Limitations of conventional MD and short range potentials}
+\label{subsection:md:limit}
At first the formation of an amorphous SiC-like phase is unexpected since IBS experiments show crystalline 3C-SiC precipitates at prevailing temperatures.
On closer inspection, however, reasons become clear, which are discussed in the following.
In addition to the time scale limitation, problems attributed to the short range potential exist.
The sharp cut-off funtion, which limits the interacting ions to the next neighboured atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbour distance, is responsible for overestimated and unphysical high forces of next neighboured 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 carbon diffusion and further rearrengements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si dumbbells spread over the whole simulation volume.
The bonds of these C-Si pairs are also affected by the cut-off artifact preventing carbon diffusion and agglomeration of the dumbbells.
To conclude the obstacle needed to get passed is twofold.
The sharp cut-off of the used bond order model potential introduces overestimated high forces between next neighboured atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
+Thus, pushing the time scale to the limits of computational ressources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
+
+Instead the first 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 strengthes 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 $450\,^{\circ}\mathrm{C}$ in IBS \cite{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 occuring 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 definetly higher than the experimentally determined temperature tapped from the surface of the sample.
-{\color{blue}
-Thus, applying longer time scales in order to enable the system to undergo diffusion events, which become very unlikely to happen due to the overestimated bond strengthes, and in the end observe the agglomeration and precipitation might not be sufficient.
-On the other hand longer time scales are not accessible to simulation due to limited computational ressources.
-Alternatively the approach of using higher temperatures to speed up or actually make possible the steps involved in the precipitation mechanism is applied.
+\subsection{Increased temperature simulations}
+\label{subsection:md:inct}
+
+Due to the limitations of short range potentials and conventional MD as discussed above elevated temperatures are used in the following.
+The simulation sequence and other parameters aside system temperature remain unchanged as in section \ref{subsection: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 refered to as low carbon and high carbon concentration simulations.
+Temperatures ranging from $450\,^{\circ}\mathrm{C}$ up to $2050\,^{\circ}\mathrm{C}$ are used.
+
+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 neighboured Si atoms and every Si atom four next neighboured 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 on hand 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 silicon 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 neighboured 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 lengthes, 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.
+
+\subsubsection{Low carbon concetration simulations}
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{tot_pc_thesis.ps}\\
+\includegraphics[width=12cm]{tot_ba.ps}
+\end{center}
+\caption[Si-C radial distribution and quality evolution for the low concentration simulations at different elevated temperatures.]{Si-C radial distribution and quality evolution for the low concentration simulations at different elevated temperatures. All structures are cooled down to $20\,^{\circ}\mathrm{C}$. The grey line shows resulting Si-C bonds in a configuration of substitutional C in c-Si (C$_\text{sub}$) at zero temperature. Arrows in the quality plot mark the end of carbon 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}
+Figure \ref{fig:md:tot_si-c_q} shows the radial distribution of Si-C bonds for different temperatures and the corresponding quality evolution as defined earlier for the low concentration simulaton, that is the $V_1$ 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 $1650\,^{\circ}\mathrm{C}$.
+The system provides enough kinetic energy to affected atoms, which are able to escape the cut-off region.
+Another important observation in structural change is exemplified in the two shaded areas.
+In the grey shaded region a decrease of the peak at 0.186 nm and the bump at 0.175 nm and a concurrent increase of the peak at 0.197 nm with increasing temperature is visible.
+Similarly the peaks at 0.335 nm and 0.386 nm shrink in contrast to a new peak forming at 0.372 nm as can be seen in the yellow shaded region.
+Obviously the structure obtained from the $450\,^{\circ}\mathrm{C}$ simulations, which is dominated by the existence of \hkl<1 0 0> C-Si dumbbells transforms into a different structure with increasing simulation temperature.
+Investigations of the atomic data reveal substitutional carbon to be responsible for the new Si-C bonds.
+The peak at 0.197 nm corresponds to the distance of a substitutional carbon atom to the next neighboured silicon atoms.
+The one at 0.372 nm is the distance of a substitutional carbon atom to the second next silicon neighbour along a \hkl<1 1 0> direction.
+Comparing the radial distribution for the Si-C bonds at $2050\,^{\circ}\mathrm{C}$ to the resulting Si-C bonds in a configuration of a substitutional carbon atom in crystalline silicon 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 \hkl<1 0 0> C-Si dumbbell characterized structure transforms into a structure dominated by substitutional C.
+
+This is also reflected in the quality values obtained for different temperatures.
+While simulations at $450\,^{\circ}\mathrm{C}$ exhibit 10 \% of fourfold coordinated carbon simulations at $2050\,^{\circ}\mathrm{C}$ exceed the 80 \% range.
+Since substitutional carbon has four next neighboured silicon 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 behaviour, yet before the cooling process starts, can be expected.
+The evolution of the quality value of the simulation at $2050\,^{\circ}\mathrm{C}$ inside the range in which the simulation is continued at constant temperature for 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 93 \%.
+Obviously the decrease in temperature accelerates the saturation and inhibits further formation of substitutional carbon.
+\label{subsubsection:md:ep}
+Conclusions drawn from investigations of the quality evolution correlate well with the findings of the radial distribution results.
-TAD, correcting time and undo changes for real temperature.
-Since the shot cut-off in the used potential introduces unphysical high forces use of higher temperatures in order to get the system to escape local minima and transform into the crystalline phase is the first approach followed.
-Anyways there is now conflict to experiments applying higher temepratures without the TAD corrections, since crystalline 3C-SiC is also expected for higher temperatures on the one hand and on the other hand the exact temperature inside the implantation volume is definetly higher than the temperature meassured at the surface of the sample.
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{tot_pc2_thesis.ps}\\
+\includegraphics[width=12cm]{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> dumbbell 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 neighboured \hkl<1 0 0> dumbbell.}
+\label{fig:md:tot_c-c_si-si}
+\end{figure}
+The formation of substitutional carbon also affects the Si-Si radial distribution displayed in the lower part of figure \ref{fig:md:tot_c-c_si-si}.
+Investigating the atomic strcuture indeed shows that the peak arising at 0.325 nm with increasing temperature is due to two Si atoms directly bound to a C substitutional.
+It corresponds to the distance of second next neighboured Si atoms along a \hkl<1 1 0>-equivalent direction with substitutional C inbetween.
+Since the expected distance of these Si pairs in 3C-SiC is 0.308 nm the existing SiC structures embedded in the c-Si host are stretched.
+
+In the upper part of figure \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 neccessary 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 neighboured C pairs can be observed.
+This is a promising result gained by the high temperature simulations since the breaking of these diomand and graphite like bonds is mandatory for the formation of 3C-SiC.
+A slight shift towards higher distances can be observed for the maximum above 0.3 nm.
+Arrows with dashed lines mark C-C distances resulting from \hkl<1 0 0> dumbbell combinations while the arrows with solid lines mark distances arising from combinations of substitutional C.
+The continuous dashed line corresponds to the distance of a substitutional C with a next neighboured \hkl<1 0 0> dumbbell.
+By comparison with the radial distribution it becomes evident that the shift accompanies the advancing transformation of \hkl<1 0 0> dumbbells into substitutional C.
+Next to combinations of two substitutional C atoms and two \hkl<1 0 0> dumbbells respectively also combinations of \hkl<1 0 0> dumbbells with a substitutional C atom arise.
+In addition, structures form that result in distances residing inbetween the ones obtained from combinations of mixed defect types and the ones obtained by substitutional C configurations, as can be seen by quite high g(r) values to the right of the continuous dashed line and to the left of the first arrow with a solid line.
+For the most part these structures can be identified as configurations of one substitutional C atom with either another C atom that practically occupies a Si lattice site but with a Si interstitial 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 dumbbells agglomerated.
+
+To summarize, results of low concentration simulations show a phase transition in conjunction with an increase in temperature.
+The C-Si \hkl<1 0 0> dumbbell dominated struture turns into a structure characterized by the occurence of an increasing amount of substitutional C with respect to temperature.
+Although diamond and graphite like bonds are reduced no agglomeration of C is observed within the simulated time resulting in the formation of isolated structures of stretched SiC, which are adjusted to the c-Si host with respect to the lattice constant and alignement.
+It would be conceivable that by agglomeration of further substitutional C atoms the interfacial energy could be overcome and a transition into an incoherent SiC precipitate could occur.
+
+{\color{red}Todo: Results reinforce the assumption of an alternative precipitation model as already pointed out in the defects chapter.}
+
+\subsubsection{High carbon concetration simulations}
+
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{12_pc_thesis.ps}\\
+\includegraphics[width=12cm]{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 $20\,^{\circ}\mathrm{C}$.}
+\label{fig:md:12_pc}
+\end{figure}
+Figure \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 neighbours indicating an increase in the long range order.
+However this change is rather small and no significant structural change is observeable.
+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 succesively 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 distance.
+The increase of the amount of Si-C pairs at 0.186 nm could be positively interpreted since this type of bond also exists in 3C-SiC.
+On the other hand the amount of next neighboured C atoms with a distance of approximately 0.15 nm, which is the distance of C in graphite or diamond, is likewise increased.
+Thus, higher temperatures seem to additionally enhance a conflictive process, that is the formation of C agglomerates, instead of the desired process of 3C-SiC formation.
+This is supported by the C-C peak at 0.252 nm, which corresponds to the second next neighbour 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 0.31 nm, wich is slightly shifted to higher distances (0.317 nm) with increasing temperature still corresponds quite well to the next neighbour distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
+The Si-C peak at 0.282 nm, 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 0.35 nm.
+In this case, the Si and the C atom are bound to a central Si atom.
+To summarize, the amorphous phase remains 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.
+
+\subsubsection{Conclusions concerning the usage of increased temperatures}
+
+Regarding the outcome of both, high and low 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 distributions 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 carbon concentration simulations.
+
+{\color{blue}
+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} on page~\pageref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02} on page~\pageref{section:defects:noneq_process_02} IBS is a nonequilibrium process, which might result in the formation of the thermodynamically less stable substitutional carbon and Si self-interstitital configuration.
+Indeed 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and/or low temperatures to stabilize the 3C polytype \cite{}.
+In IBS highly energetic C atoms are able to generate vacant sites, which in turn can be occupied by highly mobile C atoms.
+This is found to be favorable in the absence of the Si self-interstitial, which turned out to have a low interaction capture radius with a substitutional C atom very likely preventing the recombination into thermodynamically stable C-Si dumbbell interstitials for appropriate separations of the defect pair.
+Results gained in this chapter show preferential occupation of Si lattice sites by substitutional C enabled by increased temperatures supporting the assumptions drawn from the defect studies of the last chapter.
+
+Thus, employing increased temperatures is not exclusively usefull 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 obviously realized by a successive agglomeration of substitutional C.
}
-\subsection{Increased temperature simulations}
+\subsection{Valuation of a practicable temperature limit}
+\label{subsection:md:tval}
+
+The assumed applicability of increased temperature simulations as discussed above and the remaining absence of either agglomeration of substitutional C in low concentration simulations or amorphous to crystalline transition in high concentration simulations suggests to further increase the system temperature.
+So far, the highest temperature applied corresponds to 95 \% of the absolute silicon melting temperature, which is 2450 K and specific to the Erhart/Albe potential.
+However, melting is not predicted to occur instantly after exceeding the melting point due to additionally required transition enthalpy and hysteresis behaviour.
+To check for the possibly highest temperature at which a transition fails to appear plain silicon is heated up using a heating rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
+Figure \ref{fig:md:fe_and_t} shows the free energy and temperature evolution in the region around the transition temperature.
+Indeed a transition and the accompanying critical behaviour of the free energy is first observed at approximately 3125 K, which corresponds to 128 \% of the silicon melting temperature.
+The difference in free energy is 0.58 eV per atom corresponding to $55.7 \text{ kJ/mole}$, which compares quite well to the silicon enthalpy of melting of $50.2 \text{ kJ/mole}$.
+The late transition probably occurs due to the high heating rate and, thus, a large hysteresis behaviour extending the temperature of transition.
+To avoid melting transitions in further simulations system temperatures well below the transition point are considered safe.
+According to this study temperatures of 100 \% and 120 \% of the silicon melting point could be used.
+However, defects, which are introduced due to the insertion of C atoms are known to lower the transition point.
+Indeed simulations show melting transitions already at the melting point whenever C is inserted.
+Thus, the system temperature of 95 \% of the silicon melting point is considered the maximum limit.
+\begin{figure}[!t]
+\begin{center}
+\includegraphics[width=12cm]{fe_and_t.ps}
+\end{center}
+\caption{Free energy and temperature evolution of plain silicon at temperatures in the region around the melting transition.}
+\label{fig:md:fe_and_t}
+\end{figure}
-{\color{red}Todo: We go for higher temepratures first due to the potential cut-off artifact discussed above.}
+\subsection{Long time scale simulations at maximum temperature}
-\subsection{Constructed 3C-SiC precipitate in crystalline silicon}
+As discussed in section~\ref{subsection:md:limit} and~\ref{subsection: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.
-{\color{red}Todo: We want to know where we want to go ...}
+A maximum temperature to avoid melting is determined in section \ref{subsection: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 usefull.
+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 usefull 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 liekyl to occur if the simulation is proceeded at maximum temperature.
-In the following a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed as it is expected from IBS experiments and from simulations that finally succeed simulating the precipitation event.
-On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of energy for configurations that are aimed to be reproduced by simulation possibly enabling the prediction of conditions necessary for the simulation of the precipitation process.
-On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, investigations of the behaviour of such precipitates and the surrounding can be made.
+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 100 ps sequence after C insertion intended for structural evolution is exchanged by a 10 ns sequence, which is hoped to result in the occurence of infrequent processes and a subsequent phase transition.
+The return to lower temperatures is considered seperately.
-To construct a spherical 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
-A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is used.
-To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
-The initial precipitate configuration is constructed in two steps.
-In the first step the surrounding silicon matrix is created.
-This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
-The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
-In a second step 3C-SiC is created inside the empty sphere of radius $x$.
-The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated.
-This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
-\begin{equation}
-\frac{8}{a_{\text{Si}}^3}(
-\underbrace{21^3 a_{\text{Si}}^3}_{=V}
--\frac{4}{3}\pi x^3)+
-\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
-=21^3\cdot 8
-\label{eq:md:constr_sic_01}
-\text{ ,}
-\end{equation}
-which can be simplified to read
-\begin{equation}
-\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
-\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
-\label{eq:md:constr_sic_02}
-\end{equation}
-and
-\begin{equation}
-%x^3=\frac{16\pi}{5500 \cdot 3}y^3=
-%\frac{16\pi}{5500 \cdot 3}\frac{5500 \cdot 3}{32 \pi}a_{\text{Si}}^3
-%\Rightarrow
-y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
-\text{ .}
-\label{eq:md:constr_sic_03}
-\end{equation}
-By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
-Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
-However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
-\begin{table}[!ht]
+\begin{figure}[!t]
\begin{center}
-\begin{tabular}{l c c c c}
-\hline
-\hline
- & C in 3C-SiC & Si in 3C-SiC & Si in c-Si surrounding & total amount of Si\\
-\hline
-Expected & 5500 & 5500 & 68588 & 74088\\
-Obtained & 5495 & 5486 & 68591 & 74077\\
-Difference & 5 & 14 & -3 & 11\\
-\hline
-\hline
-\end{tabular}
-\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.}
-\label{table:md:sic_prec}
+\includegraphics[width=12cm]{c_in_si_95_v1_si-c.ps}\\
+\includegraphics[width=12cm]{c_in_si_95_v1_c-c.ps}
\end{center}
-\end{table}
+\caption{Si-C (top) and C-C (bottom) radial distribution for low concentration simulations at 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}[!t]
+\begin{center}
+\includegraphics[width=12cm]{c_in_si_95_v2.ps}
+\end{center}
+\caption{Si-C and C-C radial distribution for high concentration simulations at 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}
+
+Figure \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 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 neighbours occupying vacant Si sites.
+However, no increase of the amount of total C-C pairs within the observed region can be identified.
+Carbon, whether substitutional or as a dumbbell does not agglomerate within the simulated period of time visible by the unchanging area beneath the graphs.
+
+Figure \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 preferes 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 carbon 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.
+
+{\color{red}Todo: Cooling down to $20\,^{\circ}\mathrm{C}$ by $1\,^{\circ}\mathrm{C/s}$ in progress.}
+
+{\color{red}Todo: Remember NVE simulations (prevent melting).}
+
+\subsection{Further accelerated dynamics approaches}
-After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
-Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
-Once the main part of th excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+Since longer time scales are not sufficient \ldots
-PC and energy of that one.
+{\color{red}Todo: self-guided MD?}
-Now let's see, whether annealing will lead to some energetically more favorable configurations.
+{\color{red}Todo: other approaches?}
-Estimate surface energy ...
+{\color{red}Todo: ART MD?\\
+How about forcing a migration of a $V_2$ configuration to a constructed prec configuration, determine the saddle point configuration and continue the simulation from this configuration?
+}
-\subsection{Simulations at temperatures exceeding the silicon melting point}
+\section{Conclusions concerning the SiC conversion mechanism}
-LL Cool J is hot as hell!
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