-\chapter{Molecular dynamics simulations}
+\chapter{Silicon carbide precipitation 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.
$V_2$ 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
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.
+{\color{red}Todo: Mention somewhere(!) that the distance is due to neighboured differently oriented C-Si \hkl<1 0 0> dumbbells!}
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.
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.
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.
\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}.
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 the solid line mark distances arising from combinations of substitutional C.
+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.
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.
+{\color{red}Todo: Summarize again! Mention, that the agglomeration necessary in order to form 3C-SiC is missing.}
+
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=12cm]{12_pc_thesis.ps}\\
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 for the long range order.
+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 investigations of atomic configurations below remain hard to identify even for the highest temperature.
-Other than in the low concentration simulations 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.
+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 pe positively interpreted since this type of bond also exists in 3C-SiC.
+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 interconnect by a third C atom to be responsible for this distance.
+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.
-Regarding these findings there is clear evidence ...
-
-This said, there is clear evidence that this is amorphous SiC
-However there is no significant change in structure.
-But there is a decrease in the artifacts of the potential.
-So, first limitations might be condiered as
-Now, more temperature to increase infrequent events ...
+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.
+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.
+In the first case these occupied states would be expected to be higher in energy than the states occupied at low temperatures.
+Since substitutional C without the presence of a Si self-interstitial is energetically more favorable than the lowest defect structure obtained without removing a Si atom, that is the \hkl<1 0 0> dumbbell interstitial, and the migration of Si self-interstitials towards the sample surface can be assumed for real life experiments \cite{}, this approach is accepted as an accelerated way of approximatively describing the structural evolution.
+{\color{red}Todo: C sub and Si self-int is energetically less favorable! Maybe fast migration of Si (mentioned in another Todo)? If true, we have to care about Si removal in simulations? In any way these findings suggest a different prec model.}
-\subsection{Constructed 3C-SiC precipitate in crystalline silicon}
+\subsection{Valuation of a practicable temperature limit}
+\label{subsection:md:tval}
-{\color{red}Todo: We want to know where we want to go ...}
-
-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.
+\begin{figure}[!ht]
+\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}
+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 Erhard/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.
+Thus, in the following system temperatures of 100 \% and 120 \% of the silicon melting point are used.
+
+\subsection{Long time scale simulations at maximum temperature}
+
+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.
+A maximum temperature to avoid melting was determined in section \ref{subsection:md:tval}, which is 120 \% of the Si melting point.
+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 the first MD simulations to now match the conditions given in the simulations of the self-constructed precipitate configuration for reasons of comparability.
+To quantify, the initial simulation volume now consists of 21 Si unit cells in each direction and 5500 C atoms are inserted in either the whole volume or in a sphere with a radius of 3 nm.
+Since the investigated temperatures exceed the Si melting point the initial Si bulk material is heated up slowly by $1\,^{\circ}\mathrm{C}/\text{ps}$ starting from $1650\,^{\circ}\mathrm{C}$ before the C insertion sequence is started.
+The 100 ps sequence at the respective temperature intended for the structural evolution is exchanged by a 10 ns sequence, which will hopefully result in the occurence of infrequent processes.
+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}[!ht]
\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]{fe_100.ps}
+\includegraphics[width=12cm]{q_100.ps}
\end{center}
-\end{table}
+\caption[Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 100 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
+\label{fig:md:exceed100}
+\end{figure}
+\begin{figure}[!ht]
+\begin{center}
+\includegraphics[width=12cm]{fe_120.ps}
+\includegraphics[width=12cm]{q_120.ps}
+\end{center}
+\caption[Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature.]{Evolution of the free energy and quality of a simulation at 120 \% of the Si melting temperature. Matt colored parts of the graphs represent the C insertion sequence.}
+\label{fig:md:exceed120}
+\end{figure}
+Figure \ref{fig:md:exceed100} and \ref{fig:md:exceed120} show the evolution of the free energy per atom and the quality at 100 \% and 120 \% of the Si melting temperature.
+
+{\color{red}Todo: Melting occurs, show and explain it and that it's due to the defects created.}
+
+{\color{red}Todo: Due to melting, after insertion, simulation is continued NVE, so melting hopefully will not occur, before it will be cooled down later on.}
-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.
+{\color{red}Todo: In additions simulations at 95 \% of the Si melting temperature are started again for longer times.}
-PC and energy of that one.
+\subsection{Further accelerated dynamics approaches}
-Now let's see, whether annealing will lead to some energetically more favorable configurations.
+{\color{red}Todo: self-guided MD!}
-Estimate surface energy ...
+{\color{red}Todo: other approaches?}
-\subsection{Simulations at temperatures exceeding the silicon melting point}
+{\color{red}
+Todo: ART MD?
+Also, how about forcing a migration of a $V_2$ configuration to a constructed prec configuration, detrmine the maximum saddle point and let the simulation run.
+}
-LL Cool J is hot as hell!
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