-\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}
$V_2$ 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 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.
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}\\
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 to the next neighboured silicon atoms.
-The one at 0.372 is the distance of the substitutional carbon atom to the second next silicon neighbour along the \hkl<1 1 0> direction.
+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.
\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.
\begin{figure}[!ht]
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.
-With increasing temperature a decrease of the amount of next neighboured C pairs can be observed.
+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.
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.}
+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}
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.
-{\color{red}Todo: Formation energy of C sub and nearby Si self-int, to see whether this is a preferable state!}
-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: If C sub and Si self-int is energetically more favorable, the migration towards the surface can be kicked out. Otherwise we should actually care about removal of Si! In any way these findings suggest a different prec model.}
+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{Valuation of a practicable temperature limit}
\label{subsection:md:tval}
-\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.
+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.
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{Constructed 3C-SiC precipitate in crystalline silicon}
-
-Before proceeding with simulations at temperatrures around the silicon melting point 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 in simulating the precipitation event.
-On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of free energy for such a configuration that is aimed to be reproduced by simulation.
-On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, properties of such precipitates and the surrounding as well as the interface can be investiagted.
-Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.
-
-To construct a spherical and topotactically aligned 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 created.
-To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
-This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
-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]
+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}
-\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
-Obtained & 5495 & 5486 & 68591 & 74077\\
-Expected & 5500 & 5500 & 68588 & 74088\\
-Difference & -5 & -14 & 3 & -11\\
-Notation & $N^{\text{3C-SiC}}_{\text{C}}$ & $N^{\text{3C-SiC}}_{\text{Si}}$
- & $N^{\text{c-Si}}_{\text{Si}}$ & $N^{\text{total}}_{\text{Si}}$ \\
-\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_and_t.ps}
\end{center}
-\end{table}
+\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}
-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 the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+\subsection{Long time scale simulations at maximum temperature}
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{pc_0.ps}
-\end{center}
-\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
-\label{fig:md:pc_sic-prec}
-\end{figure}
-Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
-The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
-Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
-Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
-A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
-It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
-The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
-They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
-It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
-The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
-A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
-This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
-Thus, the precipitate structure is slightly compressed compared to the bulk phase.
-This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
-However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
-
-The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
-Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
-If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
-\begin{equation}
- \frac{V}{V_0}=
- \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
- \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
- {\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
-\end{equation}
-with the notation used in table \ref{table:md:sic_prec}.
-The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
-By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
-The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
-Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%.
-However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$.
-Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
-This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
-As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
-Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
-
-In the following the 3C-SiC/c-Si interface is described in further detail.
-One important size analyzing the interface is the interfacial energy.
-It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
-Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
-\begin{equation}
- E_{\text{f}}=E-
- N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
- \left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
- \mu_{\text{Si}} \text{ ,}
-\label{eq:md:ife}
-\end{equation}
-with $E$ being the free energy of the precipitate configuration at zero temperature.
-An interfacial energy of 2267.28 eV is obtained.
-The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
-Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
-This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
-
-Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
-Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
-The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
-From that point on the heating rate is reduced to $1\,^{\circ}\mathrm{C}/\text{ps}$ and heating is continued to 120 \% of the Si melting temperature, that is 2940 K.
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{fe_and_t_sic.ps}
-\end{center}
-\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
-\label{fig:md:fe_and_t_sic}
-\end{figure}
-Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
-The sudden increase of the free energy indicates possible melting occuring around 2840 K.
-\begin{figure}[!ht]
-\begin{center}
-\includegraphics[width=12cm]{pc_500-fin.ps}
-\end{center}
-\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
-\label{fig:md:pc_500-fin}
-\end{figure}
-Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
-However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
-Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
-This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in the last section.
-Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
-For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
-To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to $20\,^{\circ}\mathrm{C}$ with a cooling rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
-{\color{blue}Todo: Wait for results and then compare structure (PC) and interface energy, maybe a energetically more favorable configuration arises.}
-{\color{red}Todo: Mention the fact, that the precipitate is stable for eleveated temperatures, even for temperatures where the Si matrix is melting.}
-{\color{red}Todo: Si starts to melt at the interface, show pictures and explain, it is due to the defective interface region.}
-
-\subsection{Simulations at temperatures around the silicon melting point}
-
-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.
+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.
+
+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.
+
+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.
-\begin{figure}[!ht]
+\begin{figure}[!t]
\begin{center}
-\includegraphics[width=12cm]{fe_100.ps}
-\includegraphics[width=12cm]{q_100.ps}
+\includegraphics[width=12cm]{c_in_si_95_v1_si-c.ps}\\
+\includegraphics[width=12cm]{c_in_si_95_v1_c-c.ps}
\end{center}
-\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}
+\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}[!ht]
+\begin{figure}[!t]
\begin{center}
-\includegraphics[width=12cm]{fe_120.ps}
-\includegraphics[width=12cm]{q_120.ps}
+\includegraphics[width=12cm]{c_in_si_95_v2.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}
+\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: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.}
-{\color{red}Todo: In additions simulations at 95 \% of the Si melting temperature are started again for longer times.}
+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}
-{\color{red}Todo: self-guided MD!}
+Since longer time scales are not sufficient \ldots
+
+{\color{red}Todo: self-guided MD?}
{\color{red}Todo: other approaches?}
-{\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.
+{\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?
}
+\section{Conclusions concerning the SiC conversion mechanism}
+
projects / lectures / latex.git / blobdiff