From 2e508e9957ccdd829e470d4e7855ae019e737c71 Mon Sep 17 00:00:00 2001
From: hackbard
Date: Mon, 23 May 2011 14:12:59 +0200
Subject: [PATCH] basically finished simulation chapter

posic/thesis/simulation.tex  264 ++++++++++++++++++++
1 file changed, 147 insertions(+), 117 deletions()
diff git a/posic/thesis/simulation.tex b/posic/thesis/simulation.tex
index 943882c..a3917a4 100644
 a/posic/thesis/simulation.tex
+++ b/posic/thesis/simulation.tex
@@ 39,6 +39,7 @@ The electronic ground state is calculated by an interative Davidson scheme \cite
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
The conjugate gradiant algorithm is used for ionic relaxation.
+Migration paths are determined by the modified version of the CRT method as explained in section \ref{section:basics:migration}.
The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}.
@@ 169,52 +170,76 @@ Clearly, a competent parameter set is found, which is capabale of describing the
\section{Classical potential MD}
fast method, amoun tof atoms ...

\subsection{Tersoff vs. ErhartAlbe SiC potential}

\subsection{Temperature and volume control}

+The classical potential MD method is much less computationally costly compared to the highly accurate quantummechanical method.
+Thus, the method is capable of performing structural optimizations on large systems and MD calulations may be used to model a system over long time scales.
+Defect structures are modeled in a cubic supercell (type 3) of nine Si lattice constants in each direction containing 5832 Si atoms.
+Reproducing the SiC precipitation was attempted in cubic cSi supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
+A Tersofflike bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} is used to describe the atomic interaction.
+Constant pressure simulations are realized by the Berendsen barostat \cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
+The temperature is controlled by the Berendsen thermostat \cite{berendsen84} with a time constant of \unit[100]{fs}.
+Integration of the equations of motion is realized by the velocity Verlet algorithm \cite{verlet67} using a fixed time step of \unit[1]{fs}.
+For structural relaxation of defect structures the same algorithm is utilized with the temperature set to zero Kelvin.
+This also applies for the relaxation of structures within the CRT calculations to find migration pathways.
+In the latter case the time constant of the Berendsen thermostat is set to \unit[1]{fs} in order to achieve direct velocity scaling, which corresponds to a steepest descent minimazation driving the system into a local minimum, if the temperature is set to zero Kelvin.
+However, in some cases a time constant of \unit[100]{fs} turned out to result in lower barriers.
+
+In addition to the bond order formalism the EA potential provides a set of parameters to describe the interaction in the C/Si system, as discussed in section \ref{subsection:interact_pot}.
+There are basically no free parameters, which could be set by the user and the properties of the potential and its parameters are well known and have been extensively tested by the authors \cite{albe_sic_pot}.
+Therefore, test calculations are restricted to the time step used in the Verlet algorithm to integrate the equations of motion.
+Nevertheless, a further and rather uncommon test is carried out to roughly estimate the capabilities of the EA potential regarding the description of 3CSiC precipitation in cSi.
+% todo
+% rather a first investigation than a test
\subsection{Test calculations}
+\subsection{Time step}
Give cohesive energies of Si, C (Dia) and (3C)SiC and the respective lattice parameters ...
+The quality of the integration algorithm and the occupied time step is determined by the ability to conserve the total energy.
+Therefore, simulations of a $9\times9\times9$ 3CSiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
+The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e. \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{verlet_e.ps}
+\end{center}
+\caption{Evolution of the total energy of 3CSiC in the $NVE$ ensemble for two different initial temperatures.}
+\label{fig:simulation:verlet_e}
+\end{figure}
+The evolution of the total energy is displayed in Fig. \ref{fig:simulation:verlet_e}.
+Almost no shift in energy is observable for the simulation at \unit[0]{$^{\circ}$C}.
+Even for \unit[1000]{$^{\circ}$C} the shift is as small as \unit[0.04]{eV}, which is a quite acceptable error for $10^5$ integration steps.
+Thus, using a time step of \unit[100]{ps} is considered small enough.
\subsection{3CSiC precipitate in crystalline silicon}
+\subsection{3CSiC precipitate in cSi}
\label{section:simulation:prec}
A spherical 3CSiC precipitate enclosed in a cSi 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 cSi 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.
+Below, a spherical 3CSiC precipitate enclosed in a cSi surrounding is investigated by means of MD.
+On the one hand, these investigations are meant to draw conclusions on the capabilities of the potential for modeling the respective tasks in the C/Si system.
+Since, on the other hand, properties of the 3CSiC precipitate, the surrounding and the interface can be obtained, the calculations could be considered to constitute a first investigation rather than a test of the capabilities of the potential.
To construct a spherical and topotactically aligned 3CSiC precipitate in cSi, the approach illustrated in the following is applied.
A total simulation volume $V$ consisting of 21 unit cells of cSi in each direction is created.
To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
This corresponds to a spherical 3CSiC precipitate with a radius of approximately 3 nm.
+To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary according to experimental results as discussed in section \ref{subsection:ibs} and \ref{section:assumed_prec}.
+This corresponds to a spherical 3CSiC precipitate with a radius of approximately \unit[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 cSi matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
+In the first step the surrounding Si matrix is created.
+This is realized by just skipping the generation of Si atoms inside a sphere of radius $x$, which is the first unknown variable.
+The Si lattice constant $a_{\text{Si}}$ of the surrounding cSi matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
In a second step 3CSiC 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 and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
+This is entirely described by the equation
\begin{equation}
\frac{8}{a_{\text{Si}}^3}(
\underbrace{21^3 a_{\text{Si}}^3}_{=V}
+V
\frac{4}{3}\pi x^3)+
\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
+\frac{4}{y^3}\frac{4}{3}\pi x^3
=21^3\cdot 8
\label{eq:md:constr_sic_01}
\text{ ,}
+\label{eq:simulation:constr_sic_01}
\end{equation}
which can be simplified to read
+where the volume is given by $V=21^3 a_{\text{Si}}^3$ and the the additional condition $\frac{4}{y^3}\frac{4}{3}\pi x^3=5500$.
+This 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}
+\label{eq:simulation:constr_sic_02}
\end{equation}
and
\begin{equation}
@@ 223,12 +248,12 @@ and
%\Rightarrow
y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
\text{ .}
\label{eq:md:constr_sic_03}
+\label{eq:simulation: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 3CSiC.
+By this means values of \unit[2.973]{nm} and \unit[4.309]{\AA} are obtained for the initial precipitate radius and lattice constant of 3CSiC.
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]
+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:simulation:sic_prec}.
+\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c}
\hline
@@ 243,123 +268,128 @@ Notation & $N^{\text{3CSiC}}_{\text{C}}$ & $N^{\text{3CSiC}}_{\text{Si}}$
\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 3CSiC precipitate construction approach.}
\label{table:md:sic_prec}
+\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:simulation:constr_sic_02} and \eqref{eq:simulation:constr_sic_03} in the 3CSiC precipitate construction approach.}
+\label{table:simulation:sic_prec}
\end{center}
\end{table}
After the initial configuration is constructed some of the atoms located at the 3CSiC/cSi 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.
+Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be \unit[20]{$^{\circ}$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 \unit[10]{ps}.
\begin{figure}[!ht]
+\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm]{pc_0.ps}
+\includegraphics[width=0.7\textwidth]{pc_0.ps}
\end{center}
\caption[Radial distribution of a 3CSiC precipitate embeeded in cSi at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3CSiC precipitate embeeded in cSi at $20\,^{\circ}\mathrm{C}$. The SiSi radial distribution of plain cSi is plotted for comparison. Green arrows mark bumps in the SiSi distribution of the precipitate configuration, which do not exist in plain cSi.}
\label{fig:md:pc_sicprec}
+\caption[Radial distribution of a 3CSiC precipitate embedded in cSi at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3CSiC precipitate embedded in cSi at \unit[20]{$^{\circ}$C}. The SiSi radial distribution of plain cSi is plotted for comparison. Green arrows mark bumps in the SiSi distribution of the precipitate configuration, which do not exist in plain cSi.}
+\label{fig:simulation:pc_sicprec}
\end{figure}
Figure \ref{fig:md:pc_sicprec} shows the radial distribution of the obtained precipitate configuration.
The SiSi radial distribution for both, plain cSi and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in cSi.
Although no significant change of the lattice constant of the surrounding cSi matrix was assumed, surprisingly there is no change at all within observational accuracy.
+Figure \ref{fig:simulation:pc_sicprec} shows the radial distribution of the obtained precipitate configuration.
+The SiSi radial distribution for both, plain cSi and the precipitate configuration show a maximum at a distance of \unit[0.235]{nm}, which is the distance of next neighboured Si atoms in cSi.
+Although no significant change of the lattice constant of the surrounding cSi matrix was assumed, surprisingly, there is no change at all within observational accuracy.
Looking closer at higher order SiSi peaks might even allow the guess of a slight increase of the lattice constant compared to the plain cSi structure.
A new SiSi peak arises at 0.307 nm, which is identical to the peak of the CC distribution around that value.
It corresponds to second next neighbours in 3CSiC, which applies for Si as well as C pairs.
+A new SiSi peak arises at \unit[0.307]{nm}, which is identical to the peak of the CC distribution around that value.
+It corresponds to second next neighbors in 3CSiC, which applies for Si as well as C pairs.
The bumps of the SiSi 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 3CSiC.
+They correspond to the fourth and sixth next neighbor distance in 3CSiC.
It is easily identifiable how these CC peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the SiSi distribution.
The SiSi and CC peak at 0.307 nm enables the determination of the lattic constant of the embedded 3CSiC precipitate.
A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3CSiC is obtained.
This is in accordance with the peak of SiC pairs at a distance of 0.188 nm.
+The SiSi and CC peak at \unit[0.307]{nm} enables the determination of the lattic constant of the embedded 3CSiC precipitate.
+A lattice constant of \unit[4.34]{\AA} compared to \unit[4.36]{\AA} for bulk 3CSiC is obtained.
+This is in accordance with the peak of SiC pairs at a distance of \unit[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 cSi surrounding a nonnegligible impact of the precipitate on the materializing cSi lattice constant especially near the precipitate could be assumed.
However, it seems that the size of the cSi host matrix is chosen large enough to even find the precipitate in a compressed state.
The absence of a compression of the cSi 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 cSi surrounding.
If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the cSi surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
+Otherwise the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the cSi surrounding.
+If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the cSi 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{cSi}}_{\text{Si}}}{8/a_{\text{cSi of precipitate configuration}}}+
 \frac{N^{\text{3CSiC}}_{\text{Si}}}{4/a_{\text{3CSiC of precipitate configuration}}}}
+ \frac{\frac{N^{\text{cSi}}_{\text{Si}}}{8/a_{\text{cSi prec}}}+
+ \frac{N^{\text{3CSiC}}_{\text{Si}}}{4/a_{\text{3CSiC prec}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain cSi}}}}
\end{equation}
with the notation used in table \ref{table:md:sic_prec}.
The lattice constant of plain cSi 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 $a_{\text{plain cSi}}=5.439\text{ \AA}$ is obtained.
The same lattice constant is assumed for the cSi surrounding in the precipitate configuration $a_{\text{cSi of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain cSi.
Using $a_{\text{3CSiC of precipitate configuration}}=4.34\text{ \AA}$ 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 cSi at $20\,^{\circ}\mathrm{C}$.
+with the notation used in Table \ref{table:simulation:sic_prec}.
+Here, $a_{\text{cSi prec}}$ denotes the lattice constant of the surrounding crystalline Si and $a_{\text{3CSiC prec}}$ is the lattice constant of the precipitate.
+The lattice constant of plain cSi at \unit[20]{$^{\circ}$C} can be determined more accurately by the side lengthes of the simulation box of an equilibrated structure instead of using the radial distribution data.
+By this, a value of $a_{\text{plain cSi}}=5.439\,\text{\AA}$ is obtained.
+The same lattice constant is assumed for the cSi surrounding in the precipitate configuration $a_{\text{cSi prec}}$ since peaks in the radial distribution match the ones of plain cSi.
+Using $a_{\text{3CSiC prec}}=4.34\,\text{\AA}$ as observed from the radial distribution finally results in an increase of the initial volume by \unit[0.12]{\%}.
+However, each side length and the total volume of the simulation box is increased by \unit[0.20]{\%} and \unit[0.61]{\%} respectively compared to plain cSi at \unit[20]{$^{\circ}$C}.
Since the cSi 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 3CSiC/cSi interface region.
This also explains the possibly identified slight increase of the cSi 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 cSi in the surrounding.
In the following the 3CSiC/cSi interface is described in further detail.
+To finally draw some conclusions concerning the capabilities of the potential, the 3CSiC/cSi interface is now addressed 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 $N^{\text{3CSiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+By simulation, it can be determined exactly in the same way than the formation energy as described in equation \eqref{eq:basics:ef2}.
+Using the notation of Table \ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3CSiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E
N^{\text{3CSiC}}_{\text{C}} \mu_{\text{SiC}}
\left(N^{\text{total}}_{\text{Si}}N^{\text{3CSiC}}_{\text{C}}\right)
\mu_{\text{Si}} \text{ ,}
\label{eq:md:ife}
+\label{eq:simulation: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 $28\times 10^{4}\,\frac{\text{J}}{\text{cm}^2}$ \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 3CSiC precipitate embedded in cSi 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_500fin.ps}
\end{center}
\caption{Radial distribution of the constructed 3CSiC precipitate embedded in cSi at temperatures below and above the Si melting transition point.}
\label{fig:md:pc_500fin}
\end{figure}
Investigating the radial distribution function shown in figure \ref{fig:md:pc_500fin}, 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 SiC and CC distribution, which essentially stays the same for both temperatures.
Thus, it is only the cSi surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two SiSi distributions.
This is surprising since the melting transition of plain cSi is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}.
Obviously the precipitate lowers the transition point of the surrounding cSi matrix.
This is indeed verified by visualizing the atomic data.
% ./visualize w 640 h 480 d saves/sic_prec_120Tm_cnt1 nll 11.56 0.56 11.56 fur 11.56 0.56 11.56 c 0.2 24.0 0.6 L 0 0 0.2 r 0.6 B 0.1
\begin{figure}[!ht]
\begin{center}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps}
\end{minipage}
\begin{minipage}{7cm}
\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
\end{minipage}
\end{center}
\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3CSiC precipitate in cSi at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).}
\label{fig:md:sic_melt}
\end{figure}
Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
As can be seen from the image at 520 ps melting of the Si surrounding in fact starts in the defective interface region of the 3CSiC precipitate and the cSi surrounding propagating outwards until the whole Si matrix is affected at 720 ps.
As predicted from the radial distribution data the precipitate itself remains stable.

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 $1\,^{\circ}\mathrm{C}/\text{ps}$.
However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.
+where $E$ is the total energy of the precipitate configuration at zero temperature.
+An interfacial energy of \unit[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 \unit[29.93]{\AA}.
+Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{4}$]{J/cm$^2$}.
+This value perfectly fits within the eperimentally estimated range of \unit[$28\times10^{4}$]{J/cm$^2$} \cite{taylor93}.
+Thus, the EA potential is considered an appropriate choice for the current study properly describing the energetics of interfaces.
+
+% todo
+% nice to reproduce this value!
+
+%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 \unit[2050]{$^{\circ}$C} with a heating rate of approximately \unit[75]{$^{\circ}$C/ps}.
+%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e. \unit[2940]{K}.
+%\begin{figure}[t]
+%\begin{center}
+%\includegraphics[width=0.7\textwidth]{fe_and_t_sic.ps}
+%\end{center}
+%\caption{Total energy and temperature evolution of a 3CSiC precipitate embedded in cSi at temperatures above the Si melting point.}
+%\label{fig:simulation:fe_and_t_sic}
+%\end{figure}
+%Figure \ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
+%The sudden increase of the total energy indicates possible melting occuring around \unit[2840]{K}.
+%\begin{figure}[ht]
+%\begin{center}
+%\includegraphics[width=0.7\textwidth]{pc_500fin.ps}
+%\end{center}
+%\caption{Radial distribution of a 3CSiC precipitate embedded in cSi at temperatures below and above the Si melting transition point.}
+%%\label{fig:simulation:pc_500fin}
+%\end{figure}
+%Investigating the radial distribution function shown in figure \ref{fig:simulation:pc_500fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the total energy plot in Fig. \ref{fig:simulation:fe_and_t_sic}.
+%However, the precipitate itself is not involved, as can be seen from the SiC and CC distribution, which essentially stays the same for both temperatures.
+%Thus, it is only the cSi surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two SiSi distributions.
+%This is surprising since the melting transition of plain cSi for the same heating conditions is expected at temperatures around \unit[3125]{K}, as will be discussed later in section \ref{subsection:md:tval}.
+%Obviously the precipitate lowers the transition point of the surrounding cSi matrix.
+%This is indeed verified by visualizing the atomic data.
+%% ./visualize w 640 h 480 d saves/sic_prec_120Tm_cnt1 nll 11.56 0.56 11.56 fur 11.56 0.56 11.56 c 0.2 24.0 0.6 L 0 0 0.2 r 0.6 B 0.1
+%\begin{figure}[t]
+%%\begin{center}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
+%\end{minipage}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps}
+%\end{minipage}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
+%\end{minipage}
+%\end{center}
+%\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3CSiC precipitate in cSi at \unit[200]{ps} (top left), \unit[520]{ps} (top right) and \unit[720]{ps} (bottom).}
+%\label{fig:simulation:sic_melt}
+%\end{figure}
+%Fig. \ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
+%As can be seen from the image at \unit[520]{ps} melting of the Si surrounding in fact starts in the defective interface region of the 3CSiC precipitate and the cSi surrounding propagating outwards until the whole Si matrix is affected at \unit[720]{ps}.
+%As predicted from the radial distribution data the precipitate itself indeed remains stable.
+%
+%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 \unit[100]{\%} of the Si melting temperature is chosen and cooled down to \unit[20]{$^{\circ}$C} with a cooling rate of \unit[1]{$^{\circ}$C/ps}.
+%However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
+%Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.

2.20.1