Modelling the processes mentioned above requires the simulation of a large amount of atoms ($\approx 10^5-10^6$), which inevitably dictates the atomic interaction to be described by computationally efficient classical potentials.
These are, however, less accurate compared to quantum-mechnical methods and theire applicability for the description of the physical problem has to be verified first.
The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic potential (EDIP)\cite{bazant96,bazant97,justo98}.
+These potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first principles methods.
Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials, e.g. the one by Gao and Weber\cite{gao02} as well as the one by Erhart and Albe\cite{albe_sic_pot}.
All these potentials are short range potentials employing a cut-off function, which drops the atomic interaction to zero inbetween the first and second next neighbor distance.
In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two next neighbored atoms.
-In a previous study\cite{zirkelbach10a} we approved ... influence on migration ... crucial for problem under study.
-However ... considered good, especially non-zero temperatures(?) ...
-
-HIER WEITER ...
-
-An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.
-Despite their shortcomings these potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first principles methods.
-Remaining shortcomings have frequently been resolved by modifying the interaction\cite{tang95,gao02a,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.
+In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
+Using the Erhart/Albe (EA) potential\cite{albe_sic_pot} an overestimated barrier height compared to ab initio calculations and experiment is obtained.
+A proper edscription of C diffusion, however, is crucial for the problem under study.
In this work, a combined ab initio and empirical potential simulation study on the initially mentioned SiC precipitation mechanism has been performed.
-High accurate quantum-mechanical results have been used to identify shortcomings of the classical potentials, which then are taken into account in these type of simulations.
+High accurate quantum-mechanical results have been used to identify shortcomings of the classical potentials, which are then taken into account in these type of simulations.
% --------------------------------------------------------------------------------
\section{Methodology}
% ----- DFT ------
-The first principles DFT calculations have been performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
-The Kohn-Sham equations were solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\cite{perdew86,perdew92}.
+The first principles DFT calculations have been performed with the plane-wave based Vienna ab initio Simulation package (VASP)\cite{kresse96}.
+The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
Throughout this work, an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
-The defect structures and the migration paths have been modelled in cubic supercells containing 216 Si atoms.
-The ions and cell shape were allowed to change in order to realize a constant pressure simulation. Spin polarization has been fully accounted for.
+The defect structures and the migration paths have been modeled in cubic supercells containing 216 Si atoms.
+The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
+Spin polarization has been fully accounted for.
% ------ Albe potential ---------
-For the classical potential calculations, a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms has been used.
+For the classical potential calculations, defect structures were modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms.
+Reproducing the SiC precipitation was attempted by the successive insertion of 6000 C atoms (the number necessary to form a 3C-SiC precipitate with a radius of $\approx 3.1$ nm) into the Si host, which has a size of 31 Si unit cells in each direction consisting of 238328 Si atoms.
+At constant temperature 10 atoms are inserted at a time.
+Three different regions within the total simulation volume are considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, whih holds the necessary amount of Si atoms of the precipitate.
A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
-% ref mod: extension for short distances
The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
-% ref mod: time constants
-%Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84}.
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.
-% ref mod: time constants + language (Verlet)
-%Structural relaxation in the MD run is achieved by the velocity Verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[1]{fs} resulting in direct velocity scaling and the temperature set to zero Kelvin.
-Structural relaxation in the MD run is achieved by the Velocity Verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs} and the temperature set to zero Kelvin.
-Additionally, a time constant of \unit[1]{fs} resulting in direct velocity scaling was used for relaxation within the mobility calculations.
-% ref mod: time step
-A fixed time step of \unit[1]{fs} for integrating the equations of motion was used.
+The temperature is kept constant by the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs}.
+Integration of equations of motion is realized by the velocity Verlet algorithm\cite{verlet67} and a fixed time step of \unit[1]{fs}.
+For structural relaxation of defect structures the same algorith is used with the temperature set to 0 K.
\section{Results}
-\subsection{Carbon interstitials in various geometries}
+\subsection{Carbon and silicon defect configurations}
-Table~\ref{tab:defects} summarizes the formation energies of defect structures for the EA and DFT calculations performed in this work as well as further results from literature.
-The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined in the same way as in the articles used for comparison\cite{tersoff90,dal_pino93} chosing SiC as a reservoir for the carbon impurity in order to determine $\mu_{\text{C}}$.
-Relaxed geometries are displayed in Fig.~\ref{fig:defects}.
+Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations.
+The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing SiC as a reservoir for the carbon impurity in order to determine $\mu_{\text{C}}$.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
- & T & H & \hkl<1 0 0> dumbbell & \hkl<1 1 0> dumbbell & S & BC \\
+ & C-Si \hkl<1 0 0> dumbbell & C$_{\text{s}}$ & C-Si \hkl<1 1 0> dumbbell & C$_{\text{i}}$ bond-centered & Si$_{\text{i}}$ \hkl<1 1 0> dumbbell & Si$_{\text{i}}$ T\\
\hline
Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
- Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
- ab initio\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\end{tabular}
\end{ruledtabular}
-\caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to substitutional C. Formation energies for unstable configurations obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
+\caption{Formation energies of C and Si point defects in c-Si determined by classical potential and ab initio methods. The formation energies are given in electron Volt. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to substitutional C. Formation energies for unstable configurations obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects}
\end{table*}
\begin{figure}
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