+All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unti cells and periodic boundary conditions are imposed on the boundaries of that simulation cell.
+Strictly, these supercells become the unit cells, which, by a periodic sequence, compose the bulk material that is actually investigated by this approach.
+Thus, importance need to be attached to the construction of the supercell.
+Three basic types of supercells to compose the initial Si bulk lattice, which can be scaled by integers in the different directions, are considered.
+The basis vectors of the supercells are shown in Fig. \ref{fig:simulation:sc}.
+\begin{figure}[t]
+\begin{center}
+\subfigure[]{\label{fig:simulation:sc1}\includegraphics[width=0.3\textwidth]{sc_type0.eps}}
+\subfigure[]{\label{fig:simulation:sc2}\includegraphics[width=0.3\textwidth]{sc_type1.eps}}
+\subfigure[]{\label{fig:simulation:sc3}\includegraphics[width=0.3\textwidth]{sc_type2.eps}}
+\end{center}
+\caption{Basis vectors of three basic types of supercells used to create the initial Si bulk lattice.}
+\label{fig:simulation:sc}
+\end{figure}
+Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell.
+The basis is face-centered cubic (fcc) and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
+Two atoms, one at $(0,0,0)$ and the other at $(0.25,0.25,0.25)$ with respect to the basis, generate the Si diamond primitive cell.
+Type 2 (Fig. \ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
+The basis is given by $x_1=(0.5,-0.5,0)$, $x_2=(0.5,0.5,0)$ and $x_3=(0,0,1)$.
+Type 3 (Fig. \ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig. \ref{fig:sic:unit_cell}.
+The basis is simple cubic.
+
+In the following an overview of the different simulation procedures and respective parameters is presented.
+These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used and on what is going to be investigated.
+
+\section{DFT calculations}
+\label{section:simulation:dft_calc}
+
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}.
+The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}.
+An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
+Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
+Spin polarization has been fully accounted for.
+The electronic ground state is calculated by an interative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
+
+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.
+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}.
+
+% todo
+% All point defects are calculated for the neutral charge state.
+
+Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed.
+These parameters include the size of the supercell, cut-off energy and $k$ point mesh.
+The choice of these parameters is considered to reflect a reasonable treatment with respect to both, computational efficiency and accuracy, as will be shown in the next sections.
+Furthermore, criteria concerning the choice of the potential and the exchange-correlation (XC) functional are being outlined.
+Finally, the utilized parameter set is tested by comparing the calculated values of the cohesive energy and the lattice constant to experimental data.
+
+\subsection{Supercell}
+
+Describing defects within the supercell approach runs the risk of a possible interaction of the defect with its periodic, artificial images.
+Obviously, the interaction reduces with increasing system size and will be negligible from a certain size on.
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{si_self_int_thesis.ps}
+\end{center}
+\caption{Defect formation energies of several defects in c-Si with respect to the size of the supercell.}
+\label{fig:simulation:ef_ss}
+\end{figure}
+To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated.
+An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh \cite{monkhorst76} is used.
+The results are displayed in Fig. \ref{fig:simulation:ef_ss}.
+The formation energies converge fast with respect to the system size.
+Thus, investigating supercells containing more than 56 primitive cells or $112\pm1$ atoms should be reasonably accurate.
+
+\subsection{Brillouin zone sampling}
+
+Throughout this work sampling of the BZ is restricted to the $\Gamma$ point.
+The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since $c_{i,q}=c_{i,-q}^*$, where the $c_{i,q}$ are the Fourier coefficients of the wave function.
+As discussed in section \ref{subsection:basics:bzs} this does not pose a severe limitation if the supercell is large enough.
+Indeed, it was shown \cite{dal_pino93} that already for calculations involving only 32 atoms energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point \cite{baldereschi73}, which constitutes a mean-value point in the BZ.
+Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $k$-point mesh.
+
+\subsection{Energy cut-off}
+
+To determine an appropriate cut-off energy of the plane-wave basis set a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA.
+% todo
+% mention that results are within lda
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps}
+\end{center}
+\caption{Lattice constants of 3C-SiC with respect to the cut-off energy used for the plane-wave basis set.}
+\label{fig:simulation:lc_ce}
+\end{figure}
+Fig. \ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
+As can be seen, convergence is reached already for low energies.
+Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, is sufficient for the plane-wave expansion.
+
+\subsection{Potential and exchange-correlation functional}
+
+To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
+To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
+Next to the ultra-soft pseudopotentials \cite{vanderbilt90} {\textsc vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
+The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in {\textsc vasp}.
+
+\begin{table}[t]
+\begin{center}
+\begin{tabular}{l r c c c c c}
+\hline
+\hline
+ & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\
+\hline
+Si (dia) & $a$ [\AA] & 5.389 & 5.455 & - & - & 5.429 \\
+ & $\Delta_a$ [\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\
+ & $E_{\text{coh}}$ [eV] & -5.277 & -4.591 & - & - & -4.63 \\
+ & $\Delta_E$ [\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\
+\hline
+C (dia) & $a$ [\AA] & 3.527 & 3.567 & - & - & 3.567 \\
+ & $\Delta_a$ [\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\
+ & $E_{\text{coh}}$ [eV] & -8.812 & -7.703 & - & - & -7.374 \\
+ & $\Delta_E$ [\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\
+\hline
+3C-SiC & $a$ [\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\
+ & $\Delta_a$ [\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\
+ & $E_{\text{coh}}$ [eV] & -7.318 & -6.426 & -7.371 & -6.491 & -6.340 \\
+ & $\Delta_E$ [\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultr-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
+\label{table:simulation:potxc}
+\end{table}
+Table \ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional.
+As expected, cohesive energies are poorly reproduced by the LDA whereas the equilibrium lattice constants are in good agreement.
+Using GGA together with the ultra-soft pseudopotential yields improved lattice constants and, more importantly, a very nice agreement of the cohesive energies to the experimental data.
+The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e. overestimated binding energies.
+Thus, the PAW \& LDA combination is not pursued.
+Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped.
+To conclude, the combination of ultr-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
+
+\subsection{Lattice constants and cohesive energies}
+
+As a last test, the equilibrium lattice parameters and cohesive energies of Si, C (diamond) and 3C-SiC are again compared to experimental data.
+However, in the current calculations, the entire parameter set as determined in the beginning of this section is applied.
+\begin{table}[t]
+\begin{center}
+\begin{tabular}{l r c c c c c}
+\hline
+\hline
+ & Si (dia) & C (dia) & 3C-SiC \\
+$a$ [\AA] & 5.458 & 3.562 & 4.365 \\
+$\Delta_a$ [\%] & 0.5 & 0.1 & 0.1 \\
+\hline
+$E_{\text{coh}}$ [eV] & -4.577 & -7.695 & -6.419 \\
+$\Delta_E$ [\%] & 1.1 & 4.4 & 1.2 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\caption{Equilibrium lattice constants and cohesive energies of Si, C (diamond) and 3C-SiC using the entire parameter set as determined in the beginning of this section.}
+\label{table:simulation:paramf}
+\end{table}
+Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment.
+A nice agreement with experimental results is achieved.
+Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations.
+
+
+% todo
+% rewrite dft chapter
+% ref for experimental values!
+