+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 c c c}
+\hline
+\hline
+ & Si (dia) & C (dia) & 3C-SiC \\
+\hline
+$a$ [\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!
+
+\section{Classical potential MD}
+\label{section:classpotmd}
+
+The classical potential MD method is much less computationally costly compared to the highly accurate quantum-mechanical 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 c-Si supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
+A Tersoff-like 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.
+Defect structures as well as the simulations modeling the SiC precipitation are performed in the isothermal-isobaric $NpT$ ensemble.
+
+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 3C-SiC precipitation in c-Si.
+% todo
+% rather a first investigation than a test
+
+\subsection{Time step}
+
+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 $9\times9\times9$ 3C-SiC 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 3C-SiC 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{3C-SiC precipitate in c-Si}
+\label{section:simulation:prec}
+
+Below, a spherical 3C-SiC precipitate enclosed in a c-Si 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 3C-SiC 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 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 according to experimental results as discussed in section \ref{subsection:ibs} and \ref{section:assumed_prec}.
+This corresponds to a spherical 3C-SiC precipitate with a radius of approximately \unit[3]{nm}.
+The initial precipitate configuration is constructed in two steps.
+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 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 and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
+This is entirely described by the equation
+\begin{equation}
+\frac{8}{a_{\text{Si}}^3}(
+V
+-\frac{4}{3}\pi x^3)+
+\frac{4}{y^3}\frac{4}{3}\pi x^3
+=21^3\cdot 8
+\text{ ,}
+\label{eq:simulation:constr_sic_01}
+\end{equation}
+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:simulation: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:simulation:constr_sic_03}
+\end{equation}
+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 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:simulation:sic_prec}.
+\begin{table}[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:simulation:constr_sic_02} and \eqref{eq:simulation:constr_sic_03} in the 3C-SiC 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 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 \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}.