+Details of the Tersoff potential derivative are presented in appendix \ref{app:d_tersoff}.
+
+\subsubsection{Improved analytical bond order potential}
+
+Although the Tersoff potential is one of the most widely used potentials there are some shortcomings.
+Describing the Si-Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
+Due to this and since the first approach labeled T1 \cite{tersoff_si1} turned out to be unstable \cite{dodson87}, two further parametrizations exist, T2 \cite{tersoff_si2} and T3 \cite{tersoff_si3}.
+While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
+However, T3, which is used in the Si/C potential, suffers from an underestimation of the dimer binding energy.
+Similar behavior is found for the C-C interaction.
+
+For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction \cite{albe_sic_pot}.
+The functional form is similar to the one proposed by Tersoff.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
+The new parameter set for the C-C interaction yields improved dimer properties while at the same time delivers a description of the bulk phase similar to the Tersoff potential.
+The potential succeeds in the description of the low as well as high coordinated structures.
+The description of elastic properties of SiC is improved with respect to the potentials available in literature.
+Defect properties are only fairly reproduced but the description is comparable to previously published potentials.
+It is claimed that the potential enables modeling of widely different configurations and transitions among these and has recently been used to simulate the inert gas condensation of Si-C nanoparticles \cite{erhart04}.
+Therefore the Erhart/Albe (EA) potential is considered the superior analytical bond order potential to study the SiC precipitation and associated processes in Si.
+
+\subsection{Verlet integration}
+\label{subsection:integrate_algo}
+
+A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
+The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
+Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
+\begin{equation}
+\vec{r}_i(t+\delta t)=
+\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)+
+\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
+\label{basics:verlet:taylor1}
+\end{equation}
+\begin{equation}
+\vec{r}_i(t-\delta t)=
+\vec{r}_i(t)-\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)-
+\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
+\label{basics:verlet:taylor2}
+\end{equation}
+where $\vec{v}_i=\frac{d}{dt}\vec{r}_i$ are the velocities, $\vec{f}_i=m\frac{d}{dt^2}\vec{r}_i$ are the forces and $\vec{b}_i=\frac{d}{dt^3}\vec{r}_i$ are the third derivatives of the particle positions with respect to time.
+The Verlet algorithm is obtained by summarizing and substracting equations \eqref{basics:verlet:taylor1} and \eqref{basics:verlet:taylor2}
+\begin{equation}
+\vec{r}_i(t+\delta t)=
+2\vec{r}_i(t)-\vec{r}_i(t-\delta t)+\frac{\delta t^2}{m_i}\vec{f}_i(t)+
+\mathcal{O}(\delta t^4)
+\end{equation}
+\begin{equation}
+\vec{v}_i(t)=\frac{1}{2\delta t}[\vec{r}_i(t+\delta t)-\vec{r}_i(t-\delta t)]+
+\mathcal{O}(\delta t^3)
+\end{equation}
+the truncation error of which is of order $\delta t^4$ for the positions and $\delta t^3$ for the velocities.
+The velocities, although not used to update the particle positions, are not synchronously determined with the positions but drag behind one step of discretization.
+The Verlet algorithm can be rewritten into an equivalent form, which updates the velocities and positions in the same step.
+The so-called velocity Verlet algorithm is obtained by combining \eqref{basics:verlet:taylor1} with equation \eqref{basics:verlet:taylor2} displaced in time by $+\delta t$
+\begin{equation}
+\vec{v}_i(t+\delta t)=
+\vec{v}_i(t)+\frac{\delta t}{2m_i}[\vec{f}_i(t)+\vec{f}_i(t+\delta t)]
+\end{equation}
+\begin{equation}
+\vec{r}_i(t+\delta t)=
+\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t) \text{ .}
+\end{equation}
+Since the forces for the new positions are required to update the velocity the determination of the forces has to be carried out within the integration algorithm.
+
+\subsection{Statistical ensembles}
+\label{subsection:statistical_ensembles}
+
+Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
+Providing a stable integration algorithm the total energy $E$, i.e. the kinetic and configurational energy of the paticles, is conserved.
+This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
+
+However, the successful formation of SiC dictates precise control of temperature by external heating.
+While the temperature of such a system is well defined, the energy is no longer conserved.
+The microscopic states of a system, which is in thermal equilibrium with an external thermal heat bath, are represented by the $NVT$ ensemble.
+In the so-called canonical ensemble the temperature $T$ is related to the expactation value of the kinetic energy of the particles, i.e.
+\begin{equation}
+\langle E_{\text{kin}}\rangle = \frac{3}{2}Nk_{\text{B}}T \text{, }
+E_{\text{kin}}=\sum_i \frac{\vec{p}^2_i}{2m_i} \text{ .}
+\label{eq:basics:ts}
+\end{equation}
+The volume of the synthesized material can hardly be controlled in experiment.
+Instead the pressure can be adjusted.
+Holding constant the pressure in addition to the temperature of the system its states are represented by the isothermal-isobaric $NpT$ ensemble.
+The expression for the pressure of a system derived from the equipartition theorem is given by
+\begin{equation}
+pV=Nk_{\text{B}}T+\langle W\rangle\text{, }W=-\frac{1}{3}\sum_i\vec{r}_i\nabla_{\vec{r}_i}U
+\text{, }
+\label{eq:basics:ps}
+\end{equation}
+where $W$ is the virial and $U$ is the configurational energy.
+
+Berendsen~et~al.~\cite{berendsen84} proposed a method, which is easy to implement, to couple the system to an external bath with constant temperature $T_0$ or pressure $p_0$ with adjustable time constants $\tau_T$ and $\tau_p$ determining the strength of the coupling.
+Control of the respective variable is based on the relations given in equations \eqref{eq:basics:ts} and \eqref{eq:basics:ps}.
+The thermostat is achieved by scaling the velocities of all atoms in every time step $\delta t$ from $\vec{v}_i$ to $\lambda \vec{v}_i$, with
+\begin{equation}
+\lambda=\left[1+\frac{\delta t}{\tau_T}(\frac{T_0}{T}-1)\right]^\frac{1}{2}
+\text{ ,}
+\end{equation}
+where $T$ is the current temperature according to equation \eqref{eq:basics:ts}.
+The barostat adjusts the pressure by changing the virial through scaling of the particle positions $\vec{r}_i$ to $\mu \vec{r}_i$ and the volume $V$ to $\mu^3 V$, with
+\begin{equation}
+\mu=\left[1-\frac{\beta\delta t}{\tau_p}(p_0-p)\right]^\frac{1}{3}\text{ ,}
+\end{equation}
+where $\beta$ is the isothermal compressibility and $p$ corresponds to the current pressure, which is determined by equation \eqref{eq:basics:ps}.
+
+Using this method the system does not behave like a true $NpT$ ensemble.
+On average $T$ and $p$ correspond to the expected values.
+For large enough time constants, i.e. $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
+The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
+It provides a stable algorithm that allows smooth changes of the system to new values of temperature or pressure, which is ideal for the investigated problem.
+
+\section{Denstiy functional theory}
+\label{section:dft}
+
+In quantum-mechanical modeling the problem of describing a many-body problem is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of the nuclei and electrons.
+The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction.
+This cannot be solved exactly and there are several layers of approximations to reduce the number of parameters.
+In density functional theory (DFT) the problem is recasted to the charge density $n(\vec{r})$ instead of using the description by a wave function.
+Formally DFT can be regarded as an exactification of both, the Thomas Fermi and Hartree theory.
+
+Since {\textsc vasp} \cite{kresse96} is used in this work, theory and implementation of sophisticated algorithms of DFT codes is not subject of this work.
+Thus, the content of the following sections is restricted to the very basic idea of DFT.
+
+\subsection{Born-Oppenheimer approximation}
+
+The first approximation ...
+
+\subsection{Hohenberg-Kohn theorem}
+
+\subsection{Effective potential}
+
+\subsection{Kohn-Sham system}
+
+\subsection{Approximations for exchange and correlation}
+
+\subsection{Pseudopotentials}
+
+\section{Modeling of defects}
+\label{section:basics:defects}
+
+\section{Migration paths and diffusion barriers}
+\label{section:basics:migration}