+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}
+