The basic idea is that, in real systems, the bond order, i.e.\ the strength of the bond, depends upon the local environment~\cite{abell85}.
Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
Although the bond strength intricately depends on geometry, the focus on coordination, i.e.\ the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
-If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
+If the energy per bond decreases rapidly enough with increasing coordination, the most stable structure will be the dimer.
In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
Based on pseudopotential theory, the bond order term $b_{ijk}$ limiting the attractive pair interaction is of the form $b_{ijk}\propto Z^{-\delta}$ where $Z$ is the coordination number and $\delta$ a constant~\cite{abell85}, which is $\frac{1}{2}$ in the second-moment approximation within the tight binding scheme~\cite{horsfield96}.
\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 particles, 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.
+Providing a stable integration algorithm, the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, 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.
\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.
+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
\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.
+Berendsen~et~al.\ proposed a method~\cite{berendsen84}, 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}
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.
+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.
Following the path of Schr\"odinger, the problem in quantum-mechanical modeling of describing the many-body problem, i.e.\ a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all 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 finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters.
-Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinants to be evaluated massively increases.
+Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy, the amount of Slater determinants to be evaluated massively increases.
In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates.
In the following sections, the basic idea of DFT will be outlined.
An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$~\cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff~\cite{tersoff90} and Dal Pino et al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
-This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
+Tersoff~\cite{tersoff90} and Dal~Pino~et~al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
+This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al.\ is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
In summary, pairs of C$_{\text{i}}$ DBs and vacancies, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.
Furthermore, small activation energies, even for transitions into the ground state exist.
-If the vacancy is created at position 1 the system will end up in a configuration of C$_{\text{s}}$ anyways.
+If the vacancy is created at position 1, the system will end up in a configuration of C$_{\text{s}}$ anyways.
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.
\subsection{Combinations of \si{} and \cs}
Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects~\cite{leung99,al-mushadani03} as well as for C point defects~\cite{dal_pino93,capaz94}.
The ground-state configurations of these defects, i.e.\ the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$~\cite{leung99,al-mushadani03} as well as theoretical~\cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental~\cite{watkins76,song90} studies on C$_{\text{i}}$.
-A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values~\cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
+A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et~al.~\cite{capaz94} to experimental values~\cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
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