Thus, in order to obtain more accurate results quantum-mechanical calculations from first principles based on density functional theory (DFT) were performed.
The Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96} is used for this purpose.
The relevant basics of DFT are described in section \ref{section:dft} while an overview of utilities mainly used to create input or parse output data of {\textsc vasp} is given in appendix \ref{app:code}.
-The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the system to be described to be much smaller in size.
+The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the simulated system to be much smaller in size.
Thus, investigations based on DFT are restricted to single defects or combinations of two defects in a rather small Si supercell, their structural relaxation as well as some selected diffusion processes.
Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section \ref{section:basics:defects}.
The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section \ref{section:basics:migration} and modifications to the {\textsc vasp} code implementing this method are presented in appendix \ref{app:patch_vasp}.
\subsection{Interaction potentials for silicon and carbon}
\label{subsection:interact_pot}
-% todo
-% rewrite!
-
The potential energy of $N$ interacting atoms can be written in the form
\begin{equation}
U(\{{\bf r}\}) = \sum_i U_1({\bf r}_i) + \sum_i \sum_{j>i} U_2({\bf r}_i,{\bf r}_j) + \sum_i \sum_{j>i} \sum_{k>j>i} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) \ldots
The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increases in computational efficiency for the evaluation of forces and energy based on the short-range potential.
Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
-...
-The basic idea is that, in real systems, the bond order depends upon the local environment.
-An atom with many neighbors forms weaker bonds than an atom with few neighbors since .
-
-Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots
-
+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.
+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 $b_{ijk}$ limitting 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 seond-moment approximation within the tight binding scheme \cite{horsfield96}.
+
+Tersoff incorporated the concept of bond order in a three-body potential formalism.
The interatomic potential is taken to have the form
\begin{eqnarray}
E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
\end{array} \right.
\label{eq:basics:fc}
\end{equation}
-The function $b_{ij}$ represents a measure of the bond order, monotonically decreasing with the coordination of atoms $i$ and $j$.
+As discussed above, $b_{ij}$ represents a measure of the bond order, monotonously decreasing with the coordination of atoms $i$ and $j$.
It is of the form:
\begin{eqnarray}
b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\
\caption{Angle between bonds of atoms $i,j$ and $i,k$.}
\label{img:tersoff_angle}
\end{figure}
+The angular dependence does not give a fixed minimum angle between bonds since the expression is embedded inside the bond order term.
+The relation to the above discussed bond order potential becomes obvious if $\chi=1, \beta=1, n=1, \omega=1$ and $c=0$.
+Parameters with a single subscript correspond to the parameters of the elemental system \cite{tersoff_si3,tersoff_c} while the mixed parameters are obtained by interpolation from the elemental parameters by the arithmetic or geometric mean.
+The elemental parameters were obtained by fit with respect to the cohesive energies of real and hypothetical bulk structures and the bulk modulus and bond length of the diamond structure.
+New parameters for the mixed system are $\chi$, which is used to finetune the strength of heteropolar bonds, and $\omeag$, which is set to one for the C-Si interaction but is available as a feature to permit the application of the potential to more drastically different types of atoms in the future.
The force acting on atom $i$ is given by the derivative of the potential energy.
For a three body potential ($V_{ij} \neq V{ji}$) the derivation is of the form
\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 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 interaction.
+
+For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted potentials for the Si-Si, C-C and Si-C interaction \cite{albe_sic_pot}.
+The functional form is nearly the same like the one proposed by Tersoff.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+For Si ...
\subsection{Statistical ensembles}
\label{subsection:statistical_ensembles}
+By default ... NVE
+
+However, we need to control T -> NVT
+.. and p -> NpT ...
+
+
\section{Denstiy functional theory}
\label{section:dft}
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