In the following the simulation methods used within the scope of this study are introduced.
Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
-For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}}\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic/posic.tar.bz2} including a library collecting respective MD subroutines was developed from scratch.
+For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/posic.tar.bz2}.
The basic ideas of MD in general and the adopted techniques as implemented in {\textsc posic} in particular are outlined in section \ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix \ref{app:d_tersoff}.
An overview of the most important tools within the MD package is given in appendix \ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2]
\end{eqnarray}
where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
-This is illustrated in Figure \ref{img:tersoff_angle}.
+This is illustrated in Fig. \ref{img:tersoff_angle}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{tersoff_angle.eps}
At this site the crystalline periodicity is interrupted.
An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
+A substitutional defect exists, if an atom belonging to the perfect crystal is replaced with an atom of another species.
The disturbance caused by these defects may result in the distortion of the surrounding atomic structure and is accompanied by an increase in configurational energy.
-Thus, next to the structure of the defect, the energy needed to create such a defect, i.e. the defect formation energy, is an important value characterizing the defect and its probability of occurence.
+Thus, next to the structure of the defect, the energy needed to create such a defect, i.e. the defect formation energy, is an important value characterizing the defect and likewise determining its relative stability.
-The methods presented in the last two chapters enable the investigation of defects.
-...
-constructing defect configuration intuitively followed by relaxation procedure
+The formation energy of a defect is defined by
+\begin{equation}
+E_{\text{f}}=E-\sum_i N_i\mu_i
+\text{ ,}
+\label{eq:basics:ef2}
+\end{equation}
+where $E$ is the total energy of the interstitial structure involving $N_i$ atoms of type $i$ with chemical potential $\mu_i$.
+Here, the chemical potentials are determined by the chemical potential of the respective equilibrium bulk structure, i.e. the cohesive energy per atom for the fully relaxed structure at zero temperature and pressure.
+Considering C interstitial defects in Si, the chemical potential for C could also be determined by the cohesive energies of Si and SiC according to the relation $\mu_{\text{C}}=\mu_{\text{SiC}}-\mu_{\text{Si}}$ of the chemical potentials.
+In this way, SiC is chosen as a reservoir for the C impurity.
+For defect configurations consisting of a single atom species the formation energy reduces to
+\begin{equation}
+E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
+ -E_{\text{coh}}^{\text{defect-free}}\right)N
+\text{ ,}
+\label{eq:basics:ef1}
+\end{equation}
+where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
+Clearly, for a single atom species equation \eqref{eq:basics:ef2} is equivalent to equation \eqref{eq:basics:ef1} since $NE_{\text{coh}}^{\text{defect}}$ is equal to the total energy of the defect structure and $NE_{\text{coh}}^{\text{defect-free}}$ corresponds to $N\mu$, provided the structure is fully relaxed at zero temperature.
+
+The methods presented in the last two chapters can be used to investigate defect structures and energetics.
+Therefore, a supercell containing the perfect crystal is generated in an initial process.
+If not by construction, the system should be fully relaxed.
+The substitutional or vacancy defect is realized by replacing or removing one atom contained in the supercell.
+Interstitial defects are created by adding an atom at positions located in the space between regular lattice sites.
+Once the intuitively created defect structure is generated structural relaxation methods will yield the respective local minimum configuration.
+Since the supercell approach applies periodic boundary conditions enough bulk material surrounding the defect is required to exclude possible interaction of the defect with its periodic image.
+
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=9cm]{unit_cell_e.eps}
+\end{center}
+\caption[Insertion positions for interstitial defect atoms in the diamond lattice.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial defect atom in the diamond lattice. The black dots correspond to the lattice atoms and the blue lines indicate the covalent bonds of the perfect diamond structure.}
+\label{fig:basics:ins_pos}
+\end{figure}
+The respective estimated interstitial insertion positions for various interstitial structures in a diamond lattice are displayed in Fig. \ref{fig:basics:ins_pos}.
+The labels of the interstitial types indicate their positions in the interstitial lattice.
+In a dumbbell (DB) configuration two atoms share a single lattice site along a certain direction that is also comprehended in the label of the defect.
+For the DB configurations the nearest atom of the bulk lattice is slightly displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ of the unit cell length respectively.
+This is indicated by the dashed, unfilled circles.
+By this, high forces, which might enable the system to overcome barriers of the local minimum configuration and, thus, result in a different structure, are avoided.
\section{Migration paths and diffusion barriers}
\label{section:basics:migration}
+Investigating diffusion mechanisms is based on determining migration paths inbetween two local minimum configurations of an atom at different locations in the lattice.
+During migration, the total energy of the system increases, traverses at least one maximum of the configurational energy and finally decreases to a local minimum value.
+The maximum difference in energy is the barrier necessary for the respective migration process.
+The path exhibiting the minimal energy difference determines the diffusion path and associated diffusion barrier and the maximum configuration turns into a saddle point configuration.
+
+\begin{figure}[t]
+\begin{center}
+\subfigure[]{\label{fig:basics:crto}
+\includegraphics[width=0.5\textwidth]{crt_orig.eps}}
+\subfigure[]{\label{fig:basics:crtm}
+\includegraphics[width=0.5\textwidth]{crt_mod.eps}}
+\end{center}
+\caption{Schematic of the constrained relaxation technique (a) and of a modified version (b) used to obtain migration pathways and corresponding configurational energies.}
+\label{fig:basics:crt}
+\end{figure}
+One possibility to compute the migration path from one stable cofiguration into another one is provided by the constrained relaxation technique (CRT) \cite{kaukonen98}.
+The atoms involving great structural changes in the diffusion process are moved stepwise from the starting to the final position and relaxation after each step is only allowed in the plane perpendicular to the direction of the vector connecting its starting and final position.
+This is illustrated in Fig. \ref{fig:basics:crto}.
+The number of steps required for smooth transitions depends on the shape of the potential energy surface.
+No constraints are applied to the remaining atoms to allow for the relaxation of the surrounding lattice.
+To prevent the remaining lattice to shift according to the displacement of the defect, ohowever, some atoms far away from the defect region should be fixed in all three coordinate directions.
+However, for the present study, the method tremendously failed.
+Abrupt changes in structure and configurational energy occured among relaxed structures of two successive displacement steps.
+For some structures even the expected final configurations are not obtained.
+Thus, the method mentioned above is adjusted adding further constraints in order to obtain smooth transitions with repsect to energy and structure.
+In the modified method all atoms are stepwise displaced towards their final positions.
+In addition to this, relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in Fig. \ref{fig:basics:crtm}.
+In the modified version respective energies could be higher than the real ones due to the additional constraints that prevent a more adequate relaxation until the final copnfiguration is reached.
+
+Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i .e. the method does not guarantee to find the true minimum energy path.
+Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by caculating the respective vibrational modes.
+
+Modifications used to add the CRT feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
+Due to these constraints obtained activation energies can effectively be higher.
+
% todo
% advantages of pw basis with respect to hellmann feynman forces / pulay forces
Periodic boundary conditions in each direction are applied.
All point defects are calculated for the neutral charge state.
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=9cm]{unit_cell_e.eps}
-\end{center}
-\caption[Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.]{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
-\label{fig:defects:ins_pos}
-\end{figure}
-
-The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the
-\begin{itemize}
- \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
- \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
- \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
- \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
- \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
-\end{itemize}
-interstitial position.
-For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
-A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
-
-From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
-For defect configurations consisting of a single atom species the formation energy is defined as
-\begin{equation}
-E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
- -E_{\text{coh}}^{\text{defect-free}}\right)N
-\label{eq:defects:ef1}
-\end{equation}
-where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure.
-The formation energy of defects consisting of two or more atom species is defined as
-\begin{equation}
-E_{\text{f}}=E-\sum_i N_i\mu_i
-\label{eq:defects:ef2}
-\end{equation}
-where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
-The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
-
\section{Silicon self-interstitials}
Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
The silicon dumbbell partner remains the same.
The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-\begin{figure}[t!h!]
-\begin{center}
-\begin{minipage}{6cm}
-\underline{Original}\\
-\includegraphics[width=6cm]{crt_orig.eps}
-\end{minipage}
-\begin{minipage}{1cm}
-\hfill
-\end{minipage}
-\begin{minipage}{6cm}
-\underline{Modified}\\
-\includegraphics[width=6cm]{crt_mod.eps}
-\end{minipage}
-\end{center}
-\caption{Schematic of the constrained relaxation technique (CRT) (left) and of the modified version (right) used to obtain migration pathways and corresponding activation energies.}
-\label{fig:defects:crt}
-\end{figure}
-Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
-One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
-This is called the constrained relaxation technique (CRT), which is schematically displayed in the left part of figure \ref{fig:defects:crt}.
-No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
-To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
-However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
-Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
-For some structures even the expected final configurations were never obtained.
-Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
-In this new method all atoms are stepwise displaced towards their final positions.
-Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in the right part of figure \ref{fig:defects:crt}.
-The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
-{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
-
\subsection{Migration barriers obtained by quantum-mechanical calculations}
In the following migration barriers are investigated using quantum-mechanical calculations.
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