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.
m_i \ddot{\bf r}_i = {\bf F}_i \Leftrightarrow
m_i \dot{\bf r}_i = {\bf p}_i\textrm{, }
\dot{\bf p}_i = {\bf F}_i\textrm{ .}
+\label{eq:basics:newton}
\end{equation}
The forces ${\bf F}_i$ are obtained from the potential energy $U(\{{\bf r}\})$:
\begin{equation}
{\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
+\label{eq:basics:force}
\end{equation}
Given the initial conditions ${\bf r}_i(t_0)$ and $\dot{\bf r}_i(t_0)$ the equations can be integrated by a certain integration algorithm.
The solution of these equations provides the complete information of a system evolving in time.
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}
In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space.
In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictious anyway.
-\subsection{Structural relaxation and the Hellmann-Feynman theorem}
+\subsection{Structural relaxation and Hellmann-Feynman theorem}
-what is relaxed, ions!
-if md based relaxaion followed the respective equations of motion are valid ...
-of course better cg than simple moveing ions according to force ...
-force as a derivative of total energy with respect to positions of the ions ...
+Up to this point, the system is in the ground state with respect to the electronic subsystem, while the positions of the ions as well as size and shape of the supercell are fixed.
+To investigate equilibrium structures, however, the ionic subsystem must also be allowed to relax into a minimum energy configuration.
+Local minimum configurations can be easily obtained in a MD-like way by moving the nuclei over small distances along the directions of the forces, as discussed in the MD chapter above.
+Clearly, the conjugate gradient method constitutes a more sophisticated scheme, which will locate the equilibrium positions of the ions more rapidly.
+To find the global minimum, i.e. the absolute ground state, methods like simulated annealing or the Monte Carlo technique, which allow the system to escape local minima, have to be used for the search.
+The force on an ion is given by the negative derivative of the total energy with respect to the position of the ion.
+However, moving an ion, i.e. altering its position, changes the wave functions to the KS eigenstates corresponding to the new ionic configuration.
+Writing down the derivative of the total energy $E$ with respect to the position $\vec{R}_i$ of ion $i$
+\begin{equation}
+\frac{dE}{d\vec{R_i}}=
+ \sum_j \Phi_j^* \frac{\partial H}{\partial \vec{R}_i} \Phi_j
++\sum_j \frac{\partial \Phi_j^*}{\partial \vec{R}_i} H \Phi_j
++\sum_j \Phi_j^* H \frac{\partial \Phi_j}{\partial \vec{R}_i}
+\text{ ,}
+\end{equation}
+indeed reveals a contributon to the chnage in total energy due to the change of the wave functions $\Phi_j$.
+However, provided that the $\Phi_j$ are eigenstates of $H$, it is easy to show that the last two terms cancel each other and in the special case of $H=T+V$ the force is given by
+\begin{equation}
+\vec{F}_i=-\sum_j \Phi_j^*\Phi_j\frac{\partial V}{\partial \vec{R}_i}
+\text{ .}
+\end{equation}
+This is called the Hellmann-Feynman theorem \cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.
-\section{Modeling of defects}
+\section{Modeling of point defects}
\label{section:basics:defects}
-...
-constructing defect configuration intuitively followed by relaxation procedure
+Point defects are defects that affect a single lattice site.
+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 likewise determining its relative stability.
+
+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
+
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