-\chapter{Basics}
+\chapter{Basic principles of utilized simulation techniques}
+\label{chapter:basics}
+
+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 simulations were chosen in order to model 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} an appropriate MD code called {\em posic ({\bf P}recipitation {\bf o}f {\bf SiC})}\footnote{The source code will be made available for download at http://www.physik.uni-augsburg.de/~zirkelfr/posic/posic.tar.bz2} including a library collecting respective MD subroutines was developed from scratch.
+The basic ideas of MD in general and the adopted techniques as implemented in {\em 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}.
+In addition to the classical potential approach ...
+Highly accurate technique DFT, VASP code, VASP mods, tools in appendix ...
+
+Determining the formation energies if defects is ...
+
+Migration pathways were investigated using the , which is explained in cloder detail in ...
+
+
+\section{Molecular dynamics simulations}
+\label{section:md}
\begin{quotation}
\dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{}
-\cite{laplace}
\begin{flushright}
-{\em Marquis Pierre Simon de Laplace, 1814.}
+{\em Marquis Pierre Simon de Laplace, 1814, \cite{laplace}.}
\end{flushright}
\end{quotation}
-\section{Molecular dynamics simulations}
-
Pierre Simon de Laplace phrased this vision in terms of a controlling, omniscient instance - the {\em Laplace demon} - which would be able to look into the future as well as into the past due to the deterministic nature of processes, governed by the solution of differential equations.
Although Laplace's vision is nowadays corrected by chaos theory and quantum mechanics, it expresses two main features of classical mechanics, the determinism of processes and time reversibility of the fundamental equations.
This understanding was one of the first ideas for doing molecular dynamics simulations, considering an isolated system of particles, the behaviour of which is fully determined by the solution of the classical equations of motion.
\subsection{Introduction to molecular dynamics simulations}
Basically, molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, with their positions, volocities and forces among each other evolving in time.
-The MD method was first introduced by Alder and Wainwright in 1957 \cite{alder1,alder2} to study the interactions of hard spheres.
+The MD method was first introduced by Alder and Wainwright in 1957 \cite{alder57,alder59} to study the interactions of hard spheres.
The basis of the approach are Newton's equations of motion to describe classicaly the many-body system.
MD simulation is the numerical way of solving the $N$-body problem which cannot be solved analytically ($N>3$).
Quantum mechanical effects are taken into account by an analytical interaction potential between the nuclei.
In the following, relevant potentials for this work are discussed.
-\subsubsection{The Lennard-Jones potential}
-
-The L-J potential is a realistic two body pair potential and is of the form
-\begin{equation}
-U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,}
-\label{eq:lj-p}
-\end{equation}
-where $r$ denotes the distance between the two atoms.
-
-The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction.
-It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way.
-
-The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds.
-It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances.
-Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way.
-
-The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
-$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
-
-Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
-\begin{equation}
-\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
-\label{eq:lj-d}
-\end{equation}
-one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$.
-Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth.
-The $i$th component of the force $F^j$ on particle $j$ is obtained by
-\begin{equation}
-F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
-\label{eq:lj-f}
-\end{equation}
-
-\subsubsection{The Stillinger Weber potential}
-
-The Stillinger Weber potential \cite{stillinger_weber} \ldots
-
-\begin{equation}
-U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k)
-\end{equation}
-
-\begin{equation}
-U_2(r_{ij}) = \left\{
- \begin{array}{ll}
- \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1} \Big] & r_{ij} < a \sigma \\
- 0 & r_{ij} \ge a \sigma
- \end{array} \right.
-\end{equation}
-
-\begin{equation}
-U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) =
-\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big]
-\end{equation}
-
-\begin{equation}
-h(r_{ij},r_{ik},\theta_{jik}) =
-\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2
-\end{equation}
+%\subsubsection{The Lennard-Jones potential}
+%
+%The L-J potential is a realistic two body pair potential and is of the form
+%\begin{equation}
+%U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,}
+%\label{eq:lj-p}
+%\end{equation}
+%where $r$ denotes the distance between the two atoms.
+%
+%The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction.
+%It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way.
+%
+%The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds.
+%It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances.
+%Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way.
+%
+%The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data.
+%$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius.
+%
+%Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$)
+%\begin{equation}
+%\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big)
+%\label{eq:lj-d}
+%\end{equation}
+%one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$.
+%Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth.
+%The $i$th component of the force is given by
+%\begin{equation}
+%F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.}
+%\label{eq:lj-f}
+%\end{equation}
+%
+%\subsubsection{The Stillinger Weber potential}
+%
+%The Stillinger Weber potential \cite{stillinger_weber} \ldots
+%
+%\begin{equation}
+%U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k)
+%\end{equation}
+%
+%\begin{equation}
+%U_2(r_{ij}) = \left\{
+% \begin{array}{ll}
+% \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1} \Big] & r_{ij} < a \sigma \\
+% 0 & r_{ij} \ge a \sigma
+% \end{array} \right.
+%\end{equation}
+%
+%\begin{equation}
+%U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) =
+%\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big]
+%\end{equation}
+%
+%\begin{equation}
+%h(r_{ij},r_{ik},\theta_{jik}) =
+%\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2
+%\end{equation}
\subsubsection{The Tersoff potential}
Tersoff proposed an empirical interatomic potential for covalent systems.
The Tersoff potential explicitly incorporates the dependence of bond order on local envirenments, permitting an improved description of covalent materials.
-Tersoff applied the potential to silicon \cite{tersoff_silicon1,tersoff_silicon2,tersoff_silicon3}, carbon \cite{tersoff_carbon} and also to multicomponent systems like $SiC$ \cite{tersoff_multi}.
+Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like $SiC$ \cite{tersoff_m}.
The basic idea is that, in real systems, the bond order depends upon the local environment.
An atom with many neighbours forms weaker bonds than an atom with few neighbours.
\frac{1}{2} + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], & R_{ij} < r_{ij} < S_{ij} \\
0, & r_{ij} > S_{ij}
\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 $j$.
b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\
\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \\
g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \\
-b_{ij} & = & \chi_{ij} \Big( 1 + \beta_i^{n_i} \Big[ \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} \big[ 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \big] \Big] \Big)^{-1/2n_i}
\end{eqnarray}
where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$.
This is illustrated in Figure \ref{img:tersoff_angle}.
\printimg{!h}{width=8cm}{tersoff_angle.eps}{Angle between bonds of atoms $i,j$ and $i,k$.}{img:tersoff_angle}
-In order to calculate the forces the derivation of the potential with respect to $x^i_n$ (the $n$th component of the position vector of atom $i$ $\equiv$ ${\bf r}_i$) has to be known.
-This is gradually done in the following.
-The $n$th component of the force acting on atom $i$ is
-\begin{eqnarray}
-F_n^i & = & - \frac{\partial}{\partial x_n^i} \sum_{j \neq i} V_{ij} \nonumber\\
- & = & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber\\
-& & + f_C(r_{ij}) \big[ \partial_{x_n^i} f_R(r_{ij}) + b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big) \textrm{ .}
-\end{eqnarray}
-For the implementation it is helpful to seperate the two and three body terms.
-\begin{eqnarray}
-F_n^i & = & \sum_{j \neq i} \Big( f_R(r_{ij}) \partial_{x_n^i} f_C(r_{ij}) + f_C(r_{ij}) \partial_{x_n^i} f_R(r_{ij}) \Big) + \nonumber\\
-& + & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) b_{ij} f_A(r_{ij}) + f_C(r_{ij}) \big[ b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big)
-\end{eqnarray}
-The cutoff function $f_C$ derivated with repect to $x^i_n$ is
-\begin{equation}
-\partial_{x^i_n} f_C(r_{ij}) =
- - \frac{1}{2} \sin \Big( \pi (r_{ij} - R_{ij}) / (S_{ij} - R_{ij}) \Big) \frac{\pi (x^i_n - x^j_n)}{(S_{ij} - R_{ij}) r_{ij}}
-\label{eq:d_cutoff}
-\end{equation}
-for $R_{ij} < r_{ij} < S_{ij}$ and otherwise zero.
-The derivations of the repulsive and attractive part are:
-\begin{eqnarray}
-\partial_{x_n^i} f_R(r_{ij}) & = & - \lambda_{ij} \frac{x_n^i - x_n^j}{r_{ij}} A_{ij} \exp (-\lambda_{ij} r_{ij})\\
-\partial_{x_n^i} f_A(r_{ij}) & = & \mu_{ij} \frac{x_n^i - x_n^j}{r_{ij}} B_{ij} \exp (-\mu_{ij} r_{ij}) \textrm{ .}
-\end{eqnarray}
-The cosine of the angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines
-\begin{equation}
-\cos \theta_{ijk} = \Big( (r_{ij}^2 + r_{ik}^2 - r_{jk}^2)/(2 r_{ij} r_{ik}) \Big)
-\end{equation}
-or by the definition of the scalar product
+Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots
+
+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
\begin{equation}
-\cos \theta_{ijk} = \frac{\vec{r}_{ij} \vec{r}_{ik}}{r_{ij} r_{ik}} \textrm{ .}
+\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
-The derivation of the angle $\theta_{ijk}$ with respect to $x^i_n$ is given by
+The force is then given by
\begin{equation}
-\partial_{x^i_n} \cos \theta_{ijk} = \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ik}}{r_{ij}} \Big) (x_n^i - x_n^j) + \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ij}}{r_{ik}} \Big) (x_n^i - x_n^k)
-\label{eq:d_costheta}
+F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
\end{equation}
+The details of the Tersoff potential derivative can be seen in appendix \ref{app:d_tersoff}.
-Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_costheta} the derivation of $b_{ij}$ with respect to $x^i_n$ can be written as:
-\begin{eqnarray}
-\partial_{x^i_n} b_{ij} & = &
-- \frac{1}{2n_i} \chi_{ij} \Bigg( 1 + \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} \bigg( f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \bigg)^{n_i} \Bigg] \Bigg)^{-\frac{1}{2n_i} - 1} \times \nonumber\\
-&& \times n_i \beta_i^{n_i} \sum_{k \ne i,j} \Bigg( \Bigg[ f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \Bigg]^{n_i -1} \times \nonumber\\
-&& \times \Bigg[ \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \partial_{x^i_n} f_C(r_{ik}) - \nonumber\\
-&& - f_C(r_{ik}) \omega_{ik} \frac{2 c_i^2 (h_i - \cos \theta_{ijk})}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \partial_{x^i_n} \cos \theta_{ijk} \Bigg]
-\end{eqnarray}
+\subsubsection{A reparametrized Tersoff-like bond order potential}
-
-\subsubsection{The Brenner potential}
+Erhart-Albe potential ...
\subsection{Statistical ensembles}
\label{subsection:statistical_ensembles}
+\section{Denstiy functional theory}
+\label{section:dft}
+
+\subsection{Hohenberg-Kohn theorem}
+
+\subsection{Born-Oppenheimer (adiabatic) approximation}
+
+\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}
+
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