\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.
+Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) 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.
+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.
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 ...
-
+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.
+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.
+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}.
\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{}
\begin{flushright}
-{\em Marquis Pierre Simon de Laplace, 1814, \cite{laplace}.}
+{\em Marquis Pierre Simon de Laplace, 1814.} \cite{laplace}
\end{flushright}
\end{quotation}
+\noindent
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.
+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{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.
-
-By MD simulation techniques a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
-This microscopic information has to be translated to macroscopic observables by means of statistical mechanics.
+MD is the numerical way of solving the $N$-body problem which cannot be solved analytically for $N>3$.
+A potential is necessary describing the interaction of the particles.
+By MD a complete description of the system in the sense of classical mechanics on the microscopic level is obtained.
+The microscopic information can then be translated to macroscopic observables by means of statistical mechanics.
The basic idea is to integrate Newton's equations numerically.
A system of $N$ particles of masses $m_i$ ($i=1,\ldots,N$) at positions ${\bf r}_i$ and velocities $\dot{{\bf r}}_i$ is given by
\begin{equation}
{\bf F}_i = - \nabla_{{\bf r}_i} U({\{\bf r}\}) \, \textrm{.}
\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.
-The following chapters cover the tools of the trade necessary for the MD simulation technique.
+The following sections cover the tools of the trade necessary for the MD simulation technique.
Three ingredients are required for an MD simulation:
\begin{enumerate}
\item A model for the interaction between system constituents is needed.
- Interaction potentials and their accuracy for describing certain systems of elements will be outlined in chapter \ref{subsection:interact_pot}.
+ Interaction potentials and their accuracy for describing certain systems of elements will be outlined in section \ref{subsection:interact_pot}.
\item An integrator is needed, which propagtes the particle positions and velocities from time $t$ to $t+\delta t$, realised by a finite difference scheme which moves trajectories discretely in time.
- In chapter \ref{subsection:integrate_algo} a detailed overview of the available integration algorithms is given, including their advantages and disadvantages.
+ This is explained in section \ref{subsection:integrate_algo}.
\item A statistical ensemble has to be chosen, which allows certain thermodynamic quantities to be controlled or to stay constant.
- This is discussed in chapter \ref{subsection:statistical_ensembles}
+ This is discussed in section \ref{subsection:statistical_ensembles}.
\end{enumerate}
-
-In addition special techniques will be outlined which reduce the complexity of the MD algorithm, though the force/energy evaluation almost inevitably dictates the overall speed.
+Furthermore special techniques will be outlined which reduce the complexity of the MD algorithm, though the evaluation of the energy and force almost inevitably dictates the overall speed.
\subsection{Integration algorithms}
\label{subsection:integrate_algo}
Ordinary pair potentials have a close-packed structure like face-centered cubic (FCC) or hexagonal close-packed (HCP) as a ground state.
A pair potential is thus unable to describe properly elements with other structures than FCC or HCP.
-Silicon and carbon for instance, have a diamand/zincblende structure with four covalent bonded neighbours, which is far from a close-packed structure.
+Silicon and carbon for instance, have a diamand and zincblende structure with four covalent bonded neighbours, which is far from a close-packed structure.
A three body potential has to be included for these types of elements.
-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
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