\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.
+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}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
+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/posic}.
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.
\subsection{Statistical ensembles}
\label{subsection:statistical_ensembles}
-Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
+Using the above mentioned algorithms, the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
Providing a stable integration algorithm the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, is conserved.
This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
\end{equation}
where $\beta$ is the isothermal compressibility and $p$ corresponds to the current pressure, which is determined by equation \eqref{eq:basics:ps}.
-Using this method the system does not behave like a true $NpT$ ensemble.
+Using this method, the system does not behave like a true $NpT$ ensemble.
On average $T$ and $p$ correspond to the expected values.
For large enough time constants, i.e.\ $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinants to be evaluated massively increases.
In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates.
-In the following sections the basic idea of DFT will be outlined.
+In the following sections, the basic idea of DFT will be outlined.
As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory~\cite{thomas27,fermi27} and the self-consistent Hartree equations~\cite{hartree28}.
A nice review is given in the Nobel lecture of Kohn~\cite{kohn99}, one of the inventors of DFT.
The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are equal to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius determining the core region.
Modified methods to generate ultra-soft pseudopotentials were proposed, which address the rapid convergence with respect to the size of the plane wave basis set~\cite{vanderbilt90,troullier91}.
-Using PPs the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
+Using PPs, the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.
\subsection{Brillouin zone sampling}
Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
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.
+Therefor, 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.
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