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/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.
\section{Molecular dynamics simulations}
\label{section:md}
-% todo
-% rewrite!
+% todo - rewrite md intro chapter
\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{}
\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.
+This understanding may be regarded as the basic principle of molecular dynamics, 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}
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 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.
+A potential is necessary to describe 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 assume that the particles can be described classically by Newton's equations of motion, which are integrated numerically.
{\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.
+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 sections cover the tools of the trade necessary for the MD simulation technique.
Three ingredients are required for a MD simulation:
f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
\end{eqnarray}
-The function $f_C$ is the a cutoff function to limit the range of interaction to nearest neighbors.
+The function $f_C$ is a cutoff function to limit the range of interaction to nearest neighbors.
It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
\begin{equation}
f_C(r_{ij}) = \left\{
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}
\subsection{Verlet integration}
\label{subsection:integrate_algo}
-A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
+A numerical method to integrate Newton's equations of motion was presented by Verlet in 1967 \cite{verlet67}.
The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
\begin{equation}
Expressing $n(\tilde{\vec{r}})$ in a Taylor series, $\epsilon_{\text{xc}}$ can be thought of as a function of coefficients, which correspond to the respective terms of the expansion.
Neglecting all terms of order $\mathcal{O}(\nabla n(\vec{r}))$ results in the functional equal to LDA, which requires the function of variable $n$.
Including the next element of the Taylor series introduces the gradient correction to the functional, which requires the function of variables $n$ and $|\nabla n|$.
-This is called the generalized gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
+This is called the generalized-gradient approximation (GGA), which expresses the exchange-correlation energy density as a function of the local density and the local gradient of the density
\begin{equation}
E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
\text{ .}
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}
+\label{subsection:basics:bzs}
Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
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.
+
+However, there is hardly ever only one defect in a crystal, not even only one kind of defect.
+Again, energetic considerations can be used to investigate the existing interaction of two defects.
+The binding energy $E_{\text{b}}$ of a defect pair is given by the difference of the formation energy of the defect combination $E_{\text{f}}^{\text{comb}} $ and the sum of the two separated defect configurations $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$.
+This can be expressed by
+\begin{equation}
+E_{\text{b}}=
+E_{\text{f}}^{\text{comb}}-
+E_{\text{f}}^{1^{\text{st}}}-
+E_{\text{f}}^{2^{\text{nd}}}
+\label{eq:basics:e_bind}
+\end{equation}
+where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above.
+Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
+The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
+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.
+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}
-% todo
-% advantages of pw basis with respect to hellmann feynman forces / pulay forces
+Investigating diffusion mechanisms is based on determining migration paths in between 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=0.45\textwidth]{crt_orig.eps}}
+\subfigure[]{\label{fig:basics:crtm}\includegraphics[width=0.45\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}.
+
+% todo - advantages of pw basis concenring hf forces
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