\addcontentsline{toc}{chapter}{Acknowledgment}
\chapter*{Acknowledgment\markboth{Acknowledgment}{}}
-First of all, I would like to thank my official advisers Prof. Dr. Bernd Stritzker and Prof. Dr. Kai Nordlund for accepting me as a doctoral candidate at their chairs at the University of Augsburg and the University of Helsinki.
-I am grateful to Prof. Dr. Bernd Stritzker who, although being an experimental scientist, gave me the opportunity to work on a rather theoretical field.
+First of all, I would like to thank my official advisers Prof.\ Dr.\ Bernd Stritzker and Prof.\ Dr.\ Kai Nordlund for accepting me as a doctoral candidate at their chairs at the University of Augsburg and the University of Helsinki.
+I am grateful to Prof.\ Dr.\ Bernd Stritzker who, although being an experimental scientist, gave me the opportunity to work on a rather theoretical field.
The working environment providing insight on problems dealt with in this thesis from the point of view of an experimentalist were of great importance contributing to the success of this work.
-Furthermore, I would like to thank Prof. Dr. Kai Nordlund for his expertise in the field of atomistic simulations and the possibility to repeatedly visit his group in Helsinki.
+Furthermore, I would like to thank Prof.\ Dr.\ Kai Nordlund for his expertise in the field of atomistic simulations and the possibility to repeatedly visit his group in Helsinki.
Much progress of this study is owed to the regrettably few but fruitful discussions and explanations by electronic mail, within short meetings on conferences and, of course, during my long-term stays in the nice capital of Finland.
Due to the official supervising of both of them, participation in a special research fellowship of the Bayerische Forschungsstiftung was enabled providing financial support for the first three years.
Next to living costs, travel expenses for scientific conferences as well as my stays in Helsinki were unbureaucratically taken over by the research foundation.
-I owe my deepest gratitude to Prof. Dr. J\"org K. N. Lindner guiding me through my first time of scientific working already beginning with the diploma thesis.
+I owe my deepest gratitude to Prof.\ Dr.\ J\"org K. N. Lindner guiding me through my first time of scientific working already beginning with the diploma thesis.
I am heartily thankful for his frequent encouragements and the confidence he has shown enabling a greatly independent research and the improvement of respective skills.
Furthermore, his experiences in the materials system covered in this study and fine grasp with regard to scientific routines were of great help for this work.
-Getting appointed to a professorship at the University of Paderborn, it was also him, who initiated a collaboration with the local theory group under the direction of Prof. Dr. Wolf Gero Schmidt.
+Getting appointed to a professorship at the University of Paderborn, it was also him, who initiated a collaboration with the local theory group under the direction of Prof.\ Dr.\ Wolf Gero Schmidt.
I would like to thank him for six month of financial support, which allowed me to extend my research period, as well as for his helpful contributions to common articles and respective publishing procedures.
More importantly, the collaboration involved further investigations based on first-principles calculations, which improved the quality of this work to a great extent.
-At this point, I would like to express special thanks to Dr. Eva Rauls.
+At this point, I would like to express special thanks to Dr.\ Eva Rauls.
The present thesis would not have been possible without her assistance and mentoring with respect to the utilized methods required by the new approach of investigation as well as her expertise in the materials system.
I am greatly thankful for the possibility to repeatedly visit the theory group in Paderborn.
-In this context, Dr. Simone Sanna is acknowledged for respective technical support and Michael Weinl, doctoral student of Prof. J\"org K. N. Lindner back then, for accommodation.
+In this context, Dr.\ Simone Sanna is acknowledged for respective technical support and Michael Weinl, doctoral student of Prof. J\"org K. N. Lindner back then, for accommodation.
-I am grateful to Priv.-Doz. Dr. habil. Volker Eyert for writing one of the certificates of this work.
+I am grateful to Priv.-Doz.\ Dr.\ habil.\ Volker Eyert for writing one of the certificates of this work.
Furthermore, his lectures on computational physics and the electronic structure of materials, which I attended during my academic studies, influenced me to pursue scientific research in the field of computational physics.
-One more time, I would like to thank Prof. Dr. Bernd Stritzker for another two-month position as a member of his research staff and various long-term employments as a research assistant, which not only ensured a minimum of financial supply but also involved tutorships in the field of solid state physics that could be carried out in a more or less free and autonomous way.
+One more time, I would like to thank Prof.\ Dr.\ Bernd Stritzker for another two-month position as a member of his research staff and various long-term employments as a research assistant, which not only ensured a minimum of financial supply but also involved tutorships in the field of solid state physics that could be carried out in a more or less free and autonomous way.
I am grateful to Ralf Utermann, responsible for the computing infrastructure in the physics department, for providing access and support with the excellently maintained high performance units available in the Augsburg Linux Compute Cluster.
Furthermore, being employed as an assistant under his direction during the first times of my studies, he provided insight into modern computing technology and, in doing so, sparked my interest in computational physics.
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}.
-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}.
+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.
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 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 simulated system 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}.
+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}
\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.
+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 may be regarded as the basic principle of molecular dynamics, considering an isolated system of particles, the behavior of which is fully determined by the solution of the classical equations of motion.
Tersoff proposed an empirical interatomic potential for covalent systems \cite{tersoff_si1,tersoff_si2}.
The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
-Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increases in computational efficiency for the evaluation of forces and energy based on the short-range potential.
+Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increase in computational efficiency for the evaluation of forces and energy based on the short-range potential.
Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
The basic idea is that, in real systems, the bond order, i.e. the strength of the bond, depends upon the local environment \cite{abell85}.
Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
\frac{\delta t^3}{6}\vec{b}_i(t) + \mathcal{O}(\delta t^4)
\label{basics:verlet:taylor2}
\end{equation}
-where $\vec{v}_i=\frac{d}{dt}\vec{r}_i$ are the velocities, $\vec{f}_i=m\frac{d}{dt^2}\vec{r}_i$ are the forces and $\vec{b}_i=\frac{d}{dt^3}\vec{r}_i$ are the third derivatives of the particle positions with respect to time.
+where $\vec{v}_i=\frac{d}{dt}\vec{r}_i$ are the velocities, $\vec{f}_i=m\frac{d^2}{dt^2}\vec{r}_i$ are the forces and $\vec{b}_i=\frac{d^3}{dt^3}\vec{r}_i$ are the third derivatives of the particle positions with respect to time.
The Verlet algorithm is obtained by summarizing and subtracting equations \eqref{basics:verlet:taylor1} and \eqref{basics:verlet:taylor2}
\begin{equation}
\vec{r}_i(t+\delta t)=
the truncation error of which is of order $\delta t^4$ for the positions and $\delta t^3$ for the velocities.
The velocities, although not used to update the particle positions, are not synchronously determined with the positions but drag behind one step of discretization.
The Verlet algorithm can be rewritten into an equivalent form, which updates the velocities and positions in the same step.
-The so-called velocity Verlet algorithm is obtained by combining \eqref{basics:verlet:taylor1} with equation \eqref{basics:verlet:taylor2} displaced in time by $+\delta t$
+The so-called velocity Verlet algorithm is obtained by combining equation \eqref{basics:verlet:taylor1} with equation \eqref{basics:verlet:taylor2} displaced in time by $+\delta t$
\begin{equation}
\vec{v}_i(t+\delta t)=
\vec{v}_i(t)+\frac{\delta t}{2m_i}[\vec{f}_i(t)+\vec{f}_i(t+\delta t)]
++\mathcal{O}(\delta t^3)
\end{equation}
\begin{equation}
\vec{r}_i(t+\delta t)=
-\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t) \text{ .}
+\vec{r}_i(t)+\delta t\vec{v}_i(t)+\frac{\delta t^2}{2m_i}\vec{f}_i(t)
++\mathcal{O}(\delta t^3)
+ \text{ .}
\end{equation}
Since the forces for the new positions are required to update the velocity the determination of the forces has to be carried out within the integration algorithm.
\subsection{Kohn-Sham system}
Inspired by the Hartree equations, i.e. a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}.
-However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$.
+However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(\vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$.
The respective Kohn-Sham equations for the effective single-particle wave functions $\Phi_i(\vec{r})$ take the form
\begin{equation}
\left[
\end{equation}
\begin{equation}
V_{\text{eff}}(\vec{r})=V(\vec{r})+\int\frac{e^2n(\vec{r}')}{|\vec{r}-\vec{r}'|}d\vec{r}'
- + V_{\text{xc}(\vec{r})}
+ + V_{\text{xc}}(\vec{r})
\text{ ,}
\label{eq:basics:kse2}
\end{equation}
Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods \cite{ceperley80} and corresponding parametrizations exist \cite{perdew81}.
Obviously exact for the homogeneous electron gas, the LDA was {\em a priori} expected to be useful only for densities varying slowly on scales of the local Fermi or TF wavelength.
Nevertheless, LDA turned out to be extremely successful in describing some properties of highly inhomogeneous systems accurately within a few percent.
-Although LDA is known to overestimate the cohesive energy in solids by \unit[10-20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
+Although LDA is known to overestimate the cohesive energy in solids by \unit[10--20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density \cite{kohn65}.
Respective considerations are based on the concept of an exchange-correlation hole density describing the depletion of the electron density in the vicinity of an electron.
However, these methods rely on the fact that the wave functions are localized and exhibit an exponential decay resulting in a sparse Hamiltonian.
Another approach is to represent the KS wave functions by plane waves.
-In fact, the employed {\textsc vasp} software is solving the KS equations within a plane-wave (PW) basis set.
+In fact, the employed \textsc{vasp} software is solving the KS equations within a plane-wave (PW) basis set.
The idea is based on the Bloch theorem \cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice.
The latter one can be expressed by a Fourier series, i.e. a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal.
Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set
In order to achieve these properties different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
Mathematically, a non-local PP, which depends on the angular momentum, has the form
\begin{equation}
-V_{\text{nl}}(\vec{r}) = \sum_{lm} \mid lm \rangle V_l(\vec{r}) \langle lm \mid
+V_{\text{nl}}(\vec{r}) = \sum_{lm} | lm \rangle V_l(\vec{r}) \langle lm |
\text{ .}
\end{equation}
Applying of the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $\mid lm \rangle$, i.e. the orbitals with azimuthal angular momentum $l$ and magnetic number $m$, which are then multiplied by the respective pseudopotential $V_l(\vec{r})$ for angular momentum $l$.
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 calculating the respective vibrational modes.
-Modifications used to add the CRT feature to the {\textsc vasp} code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
+Modifications used to add the CRT feature to the \textsc{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
The contribution of the bond order term is given by:
\begin{eqnarray}
\nabla_{{\bf r}_j}\cos\theta_{ijk} &=&
- \nabla_{{\bf r}_j}\Big(\frac{{\bf r}_{ij}{\bf }r_{ik}}{r_{ij}r_{ik}}\Big)
+ \nabla_{{\bf r}_j}\Big(\frac{{\bf r}_{ij}{\bf r}_{ik}}{r_{ij}r_{ik}}\Big)
\nonumber \\
&=& \frac{1}{r_{ij}r_{ik}}{\bf r}_{ik} -
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\item
\item LOOP k \{
\begin{itemize}
- \item set $ik$-depending values
+ \item set $ik$-dependent values
\item calculate: $r_{ik}$, $r_{ik}^2$
\item IF $r_{ik} > S_{ik}$ THEN CONTINUE
\item calculate: $\theta_{ijk}$, $\cos(\theta_{ijk})$,
& \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
\hline
\multicolumn{6}{c}{Present study} \\
-{\textsc vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
-{\textsc posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
+\textsc{vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
+\textsc{posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
\multicolumn{6}{c}{Other {\em ab initio} studies} \\
Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the EA potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
-To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the {\textsc parcas} MD code \cite{parcas_md}.
-The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by {\textsc posic}.
+To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the \textsc{parcas} MD code \cite{parcas_md}.
+The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by \textsc{posic}.
In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing fundamental problems of analytical potential models for describing defect structures.
Hence, these artifacts have a negligible influence in finite temperature simulations.
The bond-centered (BC) configuration is unstable and, thus, is not listed.
-The \si{} \hkl<1 0 0> DB constitutes the most unfavorable configuration for both, the EA and {\textsc vasp} calculations.
+The \si{} \hkl<1 0 0> DB constitutes the most unfavorable configuration for both, the EA and \textsc{vasp} calculations.
The quantum-mechanical treatment of the \si{} \hkl<1 0 0> DB demands for spin polarized calculations.
The same applies for the vacancy.
In the \si{} \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & BC \\
\hline
Present study & & & & & & \\
- {\textsc posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
- {\textsc vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
+ \textsc{posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
+ \textsc{vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
Other studies & & & & & & \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
{\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6-1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
+Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
-As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3-10]{eV}.
+As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3--10]{eV}.
Keeping these considerations in mind, the \ci{} \hkl<1 0 0> DB is the most favorable interstitial configuration for all interaction models.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
However, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
In another {\em ab initio} study, Capaz~et~al.~\cite{capaz94} in turn found the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
This is assumed to be due to the neglecting of the electron spin in these calculations.
-Another {\textsc vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
+Another \textsc{vasp} calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
It is worth to note that all other listed configurations are not affected by spin polarization.
However, in calculations performed in this work, which fully account for the spin of the electrons, the BC configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \ci{} \hkl<1 0 0> DB configuration.
\end{center}
\caption[Sketch of the \ci{} \hkl<1 0 0> dumbbell structure.]{Sketch of the \ci{} \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in Table \ref{tab:defects:100db_cmp}.}
\label{fig:defects:100db_cmp}
-\end{figure}
+\end{figure}%
\begin{table}[tp]
\begin{center}
Displacements\\
& & & & \multicolumn{3}{c}{Atom 2} & \multicolumn{3}{c}{Atom 3} \\
& $a$ & $b$ & $|a|+|b|$ & $\Delta x$ & $\Delta y$ & $\Delta z$ & $\Delta x$ & $\Delta y$ & $\Delta z$ \\
\hline
-{\textsc posic} & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
-{\textsc vasp} & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
+\textsc{posic} & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
+\textsc{vasp} & 0.109 & -0.065 & 0.174 & -0.011 & -0.011 & -0.024 & -0.014 & 0.014 & 0.025 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\hline
& $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
\hline
-{\textsc posic} & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
-{\textsc vasp} & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
+\textsc{posic} & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
+\textsc{vasp} & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 & 0.548 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\hline
& $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-{\textsc posic} & 140.2 & 109.9 & 134.4 & 112.8 \\
-{\textsc vasp} & 130.7 & 114.4 & 146.0 & 107.0 \\
+\textsc{posic} & 140.2 & 109.9 & 134.4 & 112.8 \\
+\textsc{vasp} & 130.7 & 114.4 & 146.0 & 107.0 \\
\hline
\hline
\end{tabular}\\[0.5cm]
\end{center}
-\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
+\caption[Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by \textsc{posic} and {\textsc vasp} calculations.]{Atomic displacements, distances and bond angles of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc posic} and {\textsc vasp} calculations. The displacements and distances are given in nm and the angles are given in degrees. Displacements, distances and angles are schematically displayed in Fig. \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline Si is listed.}
\label{tab:defects:100db_cmp}
-\end{table}
+\end{table}%
\begin{figure}[tp]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
-\underline{\textsc posic}
+\underline{\textsc{posic}}\\
\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
-\underline{\textsc vasp}
+\underline{\textsc{vasp}}\\
\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
\end{center}
\end{minipage}
\end{center}
-\caption{Comparison of the \ci{} \hkl<1 0 0> DB structures obtained by {\textsc posic} and {\textsc vasp} calculations.}
+\caption{Comparison of the \ci{} \hkl<1 0 0> DB structures obtained by \textsc{posic} and {\textsc vasp} calculations.}
\label{fig:defects:100db_vis_cmp}
-\end{figure}
+\end{figure}%
\begin{figure}[tp]
\begin{center}
\includegraphics[height=10cm]{c_pd_vasp/eden.eps}
\includegraphics[height=12cm]{c_pd_vasp/100_2333_ksl.ps}
\end{center}
-\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and gray spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
+\caption[Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by \textsc{vasp} calculations.]{Charge density isosurface and Kohn-Sham levels of the \ci{} \hkl<1 0 0> DB structure obtained by {\textsc vasp} calculations. Yellow and gray spheres correspond to Si and C atoms. The blue surface is the charge density isosurface. In the energy level diagram red and green lines and dots mark occupied and unoccupied states.}
\label{img:defects:charge_den_and_ksl}
-\end{figure}
-The Si atom labeled '1' and the C atom compose the DB structure.
+\end{figure}%
+The Si atom labeled `1' and the C atom compose the DB structure.
They share the lattice site which is indicated by the dashed red circle.
They are displaced from the regular lattice site by length $a$ and $b$ respectively.
The atoms no longer have four tetrahedral bonds to the Si atoms located on the alternating opposite edges of the cube.
One bond is formed to the other DB atom.
The other two bonds are bonds to the two Si edge atoms located in the opposite direction of the DB atom.
The distance of the two DB atoms is almost the same for both types of calculations.
-However, in the case of the {\textsc vasp} calculation, the DB structure is pushed upwards compared to the results using the EA potential.
+However, in the case of the \textsc{vasp} calculation, the DB structure is pushed upwards compared to the results using the EA potential.
This is easily identified by comparing the values for $a$ and $b$ and the two structures in Fig. \ref{fig:defects:100db_vis_cmp}.
Thus, the angles of bonds of the Si DB atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
On the other hand, the C atom forms an almost collinear bond ($\theta_3$) with the two Si edge atoms implying the predominance of $sp$ bonding.
In the BC interstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possible migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
-However, this fact could not be reproduced by spin polarized {\textsc vasp} calculations performed in this work.
+However, this fact could not be reproduced by spin polarized \textsc{vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section \ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
If the entire transition of the \hkl[0 0 -1] into the \hkl[0 0 1] configuration is considered a two step process passing the intermediate BC configuration, an additional activation energy of \unit[0.5]{eV} is necessary to escape the BC towards the \hkl[0 0 1] configuration.
Assuming equal preexponential factors for both diffusion steps, the total probability of diffusion is given by $\exp\left((2.2\,\text{eV}+0.5\,\text{eV})/k_{\text{B}}T\right)$.
-Thus, the activation energy should be located within the range of \unit[2.2-2.7]{eV}.
+Thus, the activation energy should be located within the range of \unit[2.2--2.7]{eV}.
\begin{figure}[tp]
\begin{center}
\caption[Position of the initial \ci{} {\hkl[0 0 -1]} DB and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB.]{Position of the initial \ci{} \hkl[0 0 -1] DB (I) (a) and of the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (\si) (b). Lattice sites for the second defect used for investigating defect pairs are numbered from 1 to 5. For black/red/blue numbers, one/two/four possible atom(s) exist for the second defect to create equivalent defect combinations.}
\label{fig:defects:combos}
\end{figure}
-Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1-5) that are used for investigating defect pairs.
+Fig.~\ref{fig:defects:combos} schematically displays the initial \ci{} \hkl[0 0 -1] DB structure (Fig.~\ref{fig:defects:combos_ci}) as well as the lattice site chosen for the initial \si{} \hkl<1 1 0> DB (Fig.~\ref{fig:defects:combos_si}) and various positions for the second defect (1--5) that are used for investigating defect pairs.
The color of the number denotes the amount of possible atoms for the second defect resulting in equivalent configurations.
Binding energies of the defect pair are determined by equation \ref{eq:basics:e_bind}.
Next to formation and binding energies, migration barriers are investigated, which allow to draw conclusions on the probability of the formation of such defect complexes by thermally activated diffusion processes.
The C atom of the second DB forms threefold coordinated bonds to its Si neighbors.
A distance of \unit[2.80]{\AA} is observed for the two C atoms.
Again, the two C atoms and its two interconnecting Si atoms form a rhomboid.
-C-C distances of \unit[2.70-2.80]{\AA} seem to be characteristic for such configurations, in which the C atoms and the two interconnecting Si atoms reside in a plane.
+C-C distances of \unit[2.70--2.80]{\AA} seem to be characteristic for such configurations, in which the C atoms and the two interconnecting Si atoms reside in a plane.
Configurations obtained by adding a \ci{} \hkl<1 0 0> DB at position 4 are characterized by minimal changes from their initial creation condition during relaxation.
There is a low interaction of the DBs, which seem to exist independent of each other.
\end{figure}
Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
Nevertheless, the C and Si DB atoms remain threefold coordinated.
-Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).
+Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198--0.209]{nm}/\unit[0.189]{nm}).
Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground-state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
\hline
& C$_{\text{i}}$ \hkl<1 0 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ \hkl<1 1 0> & C$_{\text{s}}$ \& Si$_{\text{i}}$ T\\
\hline
- {\textsc vasp} & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
- {\textsc posic} & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$\\
+ \textsc{vasp} & 3.72 & 4.37 & 4.17$^{\text{a}}$/4.99$^{\text{b}}$/4.96$^{\text{c}}$ \\
+ \textsc{posic} & 3.88 & 4.93 & 5.25$^{\text{a}}$/5.08$^{\text{b}}$/4.43$^{\text{c}}$\\
\hline
\hline
\end{tabular}
Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects \cite{leung99,al-mushadani03} as well as for C point defects \cite{dal_pino93,capaz94}.
The ground-state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
-A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
+A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
Due to the limitations of short range potentials and conventional MD as discussed above, elevated temperatures are used in the following.
Increased temperatures are expected to compensate the overestimated diffusion barriers.
These are overestimated by a factor of 2.4 to 3.5.
-Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460-2260]{$^{\circ}$C}.
-Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K}) \cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450-2050]{$^{\circ}$C} are used.
+Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460--2260]{$^{\circ}$C}.
+Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K}) \cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
Thus, higher annealing temperatures and longer annealing times were considered necessary for the formation of homogeneous SiC layers.
Indeed, for the first time, buried homogeneous and stoichiometric epitaxial 3C-SiC layers embedded in single crystalline Si were obtained by the same group consequently applying annealing temperatures of \unit[1405]{$^{\circ}$C} for \unit[90]{min} and implantation temperatures of approximately \unit[550]{$^{\circ}$C} \cite{reeson87}.
The necessity of the applied extreme temperature and time scale is attributed to the stability of substitutional C within the Si matrix being responsible for high activation energies necessary to dissolve such precipitates and, thus, allow for redistribution of the implanted C atoms.
-In order to avoid extreme annealing temperatures close to the melting temperature of Si, triple-energy implantations in the range from \unit[180-190]{keV} with stoichiometric doses at a constant target temperature of \unit[860]{$^{\circ}$C} achieved by external substrate heating were performed \cite{martin90}.
+In order to avoid extreme annealing temperatures close to the melting temperature of Si, triple-energy implantations in the range from \unit[180--190]{keV} with stoichiometric doses at a constant target temperature of \unit[860]{$^{\circ}$C} achieved by external substrate heating were performed \cite{martin90}.
It was shown that a thick buried layer of SiC is directly formed during implantation, which consists of small, only slightly misorientated but severely twinned 3C-SiC crystallites.
The authors assumed that due to the auxiliary heating rather than ion beam heating as employed in all the preceding studies, the complexity of the remaining defects in the synthesized structure is fairly reduced.
Even better qualities by direct synthesis were obtained for implantations at \unit[950]{$^{\circ}$C} \cite{nejim95}.
Since no amorphous or polycrystalline regions have been identified, twinning is considered to constitute the main limiting factor in the IBS of SiC.
Further studies revealed the possibility to form buried layers of SiC by IBS at moderate substrate and anneal temperatures \cite{lindner95,lindner96}.
-Different doses of C ions with an energy of \unit[180]{keV} were implanted at \unit[330-440]{$^{\circ}$C} and annealed at \unit[1200]{$^{\circ}$C} or \unit[1250]{$^{\circ}$C} for \unit[5-10]{h}.
+Different doses of C ions with an energy of \unit[180]{keV} were implanted at \unit[330--440]{$^{\circ}$C} and annealed at \unit[1200]{$^{\circ}$C} or \unit[1250]{$^{\circ}$C} for \unit[5--10]{h}.
For a critical dose, which was found to depend on the Si substrate orientation, the formation of a stoichiometric buried layer of SiC exhibiting a well-defined interface to the Si host matrix was observed.
In case of overstoichiometric C concentrations the excess C is not redistributed.
These investigations demonstrate the presence of an upper dose limit, which corresponds to a \unit[53]{at.\%} C concentration at the implantation peak, for the thermally induced redistribution of the C atoms from a Gaussian to a box-shaped depth profile upon annealing.
Based on these findings, a recipe was developed to form buried layers of single-crystalline SiC featuring an improved interface and crystallinity \cite{lindner99,lindner01,lindner02}.
Therefore, the dose must not exceed the stoichiometry dose, i.e. the dose corresponding to \unit[50]{at.\%} C concentration at the implantation peak.
Otherwise clusters of C are formed, which cannot be dissolved during post-implantation annealing at moderate temperatures below the Si melting point \cite{lindner96,calcagno96}.
-Annealing should be performed for \unit[5-10]{h} at \unit[1250]{$^{\circ}$C} to enable the redistribution from the as-implanted Gaussian into a box-like C depth profile \cite{lindner95}.
+Annealing should be performed for \unit[5--10]{h} at \unit[1250]{$^{\circ}$C} to enable the redistribution from the as-implanted Gaussian into a box-like C depth profile \cite{lindner95}.
The implantation temperature constitutes the most critical parameter, which is responsible for the structure after implantation and, thus, the starting point for subsequent annealing steps.
Implantations at \unit[400]{$^{\circ}$C} resulted in buried layers of SiC subdivided into a polycrystalline upper and an epitaxial lower part.
This corresponds to the region of randomly oriented SiC crystallites and epitaxially aligned precipitates surrounded by thin amorphous layers without crystalline SiC inclusions in the as-implanted state.
% increase of C at substitutional sites
Epitaxial layers with \unit[1.4]{at.\%} of substitutional C have been successfully synthesized in preamorphized Si$_{0.86}$Ge$_{0.14}$ layers, which were grown by CVD on Si substrates, using multiple-energy C implantation followed by solid-phase epitaxial regrowth at \unit[700]{$^{\circ}$C} \cite{strane93}.
The tensile strain induced by the C atoms is found to compensate the compressive strain present due to the Ge atoms.
-Studies on the thermal stability of Si$_{1-y}$C$_y$/Si heterostructures formed in the same way and equal C concentrations showed a loss of substitutional C accompanied by strain relaxation for temperatures ranging from \unit[810-925]{$^{\circ}$C} and the formation of spherical 3C-SiC precipitates with diameters of \unit[2-4]{nm}, which are incoherent but aligned to the Si host \cite{strane94}.
+Studies on the thermal stability of Si$_{1-y}$C$_y$/Si heterostructures formed in the same way and equal C concentrations showed a loss of substitutional C accompanied by strain relaxation for temperatures ranging from \unit[810--925]{$^{\circ}$C} and the formation of spherical 3C-SiC precipitates with diameters of \unit[2--4]{nm}, which are incoherent but aligned to the Si host \cite{strane94}.
During the initial stages of precipitation C-rich clusters are assumed, which maintain coherency with the Si matrix and the associated biaxial strain.
Using this technique a metastable solubility limit was achieved, which corresponds to a C concentration exceeding the solid solubility limit at the Si melting point by nearly three orders of magnitude and, furthermore, a reduction of the defect density near the metastable solubility limit is assumed if the regrowth temperature is increased by rapid thermal annealing \cite{strane96}.
Since high temperatures used in the solid-phase epitaxial regrowth method promotes SiC precipitation, other groups realized substitutional C incorporation for strained Si$_{1-y}$C$_y$/Si heterostructures \cite{iyer92,fischer95,powell93,osten96,osten99,laveant2002} or partially to fully strain-compensated (even inversely distorted \cite{osten94_2}) Si$_{1-x-y}$Ge$_x$C${_y}$ layers on Si \cite{eberl92,powell93_2,osten94,dietrich94} by MBE.
Although high-quality films of single-crystalline 3C-SiC can be produced by means of IBS the precipitation mechanism in bulk Si is not yet fully understood.
Indeed, closely investigating the large amount of literature pulled up in the last two sections and a cautious combination of some of the findings reveals controversial ideas of SiC formation, which are reviewed in more detail in the following.
-High resolution transmission electron microscopy (HREM) investigations of C-implanted Si at room temperature followed by rapid thermal annealing (RTA) show the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700-800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures \cite{werner96,werner97}.
+High resolution transmission electron microscopy (HREM) investigations of C-implanted Si at room temperature followed by rapid thermal annealing (RTA) show the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700--800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures \cite{werner96,werner97}.
The precipitates with diameters between \unit[2]{nm} and \unit[5]{nm} are incorporated in the Si matrix without any remarkable strain fields, which is explained by the nearly equal atomic density of C-Si agglomerates and the SiC unit cell.
Implantations at \unit[500]{$^{\circ}$C} likewise suggest an initial formation of C-Si dumbbells on regular Si lattice sites, which agglomerate into large clusters \cite{lindner99_2}.
The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96,werner97}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}.
\section{DFT calculations}
\label{section:simulation:dft_calc}
-The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}.
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp}) \cite{kresse96}.
The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
-The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp} \cite{vanderbilt90}.
An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
Spin polarization has been fully accounted for.
The conjugate gradient algorithm is used for ionic relaxation.
Migration paths are determined by the modified version of the CRT method as explained in section \ref{section:basics:migration}.
The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
-Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
+Due to restrictions by the \textsc{vasp} code, {\em ab initio} MD could only be performed at constant volume.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
% todo - point defects are calculated for the neutral charge state.
To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
-Next to the ultra-soft pseudopotentials \cite{vanderbilt90} {\textsc vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
-The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in {\textsc vasp}.
+Next to the ultra-soft pseudopotentials \cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
+The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
\begin{table}[t]
\begin{center}
An interfacial energy of \unit[2267.28]{eV} is obtained.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of \unit[29.93]{\AA}.
Thus, the interface tension, given by the energy of the interface divided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{-4}$]{J/cm$^2$}.
-This value perfectly fits within the experimentally estimated range of \unit[$2-8\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
+This value perfectly fits within the experimentally estimated range of \unit[2--8$\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
}
if$
}
+FUNCTION {format.zvolume}
+{ volume empty$
+{ "" }
+{ series empty$
+'skip$
+{ "" series * }
+if$
+" Vol." volume tie.or.space.connect *
+"volume and number" number either.or.check
+}
+if$
+}
FUNCTION {format.number.series}
{ volume empty$
{ number empty$
{ output.bibitem
collaboration output
format.authors "author" output.check
-%new.block
-%format.title "title" output.check
+new.block
+format.title "title" output.check
new.block
crossref missing$
-{ format.in.ed.booktitle "booktitle" output.check
-format.bvolume output
+%{ format.in.ed.booktitle "booktitle" output.check
+{ format.zvolume output
format.number.series output
format.chapter.pages output
new.sentence
\label{app:patch_vasp}
\section{Description}
-In the {\textsc vasp} code, the {\em selective dynamics} mode provides a feature to allow or constrain the change of each of the three coordinates for every single atom.
+In the \textsc{vasp} code, the {\em selective dynamics} mode provides a feature to allow or constrain the change of each of the three coordinates for every single atom.
By this, however, applied constraints are restricted to the chosen basis.
For the investigation of migration pathways utilizing the constrained relaxation technique as detailed in section~\ref{section:basics:migration}, the required constraint not necessarily corresponds to one of the coordinate axes as defined by the basis, which, in turn, is determined to enable a construction within the supercell approach.
-Thus, the functionality of the {\em selective dynamics} mode had to be extended by modifications in the particle position evaluation routine of {\textsc vasp}.
+Thus, the functionality of the {\em selective dynamics} mode had to be extended by modifications in the particle position evaluation routine of \textsc{vasp}.
These modifications allow for a rotation of all atom coordinates individually before respective constraints are applied and a following, final inverse transformation.
In that way, constraints for every single atom can be applied independently of the chosen basis.
-A patch against version 4.6 of the {\textsc vasp} code containing these modifications is available for download\footnote{http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/sd\_rot\_all-atoms.patch}.
+A patch against version 4.6 of the \textsc{vasp} code containing these modifications is available for download\footnote{http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/sd\_rot\_all-atoms.patch}.
\section{Mode of operation}
0.75000 0.25000 0.75000 T T T 0.0 0.0
0.25000 0.75000 0.75000 T T T 0.0 0.0
\end{Verbatim}
-\caption{Example {\textsc vasp} input file utilizing the {\em transformed selective dynamics} mode of operation.}
+\caption{Example \textsc{vasp} input file utilizing the {\em transformed selective dynamics} mode of operation.}
\label{fig:vasp_input}
\end{figure}
In case of the first atom, the basis is transformed by a rotation of $45^{\circ}$ and $30^{\circ}$ about the $z$ and $x'$ axis.
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