\chapter{Point defects in silicon}
+\label{chapter:defects}
-Given the conversion mechnism of SiC in crystalline silicon introduced in section \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest.
-Both types of defects are examined in the following both by classical potential as well as density functional theory calculations.
+Regarding the supposed conversion mechanisms of SiC in c-Si as introduced in section \ref{section:assumed_prec} the understanding of C and Si interstitial point defects in c-Si is of fundamental interest.
+During implantation defects such as vacancies (V), substitutional C (C$_{\text{s}}$), interstitial C (C$_{\text{i}}$) and Si self-interstitials (Si$_{\text{i}}$) are created, which are believed to play a decisive role in the precipitation process.
+In the following, these defects are systematically examined by computationally efficient, classical potential as well as highly accurate DFT calculations with the parameters and simulation conditions as outlined in chapter \ref{chapter:simulation}.
+Both methods are used to investigate selected diffusion processes of some of the defect configurations.
+While the quantum-mechanical description yields results that excellently compare to experimental findings, shortcomings of the classical potential approach are identified.
+These shortcomings are further investigated and the basis for a workaround, as proposed later on in the classical MD simulation chapter, is discussed.
-In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used.
-Calculations are performed in an isothermal-isobaric NPT ensemble.
-Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs.
-The temperature is set to zero Kelvin.
-Pressure is controlled by a Berendsen barostat \cite{berendsen84} again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon.
-To exclude surface effects periodic boundary conditions are applied.
-
-Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations.
-The ions are relaxed by a conjugate gradient method.
-The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81}.
-Periodic boundary conditions in each direction are applied.
-All point defects are calculated for the neutral charge state.
-
-\begin{figure}[th]
-\begin{center}
-\includegraphics[width=9cm]{unit_cell_e.eps}
-\end{center}
-\caption[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 configuration.]{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 configuration. The black dots ({\color{black}$\bullet$}) correspond to the silicon atoms and the blue lines ({\color{blue}-}) indicate the covalent bonds of the perfect c-Si structure.}
-\label{fig:defects:ins_pos}
-\end{figure}
-
-The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}.
-In seperated simulation runs the silicon or carbon atom is inserted at the
-\begin{itemize}
- \item tetrahedral, $\vec{r}=(0,0,0)$, ({\color{red}$\bullet$})
- \item hexagonal, $\vec{r}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$})
- \item nearly \hkl<1 0 0> dumbbell, $\vec{r}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$})
- \item nearly \hkl<1 1 0> dumbbell, $\vec{r}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$})
- \item bond-centered, $\vec{r}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$})
-\end{itemize}
-interstitial position.
-For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces.
-A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively.
-
-From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures.
-For defect configurations consisting of a single atom species the formation energy is defined as
-\begin{equation}
-E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}}
- -E_{\text{coh}}^{\text{defect-free}}\right)N
-\label{eq:defects: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.
-The formation energy of defects consisting of two or more atom species is defined as
-\begin{equation}
-E_{\text{f}}=E-\sum_i N_i\mu_i
-\label{eq:defects:ef2}
-\end{equation}
-where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$.
-The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin.
-For a defect configuration of a single atom species equation \eqref{eq:defects:ef2} is equivalent to equation \eqref{eq:defects:ef1}.
+However, the implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.
+Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.
+Thus, the study proceeds examining pairs of most probable defect configurations and related diffusion processes exclusively by first-principles methods.
+These systems can still be described by the highly accurate but computationally costly method.
+Respective results allow to draw conclusions concerning the SiC precipitation in Si.
\section{Silicon self-interstitials}
-Point defects in silicon have been extensively studied, both experimentally and theoretically \cite{fahey89,leung99}.
-Quantum-mechanical total-energy calculations are an invalueable tool to investigate the energetic and structural properties of point defects since they are experimentally difficult to assess.
-
-The formation energies of some of the silicon self-interstitial configurations are listed in table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by former studies \cite{leung99}.
-\begin{table}[th]
+For investigating the \si{} structures a Si atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
+The formation energies of \si{} configurations are listed in Table \ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies \cite{al-mushadani03,leung99}.
+\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c c}
\hline
\hline
- & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
+ & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
\hline
- Erhart/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
- VASP & 3.77 & 3.42 & 4.41 & 3.39 & 3.63 \\
- LDA \cite{leung99} & 3.43 & 3.31 & - & 3.31 & - \\
- GGA \cite{leung99} & 4.07 & 3.80 & - & 3.84 & - \\
+\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 \\
+\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 & - & - \\
+% todo cite without []
\hline
\hline
\end{tabular}
\end{center}
-\caption[Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of silicon self-interstitials in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and V the vacancy interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
+\caption[Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations.]{Formation energies of Si self-interstitials in crystalline Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral and H the hexagonal interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:si_self}
\end{table}
-The final configurations obtained after relaxation are presented in figure \ref{fig:defects:conf}.
-\begin{figure}[t!h!]
+\begin{figure}[t]
\begin{center}
-%\hrule
-%\vspace*{0.2cm}
-%\begin{flushleft}
-%\begin{minipage}{5cm}
-%\underline{\hkl<1 1 0> dumbbell}\\
-%$E_{\text{f}}=3.39\text{ eV}$\\
-%\includegraphics[width=3.0cm]{si_pd_vasp/110_2333.eps}
-%\end{minipage}
-%\begin{minipage}{5cm}
-%\underline{Hexagonal}\\
-%$E_{\text{f}}=3.42\text{ eV}$\\
-%\includegraphics[width=3.0cm]{si_pd_vasp/hex_2333.eps}
-%\end{minipage}
-%\begin{minipage}{5cm}
-%\underline{Tetrahedral}\\
-%$E_{\text{f}}=3.77\text{ eV}$\\
-%\includegraphics[width=3.0cm]{si_pd_vasp/tet_2333.eps}
-%\end{minipage}\\[0.2cm]
-%\begin{minipage}{5cm}
-%\underline{\hkl<1 0 0> dumbbell}\\
-%$E_{\text{f}}=4.41\text{ eV}$\\
-%\includegraphics[width=3.0cm]{si_pd_vasp/100_2333.eps}
-%\end{minipage}
-%\begin{minipage}{5cm}
-%\underline{Vacancy}\\
-%$E_{\text{f}}=3.63\text{ eV}$\\
-%\includegraphics[width=3.0cm]{si_pd_vasp/vac_2333.eps}
-%\end{minipage}
-%\begin{minipage}{5cm}
-%\begin{center}
-%VASP\\
-%calculations\\
-%\end{center}
-%\end{minipage}
-%\end{flushleft}
-%\vspace*{0.2cm}
-%\hrule
\begin{flushleft}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
-$E_{\text{f}}=3.40\text{ eV}$\\
+$E_{\text{f}}=3.40\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/tet.eps}
\end{minipage}
\begin{minipage}{10cm}
\underline{Hexagonal}\\[0.1cm]
\begin{minipage}{4cm}
-$E_{\text{f}}^*=4.48\text{ eV}$\\
+$E_{\text{f}}^*=4.48\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex_a.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\end{center}
\end{minipage}
\begin{minipage}{4cm}
-$E_{\text{f}}=3.96\text{ eV}$\\
+$E_{\text{f}}=3.96\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/hex.eps}
\end{minipage}
\end{minipage}\\[0.2cm]
\begin{minipage}{5cm}
\underline{\hkl<1 0 0> dumbbell}\\
-$E_{\text{f}}=5.42\text{ eV}$\\
+$E_{\text{f}}=5.42\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{\hkl<1 1 0> dumbbell}\\
-$E_{\text{f}}=4.39\text{ eV}$\\
+$E_{\text{f}}=4.39\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{5cm}
\underline{Vacancy}\\
-$E_{\text{f}}=3.13\text{ eV}$\\
+$E_{\text{f}}=3.13\,\text{eV}$\\
\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
\end{minipage}
\end{flushleft}
%\hrule
\end{center}
-\caption[Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed silicon self-interstitial defect configurations obtained by classical potential calculations. The silicon atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
+\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. The Si atoms and the bonds (only for the interstitial atom) are illustrated by yellow spheres and blue lines.}
\label{fig:defects:conf}
\end{figure}
+The final configurations obtained after relaxation are presented in Fig. \ref{fig:defects:conf}.
+The displayed structures are the results of the classical potential simulations.
-There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell followed by the hexagonal and tetrahedral defect is the lowest in energy.
+There are differences between the various results of the quantum-mechanical calculations but the consensus view is that the \hkl<1 1 0> dumbbell (DB) followed by the hexagonal and tetrahedral defect is the lowest in energy.
This is nicely reproduced by the DFT calculations performed in this work.
It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
Among the established analytical potentials only the EDIP \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
However, these potenitals show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
-In fact the Erhart/Albe potential calculations favor the tetrahedral defect configuration.
+In fact the EA potential calculations favor the tetrahedral defect configuration.
+This limitation is assumed to arise due to the cut-off.
+In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
+Indeed, an increase of the cut-off results in increased values of the formation energies \cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
+The same issue has already been discussed by Tersoff \cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
+While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the investigations of defect combinations later on in this chapter.
+
The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
-In the first two pico seconds while kinetic energy is decoupled from the system the Si interstitial seems to condense at the hexagonal site.
-The formation energy of 4.48 eV is determined by this low kinetic energy configuration shortly before the relaxation process starts.
-The Si interstitial atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
-The formation energy of 3.96 eV for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
+In the first two pico seconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
+The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
+The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
+The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
Obviously the authors did not carefully check the relaxed results assuming a hexagonal configuration.
-In figure \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
-\begin{figure}[th]
+In Fig. \ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
+\begin{figure}[t]
\begin{center}
-\includegraphics[width=10cm]{e_kin_si_hex.ps}
+\includegraphics[width=0.7\textwidth]{e_kin_si_hex.ps}
\end{center}
-\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhart/Albe classical potential.}
+\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 PARCAS MD code \cite{parcas_md}.
-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 tetdrahedral 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 basic problems of analytical potential models for describing defect structures.
+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 tetdrahedral 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.
However, the energy barrier is small.
-\begin{figure}[th]
+\begin{figure}[ht]
\begin{center}
-\includegraphics[width=12cm]{nhex_tet.ps}
+\includegraphics[width=0.7\textwidth]{nhex_tet.ps}
\end{center}
\caption{Migration barrier of the tetrahedral Si self-interstitial slightly displaced along all three coordinate axes into the exact tetrahedral configuration using classical potential calculations.}
\label{fig:defects:nhex_tet_mig}
\end{figure}
-This is exemplified in figure \ref{fig:defects:nhex_tet_mig}, which shows the change in potential energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
-The technique used to obtain the migration data is explained in a later section (\ref{subsection:100mig}).
-The barrier is less than 0.2 eV.
-Hence these artifacts should have a negligent influence in finite temperature simulations.
+This is exemplified in Fig. \ref{fig:defects:nhex_tet_mig}, which shows the change in configurational energy during the migration of the interstitial displaced along all three coordinate axes into the tetrahedral configuration.
+The barrier is smaller than \unit[0.2]{eV}.
+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 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.
+For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
+No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.
+
+In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e. if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
+The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
+The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
+A more detailed description of the chemical bonding is achieved through quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-The bond-centered configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhart/Albe and VASP calculations.
-In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, that is if the distance of two atoms is within the cutoff region $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
-For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
-The length of these bonds are, however, close to the cutoff range and thus are weak interactions not constituting actual chemical bonds.
-The same applies to the bonds between the interstitial and the upper two atoms in the \hkl<1 1 0> dumbbell configuration.
-A more detailed description of the chemical bonding is achieved by quantum-mechanical calculations by investigating the accumulation of negative charge between the nuclei.
-{\color{red}Todo: Plot the electron density for these types of defect to derive conclusions of existing bonds?}
-\section{Carbon related point defects}
+\section{Carbon point defects in silicon}
-Carbon is a common and technologically important impurity in silicon.
-Concentrations as high as $10^{18}\text{ cm}^{-3}$ occur in Czochralski-grown silicon samples.
-It is well established that carbon and other isovalent impurities prefer to dissolve substitutionally in silicon.
-However, radiation damage can generate carbon interstitials \cite{watkins76} which have enough mobility at room temeprature to migrate and form defect complexes.
+\subsection{Defect structures in a nutshell}
-Formation energies of the most common carbon point defects in crystalline silicon are summarized in table \ref{tab:defects:c_ints} and the relaxed configurations obtained by classical potential calculations visualized in figure \ref{fig:defects:c_conf}.
-The type of reservoir of the carbon impurity to determine the formation energy of the defect was chosen to be SiC.
+For investigating the \ci{} structures a C atom is inserted or removed according to Fig. \ref{fig:basics:ins_pos} of section \ref{section:basics:defects}.
+Formation energies of the most common C point defects in crystalline Si are summarized in Table \ref{tab:defects:c_ints}.
+The relaxed configurations are visualized in Fig. \ref{fig:defects:c_conf}.
+Again, the displayed structures are the results obtained by the classical potential calculations.
+The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
-Hence, the chemical potential of silicon and carbon is determined by the cohesive energy of silicon and silicon carbide.
-\begin{table}[th]
+Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
+\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\hline
- & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & BC \\
\hline
- Erhart/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
- %VASP & unstable & unstable & 3.15 & 3.60 & 1.39 & 4.10 \\
- VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
+\multicolumn{6}{c}{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 \\
+\multicolumn{6}{c}{Other studies} \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
- ab initio & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
+ {\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
\hline
\hline
\end{tabular}
\end{center}
-\caption[Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations.]{Formation energies of carbon point defects in crystalline silicon determined by classical potential molecular dynamics and density functional calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal, B the bond-centered and S the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
+\caption[Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations.]{Formation energies of C point defects in c-Si determined by classical potential MD and DFT calculations. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to the substitutional interstitial configuration. The dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations are marked by an asterisk and are determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects:c_ints}
\end{table}
-\begin{figure}[th]
+\begin{figure}[t]
\begin{center}
\begin{flushleft}
\begin{minipage}{4cm}
\underline{Hexagonal}\\
-$E_{\text{f}}^*=9.05\text{ eV}$\\
+$E_{\text{f}}^*=9.05\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/hex.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 0 0>}\\
-$E_{\text{f}}=3.88\text{ eV}$\\
+$E_{\text{f}}=3.88\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/100.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\end{minipage}
\begin{minipage}{5cm}
\underline{Tetrahedral}\\
-$E_{\text{f}}=6.09\text{ eV}$\\
+$E_{\text{f}}=6.09\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/tet.eps}
\end{minipage}\\[0.2cm]
\begin{minipage}{4cm}
\underline{Bond-centered}\\
-$E_{\text{f}}^*=5.59\text{ eV}$\\
+$E_{\text{f}}^*=5.59\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/bc.eps}
\end{minipage}
\begin{minipage}{0.8cm}
\end{minipage}
\begin{minipage}{4cm}
\underline{\hkl<1 1 0> dumbbell}\\
-$E_{\text{f}}=5.18\text{ eV}$\\
+$E_{\text{f}}=5.18\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/110.eps}
\end{minipage}
\begin{minipage}{0.5cm}
\end{minipage}
\begin{minipage}{5cm}
\underline{Substitutional}\\
-$E_{\text{f}}=0.75\text{ eV}$\\
+$E_{\text{f}}=0.75\,\text{eV}$\\
\includegraphics[width=4.0cm]{c_pd_albe/sub.eps}
\end{minipage}
\end{flushleft}
\end{center}
-\caption[Relaxed carbon point defect configurations obtained by classical potential calculations.]{Relaxed carbon point defect configurations obtained by classical potential calculations. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
+\caption[Relaxed C point defect configurations obtained by classical potential calculations.]{Relaxed C point defect configurations obtained by classical potential calculations. The Si/C atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines.}
\label{fig:defects:c_conf}
\end{figure}
-Substitutional carbon in silicon is found to be the lowest configuration in energy for all potential models.
-An experiemntal value of the formation energy of substitutional carbon 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}.
+\cs{} occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration in energy for all potential models.
+An experiemntal 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 1.6 to 1.89 eV an excellent agreement with the experimental solubility data within the entire temeprature 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 Erhart/Albe potential undervalues the formation energy roughly by a factor of two.
-
-Except for Tersoff's tedrahedral configuration results the \hkl<1 0 0> dumbbell is the energetically most favorable interstital configuration.
-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 cutoff set to 2.5 \AA{} (see ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between 3 and 10 eV.
-Keeping these considerations in mind, the \hkl<1 0 0> dumbbell is the most favorable interstitial configuration for all interaction models.
-In addition to the theoretical results compared to in table \ref{tab:defects:c_ints} there is experimental evidence of the existence of this configuration \cite{watkins76}.
-It is frequently generated in the classical potential simulation runs in which carbon is inserted at random positions in the c-Si matrix.
-In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \hkl<1 0 0> dumbbell configuration.
+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 temeprature 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 tedrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstital 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}.
+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.
+The defect is frequently generated in the classical potential simulation runs, in which C is inserted at random positions in the c-Si matrix.
+In quantum-mechanical simulations the unstable tetrahedral and hexagonal configurations undergo a relaxation into the \ci{} \hkl<1 0 0> DB configuration.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
+It should be noted that EA and DFT predict almost equal formation energies.
The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
-Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the Erhart/Albe potential.
-In both cases a relaxation towards the \hkl<1 0 0> dumbbell configuration is observed.
-In fact the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
-Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on carbon point defects in silicon \cite{tersoff90}.
-
-The tetrahedral is the second most unfavorable interstitial configuration using classical potentials and keeping in mind the abrupt cutoff effect in the case of the Tersoff potential as discussed earlier.
-Again, quantum-mechanical results reveal this configuration unstable.
-The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.
-
-Just as for the Si self-interstitial a carbon \hkl<1 1 0> dumbbell configuration exists.
-For the Erhart/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
-Similar structures arise in both types of simulations with the silicon and carbon atom sharing a silicon lattice site aligned along \hkl<1 1 0> where the carbon atom is localized slightly closer to the next nearest silicon atom located in the opposite direction to the site-sharing silicon atom even forming a bond to the next but one silicon atom in this direction.
-
-The bond-centered configuration is unstable for the Erhart/Albe potential.
-The system moves into the \hkl<1 1 0> interstitial configuration.
-This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
+Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the EA potential.
+In both cases a relaxation towards the \ci{} \hkl<1 0 0> DB configuration is observed.
+Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable \cite{tersoff90}.
+In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodifed Tersoff potential parameters.
+Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on C point defects in Si.
+
+The tetrahedral is the second most unfavorable interstitial configuration using classical potentials if the abrupt cut-off effect of the Tersoff potential is taken into account.
+Again, quantum-mechanical results reveal this configuration to be unstable.
+The fact that the tetrahedral and hexagonal configurations are the two most unstable configurations in classical potential calculations and, thus, are less likely to arise in MD simulations, acts in concert with the fact that these configurations are found to be unstable in the more accurate quantum-mechanical description.
+
+Just as for \si{}, a \ci{} \hkl<1 1 0> DB configuration exists.
+It constitutes the second most favorable configuration, reproduced by both methods.
+Similar structures arise in both types of simulations.
+The Si and C atom share a regular Si lattice site aligned along the \hkl<1 1 0> direction.
+The C atom is slightly displaced towards the next nearest Si atom located in the opposite direction with respect to the site-sharing Si atom and even forms a bond with this atom.
+
+The \ci{} \hkl<1 1 0> DB structure is energetically followed by the BC configuration.
+However, even though EA based results yield the same difference in energy with respect to the \hkl<1 1 0> defect as DFT does, the BC configuration is found to be a unstable within the EA description.
+The BC configuration descends into the \ci{} \hkl<1 1 0> DB configuration.
+Due to the high formation energy of the BC defect resulting in a low probability of occurrence of this defect, the wrong description is not posing a serious limitation of the EA potential.
+Tersoff indeed predicts a metastable BC configuration.
+However, it is not in the correct order and lower in energy than the \ci{} \hkl<1 1 0> DB.
Quantum-mechanical results of this configuration are discussed in more detail in section \ref{subsection:bc}.
-In another ab inito study Capaz et al. \cite{capaz94} determined this configuration as an intermediate saddle point structure of a possible migration path, which is 2.1 eV higher than the \hkl<1 0 0> dumbbell configuration.
-In calculations performed in this work the bond-centered configuration in fact is a real local minimum and an energy barrier is needed to reach this configuration starting from the \hkl<1 0 0> dumbbell configuration, which is discussed in section \ref{subsection:100mig}.
-
-\subsection[\hkl<1 0 0> dumbbell interstitial configuration]{\boldmath\hkl<1 0 0> dumbbell interstitial configuration}
+In another {\em ab inito} study, Capaz~et~al. \cite{capaz94} in turn found 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 neglection 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.
+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.
+This is discussed in more detail in section \ref{subsection:100mig}.
+
+To conclude, discrepancies between the results from classical potential calculations and those obtained from first principles are observed.
+Within the classical potentials EA outperforms Tersoff and is, therefore, used for further studies.
+Both methods (EA and DFT) predict the \ci{} \hkl<1 0 0> DB configuration to be most stable.
+Also the remaining defects and their energetical order are described fairly well.
+It is thus concluded that, so far, modeling of the SiC precipitation by the EA potential might lead to trustable results.
+
+\subsection[C \hkl<1 0 0> dumbbell interstitial configuration]{\boldmath C \hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}
-As the \hkl<1 0 0> dumbbell interstitial is the lowest configuration in energy it is the most probable hence important interstitial configuration of carbon in silicon.
-It was first identified by infra-red (IR) spectroscopy \cite{bean70} and later on by electron paramagnetic resonance (EPR) spectroscopy \cite{watkins76}.
+As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
+The structure was initially suspected by IR local vibrational mode absorption \cite{bean70} and finally verified by electron paramegnetic resonance (EPR) \cite{watkins76} studies on irradiated Si substrates at low temperatures.
-Figure \ref{fig:defects:100db_cmp} schematically shows the \hkl<1 0 0> dumbbell structure and table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by analytical potential and quantum-mechanical calculations.
-For comparison, the obtained structures for both methods visualized out of the atomic position data are presented in figure \ref{fig:defects:100db_vis_cmp}.
-\begin{figure}[th]
+Fig. \ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table \ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
+For comparison, the obtained structures for both methods are visualized in Fig. \ref{fig:defects:100db_vis_cmp}.
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=12cm]{100-c-si-db_cmp.eps}
\end{center}
-\caption[Sketch of the \hkl<1 0 0> dumbbell structure.]{Sketch of the \hkl<1 0 0> dumbbell structure. Atomic displacements, distances and bond angles are listed in table \ref{tab:defects:100db_cmp}.}
+\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}
-%
-\begin{table}[th]
+\begin{table}[ht]
\begin{center}
Displacements\\
\begin{tabular}{l c c c c c c c c c}
& & & & \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
-Erhart/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
-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
-Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
-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
-Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
-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 \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhart/Albe potential and 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 figure \ref{fig:defects:100db_cmp}. In addition, the equilibrium lattice constant for crystalline silicon 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}
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
-\underline{Erhart/Albe}
+\underline{\textsc posic}
\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
-\underline{VASP}
+\underline{\textsc vasp}
\includegraphics[width=5cm]{c_pd_vasp/100_cmp.eps}
\end{center}
\end{minipage}
\end{center}
-\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhart/Albe potential and 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}
-\begin{figure}[th]
+\begin{figure}[ht]
\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 C \hkl<1 0 0> dumbbell structure obtained by VASP calculations.]{Charge density isosurface and Kohn-Sham levels of the C \hkl<1 0 0> dumbbell structure obtained by VASP calculations. Yellow and grey spheres correspond to silicon and carbon 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 grey 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 silicon atom numbered '1' and the C atom compose the dumbbell structure.
-They share the lattice site which is indicated by the dashed red circle and which they are displaced from by length $a$ and $b$ respectively.
-The atoms no longer have four tetrahedral bonds to the silicon atoms located on the alternating opposite edges of the cube.
-Instead, each of the dumbbell atoms forms threefold coordinated bonds, which are located in a plane.
-One bond is formed to the other dumbbell atom.
-The other two bonds are bonds to the two silicon edge atoms located in the opposite direction of the dumbbell atom.
-The distance of the two dumbbell atoms is almost the same for both types of calculations.
-However, in the case of the VASP calculation, the dumbbell structure is pushed upwards compared to the Erhart/Albe results.
-This is easily identified by comparing the values for $a$ and $b$ and the two structures in figure \ref{fig:defects:100db_vis_cmp}.
-Thus, the angles of bonds of the silicon dumbbell atom ($\theta_1$ and $\theta_2$) are closer to $120^{\circ}$ signifying the predominance of $sp^2$ hybridization.
-On the other hand, the carbon atom forms an almost collinear bond ($\theta_3$) with the two silicon edge atoms implying the predominance of $sp$ bonding.
-This is supported by the image of the charge density isosurface in figure \ref{img:defects:charge_den_and_ksl}.
-The two lower Si atoms are $sp^3$ hybridised and form $\sigma$ bonds to the silicon dumbbell atom.
-The same is true for the upper two silicon atoms and the C dumbbell atom.
-In addition the dumbbell atoms form $\pi$ bonds.
-However, due to the increased electronegativity of the carbon atom the electron density is attracted by and thus localized around the carbon atom.
+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.
+Instead, each of the DB atoms forms threefold coordinated bonds, which are located in a plane.
+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.
+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.
+This is supported by the image of the charge density isosurface in Fig. \ref{img:defects:charge_den_and_ksl}.
+The two lower Si atoms are $sp^3$ hybridized and form $\sigma$ bonds to the Si DB atom.
+The same is true for the upper two Si atoms and the C DB atom.
+In addition the DB atoms form $\pi$ bonds.
+However, due to the increased electronegativity of the C atom the electron density is attracted by and, thus, localized around the C atom.
In the same figure the Kohn-Sham levels are shown.
There is no magnetization density.
-An acceptor level arises at approximately $E_v+0.35\text{ eV}$ while a band gap of about 0.75 eV can be estimated from the Kohn-Sham level diagram for plain silicon.
+An acceptor level arises at approximately $E_v+0.35\,\text{eV}$ while a band gap of about \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
+However, strictly speaking, the Kohn-Sham levels and orbitals do not have a direct physical meaning and, thus, these values have to be taken with care.
+% todo band gap problem
\subsection{Bond-centered interstitial configuration}
\label{subsection:bc}
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{c_pd_vasp/bc_2333.eps}\\
\caption[Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration.]{Structure, charge density isosurface and Kohn-Sham level diagram of the bond-centered interstitial configuration. Gray, green and blue surfaces mark the charge density of spin up, spin down and the resulting spin up electrons in the charge density isosurface, in which the carbon atom is represented by a red sphere. In the energy level diagram red and green lines mark occupied and unoccupied states.}
\label{img:defects:bc_conf}
\end{figure}
-In the bond-centerd insterstitial configuration the interstitial atom is located inbetween two next neighboured silicon atoms forming linear bonds.
-In former studies this configuration is found to be an intermediate saddle point configuration determining the migration barrier of one possibe migration path of a \hkl<1 0 0> dumbbel configuration into an equivalent one \cite{capaz94}.
-This is in agreement with results of the Erhart/Albe potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
-However, this fact could not be reproduced by spin polarized VASP calculations performed in this work.
-Present results suggest this configuration to be a real local minimum.
-In fact, an additional barrier has to be passed to reach this configuration starting from the \hkl<1 0 0> interstitital configuration, which is investigated in section \ref{subsection:100mig}.
-After slightly displacing the carbon 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 resulting structures relax back into the bond-centered configuration.
-As we will see in later migration simulations the same would happen to structures where the carbon atom is displaced along the migration direction, which approximately is the \hkl<1 1 0> direction.
-These relaxations indicate that the bond-cenetered configuration is a real local minimum instead of an assumed saddle point configuration.
-Figure \ref{img:defects:bc_conf} shows the structure, the charge density isosurface and the Kohn-Sham levels of the bond-centered configuration.
-The linear bonds of the carbon atom to the two silicon atoms indicate the $sp$ hybridization of the carbon atom.
-Two electrons participate to the linear $\sigma$ bonds with the silicon neighbours.
+In the BC insterstitial configuration the interstitial atom is located inbetween 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 possibe 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.
+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> interstitital 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.
+As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
+These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
+Fig. \ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
+The linear bonds of the C atom to the two Si atoms indicate the $sp$ hybridization of the C atom.
+Two electrons participate to the linear $\sigma$ bonds with the Si neighbors.
The other two electrons constitute the $2p^2$ orbitals resulting in a net magnetization.
-This is supported by the charge density isosurface and the Kohn-Sham levels in figure \ref{img:defects:bc_conf}.
-The blue torus, reinforcing the assumption of the p orbital, illustrates the resulting spin up electron density.
+This is supported by the charge density isosurface and the Kohn-Sham levels in Fig. \ref{img:defects:bc_conf}.
+The blue torus, which reinforces the assumption of the $p$ orbital, illustrates the resulting spin up electron density.
In addition, the energy level diagram shows a net amount of two spin up electrons.
+% todo smaller images, therefore add mo image
-\section{Migration of the carbon interstitials}
-\label{subsection:100mig}
+\clearpage{}
+\cleardoublepage{}
-In the following the problem of interstitial carbon migration in silicon is considered.
-Since the carbon \hkl<1 0 0> dumbbell interstitial is the most probable hence most important configuration the migration simulations focus on this defect.
+% todo migration of \si{}!
+
+\section{Migration of the carbon interstitial}
+\label{subsection:100mig}
-\begin{figure}[t!h!]
+A measure for the mobility of interstitial C is the activation energy necessary for the migration from one stable position to another.
+The stable defect geometries have been discussed in the previous subsection.
+In the following, the problem of interstitial C migration in Si is considered.
+Since the \ci{} \hkl<1 0 0> DB is the most probable, hence, most important configuration, the migration of this defect atom from one site of the Si host lattice to a neighboring site is in the focus of investigation.
+\begin{figure}[ht]
\begin{center}
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
\end{minipage}
\end{minipage}
\end{center}
-\caption{Migration pathways of the carbon \hkl<1 0 0> interstitial dumbbell in silicon.}
+\caption{Conceivable migration pathways among two \ci{} \hkl<1 0 0> DB configurations.}
\label{img:defects:c_mig_path}
\end{figure}
-Three different migration paths are accounted in this work, which are shown in figure \ref{img:defects:c_mig_path}.
-The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> dumbbell interstitial configuration.
-During this migration the carbon atom is changing its silicon dumbbell partner.
-The new partner is the one located at $\frac{a}{4}\hkl<1 1 -1>$ relative to the initial one.
-Two of the three bonds to the next neighboured silicon atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
-The carbon atom resides in the \hkl(1 1 0) plane.
-This transition involves an intermediate bond-centerd configuration.
-Results discussed in \ref{subsection:bc} indicate, that the bond-ceneterd configuration is a real local minimum.
-Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate bond-cenetered configuration constitutes a metastable configuration.
-Due to symmetry it is enough to consider the transition from the bond-centered to the \hkl<1 0 0> configuration or vice versa.
-In the second path, the carbon atom is changing its silicon partner atom as in path one.
-However, the trajectory of the carbon atom is no longer proceeding in the \hkl(1 1 0) plane.
-The orientation of the new dumbbell configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
-Again one bond is broken while another one is formed.
+Three different migration paths are accounted in this work, which are displayed in Fig. \ref{img:defects:c_mig_path}.
+The first migration investigated is a transition of a \hkl<0 0 -1> into a \hkl<0 0 1> DB interstitial configuration.
+During this migration the C atom is changing its Si DB partner.
+The new partner is the one located at $a_{\text{Si}}/4 \hkl<1 1 -1>$ relative to the initial one, where $a_{\text{Si}}$ is the Si lattice constant.
+Two of the three bonds to the next neighbored Si atoms are preserved while the breaking of the third bond and the accompanying formation of a new bond is observed.
+The C atom resides in the \hkl(1 1 0) plane.
+This transition involves the intermediate BC configuration.
+However, results discussed in the previous section indicate that the BC configuration is a real local minimum.
+Thus, the \hkl<0 0 -1> to \hkl<0 0 1> migration can be thought of a two-step mechanism in which the intermediate BC configuration constitutes a metastable configuration.
+Due to symmetry it is enough to consider the transition from the BC to the \hkl<1 0 0> configuration or vice versa.
+In the second path, the C atom is changing its Si partner atom as in path one.
+However, the trajectory of the C atom is no longer proceeding in the \hkl(1 1 0) plane.
+The orientation of the new DB configuration is transformed from \hkl<0 0 -1> to \hkl<0 -1 0>.
+Again, one bond is broken while another one is formed.
As a last migration path, the defect is only changing its orientation.
-Thus, it is not responsible for long-range migration.
-The silicon dumbbell partner remains the same.
-The bond to the face-centered silicon atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
+Thus, this path is not responsible for long-range migration.
+The Si DB partner remains the same.
+The bond to the face-centered Si atom at the bottom of the unit cell breaks and a new one is formed to the face-centered atom at the forefront of the unit cell.
-\begin{figure}[t!h!]
-\begin{center}
-\begin{minipage}{6cm}
-\underline{Original}\\
-\includegraphics[width=6cm]{crt_orig.eps}
-\end{minipage}
-\begin{minipage}{1cm}
-\hfill
-\end{minipage}
-\begin{minipage}{6cm}
-\underline{Modified}\\
-\includegraphics[width=6cm]{crt_mod.eps}
-\end{minipage}
-\end{center}
-\caption{Schematic of the constrained relaxation technique (CRT) (left) and of the modified version (right) used to obtain migration pathways and corresponding activation energies.}
-\label{fig:defects:crt}
-\end{figure}
-Since the starting and final structure, which are both local minima of the potential energy surface, are known, the aim is to find the minimum energy path from one local minimum to the other one.
-One method to find a minimum energy path is to move the diffusing atom stepwise from the starting to the final position and only allow relaxation in the plane perpendicular to the direction of the vector connecting its starting and final position.
-This is called the constrained relaxation technique (CRT), which is schematically displayed in the left part of figure \ref{fig:defects:crt}.
-No constraints are applied to the remaining atoms in order to allow relaxation of the surrounding lattice.
-To prevent the remaining lattice to migrate according to the displacement of the defect an atom far away from the defect region is fixed in all three coordinate directions.
-However, it turned out, that this method tremendously failed applying it to the present migration pathways and structures.
-Abrupt changes in structure and free energy occured among relaxed structures of two successive displacement steps.
-For some structures even the expected final configurations were never obtained.
-Thus, the method mentioned above was adjusted adding further constraints in order to obtain smooth transitions, either in energy as well as structure is concerned.
-In this new method all atoms are stepwise displaced towards their final positions.
-Relaxation of each individual atom is only allowed in the plane perpendicular to the last individual displacement vector, as displayed in the right part of figure \ref{fig:defects:crt}.
-The modifications used to add this feature to the VASP code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
-Due to these constraints obtained activation energies can effectively be higher.
-{\color{red}Todo: To refine the migration barrier one has to find the saddle point structure and recalculate the free energy of this configuration with a reduced set of constraints.}
-
-\subsection{Migration barriers obtained by quantum-mechanical calculations}
-
-In the following migration barriers are investigated using quantum-mechanical calculations.
-The amount of simulated atoms is the same as for the investigation of the point defect structures.
-Due to the time necessary for computing only ten displacement steps are used.
-
-\begin{figure}[t!h!]
+\subsection{Migration paths obtained by first-principles calculations}
+
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
\begin{picture}(0,0)(150,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to bond-centered (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to BC (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_001_mig}
\end{figure}
-In figure \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> dumbbell to the bond-ceneterd configuration is displayed.
-To reach the bond-centered configuration, which is 0.94 eV higher in energy than the \hkl<0 0 -1> dumbbell configuration, an energy barrier of approximately 1.2 eV, given by the saddle point structure at a displacement of 60 \%, has to be passed.
-This amount of energy is needed to break the bond of the carbon atom to the silicon atom at the bottom left.
-In a second process 0.25 eV of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
+In Fig. \ref{fig:defects:00-1_001_mig} results of the \hkl<0 0 -1> to \hkl<0 0 1> migration fully described by the migration of the \hkl<0 0 -1> to the BC configuration is displayed.
+To reach the BC configuration, which is \unit[0.94]{eV} higher in energy than the \hkl<0 0 -1> DB configuration, an energy barrier of approximately \unit[1.2]{eV} given by the saddle point structure at a displacement of \unit[60]{\%} has to be passed.
+This amount of energy is needed to break the bond of the C atom to the Si atom at the bottom left.
+In a second process \unit[0.25]{eV} of energy are needed for the system to revert into a \hkl<1 0 0> configuration.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[1.6cm]
\begin{picture}(0,0)(140,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition.]{Migration barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition. Bonds of the C atom are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_mig}
\end{figure}
-Figure \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> dumbbell transition.
-The resulting migration barrier of approximately 0.9 eV is very close to the experimentally obtained values of 0.73 \cite{song90} and 0.87 eV \cite{tipping87}.
+Fig. \ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
+The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV} \cite{lindner06}, \unit[0.73]{eV} \cite{song90} and \unit[0.87]{eV} \cite{tipping87}.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[1.8cm]
\begin{picture}(0,0)(140,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place.]{Migration barrier and structures of the \hkl<0 0 -1> dumbbell (left) to the \hkl<0 -1 0> dumbbell (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
+\caption[Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place.]{Reorientation barrier and structures of the \hkl<0 0 -1> DB (left) to the \hkl<0 -1 0> DB (right) transition in place. Bonds of the carbon atoms are illustrated by blue lines.}
\label{fig:defects:00-1_0-10_ip_mig}
\end{figure}
-The third migration path in which the dumbbell is changing its orientation is shown in figure \ref{fig:defects:00-1_0-10_ip_mig}.
-An energy barrier of roughly 1.2 eV is observed.
-Experimentally measured activation energies for reorientation range from 0.77 eV to 0.88 eV \cite{watkins76,song90}.
+The third migration path, in which the DB is changing its orientation, is shown in Fig. \ref{fig:defects:00-1_0-10_ip_mig}.
+An energy barrier of roughly \unit[1.2]{eV} is observed.
+Experimentally measured activation energies for reorientation range from \unit[0.77]{eV} to \unit[0.88]{eV} \cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
-Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely carbon interstitial in silicon explaining both, annealing and reorientation experiments.
-The activation energy of roughly 0.9 eV nicely compares to experimental values reinforcing the correct identification of the C-Si dumbbell diffusion mechanism.
-The theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by 35 \%.
-In addition the bond-ceneterd configuration, for which spin polarized calculations are necessary, is found to be a real local minimum instead of a saddle point configuration.
+Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely C interstitial in Si explaining both, annealing and reorientation experiments.
+The activation energy of roughly \unit[0.9]{eV} nicely compares to experimental values reinforcing the correct identification of the C-Si DB diffusion mechanism.
+Slightly increased values compared to experiment might be due to the tightend constraints applied in the modified CRT approach.
+Nevertheless, the theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
+In addition, it is finally shown that the BC configuration, for which spin polarized calculations are necessary, constitutes a real local minimum instead of a saddle point configuration due to the presence of restoring forces for displacements in migration direction.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
%\end{picture}
\end{center}
-\caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si dumbbell transition.}
+\caption{Migration barriers of the \hkl<1 1 0> DB to BC (blue), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, red) C-Si DB transition.}
\label{fig:defects:110_mig_vasp}
\end{figure}
-Further migration pathways in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration are totally conceivable.
+Further migration pathways, in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration, are totally conceivable.
+% HIER WEITER
This is investigated in the following in order to find possible migration pathways that have an activation energy lower than the ones found up to now.
The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
-The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighboured lattice site is approximately 1.35 eV.
+The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighbored lattice site is approximately 1.35 eV.
During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.
Due to applying updated constraints on all atoms the obtained migration barriers and pathes might be overestimated and misguided.
To reinforce the applicability of the employed technique the obtained activation energies and migration pathes for the \hkl<0 0 -1> to \hkl<0 -1 0> transition are compared to two further migration calculations, which do not update the constrainted direction and which only apply updated constraints on three selected atoms, that is the diffusing C atom and the Si dumbbell pair in the initial and final configuration.
Results are presented in figure \ref{fig:defects:00-1_0-10_cmp}.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/00-1_0-10_nosym_sp_cmp.ps}
\end{center}
The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
This is understandable since the update results in a more aggressive advance towards the final configuration.
In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
-The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}in progress} ...
+The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}Todo: does not work!} ...
\subsection{Migration barriers obtained by classical potential calculations}
\label{subsection:defects:mig_classical}
-The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhart/Albe potential.
+The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical EA potential.
Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
The time constant $\tau$ for the Berendsen thermostat is set to 1 fs in order to have direct velocity scaling and with the temperature set to zero Kelvin perform a steepest descent minimazation to drive the system into a local minimum.
However, in some cases a time constant of 100 fs resuls in lower barriers and, thus, is shown whenever appropriate.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{bc_00-1.ps}\\[5.6cm]
\begin{pspicture}(0,0)(0,0)
\includegraphics[height=2.2cm]{010_arrow.eps}
\end{picture}
\end{center}
-\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhart/Albe potential.}
+\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical EA potential.}
\label{fig:defects:cp_bc_00-1_mig}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20 -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.75 -1.25 -0.25 -L -0.25 -0.25 -0.25 -r 0.6 -B 0.1
% blue: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_bc_00-1_s20_tr100/ -nll -0.56 -0.56 -0.7 -fur 0.2 0.2 0.0 -c 0.0 -0.25 1.0 -L 0.0 -0.25 -0.25 -r 0.6 -B 0.1
However, the investigated pathways cover an activation energy approximately twice as high as the one obtained by quantum-mechanical calculations.
For the entire transition of the \hkl<0 0 -1> into the \hkl<0 0 1> configuration by passing the bond-centered configuration an additional activation energy of 0.5 eV is necessary to escape from the bond-centered and reach the \hkl<0 0 1> configuration.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{00-1_0-10.ps}\\[2.4cm]
\begin{pspicture}(0,0)(0,0)
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition using the classical Erhart/Albe potential.}
+\caption{Migration barrier and structures of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition using the classical EA potential.}
% red: ./visualize -w 640 -h 480 -d saves/c_in_si_mig_00-1_0-10_s20 -nll -0.56 -0.56 -0.8 -fur 0.3 0.2 0 -c -0.125 -1.7 0.7 -L -0.125 -0.25 -0.25 -r 0.6 -B 0.1
\label{fig:defects:cp_00-1_0-10_mig}
\end{figure}
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{00-1_ip0-10.ps}
\end{center}
-\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical Erhart/Albe potential.}
+\caption{Migration barrier of the \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition in place using the classical EA potential.}
\label{fig:defects:cp_00-1_ip0-10_mig}
\end{figure}
-Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighboured lattice site in the first case and a reorientation within the same lattice site in the latter case.
+Figure \ref{fig:defects:cp_00-1_0-10_mig} and \ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of \hkl<0 0 -1> to \hkl<0 -1 0> C-Si dumbbell transition, with a transition of the C atom to the neighbored lattice site in the first case and a reorientation within the same lattice site in the latter case.
Both pathways look similar.
A local minimum exists inbetween two peaks of the graph.
The corresponding configuration, which is illustrated for the migration simulation with a time constant of 1 fs, looks similar to the \hkl<1 1 0> configuration.
In the first migration path it is the configuration resulting from further relaxation of the rather unstable bond-centered configuration, which is fixed to be a transition point in the migration calculations.
The last two pathways show configurations almost identical to the \hkl<1 1 0> configuration, which constitute a local minimum within the pathway.
Thus, migration pathways with the \hkl<1 1 0> C-Si dumbbell interstitial configuration as a starting or final configuration are further investigated.
-\begin{figure}[ht!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{110_mig.ps}
\end{center}
As expected there is no maximum for the transition into the bond-centered configuration.
As mentioned earlier the bond-centered configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl<1 1 0> configuration.
An activation energy of 2.2 eV is necessary to reorientate the \hkl<0 0 -1> dumbbell configuration into the \hkl<1 1 0> configuration, which is 1.3 eV higher in energy.
-Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell configuration with the Si atom of the neighboured lattice site.
-In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV.
-{\color{red}Todo: Stress out that this is actually more probable, since BC conf is unstable!}
+Residing in this state another 0.9 eV is enough to make the C atom form a \hkl<0 0 -1> dumbbell configuration with the Si atom of the neighbored lattice site.
+In contrast to quantum-mechanical calculations, in which the direct transition is the energetically most favorable transition and the transition composed of the intermmediate migration steps is very unlikely to occur the just presented pathway is much more supposable in classical potential simulations, since the energetically most favorable transition found so far is also composed of two migration steps with activation energies of 2.2 eV and 0.5 eV, for which the intermediate state is the bond-centered configuration, which is unstable.
+Thus the just proposed migration path involving the \hkl<1 1 0> interstitial configuration becomes even more probable than path 1 involving the unstable bond-centered configuration.
Although classical potential simulations reproduce the order in energy of the \hkl<1 0 0> and \hkl<1 1 0> C-Si dumbbell interstitial configurations as obtained by more accurate quantum-mechanical calculations the obtained migration pathways and resulting activation energies differ to a great extent.
On the one hand the most favorable pathways differ.
The probability of already rare diffusion events is further decreased for this reason.
Since agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism a problem arises, which is formulated and discussed in more detail in section \ref{subsection:md:limit}.
+\clearpage{}
+\cleardoublepage{}
+
\section{Combination of point defects}
The structural and energetic properties of combinations of point defects are examined in the following.
The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
Several distances of the two defects are examined.
-\begin{figure}[th]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}{7.5cm}
\includegraphics[width=7cm]{comb_pos.eps}
\begin{minipage}{6.0cm}
\underline{Positions given in $a_{\text{Si}}$}\\[0.3cm]
Initial interstitial I: $\frac{1}{4}\hkl<1 1 1>$\\
-Relative silicon neighbour positions:
+Relative silicon neighbor positions:
\begin{enumerate}
\item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
\item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 -1>$,\\[0.2cm]
\includegraphics[height=2.2cm]{001_arrow.eps}
\end{picture}
\end{center}
-\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighboured silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
+\caption[\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect.]{\hkl<0 0 -1> dumbbell interstitial defect and positions of next neighbored silicon atoms used for the second defect. Two possibilities exist for red numbered atoms and four possibilities exist for blue numbered atoms.}
\label{fig:defects:pos_of_comb}
\end{figure}
-\begin{table}[th]
+\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption[Energetic results of defect combinations.]{Energetic results of defect combinations. The given energies in eV are defined by equation \eqref{eq:defects:e_of_comb}. Equivalent configurations are marked by identical colors. The first column lists the types of the second defect combined with the initial \hkl<0 0 -1> dumbbell interstitial. The position index of the second defect is given in the first row according to figure \ref{fig:defects:pos_of_comb}. R is the position located at $\frac{a_{\text{Si}}}{2}\hkl<3 2 3>$ relative to the initial defect, which is the maximum realizable distance due to periodic boundary conditions.}
\label{tab:defects:e_of_comb}
\end{table}
-Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighboured silicon atoms used for the second defect.
+Figure \ref{fig:defects:pos_of_comb} shows the initial \hkl<0 0 -1> dumbbell interstitial defect and the positions of next neighbored silicon atoms used for the second defect.
Table \ref{tab:defects:e_of_comb} summarizes energetic results obtained after relaxation of the defect combinations.
The energy of interest $E_{\text{b}}$ is defined to be
\begin{equation}
Energies below zero and below the reference value indicate configurations favored compared to configurations in which these point defects are separated far away from each other.
Investigating the first part of table \ref{tab:defects:e_of_comb}, namely the combinations with another \hkl<1 0 0>-type interstitial, most of the combinations result in energies below zero.
-Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbour and a change in orientation compared to the initial one.
+Surprisingly the most favorable configurations are the ones with the second defect created at the very next silicon neighbor and a change in orientation compared to the initial one.
This leads to the conclusion that an agglomeration of C-Si dumbbell interstitials as proposed by the precipitation model introduced in section \ref{section:assumed_prec} is indeed an energetically favored configuration of the system.
The reason for nearby interstitials being favored compared to isolated ones is most probably the reduction of strain energy enabled by combination in contrast to the strain energy created by two individual defects.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{$E_{\text{b}}=-2.25\text{ eV}$}
Structure \ref{fig:defects:comb_db_01} b) is the energetically most favorable configuration.
After relaxation the initial configuration is still evident.
As expected by the initialization conditions the two carbon atoms form a bond.
-This bond has a length of 1.38 \AA{} close to the nex neighbour distance in diamond or graphite, which is approximately 1.54 \AA.
+This bond has a length of 1.38 \AA{} close to the nex neighbor distance in diamond or graphite, which is approximately 1.54 \AA.
The minimum of binding energy observed for this configuration suggests prefered C clustering as a competing mechnism to the C-Si dumbbell interstitial agglomeration inevitable for the SiC precipitation.
-{\color{red}Todo: Activation energy to obtain a configuration of separated C atoms again or vice versa to obtain this configuration from separated C confs?}
+{\color{red}Todo: Activation energies to obtain separated C confs FAILED (again?) - could be added in the combined defect migration chapter and mentioned here, too!}
However, for the second most favorable configuration, presented in figure \ref{fig:defects:comb_db_01} a), the amount of possibilities for this configuration is twice as high.
In this configuration the initial Si (I) and C (I) dumbbell atoms are displaced along \hkl<1 0 0> and \hkl<-1 0 0> in such a way that the Si atom is forming tetrahedral bonds with two silicon and two carbon atoms.
-The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbours, a configuration expected in silicon carbide.
+The carbon and silicon atom constituting the second defect are as well displaced in such a way, that the carbon atom forms tetrahedral bonds with four silicon neighbors, a configuration expected in silicon carbide.
The two carbon atoms spaced by 2.70 \AA{} do not form a bond but anyhow reside in a shorter distance as expected in silicon carbide.
The Si atom numbered 2 is pushed towards the carbon atom, which results in the breaking of the bond to atom 4.
The breaking of the $\sigma$ bond is indeed confirmed by investigating the charge density isosurface of this configuration.
{\color{red}Todo: Is this conf really benificial for SiC prec?}
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{5cm}
a) \underline{$E_{\text{b}}=-2.16\text{ eV}$}
After relaxation the second dumbbell is aligned along \hkl<1 1 0>.
The bond of the silicon atoms 1 and 2 does not persist.
Instead the silicon atom forms a bond with the initial carbon interstitial and the second carbon atom forms a bond with silicon atom 1 forming four bonds in total.
-The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbour distance of 3.08 \AA{} in silicon carbide.
+The carbon atoms are spaced by 3.14 \AA, which is very close to the expected C-C next neighbor distance of 3.08 \AA{} in silicon carbide.
Figure \ref{fig:defects:comb_db_02} c) displays the results of another \hkl<0 0 1> dumbbell inserted at position 3.
The binding energy is -2.05 eV.
Both dumbbells are tilted along the same direction remaining parallely aligned and the second dumbbell is pushed downwards in such a way, that the four dumbbell atoms form a rhomboid.
In figure \ref{fig:defects:comb_db_02} b) a second \hkl<0 1 0> dumbbell is constructed at position 2.
An energy of -1.90 eV is observed.
The initial dumbbell and especially the carbon atom is pushed towards the silicon atom of the second dumbbell forming an additional fourth bond.
-Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighboured silicon atom in \hkl<1 1 -1> direction.
-The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbours.
+Silicon atom number 1 is pulled towards the carbon atoms of the dumbbells accompanied by the disappearance of its bond to silicon number 5 as well as the bond of silicon number 5 to its next neighbored silicon atom in \hkl<1 1 -1> direction.
+The carbon atom of the second dumbbell forms threefold coordinated bonds to its silicon neighbors.
A distance of 2.80 \AA{} is observed for the two carbon atoms.
Again, the two carbon atoms and its two interconnecting silicon atoms form a rhomboid.
C-C distances of 2.70 to 2.80 \AA{} seem to be characteristic for such configurations, in which the carbon atoms and the two interconnecting silicon atoms reside in a plane.
There is a low interaction of the dumbbells, which seem to exist independent of each other.
This, on the one hand, becomes evident by investigating the final structure, in which both of the dumbbells essentially retain the structure expected for a single dumbbell and on the other hand is supported by the observed binding energies which vary only slightly around zero.
This low interaction is due to the larger distance and a missing direct connection by bonds along a crystallographic direction.
-Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbours.
+Both carbon and silicon atoms of the dumbbells form threefold coordinated bonds to their next neighbors.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\text{ eV}$) is obtained for the \hkl<0 0 1> interstitial oppositely orientated to the initial one.
A dumbbell taking the same orientation as the initial one is less unfavorble ($E_{\text{b}}=0.04\text{ eV}$).
Both configurations are unfavorable compared to far-off isolated dumbbells.
Nonparallel orientations, that is the \hkl<0 1 0>, \hkl<0 -1 0> and its equivalents, result in binding energies of -0.12 eV and -0.27 eV, thus, constituting energetically favorable configurations.
The reduction of strain energy is higher in the second case where the carbon atom of the second dumbbell is placed in the direction pointing away from the initial carbon atom.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{$E_{\text{b}}=-1.53\text{ eV}$}
The symmetric configuration is energetically more favorable ($E_{\text{b}}=-1.66\text{ eV}$) since the displacements of the atoms is less than in the antiparallel case ($E_{\text{b}}=-1.53\text{ eV}$).
In figure \ref{fig:defects:comb_db_03} c) and d) the nonparallel orientations, namely the \hkl<0 -1 0> and \hkl<1 0 0> dumbbells are shown.
Binding energies of -1.88 eV and -1.38 eV are obtained for the relaxed structures.
-In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbours.
+In both cases the silicon atom of the initial interstitial is pulled towards the near by atom of the second dumbbell so that both atoms form fourfold coordinated bonds to their next neighbors.
In case c) it is the carbon and in case d) the silicon atom of the second interstitial which forms the additional bond with the silicon atom of the initial interstitial.
The atom of the second dumbbell, the carbon atom of the initial dumbbell and the two interconnecting silicon atoms again reside in a plane.
A typical C-C distance of 2.79 \AA{} is, thus, observed for case c).
The far-off atom of the second dumbbell resides in threefold coordination.
Assuming that it is possible for the system to minimize free energy by an in place reorientation of the dumbbell at any position the minimum energy orientation of dumbbells along the \hkl<1 1 0> direction and the resulting C-C distance is shown in table \ref{tab:defects:comb_db110}.
-\begin{table}[t!h!]
+\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption{Binding energy and type of the minimum energy configuration of an additional dumbbell with respect to the separation distance in bonds along the \hkl<1 1 0> direction and the C-C distance.}
\label{tab:defects:comb_db110}
\end{table}
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=12.5cm]{db_along_110.ps}\\
\includegraphics[width=12.5cm]{db_along_110_cc.ps}
Thus, far-off located dumbbells show an interaction proportional to the reciprocal cube of the distance and the amount of bonds along \hkl<1 1 0> respectively.
This behavior is no longer valid for the immediate vicinity revealed by the saturating binding energy of a second dumbbell at position 1, which is ignored in the fitting procedure.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{5cm}
a) \underline{Pos: 1, $E_{\text{b}}=0.26\text{ eV}$}
\caption{Relaxed structures of defect complexes obtained by creating a carbon substitutional at position 1 (a)), 3 (b)) and 5 (c)).}
\label{fig:defects:comb_db_04}
\end{figure}
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{Pos: 2, $E_{\text{b}}=-0.51\text{ eV}$}
On the one hand, C-C distances around 3.1 \AA{} exist for substitution positions 2 and 3, which are close to the C-C distance expected in silicon carbide.
On the other hand stretched silicon carbide is obtained by the transition of the silicon dumbbell atom into a silicon self-interstitial located somewhere in the silicon host matrix and the transition of the carbon dumbbell atom into another substitutional atom occupying the dumbbell lattice site.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}[t]{7cm}
a) \underline{Pos: 2, $E_{\text{b}}=-0.59\text{ eV}$}
The ground state of a single Si self-interstitial was found to be the Si \hkl<1 1 0> self-interstitial configuration.
For the follwoing study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with substitutional C.
-\begin{table}[ht!]
+\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c}
\hline
\caption{Equivalent configurations of \hkl<1 1 0>-type Si self-interstitials created at position I of figure \ref{fig:defects:pos_of_comb} and substitutional C created at positions 1 to 5.}
\label{tab:defects:comb_csub_si110}
\end{table}
-\begin{table}[ht!]
+\begin{table}[ht]
\begin{center}
\begin{tabular}{l c c c c c c c c c c}
\hline
In addition the separation distance of the ssubstitutional C atom and the Si \hkl<1 1 0> dumbbell interstitial, which is defined to reside at $\frac{a_{\text{Si}}}{4} \hkl<1 1 1>$ is given.
In total 10 different configurations exist within the investigated range.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=12cm]{c_sub_si110.ps}
\end{center}
-\caption{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.}
+\caption[Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance.]{Binding energy of combinations of a substitutional C and a Si \hkl<1 1 0> dumbbell self-interstitial with respect to the separation distance. The binding energy of the defect pair is well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}
\label{fig:defects:csub_si110}
\end{figure}
According to the formation energies none of the investigated structures is energetically preferred over the C-Si \hkl<1 0 0> dumbbell interstitial, which exhibits a formation energy of 3.88 eV.
Thus, the C-Si \hkl<1 0 0> dumbbell structure remains the ground state configuration of a C interstitial in c-Si with a constant number of Si atoms.
{\color{blue}
-However the binding energy quickly drops to zero with respect of the distance indicating a possibly low interaction capture radius of the defect pair.
+However the binding energy quickly drops to zero with respect to the distance, which is reinforced by the Lennard-Jones fit estimating almost zero interaction energy already at 0.6 nm.
+This indicates a possibly low interaction capture radius of the defect pair.
Highly energetic collisions in the IBS process might result in separations of these defects exceeding the capture radius.
For this reason situations most likely occur in which the configuration of substitutional C can be considered without a nearby interacting Si self-interstitial and, thus, unable to form a thermodynamically more stable C-Si \hkl<1 0 0> dumbbell configuration.
}
+\label{section:defects:noneq_process_01}
The energetically most favorable configuration of the combined structures is the one with the substitutional C atom located next to the \hkl<1 1 0> interstitial along the \hkl<1 1 0> direction (configuration \RM{1}).
-Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighboured Si lattice site.
+Compressive stress along the \hkl<1 1 0> direction originating from the Si \hkl<1 1 0> self-intesrtitial is partially compensated by tensile stress resulting from substitutional C occupying the neighbored Si lattice site.
In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
The substitutional C is located next to the lattice site shared by the \hkl<1 1 0> Si self-interstitial along the \hkl<1 -1 0> direction.
Thus, the compressive stress along \hkl<1 1 0> of the Si \hkl<1 1 0> interstitial is not compensated but intensified by the tensile stress of the substitutional C atom, which is no longer loacted along the direction of stress.
-{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 int conf.}
+{\color{red}Todo: EA calc for most and less favorable configuration!}
+
+{\color{red}Todo: Mig of C-Si DB conf to or from C sub + Si 110 in progress.}
+
+{\color{red}Todo: Mig of Si DB located next to a C sub (also by MD!).}
+
+\clearpage{}
+\cleardoublepage{}
\section{Migration in systems of combined defects}
This is especially important for the case, in which the vacancy is created at position 3, as displayed in figure \ref{fig:defects:comb_db_06} b).
Due to the low binding energy this configuration might constitute a trap, which it is hard to escape from.
However, migration simulations show that only a low amount of energy is necessary to transform the system into the energetically most favorable configuration.
-\begin{figure}[!t!h]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[2.0cm]
\begin{picture}(0,0)(170,0)
\caption{Transition of the configuration of the C-Si dumbbell interstitial in combination with a vacancy created at position 2 into the configuration of substitutional carbon.}
\label{fig:defects:comb_mig_01}
\end{figure}
-\begin{figure}[!t!h]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[1.0cm]
\begin{picture}(0,0)(150,0)
This energy is low enough to constitute a feasible mechanism in SiC precipitation.
To reverse this process 5.4 eV are needed, which make this mechanism very unprobable.
The migration path is best described by the reverse process.
-Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighboured silicon atoms and that of the carbon atom to silicon atom number 5.
+Starting at 100 \% energy is needed to break the bonds of silicon atom 1 to its neighbored silicon atoms and that of the carbon atom to silicon atom number 5.
At a displacement of 60 \% these bonds are broken.
-Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbour in the bottom left, a less steep increase of free energy is observed.
+Due to this and due to the formation of new bonds, that is the bond of silicon atom number 1 to silicon atom number 5 and the bond of the carbon atom to its siliocn neighbor in the bottom left, a less steep increase of free energy is observed.
At a displacement of approximately 30 \% the bond of silicon atom number 1 to the just recently created siliocn atom is broken up again, which explains the repeated boost in energy.
Finally the system gains energy relaxing into the configuration of zero displacement.
+{\color{red}Todo: Direct migration of C in progress.}
Due to the low binding energy observed, the configuration of the vacancy created at position 3 is assumed to be stable against transition.
However, a relatively simple migration path exists, which intuitively seems to be a low energy process.
The migration path and the corresponding differences in free energy are displayed in figure \ref{fig:defects:comb_mig_02}.
In fact, migration simulations yield a barrier as low as 0.1 eV.
This energy is needed to tilt the dumbbell as the displayed structure at 30 \% displacement shows.
-Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighboured silicon atoms.
+Once this barrier is overcome, the carbon atom forms a bond to the top left silicon atom and the interstitial silicon atom capturing the vacant site is forming new tetrahedral bonds to its neighbored silicon atoms.
These new bonds and the relaxation into the substitutional carbon configuration are responsible for the gain in free energy.
For the reverse process approximately 2.4 eV are needed, which is 24 times higher than the forward process.
Thus, substitutional carbon is assumed to be stable in contrast to the C-Si dumbbell interstitial located next to a vacancy.
+\clearpage{}
+\cleardoublepage{}
+
\section{Conclusions concerning the SiC conversion mechanism}
-The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration.
-The threefold coordinated carbon and silicon atom share a usual silicon lattice site.
-Migration simulations reveal the carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+The ground state configuration of a carbon interstitial in crystalline siliocn is found to be the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
+This supports the assumption of C-Si \hkl<1 0 0>-type dumbbel interstitial formation in the first steps of the IBS process as proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+
+Migration simulations reveal this carbon interstitial to be mobile at prevailing implantation temperatures requireing an activation energy of approximately 0.9 eV for migration as well as reorientation processes.
+This enables possible migration of the defects to form defect agglomerates as demanded by the model.
+Unfortunately classical potential simulations show tremendously overestimated migration barriers indicating a possible failure of the necessary agglomeration of such defects.
Investigations of two carbon interstitials of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
Depending on orientation, energetically favorable configurations are found in which these two interstitials are located close together instead of the occurernce of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
For dumbbells oriented along the \hkl<1 1 0> direction and the assumption that there is the possibility of free orientation, an interaction energy proportional to the reciprocal cube of the distance in the far field regime is found.
-These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model introduced in section \ref{section:assumed_prec}.
+These findings support the assumption of the C-Si dumbbell agglomeration proposed by the precipitation model.
Next to the C-Si \hkl<1 0 0> dumbbell interstitial configuration, in which the C atom is sharing a Si lattice site with the corresponding Si atom the C atom could occupy the site of the Si atom, which in turn forms a Si self-interstitial.
Combinations of substitutional C and a \hkl<1 1 0> Si self-interstitial, which is the ground state configuration for a Si self-interstitial and, thus, assumed to be the energetically most favorable configuration for combined structures, show formation energies 0.5 eV to 1.5 eV greater than that of the C-Si \hkl<1 0 0> interstitial configuration, which remains the energetically most favorable configuration.
However, the binding energy of substitutional C and the Si self-interstitial quickly drops to zero already for short separations indicating a low interaction capture radius.
-Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitials substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
+Thus, due to missing attractive interaction forces driving the system to form C-Si \hkl<1 0 0> dumbbell interstitial complexes substitutional C, while thermodynamically not stable, constitutes a most likely configuration occuring in IBS, a process far from equlibrium.
Due to the low interaction capture radius substitutional C can be treated independently of the existence of separated Si self-interstitials.
This should be also true for combinations of C-Si interstitials next to a vacancy and a further separated Si self-interstitial excluded from treatment, which again is a conveivable configuration in IBS.
-By combination of the \hkl<1 0 0> dumbbell with a vacancy it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
+By combination of a \hkl<1 0 0> dumbbell with a vacancy in the absence of the Si self-interstitial it is found that the configuration of substitutional carbon occupying the vacant site is the energetically most favorable configuration.
Low migration barriers are necessary to obtain this configuration and in contrast comparatively high activation energies necessary for the reverse process.
Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide.
-While first results point to ...
-
-In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation.
-
-0 K simulations -> C-Si DB, however non-zero temperatures and the IBS, process far from equilibrium, so sub C should be feasible ...
+While first results support the proposed precipitation model the latter suggest the formation of silicon carbide by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt transition.
+Prevailing conditions in the IBS process at elevated temperatures and the fact that IBS is a nonequilibrium process reinforce the possibility of formation of substitutional C instead of the thermodynamically stable C-Si dumbbell interstitials predicted by simulations at zero Kelvin.
+\label{section:defects:noneq_process_02}
-{\color{red}Todo: Explain that formation of SiC by substitutional C is more likely than the supposed C-Si agglomeration, at least in the absence of the accompanied Si self-interstitial.}
+{\color{blue}
+In addition, there are experimental findings, which might be exploited to reinforce the non-validity of the proposed precipitation model.
+High resolution TEM shows equal orientation of \hkl(h k l) planes of the c-Si host matrix and the 3C-SiC precipitate.
+Formation of 3C-SiC realized by successive formation of substitutional C, in which the atoms belonging to one of the two fcc lattices are substituted by C atoms perfectly conserves the \hkl(h k l) planes of the initial c-Si diamond lattice.
+Silicon self-interstitials consecutively created to the same degree are able to diffuse into the c-Si host one after another.
+Investigated combinations of C interstitials, however, result in distorted configurations, in which C atoms, which at some point will form SiC, are no longer aligned to the host.
+It is easily understandable that the mismatch in alignement will increase with increasing defect density.
+In addition, the amount of Si self-interstitials equal to the amount of agglomerated C atoms would be released all of a sudden probably not being able to diffuse into the c-Si host matrix without damaging the Si surrounding or the precipitate itself.
+In addition, IBS results in the formation of the cubic polytype of SiC only.
+As this result conforms well with the model of precipitation by substitutional C there is no obvious reason why hexagonal polytypes should not be able to form or an equal alignement would be mandatory assuming the model of precipitation by C-Si dumbbell agglomeration.
+}
-{\color{red}Todo: Si \hkl<1 1 0> migration barriers. If Si can go away fast, formation of substitutional C (and thus formation of SiC) might be a more probable process than C-Si dumbbell agglomeration.}
+{\color{red}Todo: C mobility higher than Si mobility? -> substitutional C is more likely to arise, since it migrates 'faster' to vacant sites?}
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