\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.
-
-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.
+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.
+
+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 configuration is unstable and the \hkl<1 0 0> dumbbell interstitial is the most unfavorable configuration for both, the Erhart/Albe and VASP calculations.
+The bond-centered 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.
-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}.
+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 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?}
+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.
-\section{Carbon related point defects}
+%\clearpage{}
+%\cleardoublepage{}
-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.
+\section{Carbon point defects in silicon}
-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.
+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.
+For the EA potential the formation energy is situated in the same order as found by quantum-mechanical results.
+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 bond-centered 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 bond-centered configuration is found to be a unstable within the EA description.
+The bond-centered configuration relaxes into the \ci{} \hkl<1 1 0> DB configuration.
This, like in the hexagonal case, is also true for the unmodified Tersoff potential and the given relaxation conditions.
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}.
+In another {\em ab inito} study Capaz et al. \cite{capaz94} determined this configuration as 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.
+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 \ci{} \hkl<1 0 0> DB 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}
+\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 numbered '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 silicon.
+However, these values have to be ...
\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}\\
\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.
+This is in agreement with results of the EA 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 {\textsc 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.
The blue torus, reinforcing 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.
+\clearpage{}
+\cleardoublepage{}
+
\section{Migration of the carbon interstitials}
\label{subsection:100mig}
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.
-\begin{figure}[t!h!]
+\begin{figure}[ht]
\begin{center}
\begin{minipage}{15cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
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!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[1.5cm]
\begin{picture}(0,0)(150,0)
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.
-\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)
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}.
-\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)
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.
-\begin{figure}[th!]
+\begin{figure}[ht]
\begin{center}
\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
%\begin{picture}(0,0)(140,0)
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}
\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.
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}
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}
\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.}
\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
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.
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{$E_{\text{b}}=-2.25\text{ eV}$}
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{$E_{\text{b}}=-2.16\text{ eV}$}
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 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}
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: Erhart/Albe calc for most and less favorable configuration!}
+{\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}
As already pointed out in the previous section energetic carbon atoms may kick out silicon atoms from their lattice sites during carbon implantation into crystalline silicon.
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)
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, in which the threefold coordinated carbon and silicon atom share a usual silicon lattice site.
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