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 that are defined in chapter~\ref{chapter:simulation}.
+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 that are defined 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.
+Thus, investigations proceed 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.
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}.
-\bibpunct{}{}{,}{n}{}{}
-\begin{table}[tp]
-\begin{center}
-\begin{tabular}{l c c c c c}
-\hline
-\hline
- & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
-\hline
-\multicolumn{6}{c}{Present study} \\
-\textsc{vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
-\textsc{posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
-\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 & - & - \\
-\hline
-\hline
-\end{tabular}
-\end{center}
-\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}
-\bibpunct{[}{]}{,}{n}{}{}
-\begin{figure}[tp]
-\begin{center}
-\begin{flushleft}
-\begin{minipage}{5cm}
-\underline{Tetrahedral}\\
-$E_{\text{f}}=3.40\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/tet_bonds.eps}
-\end{minipage}
-\begin{minipage}{10cm}
-\underline{Hexagonal}\\[0.1cm]
-\begin{minipage}{4cm}
-$E_{\text{f}}^*=4.48\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/hex_a_bonds.eps}
-\end{minipage}
-\begin{minipage}{0.8cm}
-\begin{center}
-$\Rightarrow$
-\end{center}
-\end{minipage}
-\begin{minipage}{4cm}
-$E_{\text{f}}=3.96\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/hex_bonds.eps}
-\end{minipage}
-\end{minipage}\\[0.2cm]
-\begin{minipage}{5cm}
-\underline{\hkl<1 0 0> dumbbell}\\
-$E_{\text{f}}=5.42\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/100_bonds.eps}
-\end{minipage}
-\begin{minipage}{5cm}
-\underline{\hkl<1 1 0> dumbbell}\\
-$E_{\text{f}}=4.39\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/110_bonds.eps}
-\end{minipage}
-\begin{minipage}{5cm}
-\underline{Vacancy}\\
-$E_{\text{f}}=3.13\,\text{eV}$\\
-\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}
-\end{minipage}
-\end{flushleft}
-%\hrule
-\end{center}
-\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. Si atoms and bonds are illustrated by yellow spheres and blue lines. Bonds of the defect atoms are drawn in red color.}
-\label{fig:defects:conf}
-\end{figure}
+\bibpunct{}{}{,}{n}{}{}%
+\begin{table}[tp]%
+\begin{center}%
+\begin{tabular}{l c c c c c}%
+\hline%
+\hline%
+ & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\%
+\hline%
+\multicolumn{6}{c}{Present study} \\%
+\textsc{vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\%
+\textsc{posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\%
+\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 & - & - \\%
+\hline%
+\hline%
+\end{tabular}%
+\end{center}%
+\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}%
+\bibpunct{[}{]}{,}{n}{}{}%
+\begin{figure}[tp]%
+\begin{center}%
+\begin{flushleft}%
+\begin{minipage}{5cm}%
+\underline{Tetrahedral}\\%
+$E_{\text{f}}=3.40\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/tet_bonds.eps}%
+\end{minipage}%
+\begin{minipage}{10cm}%
+\underline{Hexagonal}\\[0.1cm]%
+\begin{minipage}{4cm}%
+$E_{\text{f}}^*=4.48\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/hex_a_bonds.eps}%
+\end{minipage}%
+\begin{minipage}{0.8cm}%
+\begin{center}%
+$\Rightarrow$%
+\end{center}%
+\end{minipage}%
+\begin{minipage}{4cm}%
+$E_{\text{f}}=3.96\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/hex_bonds.eps}%
+\end{minipage}%
+\end{minipage}\\[0.2cm]%
+\begin{minipage}{5cm}%
+\underline{\hkl<1 0 0> dumbbell}\\%
+$E_{\text{f}}=5.42\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/100_bonds.eps}%
+\end{minipage}%
+\begin{minipage}{5cm}%
+\underline{\hkl<1 1 0> dumbbell}\\%
+$E_{\text{f}}=4.39\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/110_bonds.eps}%
+\end{minipage}%
+\begin{minipage}{5cm}%
+\underline{Vacancy}\\%
+$E_{\text{f}}=3.13\,\text{eV}$\\%
+\includegraphics[width=4.0cm]{si_pd_albe/vac.eps}%
+\end{minipage}%
+\end{flushleft}%
+%\hrule%
+\end{center}%
+\caption[Relaxed Si self-interstitial defect configurations obtained by classical potential calculations.]{Relaxed Si self-interstitial defect configurations obtained by classical potential calculations. Si atoms and bonds are illustrated by yellow spheres and blue lines. Bonds of the defect atoms are drawn in red color.}%
+\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.
To exclude failures in the implementation of the potential or the MD code itself, the hexagonal defect structure was double-checked with the \textsc{parcas} MD code~\cite{parcas_md}.
The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by \textsc{posic}.
In fact, the same type of interstitial arises using random insertions.
-In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetrahedral configuration and formation energy.
+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> axis ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetrahedral configuration and formation energy.
The existence of these local minima located near the tetrahedral configuration seems to be an artifact of the analytical potential without physical authenticity revealing fundamental problems of analytical potential models for describing defect structures.
However, the energy barrier required for a transition into the tetrahedral configuration is small.
\begin{figure}[tp]
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.
+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}.
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.
+It constitutes the second most favorable configuration, which is 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.
+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 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.
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.
+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 \unit[0.75]{eV} can be estimated from the Kohn-Sham level diagram for plain Si.
\label{img:defects:c_mig_path}
\end{figure}
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.
+The first investigated migration 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.
Figures~\ref{fig:defects:cp_00-1_0-10_mig} and~\ref{fig:defects:cp_00-1_ip0-10_mig} show the migration barriers of the \ci{} \hkl[0 0 -1] to \hkl[0 -1 0] DB transition.
In the first case, the transition involves a change in the lattice site of the C atom whereas in the second case, a reorientation at the same lattice site takes place.
In the first case, the pathways for the two different time constants look similar.
-A local minimum exists in between two peaks of the graph.
+A local minimum exists in between the two peaks of the graph.
The corresponding configuration, which is illustrated for the results obtained for a time constant of \unit[1]{fs}, looks similar to the \ci{} \hkl[1 1 0] configuration.
-Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near the minimum.
+Indeed, this configuration is obtained by relaxation simulations without constraints of configurations near this local minimum.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
The \ci{} \hkl[1 1 0] configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
\subsection{Combinations of \si{} and \cs}
\label{subsection:si-cs}
-So far the C-Si \hkl<1 0 0> DB interstitial was found to be the energetically most favorable configuration.
+So far, the C-Si \hkl<1 0 0> DB interstitial was found to be the energetically most favorable configuration.
In fact, substitutional C exhibits a configuration more than \unit[3]{eV} lower with respect to the formation energy.
However, the configuration does not account for the accompanying Si self-interstitial that is generated once a C atom occupies the site of a Si atom.
With regard to the IBS process, in which highly energetic C atoms enter the Si target being able to kick out Si atoms from their lattice sites, such configurations are absolutely conceivable and a significant influence on the precipitation process might be attributed to them.
Thus, combinations of \cs{} and an additional \si{} are examined in the following.
The ground-state of a single \si{} was found to be the \si{} \hkl<1 1 0> DB configuration.
-For the following study the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.
+For the following study, the same type of self-interstitial is assumed to provide the energetically most favorable configuration in combination with \cs.
\begin{table}[tp]
\begin{center}
%
The relaxed structure is displayed in the bottom right of Fig.~\ref{fig:162-097}.
Compressive strain originating from the Si$_{\text{i}}$ is compensated by tensile strain inherent to the C$_{\text{s}}$ configuration.
-The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si$_{\text{i}}$ DB atom closest to the C atom does no longer form bonds to its top Si neighbors, but to the next neighbored Si atom along \hkl[1 1 0].
+The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si$_{\text{i}}$ DB atom closest to the C atom does no longer form bonds to its top Si neighbors but to the next neighbored Si atom along \hkl[1 1 0].
%
In the same way the energetically most unfavorable configuration can be explained, which is configuration \RM{3}.
The \cs{} is located next to the lattice site shared by the \si{} \hkl[1 1 0] DB in \hkl[1 -1 1] direction.
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
However, only \unit[0.77]{eV} are needed for the reverse process, i.e.\ the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
-Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
+Due to the low activation energy, this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
\begin{figure}[tp]
\begin{center}
The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.
Unable to model possible positive values of the binding energy, i.e.\ unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e.\ the absolute value of the binding energy, with increasing separation distance.
The binding energy quickly drops to zero.
-The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}.
- indicating a low interaction capture radius of the defect pair.
+The LJ fit estimates almost zero interaction already at \unit[0.6]{nm} indicating a low interaction capture radius of the defect pair.
%As can be seen, the interaction strength, i.e.\ the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
%Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS, highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
-For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.
-In particular in IBS, which constitutes a system driven far from equilibrium, respective defect configurations might exist that do not combine into the ground-state configuration.
+For this reason, C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.
+Particularly in IBS, which constitutes a system driven far from equilibrium, respective defect configurations might exist that do not combine into the ground-state configuration.
Thus, the existence of C$_{\text{s}}$ is very likely.
\label{section:defects:noneq_process_01}
% the ab initio md, where to put
-Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.
+Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be eminently important at higher temperatures due to the low activation energy necessary for its formation.
At higher temperatures, the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.
Indeed, an {\em ab initio} MD run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:defects:md}.
\end{figure}
The barrier, which is even lower than the one for \ci{}, indeed indicates highly mobile \si.
In fact, a similar transition is expected if the \si{} atom, which does not change the lattice site during transition, is located next to a \cs{} atom.
-Due to the low barrier the initial separation of the \cs{} and \si{} atom are very likely to occur.
+Due to the low barrier, the initial separation of the \cs{} and \si{} atom are very likely to occur.
Further investigations revealed transition barriers of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the hexagonal Si$_{\text{i}}$, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to the tetrahedral Si$_{\text{i}}$ and \unit[0.35]{eV} for the hexagonal Si$_{\text{i}}$ to the tetrahedral Si$_{\text{i}}$ configuration.
The respective configurational energies are shown in Fig.~\ref{fig:defects:si_mig2}.
\begin{figure}[tp]
\section{Applicability: Competition of \ci{} and \cs-\si{}}
\label{section:ea_app}
-As has been shown, the energetically most favorable configuration of \cs{} and \si{} is obtained for \cs{} located at the neighbored lattice site along the \hkl<1 1 0> bond chain of a Si$_{\text{i}}$ \hkl<1 1 0> DB.
+As has been shown, the energetically most favorable configuration of \cs{} and \si{} is obtained for \cs{} located at the neighbored lattice site along the \hkl[1 1 0] bond chain of a Si$_{\text{i}}$ \hkl[1 1 0] DB.
However, the energy of formation is slightly higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state for a C impurity introduced into otherwise perfect c-Si.
For a possible clarification of the controversial views on the participation of C$_{\text{s}}$ in the precipitation mechanism by classical potential simulations, test calculations need to ensure the proper description of the relative formation energies of combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ compared to C$_{\text{i}}$.
Contrasts are also assumed for Si$_{\text{i}}$.
Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.
Until then, however, these regions could either be composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials, where the latter is too small to be detected in HREM.
-In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
+In both cases, Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.
Furthermore, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.
In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
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