\hline
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & V \\
\hline
- Erhard/Albe MD & 3.40 & 4.48$^*$ & 5.42 & 4.39 & 3.13 \\
+ 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 & - \\
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 Erhard/Albe potential calculations favor the tetrahedral defect configuration.
+In fact the Erhart/Albe potential calculations favor the tetrahedral defect configuration.
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.
\begin{center}
\includegraphics[width=10cm]{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 Erhard/Albe classical potential.}
+\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the Erhart/Albe classical 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 barrier is less than 0.2 eV.
Hence these artifacts should have a negligent 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 Erhard/Albe and VASP calculations.
+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.
\hline
& T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\hline
- Erhard/Albe MD & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
+ 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 \\
Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
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 Erhard/Albe potential undervalues the formation energy roughly by a factor of two.
+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.
Thus, this configuration is of great importance and discussed in more detail in section \ref{subsection:100db}.
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 Erhard/Albe potential.
+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 calculations.
+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 Erhard/Albe potential the formation energy is situated in the same order as found by quantum-mechanical results.
+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 Erhard/Albe potential.
+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 results of this configuration are discussed in more detail in section \ref{subsection:bc}.
& & & & \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
-Erhard/Albe & 0.084 & -0.091 & 0.175 & -0.015 & -0.015 & -0.031 & -0.014 & 0.014 & 0.020 \\
+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 \\
\hline
\hline
\hline
& $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ & $a_{\text{Si}}^{\text{equi}}$\\
\hline
-Erhard/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 & 0.543 \\
+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 \\
\hline
\hline
\hline
& $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
-Erhard/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
+Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
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 Erhard/Albe potential and VASP calculations.]{Atomic displacements, distances and bond angles of the \hkl<1 0 0> dumbbell structure obtained by the Erhard/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 \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.}
\label{tab:defects:100db_cmp}
\end{table}
\begin{figure}[t!h!]
\begin{center}
\begin{minipage}{6cm}
\begin{center}
-\underline{Erhard/Albe}
+\underline{Erhart/Albe}
\includegraphics[width=5cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\end{center}
\end{minipage}
\end{center}
-\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhard/Albe potential and VASP calculations.}
+\caption{Comparison of the visualized \hkl<1 0 0> dumbbel structures obtained by Erhart/Albe potential and VASP calculations.}
\label{fig:defects:100db_vis_cmp}
\end{figure}
\begin{figure}[th]
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 Erhard/Albe results.
+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.
\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 Erhard/Albe potential simulations which reveal this configuration to be unstable relaxing into the \hkl<1 1 0> configuration.
+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}.
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.
+\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.
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.
+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.}
Experimentally measured activation energies for reorientation range from 0.77 eV to 0.88 eV \cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
-{\color{red}Todo: Stress out that this is a promising result excellently matching experimental observations.}
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.
+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.
\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 Erhard/Albe potential.
+The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhart/Albe 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.
\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 Erhard/Albe potential.}
+\caption{Migration barrier and structures of the bond-centered to \hkl<0 0 -1> dumbbell transition using the classical Erhart/Albe 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
\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 Erhard/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 Erhart/Albe 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{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 Erhard/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 Erhart/Albe 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.
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!}
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 investigations of defect combinations approached in the following are still feasible within the available computational power and allow to draw conclusions on some important ongoing mechanisms during SiC precipitation.
\subsection[Combinations with a C-Si \hkl<1 0 0>-type interstitial]{\boldmath Combinations with a C-Si \hkl<1 0 0>-type interstitial}
+\label{subsection:defects:c-si_comb}
This section focuses on combinations of the \hkl<0 0 -1> dumbbell interstitial with a second defect.
The second defect is either another \hkl<1 0 0>-type interstitial occupying different orientations, a vacany or a substitutional carbon atom.
\caption{Formation $E_{\text{f}}$ and binding $E_{\text{b}}$ energies in eV of the combinational substitutional C and Si self-interstitial configurations as defined in table \ref{tab:defects:comb_csub_si110}.}
\label{tab:defects:comb_csub_si110_energy}
\end{table}
-
Table \ref{tab:defects:comb_csub_si110} shows equivalent configurations of \hkl<1 1 0>-type Si self-interstitials and substitutional C.
The notation of figure \ref{fig:defects:pos_of_comb} is used with the six possible Si self-interstitials created at the usual C-Si dumbbell position.
Substitutional C is created at positions 1 to 5.
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.
-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.
-Further separated defects are assumed to approximate the sum of the formation energies of the isolated single defects.
-This is affirmed by the plot of the binding energies with respect to the separation distance in figure \ref{fig:defects:csub_si110} approximating zero with increasing distance.
-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.
\begin{figure}[th!]
\begin{center}
\includegraphics[width=12cm]{c_sub_si110.ps}
\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.}
\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.
+Further separated defects are assumed to approximate the sum of the formation energies of the isolated single defects.
+This is affirmed by the plot of the binding energies with respect to the separation distance in figure \ref{fig:defects:csub_si110} approximating zero with increasing distance.
+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.
+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.
+}
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.
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: 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: Attraction of defect pair for large separation distances might be very low and thus, substitutional C + Si, which is diffusing somewhere else remains (out of a reaction radius)?}
\section{Migration in systems of combined defects}
-During carbon implantation into crystalline silicon the energetic carbon atoms may kick out silicon atoms from their lattice sites.
-A vacancy accompanied by a silicon self-interstitial is generated.
-The silicon self-interstitial may migrate to the surface or recombine with other vacancies.
-Once a vacancy and a carbon interstitial defect exist the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
+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.
+However configurations might arise in which C atoms do not already occupy the vacant site but instead form a C interstitial next to the vacancy as discussed shortly before in the very end of section \ref{subsection:defects:c-si_comb}.
+In the absence of the Si self-interstitial the energetically most favorable configuration is the configuration of a substitutional carbon atom, that is the carbon atom occupying the vacant site.
In addition, it is a conceivable configuration the system might experience during the silicon carbide precipitation process.
Energies needed to overcome the migration barrier of the transformation into this configuration enable predictions concerning the feasibility of a silicon carbide conversion mechanism derived from these microscopic processes.
-This is especially important for the case, in which the vacancy is created at position 3, as discussed in the last section and figure \ref{fig:defects:comb_db_06} b).
+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]
\label{fig:defects:comb_mig_02}
\end{figure}
Figure \ref{fig:defects:comb_mig_01} and \ref{fig:defects:comb_mig_02} show the migration barriers and structures for transitions of the vacancy-interstitial configurations examined in figure \ref{fig:defects:comb_db_06} a) and b) into a configuration of substitutional carbon.
+If the vacancy is created at position 1 the system will end up in a configuration of substitutional C anyways.
-In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2, while the carbon atom substitutes the site of the migrating silicon atom.
+In the first case the focus is on the migration of silicon atom number 1 towards the vacant site created at position 2 while the carbon atom substitutes the site of the migrating silicon atom.
An energy of 0.6 eV necessary to overcome the migration barrier is found.
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.
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 nedded, 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.
-
-{\color{red}Todo: DB migration calculations along 110 (at the starting of this section)?}
\section{Conclusions concerning the SiC conversion mechanism}
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.
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.
+{\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{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:
Better structure, better language, better methodology!
}
-{\color{red}Todo:
-Fit of lennard-jones and other rep + attr potentials in 110 interaction data!
-}
projects / lectures / latex.git / commitdiff