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{center}
+\includegraphics[width=13cm]{vasp_mig/110_mig_vasp.ps}
+%\begin{picture}(0,0)(140,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_b.eps}
+%\end{picture}
+%\begin{picture}(0,0)(20,0)
+%\includegraphics[width=2.2cm]{vasp_mig/00-1_ip0-10_sp.eps}
+%\end{picture}
+%\begin{picture}(0,0)(-120,0)
+%\includegraphics[width=2.2cm]{vasp_mig/0-10_b.eps}
+%\end{picture}
+\end{center}
+\caption{Migration barriers of the \hkl<1 1 0> dumbbell to bond-centered (red), \hkl<0 0 -1> (green) and \hkl<0 -1 0> (in place, blue) C-Si dumbbell transition.}
+\label{fig:defects:110_mig_vasp}
+\end{figure}
+Further migration pathways in particular those occupying other defect configurations than the \hkl<1 0 0>-type either as a transition state or a final or starting configuration are totally conceivable.
+In order to find possible migration pathways that have an activation energy lower than the ones found up to now.
+The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si dumbbell interstitial.
+Figure \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si dumbbell to the bond-centered, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
+Indeed less than 0.7 eV are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si dumbbell interstitial.
+This transition is carried out in place, that is the Si dumbbell pair is not changed and both, the Si and C atom share the same lattice site.
+Thus, this transition does not contribute to long-range diffusion.
+Once the C atom resides in the \hkl<1 1 0> interstitial configuration it can migrate into the bond-centered configuration by employing approximately 0.95 eV of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway shown in figure \ref{fig:defects:00-1_0-10_mig}.
+As already known from the migration of the \hkl<0 0 -1> to the bond-centered configuration as discussed in figure \ref{fig:defects:00-1_001_mig} another 0.25 eV are needed to turn back from the bond-centered to a \hkl<1 0 0>-type interstitial.
+However, due to the fact that this migration consists of three single transitions with the second one having an activation energy slightly higher than observed for the direct transition it is considered very unlikely to occur.
+The migration barrier of the \hkl<1 1 0> to \hkl<0 0 -1> transition, in which the C atom is changing its Si partner and, thus, moving to the neighboured lattice site is approximately 1.35 eV.
+During this transition the C atom is escaping the \hkl(1 1 0) plane approaching the final configuration on a curved path.
+This barrier is much higher than the ones found previously, which again make this transition very unlikely to occur.
+For this reason the assumption that C diffusion and reorientation is achieved by transitions of the type presented in figure \ref{fig:defects:00-1_0-10_mig} is reinforced.
+
As mentioned earlier the procedure to obtain the migration barriers differs from the usually applied procedure in two ways.
Firstly constraints to move along the displacement direction are applied on all atoms instead of solely constraining the diffusing atom.
Secondly the constrainted directions are not kept constant to the initial displacement direction.
\end{figure}
The method without updating the constraints but still applying them to all atoms shows a delayed crossing of the saddle point.
This is understandable since the update results in a more aggressive advance towards the final configuration.
-In any case the barrier obtained is slightly higher, which means that it is not the energetically most favorable pathway.
+In any case the barrier obtained is slightly higher, which means that it does not constitute an energetically more favorable pathway.
The method in which the constraints are only applied to the diffusing C atom and two Si atoms, ... {\color{red}in progress} ...
\subsection{Migration barriers obtained by classical potential calculations}
The same method for obtaining migration barriers and the same suggested pathways are applied to calculations employing the classical Erhard/Albe potential.
+Since the evaluation of the classical potential and force is less computationally intensive higher amounts of steps can be used.
\begin{figure}[th!]
\begin{center}
\label{fig:defects:cp_bc_00-1_mig}
\end{figure}
Figure \ref{fig:defects:cp_bc_00-1_mig} shows the migration barrier of the bond-centered to \hkl<0 0 -1> dumbbell transition.
-Since the bond-centered configuration is unstable within this potential the low kinetic energy state is used as a starting configuration.
-This would relax towards the \hkl<1 1 0> C-Si interstitial.
+Since the bond-centered configuration is unstable relaxing into the \hkl<1 1 0> C-Si dumbbell interstitial configuration within this potential the low kinetic energy state is used as a starting configuration.
+
\begin{figure}[th!]
\begin{center}
\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{center}
+\begin{tabular}{l c c c c c c c c c c c}
+\hline
+\hline
+Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E &F\\
+\hline
+$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.56 & 5.32 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 & 5.32 \\
+$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.03 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 & -0.03 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\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.
projects / lectures / latex.git / commitdiff