+ {\large\bf\boldmath
+ Energy cut-off for $\Gamma$-point only caclulations
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
+ $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
+ $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
+ \small
+ \begin{minipage}{10cm}
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+ & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
+ Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
+ \hline
+ Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
+ Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{2cm}
+ {\LARGE
+ ${\color{green}\surd}$
+ }
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
+ in c-Si (Albe)
+ }
+
+ \small
+
+ \begin{minipage}[t]{4.2cm}
+ \underline{Starting configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/start.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.0cm}
+ \vspace*{0.8cm}
+ $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta z=0.325\text{ \AA}$\\
+ \end{minipage}
+ \begin{minipage}[t]{4.2cm}
+ \underline{{\bf Expected} final configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/final.eps}\\
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \begin{itemize}
+ \item Fix border atoms of the simulation cell
+ \item Constraints and displacement of the C atom:
+ \begin{itemize}
+ \item along {\color{green}\hkl<1 1 0> direction}\\
+ displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
+ \item C atom {\color{red}entirely fixed in position}\\
+ displaced by
+ {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
+ \end{itemize}
+ \item Berendsen thermostat applied
+ \end{itemize}
+ {\bf\color{blue}Expected configuration not obtained!}
+ \end{minipage}
+ \begin{minipage}{0.5cm}
+ \hfill
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
+ in c-Si (Albe)
+ }
+
+ \footnotesize
+
+ \begin{minipage}{3.2cm}
+ \includegraphics[width=3cm]{c_100_mig/fixmig_50.eps}
+ \begin{center}
+ 50 \%
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{3.2cm}
+ \includegraphics[width=3cm]{c_100_mig/fixmig_80.eps}
+ \begin{center}
+ 80 \%
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{3.2cm}
+ \includegraphics[width=3cm]{c_100_mig/fixmig_90.eps}
+ \begin{center}
+ 90 \%
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{3.2cm}
+ \includegraphics[width=3cm]{c_100_mig/fixmig_99.eps}
+ \begin{center}
+ 100 \%
+ \end{center}
+ \end{minipage}
+
+ Open questions ...
+ \begin{enumerate}
+ \item Why is the expected configuration not obtained?
+ \item How to find a migration path preceding to the expected configuration?
+ \end{enumerate}
+
+ Answers ...
+ \begin{enumerate}
+ \item Simple: it is not the right migration path!
+ \begin{itemize}
+ \item (Surrounding) atoms settle into a local minimum configuration
+ \item A possibly existing more favorable configuration is not achieved
+ \end{itemize}
+ \item \begin{itemize}
+ \item Search global minimum in each step (by simulated annealing)\\
+ {\color{red}But:}
+ Loss of the correct energy needed for migration
+ \item Smaller displacements\\
+ A more favorable configuration might be achieved
+ possibly preceding to the expected configuration
+ \end{itemize}
+ \end{enumerate}
+
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
+ in c-Si (Albe)\\
+ }
+
+ Displacement step size decreased to
+ $\frac{1}{100} (\Delta x,\Delta y)$\\[0.2cm]
+
+ \begin{minipage}{7.5cm}
+ Result: (Video \href{../video/c_in_si_smig_albe.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_albe.avi}{$\rhd_{\text{remote url}}$})
+ \begin{itemize}
+ \item Expected final configuration not obtained
+ \item Bonds to neighboured silicon atoms persist
+ \item C and neighboured Si atoms move along the direction of displacement
+ \item Even the bond to the lower left silicon atom persists
+ \end{itemize}
+ {\color{red}
+ Obviously: overestimated bond strength
+ }
+ \end{minipage}
+ \begin{minipage}{5cm}
+ \includegraphics[width=6cm]{c_100_110smig_01_albe.ps}
+ \end{minipage}\\[0.4cm]
+ New approach to find the migration path:\\
+ {\color{blue}
+ Place interstitial carbon atom at the respective coordinates
+ into a perfect c-Si matrix!
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
+ in c-Si (Albe)
+ }
+
+ {\color{blue}New approach:}\\
+ Place interstitial carbon atom at the respective coordinates
+ into a perfect c-Si matrix!\\
+ {\color{blue}Problem:}\\
+ Too high forces due to the small distance of the C atom to the Si
+ atom sharing the lattice site.\\
+ {\color{blue}Solution:}
+ \begin{itemize}
+ \item {\color{red}Slightly displace the Si atom}
+ (Video \href{../video/c_in_si_pmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_albe.avi}{$\rhd_{\text{remote url}}$})
+ \item {\color{green}Immediately quench the system}
+ (Video \href{../video/c_in_si_pqmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pqmig_albe.avi}{$\rhd_{\text{remote url}}$})
+ \end{itemize}
+
+ \begin{minipage}{6.5cm}
+ \includegraphics[width=6cm]{c_100_110pqmig_01_albe.ps}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \begin{itemize}
+ \item Jump in energy corresponds to the abrupt
+ structural change (as seen in the videos)
+ \item Due to the abrupt changes in structure and energy
+ this is {\color{red}not} the correct migration path and energy!?!
+ \end{itemize}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
+ }
+
+ \small
+
+ {\color{blue}Method:}
+ \begin{itemize}
+ \item Place interstitial carbon atom at the respective coordinates
+ into perfect c-Si
+ \item \hkl<1 1 0> direction fixed for the C atom
+ \item $4\times 4\times 3$ Type 1, $198+1$ atoms
+ \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
+ \end{itemize}
+ {\color{blue}Results:}
+ (Video \href{../video/c_in_si_pmig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
+ \begin{minipage}{7cm}
+ \includegraphics[width=7cm]{c_100_110pmig_01_vasp.ps}
+ \end{minipage}
+ \begin{minipage}{5.5cm}
+ \begin{itemize}
+ \item Characteristics nearly equal to classical calulations
+ \item Approximately half of the classical energy
+ needed for migration
+ \end{itemize}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
+ }
+
+ \small
+
+ {\color{blue}Method:}
+ \begin{itemize}
+ \item Continue with atomic positions of the last run
+ \item Displace the C atom in \hkl<1 1 0> direction
+ \item \hkl<1 1 0> direction fixed for the C atom
+ \item $4\times 4\times 3$ Type 1, $198+1$ atoms
+ \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
+ \end{itemize}
+ {\color{blue}Results:}
+ (Video \href{../video/c_in_si_smig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
+ \includegraphics[width=7cm]{c_100_110mig_01_vasp.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Again: C \hkl<1 0 0> interstitial migration
+ }
+
+ \small
+
+ {\color{blue}The applied methods:}
+ \begin{enumerate}
+ \item Method
+ \begin{itemize}
+ \item Start in relaxed \hkl<1 0 0> interstitial configuration
+ \item Displace C atom along \hkl<1 1 0> direction
+ \item Relaxation (Berendsen thermostat)
+ \item Continue with configuration of the last run
+ \end{itemize}
+ \item Method
+ \begin{itemize}
+ \item Place interstitial carbon at the respective coordinates
+ into the perfect Si matrix
+ \item Quench the system
+ \end{itemize}
+ \end{enumerate}
+ {\color{blue}In both methods:}
+ \begin{itemize}
+ \item Fixed border atoms
+ \item Applied \hkl<1 1 0> constraint for the C atom
+ \end{itemize}
+ {\color{red}Pitfalls} and {\color{green}refinements}:
+ \begin{itemize}
+ \item {\color{red}Fixed border atoms} $\rightarrow$
+ Relaxation of stress not possible\\
+ $\Rightarrow$
+ {\color{green}Fix only one Si atom} (the one furthermost to the defect)
+ \item {\color{red}\hkl<1 1 0> constraint not sufficient}\\
+ $\Rightarrow$ {\color{green}Apply 11x constraint}
+ (connecting line of initial and final C positions)
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Again: C \hkl<1 0 0> interstitial migration (Albe)
+ }
+
+ Constraint applied by modifying the Velocity Verlet algorithm
+
+ {\color{blue}Results:}
+ (Video \href{../video/c_in_si_fmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_fmig_albe.avi}{$\rhd_{\text{remote url}}$})\\
+ \begin{minipage}{6.3cm}
+ \includegraphics[width=6cm]{c_100_110fmig_01_albe.ps}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \begin{center}
+ Again there are jumps in energy corresponding to abrupt
+ structural changes as seen in the video
+ \end{center}
+ \end{minipage}
+ \begin{itemize}
+ \item Expected final configuration not obtained
+ \item Bonds to neighboured silicon atoms persist
+ \item C and neighboured Si atoms move along the direction of displacement
+ \item Even the bond to the lower left silicon atom persists
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Again: C \hkl<1 0 0> interstitial migration (VASP)