+ \vspace*{0.2cm}
+ {\color{red}
+ \[
+ \Rightarrow
+ E_{\textrm{coh}}=\frac{
+ -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M)
+ -\ldots\Big)}
+ {n+m+\ldots}
+ \]
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Used types of supercells\\
+ }
+
+ \footnotesize
+
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=4cm]{sc_type0.eps}\\[0.3cm]
+ \underline{Type 0}\\[0.2cm]
+ Basis: fcc\\
+ $x_1=(0.5,0.5,0)$\\
+ $x_2=(0,0.5,0.5)$\\
+ $x_3=(0.5,0,0.5)$\\
+ 1 primitive cell / 2 atoms
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=4cm]{sc_type1.eps}\\[0.3cm]
+ \underline{Type 1}\\[0.2cm]
+ Basis:\\
+ $x_1=(0.5,-0.5,0)$\\
+ $x_2=(0.5,0.5,0)$\\
+ $x_3=(0,0,1)$\\
+ 2 primitive cells / 4 atoms
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+ \includegraphics[width=4cm]{sc_type2.eps}\\[0.3cm]
+ \underline{Type 2}\\[0.2cm]
+ Basis: sc\\
+ $x_1=(1,0,0)$\\
+ $x_2=(0,1,0)$\\
+ $x_3=(0,0,1)$\\
+ 4 primitive cells / 8 atoms
+ \end{minipage}\\[0.4cm]
+
+ {\bf\color{blue}
+ In the following these types of supercells are used and
+ are possibly scaled by integers in the different directions!
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Silicon point defects\\
+ }
+
+ \small
+
+ Calculation of formation energy $E_{\textrm{f}}$
+ \begin{itemize}
+ \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
+ cohesive energy per atom of the initial system
+ \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
+ cohesive energy per atom of the interstitial system
+ \item N: amount of atoms in the interstitial system
+ \end{itemize}
+ \vspace*{0.2cm}
+ {\color{blue}
+ \[
+ \Rightarrow
+ E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
+ -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
+ \]
+ }
+ Influence of supercell size\\
+ \begin{minipage}{8cm}
+ \includegraphics[width=7.0cm]{si_self_int.ps}
+ \end{minipage}
+ \begin{minipage}{5cm}
+ $E_{\textrm{f}}^{\textrm{110},\,{\color{red}32}\textrm{pc}}=3.38\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Questions so far ...\\
+ }
+
+ What configuration to chose for C in Si simulations?
+ \begin{itemize}
+ \item Switch to another method for the XC approximation (GGA, PAW)?
+ \item Reasonable cut-off energy
+ \item Switch off symmetry? (especially for defect simulations)
+ \item $k$-points
+ (Monkhorst? $\Gamma$-point only if cell is large enough?)
+ \item Switch to tetrahedron method or Gaussian smearing ($\sigma$?)
+ \item Size and type of supercell
+ \begin{itemize}
+ \item connected to choice of $k$-point mesh?
+ \item hence also connected to choice of smearing method?
+ \item constraints can only be applied to the lattice vectors!
+ \end{itemize}
+ \item Use of real space projection operators?
+ \item \ldots
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
+ and $k$-point mesh
+
+ \begin{itemize}
+ \item $1\times 1\times 1$ Type 0 simulations
+ \begin{itemize}
+ \item No difference in tetrahedron method and Gauss smearing
+ \item ...
+ \end{itemize}
+ \item $1\times 1\times 1$ Type 2 simulations
+ \begin{itemize}
+ \item Again, no difference in tetrahedron method and Gauss smearing
+ \item ...
+ \end{itemize}
+ \end{itemize}
+
+ {\LARGE\bf\color{red}
+ More simulations running ...
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ Symmetry (in defect simulations)
+
+ {\LARGE\bf\color{red}
+ Simulations running ...
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ Real space projection
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ Energy cut-off
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ Size and type of supercell
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Not answered (so far) ...\\
+ }
+
+\vspace{1.5cm}
+
+ \LARGE
+ \bf
+ \color{blue}
+
+ \begin{center}
+ Continue\\
+ with\\
+ US LDA?
+ \end{center}
+
+\vspace{1.5cm}