\vspace{08pt}
- June 2009
+ July 2009
\end{center}
\end{slide}
Silicon bulk properties
}
+ \begin{itemize}
+ \item Calculation of cohesive energies for different lattice constants
+ \item No ionic update
+ \item Tetrahedron method with Blöchl corrections for
+ the partial occupancies $f_{nk}$
+ \item Supercell 3 (8 atoms, 4 primitive cells)
+ \end{itemize}
+ \vspace*{0.6cm}
\begin{minipage}{6.5cm}
+ \begin{center}
+ $E_{\textrm{cut-off}}=150$ eV\\
\includegraphics[width=6.5cm]{si_lc_fit.ps}
+ \end{center}
\end{minipage}
\begin{minipage}{6.5cm}
+ \begin{center}
+ $E_{\textrm{cut-off}}=250$ eV\\
\includegraphics[width=6.5cm]{si_lc_fit_250.ps}
+ \end{center}
\end{minipage}
\end{slide}
\begin{slide}
{\large\bf
- Interstitial configurations
+ 3C-SiC bulk properties\\[0.2cm]
+ }
+
+ \begin{minipage}{6.5cm}
+ \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps}
+ \end{minipage}
+ \begin{minipage}{6.5cm}
+ \includegraphics[width=6.5cm]{sic_lc_and_ce.ps}
+ \end{minipage}\\[0.3cm]
+ \begin{itemize}
+ \item Supercell 3 (4 primitive cells, 4+4 atoms)
+ \item Error in equilibrium lattice constant: {\color{green} $0.9\,\%$}
+ \item Error in cohesive energy: {\color{red} $31.6\,\%$}
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ 3C-SiC bulk properties\\[0.2cm]
}
- <100> interstitial:
+ \small
+
\begin{itemize}
- \item
- \item
+ \item Calculation of cohesive energies for different lattice constants
+ \item No ionic update
+ \item Tetrahedron method with Blöchl corrections for
+ the partial occupancies $f_{nk}$
\end{itemize}
+ \vspace*{0.6cm}
+ \begin{minipage}{6.5cm}
+ \begin{center}
+ Supercell 3, $4\times 4\times 4$ k-points\\
+ \includegraphics[width=6.5cm]{sic_lc_fit.ps}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{6.5cm}
+ \begin{center}
+ {\color{red}
+ Non-continuous energies\\
+ for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
+ }
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ 3C-SiC bulk properties\\[0.2cm]
+ }
- Hexagonal interstitial:
+ \footnotesize
+
+\begin{picture}(0,0)(-188,80)
+ %Supercell 1, $3\times 3\times 3$ k-points\\
+ \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps}
+\end{picture}
+
+ \begin{minipage}{6.5cm}
\begin{itemize}
- \item
- \item
+ \item Supercell 1 simulations
+ \item Variation of k-points
+ \item Continuous energies for
+ $E_{\textrm{cut-off}} > 550\,\textrm{eV}$
+ \item Critical $E_{\textrm{cut-off}}$ for
+ different k-points\\
+ depending on supercell?
\end{itemize}
+ \end{minipage}\\[1.0cm]
+ \begin{minipage}{6.5cm}
+ \begin{center}
+ \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{6.5cm}
+ \begin{center}
+ \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Cohesive energies
+ }
+
+ {\bf\color{red} From now on ...}
+
+ {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
+
+ \small
+
+ \begin{itemize}
+ \item $E_{\textrm{free,sp}}$:
+ energy of spin polarized free atom
+ \begin{itemize}
+ \item $k$-points: Monkhorst $1\times 1\times 1$
+ \item Symmetry switched off
+ \item Spin polarized calculation
+ \item Interpolation formula according to Vosko Wilk and Nusair
+ for the correlation part of the exchange correlation functional
+ \item Gaussian smearing for the partial occupancies $f_{nk}$
+ ($\sigma=0.05$)
+ \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
+ \item Supercell: one atom in cubic
+ $10\times 10\times 10$ \AA$^3$ box
+ \end{itemize}
+ {\color{blue}
+ $E_{\textrm{free,sp}}(\textrm{Si},250\, \textrm{eV})=
+ -0.70036911\,\textrm{eV}$
+ },
+ {\color{gray}
+ $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})=
+ yyy\,\textrm{eV}$
+ }
+ \item $E$:
+ energy (non-polarized) of system of interest composed of\\
+ n atoms of type N, m atoms of type M, \ldots
+ \end{itemize}
+ \vspace*{0.3cm}
+ {\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}
+ $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$
+ \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 \ldots
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+
+
\end{slide}
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