\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 Tetrahedron method with Blöchl corrections for
+ the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
\item Supercell 3 (8 atoms, 4 primitive cells)
\end{itemize}
\vspace*{0.6cm}
\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 Tetrahedron method with Blöchl corrections for
+ the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
\end{itemize}
\vspace*{0.6cm}
\begin{minipage}{6.5cm}
\begin{center}
{\color{red}
Non-continuous energies\\
- for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
+ for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
+ }
+ \vspace*{0.5cm}
+ {\footnotesize
+ Does this matter in structural optimizaton simulations?
+ \begin{itemize}
+ \item Derivative might be continuous
+ \item Similar lattice constants where derivative equals zero
+ \end{itemize}
}
\end{center}
\end{minipage}
\begin{slide}
{\large\bf
- Silicon point defects
+ Cohesive energies
}
+ {\bf\color{red} From now on ...}
+
+ {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)}
+
\small
- {\color{red}\bf\LARGE
- HIER NOCH NICHT LESEN! :)\\
- }
- \begin{minipage}{6.5cm}
- <110> Si interstitial
\begin{itemize}
- \item 2 pc's:
- $E_{\textrm{f}}=-5.47342\,\textrm{eV}$
- \item 4 pc's:
- $E_{\textrm{f}}=-5.65098\,\textrm{eV}$
- \item 32 pc's: running ...
- $E_{\textrm{f}}=\,\textrm{eV}$
+ \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
+ $f(\{\epsilon_{n{\bf k}}\})$
+ ($\sigma=0.05$)
+ \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}
+ $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=
+ -0.70036911\,\textrm{eV}$
+ }\\
+ {\color{blue}
+ $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
+ -0.70021403\,\textrm{eV}$
+ },
+ {\color{gray}
+ $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
+ -1.3535731\,\textrm{eV}$
+ }
+ \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*{0.2cm}
+ {\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}{6.5cm}
- plain Si
- \begin{itemize}
- \item 2 pc's:
- $E_{\textrm{coh}}=-6.00846\,\textrm{eV}$
- \item 4 pc's:
- $E_{\textrm{coh}}=-5.97464\,\textrm{eV}$
- \item 32 pc's:
- $E_{\textrm{coh}}=-5.97633\,\textrm{eV}$
- \end{itemize}
- \end{minipage}\\[0.2cm]
- \begin{minipage}{6.5cm}
- hexagonal Si interstitial
+ \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 2 pc's:
- $E_{\textrm{f}}=-5.60654\,\textrm{eV}$
- \item 4 pc's:
- $E_{\textrm{f}}=-5.60174\,\textrm{eV}$
- \item 32 pc's: running ...
- $E_{\textrm{f}}=\,\textrm{eV}$
+ \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}
- tetrahedral Si interstitial
+ \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 2 pc's: $E_{\textrm{f}}=-5.53623\,\textrm{eV}$
- \item 4 pc's: $E_{\textrm{f}}=-5.64704\,\textrm{eV}$
- \item 32 pc's: $E_{\textrm{f}}=-5.92383\,\textrm{eV}$
+ \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}
- Si vacancy
+
+\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 2 pc's: $E_{\textrm{f}}=-5.65965\,\textrm{eV}$
- \item 4 pc's: $E_{\textrm{f}}=-5.56361\,\textrm{eV}$
- \item 32 pc's:$E_{\textrm{f}}=-5.92053\,\textrm{eV}$
+ \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}
- \end{minipage}
- \begin{minipage}{6.5cm}
+
+ {\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}
- \includegraphics[width=6.5cm]{si_self_int.ps}
+ Continue\\
+ with\\
+ US LDA?
\end{center}
- \end{minipage}
+
+\vspace{1.5cm}
\end{slide}
projects / lectures / latex.git / blobdiff