X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Ftalks%2Fupb-ua-xc.tex;h=502375bbb17bd3c4d370cb80f90d59b7450a27e0;hb=308a99aefd361f2e795db24e852574f91758a9df;hp=d277beb1202a4341713413b05c8350f106bb2d41;hpb=2784e1e2cf893049ad17688164799d4ca2f1dc42;p=lectures%2Flatex.git

diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex
index d277beb..502375b 100644
--- a/posic/talks/upb-ua-xc.tex
+++ b/posic/talks/upb-ua-xc.tex
@@ -97,7 +97,7 @@
 
  \vspace{08pt}
 
- June 2009
+ July 2009
 
 \end{center}
 \end{slide}
@@ -215,11 +215,25 @@ POTIM = 0.1
   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}
@@ -227,20 +241,257 @@ POTIM = 0.1
 \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}