X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Ftalks%2Fupb-ua-xc.tex;h=f652d80651f63c9b9953be3174c809563d5a8826;hb=6df80c58ea9ceda2479c185f218e868a42a44fe0;hp=a3d1048f9c5eadf272cfff136966e1de6a1c37fb;hpb=200984a06c22985aa9403bd9950934430d770906;p=lectures%2Flatex.git

diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex
index a3d1048..f652d80 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}
@@ -185,6 +185,7 @@ POTIM = 0.1
   \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$;
         64 atoms (32 pc)
  \end{enumerate}
+ \begin{minipage}{6cm}
  Cohesive energy / Lattice constant:
  \begin{enumerate}
   \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.378 \AA\\
@@ -197,32 +198,626 @@ POTIM = 0.1
         $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 5.975 eV / 5.390 \AA
   \item $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.977 eV / 5.389 \AA
  \end{enumerate}
+ \end{minipage}
+ \begin{minipage}{7cm}
+ \includegraphics[width=7cm]{si_lc_and_ce.ps}
+ \end{minipage}\\[0.3cm]
+ {\scriptsize
+  $^*$special settings (p. 138, VASP manual):
+  spin polarization, no symmetry, ...
+ }
+ 
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  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 $f(\{\epsilon_{n{\bf k}}\})$
+  \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
+  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
-  Interstitial configurations
+  3C-SiC bulk properties\\[0.2cm]
  }
 
- Silicon:
+ \small
+
+ \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 $f(\{\epsilon_{n{\bf k}}\})$
+ \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 $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
+ }
+ \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}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  3C-SiC bulk properties\\[0.2cm]
+ }
+
+ \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 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
+               $f(\{\epsilon_{n{\bf k}}\})$
+               ($\sigma=0.05$)
+         \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
+  Calculation of the defect formation energy\\
+ }
+
+ \small
+ 
+ {\color{blue}Method 1} (single species)
+ \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}
+ \[
+ \Rightarrow
+ E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
+               -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
+ \]
+ }\\[0.4cm]
+ {\color{magenta}Method 2} (two and more species)
+ \begin{itemize}
+  \item $E$: energy of the interstitial system
+        (with respect to the ground state of the free atoms!)
+  \item $N_{\text{Si}}$, $N_{\text{C}}$:
+        amount of Si and C atoms
+  \item $\mu_{\text{Si}}$, $\mu_{\text{C}}$:
+        chemical potential (cohesive energy) of Si and C
+ \end{itemize}
+ \vspace*{0.2cm}
+ {\color{magenta}
+ \[
+ \Rightarrow
+ E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}
+ \]
+ }
+
+\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
+
+ Influence of supercell size\\
+ \begin{minipage}{8cm}
+ \includegraphics[width=7.0cm]{si_self_int.ps}
+ \end{minipage}
+ \begin{minipage}{5cm}
+ $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
+ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$
+ \end{minipage}
+
+ Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
+ \begin{minipage}{8cm}
  \begin{itemize}
-  \item Lattice constant:
-  \item Cohesive energy: 5.95 eV, 5.99 eV, 5.96 eV, 5.98 eV
+  \item GGA and LDA
+  \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
+  \item 216 atom supercell
+  \item Gamma point only calculations
  \end{itemize}
+ \end{minipage}
+ \begin{minipage}{5cm}
+ $E_{\textrm{f}}^{\textrm{110}}=3.31 / 2.88\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
+ \end{minipage}
  
- <100> interstitial:
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Questions so far ...\\
+ }
+
+ What configuration to chose for C in Si simulations?
  \begin{itemize}
-  \item Lattice constant:
-  \item Cohesive energy:
+  \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}
 
- Hexagonal interstitial:
+\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{minipage}{4.4cm}
+  \includegraphics[width=4.4cm]{sic_smear_k.ps}
+ \end{minipage}
+ \begin{minipage}{4.4cm}
+  \includegraphics[width=4.4cm]{c_smear_k.ps}
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+  \includegraphics[width=4.4cm]{si_smear_k.ps}
+ \end{minipage}\\[0.3cm]
  \begin{itemize}
-  \item Lattice constant:
-  \item Cohesive energy:
+  \item Convergence reached at $6\times 6\times 6$ k-point mesh
+  \item No difference between Gauss ($\sigma=0.05$)
+        and tetrahedron smearing method!
  \end{itemize}
+ \begin{center}
+ $\Downarrow$\\
+ {\color{blue}\bf
+   Gauss ($\sigma=0.05$) smearing
+   and $6\times 6\times 6$ Monkhorst $k$-point mesh used
+ }
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Review (so far) ...\\
+ }
+
+ \underline{Symmetry (in defect simulations)}
+
+ \begin{center}
+ {\color{red}No}
+ difference in $1\times 1\times 1$ Type 2 defect calculations\\
+ $\Downarrow$\\
+ Symmetry precission (SYMPREC) small enough\\
+ $\Downarrow$\\
+ {\bf\color{blue}Symmetry switched on}\\
+ \end{center}
+
+ \underline{Real space projection}
+
+ \begin{center}
+ Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2):
+ $0.025\,\%$\\
+ Error in position of the 110 interstitital in Si ($1\times 1\times 1$ Type 2):
+ $0.026\,\%$\\
+ $\Downarrow$\\
+ {\bf\color{blue}
+  Real space projection used for 'large supercell' simulations}
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Review (so far) ...
+ }
+
+ Energy cut-off\\
+
+ \begin{center}
+
+ {\small
+ 3C-SiC equilibrium lattice constant and free energy\\ 
+ \includegraphics[width=7cm]{plain_sic_lc.ps}\\
+ $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
+ }
+
+ $\Downarrow$\\
+
+ {\bf\color{blue}
+  650 eV used as energy cut-off
+ }
+
+ \end{center}
+
+\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}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Final parameter choice
+ }
+
+ \footnotesize
+
+ \underline{Param 1}\\
+ My first choice. Used for more accurate calculations.
+ \begin{itemize}
+  \item $6\times 6 \times 6$ Monkhorst k-point mesh
+  \item $E_{\text{cut-off}}=650\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+ \end{itemize}
+ \vspace*{0.2cm}
+ \underline{Param 2}\\
+ After talking to the pros!
+ \begin{itemize}
+  \item $\Gamma$-point only
+  \item $E_{\text{cut-off}}=xyz\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+  \item Real space projection (Auto, Medium) for 'large' simulations
+ \end{itemize}
+ \vspace*{0.2cm}
+ {\color{blue}
+  In both parameter sets the ultra soft pseudo potential method
+  as well as the projector augmented wave method is used with both,
+  the LDA and GGA exchange correlation potential!
+ }
+\end{slide}
+
+\begin{slide}
+
+ \footnotesize
+
+ {\large\bf
+  Properties of Si, C and SiC using the new parameters\\
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
+ Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
+ Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
+ \hline
+ Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
+ Expt. [eV] & -4.63 & -7.374 & -6.340 \\
+ Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
+ \hline
+ \end{tabular}\\
+
+ \begin{minipage}{10cm}
+ $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
+ \hline
+ Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
+ Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
+ \hline
+ Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
+ Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{3cm}
+ US PP, GGA\\[0.2cm]
+ \begin{tabular}{|l|l|}
+ \hline
+ c-Si & c-C \\
+ \hline
+ 5.455 & 3.567 \\
+ {\color{green}0.5} & {\color{green}0.01} \\
+ \hline
+ -4.591 & -7.703 \\
+ {\color{green}0.8} & {\color{orange}4.5} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  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
+  C 100 interstitial migration along 110 in c-Si (Albe potential)
+ }
+
+ \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{Final configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/final.eps}\\
+ \end{minipage}
+ \begin{minipage}[t]{6cm}
+ \begin{itemize}
+  \item Constraints applied:
+        \begin{itemize}
+         \item along {\color{green}110 direction}
+         \item C atom {\color{red}entirely fixed in position}
+        \end{itemize}
+  \item C atom repeatedly displaced by\\
+        $\frac{1}{10}(\Delta x,\Delta y,\Delta z)$
+ \end{itemize}
+ \end{minipage}
+ \begin{minipage}[t]{0.5cm}
+ \hfill
+ \end{minipage}
+ \begin{minipage}[t]{6cm}
+ hier kommt mig energie rein ...
+ \end{minipage}
+
 
 \end{slide}