new params

[lectures/latex.git] / posic / talks / upb-ua-xc.tex

diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex
index e95c0fb..0419252 100644 (file)
--- a/posic/talks/upb-ua-xc.tex
+++ b/posic/talks/upb-ua-xc.tex
@@ -218,8 +218,8 @@ POTIM = 0.1
  \begin{itemize}
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
  \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}
   \item Supercell 3 (8 atoms, 4 primitive cells)
  \end{itemize}
  \vspace*{0.6cm}


@@ -269,8 +269,8 @@ POTIM = 0.1
  \begin{itemize}
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
  \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}
  \end{itemize}
  \vspace*{0.6cm}
  \begin{minipage}{6.5cm}


@@ -283,7 +283,15 @@ POTIM = 0.1
  \begin{center}
  {\color{red}
   Non-continuous energies\\
  \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}
  }
  \end{center}
  \end{minipage}


@@ -348,25 +356,30 @@ POTIM = 0.1
          \item Spin polarized calculation
          \item Interpolation formula according to Vosko Wilk and Nusair
                for the correlation part of the exchange correlation functional
          \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 $f_{nk}$
+         \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}
                ($\sigma=0.05$)
          \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})=
+        $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=

          -0.70036911\,\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}
         },
         {\color{gray}

-        $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})=
-         yyy\,\textrm{eV}$
+        $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
+         -1.3535731\,\textrm{eV}$

         }
   \item $E$:
         energy (non-polarized) of system of interest composed of\\
         n atoms of type N, m atoms of type M, \ldots
  \end{itemize}
         }
   \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}
+ \vspace*{0.2cm}

  {\color{red}
  \[
  \Rightarrow
  {\color{red}
  \[
  \Rightarrow


@@ -382,12 +395,12 @@ POTIM = 0.1
 \begin{slide}
 
  {\large\bf
 \begin{slide}
 
  {\large\bf

-  Silicon point defects\\
+  Calculation of the defect formation energy\\

  }
 
  \small
  }
 
  \small

-
- Calculation of formation energy $E_{\textrm{f}}$
+ 
+ {\color{blue}Method 1} (single species)

  \begin{itemize}
   \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
         cohesive energy per atom of the initial system
  \begin{itemize}
   \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
         cohesive energy per atom of the initial system


@@ -402,12 +415,312 @@ POTIM = 0.1
  E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
                -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
  \]
  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}
+ \[
+ \Rightarrow
+ E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}
+ \]

  }
 
  }
 

- \begin{center}
+\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}
  \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 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}
+ 
+
+\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{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 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{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! Used for 'large' simulations.
+ \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)
+ \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!
+ }
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Properties of Si, C and SiC using the new parameters\\
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 1\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.389 & 3.527 &  \\
+ Expt. [\AA] & 5.429 & 3.567 & \\
+ Error [\%] & {\color{green}0.7} & 1.1 & \\
+ \hline
+ Cohesive energy [eV] & -4.674 & -8.812 &  \\
+ Expt. [eV] & -4.63 & -7.374 & \\
+ Error [\%] & {\color{green}1.0} & {\color{red}19.5} &  \\
+ \hline
+ \end{tabular}\\
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  C interstitial in c-Si
+ }
+
+ 
+

 \end{slide}
 
 \end{document}
 \end{slide}
 
 \end{document}