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% topic

\begin{slide}
\begin{center}

 \vspace{16pt}

 {\LARGE\bf
  Atomistic simulation study\\[0.2cm]
  of the SiC precipitation in Si
 }

 \vspace{48pt}

 \textsc{F. Zirkelbach}

 \vspace{48pt}

 For the exchange among Paderborn and Augsburg

 \vspace{08pt}

 July 2009

\end{center}
\end{slide}

% start of contents

\begin{slide}

 {\large\bf
  VASP parameters
 }

 \small
 \begin{minipage}{6.5cm}
 \begin{itemize}
  \item Start from scratch
  \item $V_{xc}$: US LDA (out of ./pot directory)
  \item $k$-points: Monkhorst $4\times 4\times 4$
  \item Ionic relaxation
        \begin{itemize}
         \item Conjugate gradient method
         \item Scaling constant of 0.1 for forces
         \item Default break condition ($0.1 \cdot 10^{-2}$ eV)
         \item Maximum of 100 steps
        \end{itemize}
        {\color{blue} NVT}:
        \begin{itemize}
         \item No change in volume
        \end{itemize}
        {\color{red} NPT}:
        \begin{itemize}
         \item Change of cell volume and shape\\
               allowed
        \end{itemize}
 \end{itemize}
 \end{minipage}
 \hspace*{0.5cm}
 \begin{minipage}{6.0cm}
{\scriptsize\color{blue}
 Example INCAR file (NVT):
}
\begin{verbatim}
System = C 100 interstitial in Si

ISTART = 0

NSW = 100
IBRION = 2
ISIF = 2
POTIM = 0.1
\end{verbatim}
{\scriptsize\color{red}
 Example INCAR file (NPT):
}
\begin{verbatim}
System = C hexagonal interstitial in Si

ISTART = 0

NSW = 100
IBRION = 2
ISIF = 3
POTIM = 0.1
\end{verbatim}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Silicon bulk properties
 }

 \small

 Simulations (NPT, $\textrm{EDIFFG}=0.1\cdot 10^{-3}$ eV):
 \begin{enumerate}
  \item Supercell: $x_1=(0,0.5,0.5),\, x_2=(0.5,0,0.5),\, x_3=(0.5,0.5,0)$;
        2 atoms (1 {\bf p}rimitive {\bf c}ell)
  \item Supercell: $x_1=(0.5,-0.5,0),\, x_2=(0.5,0.5,0),\, x_3=(0,0,1)$;
        4 atoms (2 pc)
  \item Supercell: $x_1=(1,0,0),\, x_2=(0,1,0),\, x_3=(0,0,1)$;
        8 atoms (4 pc)
  \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\\
        $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.387 \AA
  \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.989 eV / 5.356 \AA
  \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.380 \AA\\
        $E_{\textrm{cut-off}}=200\, \textrm{eV}$: 5.972 eV / 5.388 \AA\\
        $E_{\textrm{cut-off}}=250\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
        $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.975 eV / 5.389 \AA\\
        $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 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
  3C-SiC bulk properties\\[0.2cm]
 }

 \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
  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}
 $E_{\textrm{f}}^{\textrm{110},\,{\color{red}32}\textrm{pc}}=3.38\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Questions so far ...\\
 }

 What configuration to chose for C in Si simulations?
 \begin{itemize}
  \item 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{itemize}
  \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}

 {\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}
 Continue\\
 with\\
 US LDA?
 \end{center}

\vspace{1.5cm}

\end{slide}

\end{document}

