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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
  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}
 $E_{\textrm{f}}^{\hkl<1 1 0>,\,32\textrm{pc}}=3.38\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
 $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
 $E_{\textrm{f}}^{\hkl<1 1 0>,\,54\textrm{pc}}=3.48\textrm{ eV}$
 \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}
 $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
 $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
 \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{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 \hkl<1 1 0> 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\boldmath
  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\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
  in c-Si (Albe)
 }

 \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{{\bf Expected} final configuration}\\
 \includegraphics[width=4cm]{c_100_mig/final.eps}\\
 \end{minipage}
 \begin{minipage}{6cm}
 \begin{itemize}
  \item Fix border atoms of the simulation cell
  \item Constraints and displacement of the C atom:
        \begin{itemize}
         \item along {\color{green}\hkl<1 1 0> direction}\\
               displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
         \item C atom {\color{red}entirely fixed in position}\\
               displaced by
               {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
        \end{itemize}
  \item Berendsen thermostat applied
 \end{itemize}
 {\bf\color{blue}Expected configuration not obtained!}
 \end{minipage}
 \begin{minipage}{0.5cm}
 \hfill
 \end{minipage}
 \begin{minipage}{6cm}
 \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
  in c-Si (Albe)
 }

 \footnotesize

 \begin{minipage}{3.2cm}
 \includegraphics[width=3cm]{c_100_mig/fixmig_50.eps}
 \begin{center}
 50 \% 
 \end{center}
 \end{minipage}
 \begin{minipage}{3.2cm}
 \includegraphics[width=3cm]{c_100_mig/fixmig_80.eps}
 \begin{center}
 80 \% 
 \end{center}
 \end{minipage}
 \begin{minipage}{3.2cm}
 \includegraphics[width=3cm]{c_100_mig/fixmig_90.eps}
 \begin{center}
 90 \% 
 \end{center}
 \end{minipage}
 \begin{minipage}{3.2cm}
 \includegraphics[width=3cm]{c_100_mig/fixmig_99.eps}
 \begin{center}
 100 \% 
 \end{center}
 \end{minipage}

 Open questions ...
 \begin{enumerate}
  \item Why is the expected configuration not obtained?
  \item How to find a migration path preceding to the expected configuration?
 \end{enumerate}

 Answers ...
 \begin{enumerate}
  \item Simple: it is not the right migration path!
        \begin{itemize}
         \item (Surrounding) atoms settle into a local minimum configuration
         \item A possibly existing more favorable configuration is not achieved
        \end{itemize}
  \item \begin{itemize}
         \item Search global minimum in each step (by simulated annealing)\\
               {\color{red}But:}
               Loss of the correct energy needed for migration
         \item Smaller displacements\\
               A more favorable configuration might be achieved
               possibly preceding to the expected configuration
        \end{itemize}
 \end{enumerate}
 

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
  in c-Si (Albe)\\
 }

 Displacement step size decreased to
 $\frac{1}{100} (\Delta x,\Delta y)$\\[0.2cm]

 \begin{minipage}{7.5cm}
 Result: (Video \href{../video/c_in_si_smig_albe.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_albe.avi}{$\rhd_{\text{remote url}}$})
 \begin{itemize}
  \item Expected final configuration not obtained
  \item Bonds to neighboured silicon atoms persist
  \item C and neighboured Si atoms move along the direction of displacement
  \item Even the bond to the lower left silicon atom persists
 \end{itemize}
 {\color{red}
  Obviously: overestimated bond strength
 }
 \end{minipage}
 \begin{minipage}{5cm}
  \includegraphics[width=6cm]{c_100_110smig_01_albe.ps}
 \end{minipage}\\[0.4cm]
 New approach to find the migration path:\\
 {\color{blue}
 Place interstitial carbon atom at the respective coordinates
 into a perfect c-Si matrix!
 }

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
  in c-Si (Albe)
 }

 {\color{blue}New approach:}\\
 Place interstitial carbon atom at the respective coordinates
 into a perfect c-Si matrix!\\
 {\color{blue}Problem:}\\
 Too high forces due to the small distance of the C atom to the Si
 atom sharing the lattice site.\\
 {\color{blue}Solution:}
 \begin{itemize}
  \item {\color{red}Slightly displace the Si atom}
  (Video \href{../video/c_in_si_pmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
  \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_albe.avi}{$\rhd_{\text{remote url}}$})
  \item {\color{green}Immediately quench the system}
  (Video \href{../video/c_in_si_pqmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
  \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pqmig_albe.avi}{$\rhd_{\text{remote url}}$})
 \end{itemize}

 \begin{minipage}{6.5cm}
 \includegraphics[width=6cm]{c_100_110pqmig_01_albe.ps}
 \end{minipage}
 \begin{minipage}{6cm}
 \begin{itemize}
  \item Jump in energy corresponds to the abrupt
        structural change (as seen in the videos)
  \item Due to the abrupt changes in structure and energy
        this is {\color{red}not} the correct migration path and energy!?!
 \end{itemize}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
 }

 \small

 {\color{blue}Method:}
 \begin{itemize}
  \item Place interstitial carbon atom at the respective coordinates
        into perfect c-Si
  \item \hkl<1 1 0> direction fixed for the C atom
  \item $4\times 4\times 3$ Type 1, $198+1$ atoms
  \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
 \end{itemize}
 {\color{blue}Results:}
 (Video \href{../video/c_in_si_pmig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_pmig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
 \begin{minipage}{7cm}
 \includegraphics[width=7cm]{c_100_110pmig_01_vasp.ps} 
 \end{minipage}
 \begin{minipage}{5.5cm}
 \begin{itemize}
  \item Characteristics nearly equal to classical calulations
  \item Approximately half of the classical energy
        needed for migration
 \end{itemize}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
 }

 \small

 {\color{blue}Method:}
 \begin{itemize}
  \item Continue with atomic positions of the last run
  \item Displace the C atom in \hkl<1 1 0> direction
  \item \hkl<1 1 0> direction fixed for the C atom
  \item $4\times 4\times 3$ Type 1, $198+1$ atoms
  \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed
 \end{itemize}
 {\color{blue}Results:}
 (Video \href{../video/c_in_si_smig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_smig_vasp.avi}{$\rhd_{\text{remote url}}$})\\
 \includegraphics[width=7cm]{c_100_110mig_01_vasp.ps} 

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration
 }

 \small

 {\color{blue}The applied methods:}
 \begin{enumerate}
  \item Method
        \begin{itemize}
          \item Start in relaxed \hkl<1 0 0> interstitial configuration
          \item Displace C atom along \hkl<1 1 0> direction
          \item Relaxation (Berendsen thermostat)
          \item Continue with configuration of the last run
        \end{itemize} 
  \item Method
        \begin{itemize}
          \item Place interstitial carbon at the respective coordinates
                into the perfect Si matrix
          \item Quench the system
        \end{itemize} 
 \end{enumerate}
 {\color{blue}In both methods:}
 \begin{itemize} 
  \item Fixed border atoms
  \item Applied \hkl<1 1 0> constraint for the C atom
 \end{itemize}
 {\color{red}Pitfalls} and {\color{green}refinements}:
 \begin{itemize}
  \item {\color{red}Fixed border atoms} $\rightarrow$
        Relaxation of stress not possible\\
        $\Rightarrow$
        {\color{green}Fix only one Si atom} (the one furthermost to the defect)
  \item {\color{red}\hkl<1 1 0> constraint not sufficient}\\
        $\Rightarrow$ {\color{green}Apply 11x constraint}
        (connecting line of initial and final C positions)
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration (Albe)
 }

 Constraint applied by modifying the Velocity Verlet algorithm

 {\color{blue}Results:}
 (Video \href{../video/c_in_si_fmig_albe.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_fmig_albe.avi}{$\rhd_{\text{remote url}}$})\\
 \begin{minipage}{6.3cm}
 \includegraphics[width=6cm]{c_100_110fmig_01_albe.ps}
 \end{minipage}
 \begin{minipage}{6cm}
 \begin{center}
  Again there are jumps in energy corresponding to abrupt
  structural changes as seen in the video
 \end{center}
 \end{minipage}
 \begin{itemize}
  \item Expected final configuration not obtained
  \item Bonds to neighboured silicon atoms persist
  \item C and neighboured Si atoms move along the direction of displacement
  \item Even the bond to the lower left silicon atom persists
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration (VASP)
 }

 Transformation for the Type 2 supercell

 \small

 \begin{minipage}[t]{4.2cm}
 \underline{Starting configuration}\\
 \includegraphics[width=3cm]{c_100_mig_vasp/start.eps}
 \end{minipage}
 \begin{minipage}[t]{4.0cm}
 \vspace*{1.0cm}
 $\Delta x=1.367\text{ \AA}$\\
 $\Delta y=1.367\text{ \AA}$\\
 $\Delta z=0.787\text{ \AA}$\\
 \end{minipage}
 \begin{minipage}[t]{4.2cm}
 \underline{{\bf Expected} final configuration}\\
 \includegraphics[width=3cm]{c_100_mig_vasp/final.eps}\\
 \end{minipage}
 \begin{minipage}{6.2cm}
 Rotation angles:
 \[
 \alpha=45^{\circ}
 \textrm{ , }
 \beta=\arctan\frac{\Delta z}{\sqrt{2}\Delta x}=22.165^{\circ}
 \]
 \end{minipage}
 \begin{minipage}{6.2cm}
 Length of migration path:
 \[
 l=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}=2.087\text{ \AA}
 \]
 \end{minipage}\\[0.3cm]
 Transformation of basis:
 \[
 T=ABA^{-1}A=AB \textrm{, mit }
 A=\left(\begin{array}{ccc}
 \cos\alpha & -\sin\alpha & 0\\
 \sin\alpha & \cos\alpha & 0\\
 0 & 0 & 1
 \end{array}\right)
 \textrm{, }
 B=\left(\begin{array}{ccc}
 1 & 0 & 0\\
 0 & \cos\beta & \sin\beta \\
 0 & -\sin\beta & \cos\beta
 \end{array}\right)
 \]
 Atom coordinates transformed by: $T^{-1}=B^{-1}A^{-1}$

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration\\
 }

 {\color{blue}Reminder:}\\
 Transformation needed since in VASP constraints can only be applied to
 the basis vectors!\\
 {\color{red}Problem:} (stupid me!)\\
 Transformation of supercell 'destroys' the correct periodicity!\\
 {\color{green}Solution:}\\
 Find a supercell with one basis vector forming the correct constraint\\
 {\color{red}Problem:}\\
 Hard to find a supercell for the $22.165^{\circ}$ rotation\\

 Another method to {\color{blue}\underline{estimate}} the migration energy:
 \begin{itemize}
  \item Assume an intermediate saddle point configuration during migration
  \item Determine the energy of the saddle point configuration
  \item Substract the saddle point configuration energy by
        the energy of the initial (final) defect configuration
 \end{itemize}
 

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  The C \hkl<1 0 0> defect configuration
 }

 Needed so often for input configurations ...\\[0.8cm]
 \begin{minipage}{7.0cm}
 \includegraphics[width=6.5cm]{100-c-si-db_light.eps}\\
 Qualitative {\color{red}and} quantitative {\color{red}difference}!
 \end{minipage}
 \begin{minipage}{5.5cm}
 \scriptsize
 \begin{center}
 \begin{tabular}{|l|l|l|}
 \hline
  & a & b \\
 \hline
 \underline{VASP} & & \\
 fractional & 0.1969 & 0.1211 \\
 in \AA & 1.08 & 0.66 \\
 \hline
 \underline{Albe} & & \\
 fractional & 0.1547 & 0.1676 \\
 in \AA & 0.84 & 0.91 \\
 \hline
 \end{tabular}
 \includegraphics[width=6.1cm]{c_100_pc_vasp.ps}
 \end{center}
 \end{minipage}


\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration (VASP)
 }

 $\hkl<0 0 -1> \rightarrow \hkl<0 0 1>$ migration:

 \small

 \begin{minipage}[t]{4.1cm}
 \underline{Starting configuration}\\
 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
 \begin{center}
 $E_{\text{f}}=3.15 \text{ eV}$
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{4.1cm}
 \underline{Intermediate configuration}\\
 \includegraphics[height=3.2cm]{c_100_mig_vasp/00-1_001_im.eps}
 \begin{center}
 $E_{\text{f}}=4.41 \text{ eV}$
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{4.1cm}
 \underline{Final configuration}\\
 \includegraphics[height=3.2cm]{c_100_mig_vasp/final.eps}
 \begin{center}
 $E_{\text{f}}=3.17 \text{ eV}$
 \end{center}
 \end{minipage}\\[0.4cm]
 \[
 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = 1.26 \text{ eV}
 \]

 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
 \begin{itemize}
  \renewcommand\labelitemi{{\color{orange}$\bullet$}}
  \item Difference in formation energy (0.02 eV)
        of the initial and final configuration
  \renewcommand\labelitemi{{\color{red}$\bullet$}}
  \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
        of Type 1 (198+1 atoms) calculations
  \renewcommand\labelitemi{{\color{black}$\bullet$}}
 \end{itemize}
 
\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Again: C \hkl<1 0 0> interstitial migration (VASP)
 }

 $\hkl<0 0 -1> \rightarrow \hkl<0 -1 0>$ migration:

 \small

 \begin{minipage}[t]{4.1cm}
 \underline{Starting configuration}\\
 \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps}
 \begin{center}
 $E_{\text{f}}=3.154 \text{ eV}$
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{4.1cm}
 \underline{Intermediate configuration}\\
 in progress ...
 \begin{center}
 $E_{\text{f}}=?.?? \text{ eV}$
 \end{center}
 \end{minipage}
 \begin{minipage}[t]{4.1cm}
 \underline{Final configuration}\\
 \includegraphics[height=3.2cm]{c_100_mig_vasp/0-10.eps}
 \begin{center}
 $E_{\text{f}}=3.157 \text{ eV}$
 \end{center}
 \end{minipage}\\[0.4cm]
 \[
 \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV}
 \]

 Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal:
 \begin{itemize}
  \renewcommand\labelitemi{{\color{orange}$\bullet$}}
  \item Difference in formation energy (0.02 eV)
        of the initial and final configuration
  \renewcommand\labelitemi{{\color{red}$\bullet$}}
  \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier
        of Type 1 (198+1 atoms) calculations
  \renewcommand\labelitemi{{\color{black}$\bullet$}}
 \end{itemize}
 
\end{slide}

\begin{slide}

 {\large\bf
  Molecular dynamics simulations (VASP)
 }

 2 C atoms in $2\times 2\times 2$ Type 2 supercell

 \small

 \begin{minipage}{7.6cm}
 Radial distribution\\
 \includegraphics[width=7.6cm]{md_02c_2222si_pc.ps}
 \end{minipage}
 \begin{minipage}{5.0cm}
 \begin{center}
 PC average from\\
 $t_1=50$ ps to $t_2=50.93$ ps
 \end{center}
 \end{minipage}
 Diffusion:
 \begin{itemize}
  \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
        boundary crossings
  \item No jumps recognized in the
 Video \href{../video/md_02c_2222si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_02c_2222si_vasp.avi}{$\rhd_{\text{remote url}}$}
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf
  Molecular dynamics simulations (VASP)
 }

 10 C atoms in $3\times 3\times 3$ Type 2 supercell

 \small

 \begin{minipage}{7.2cm}
 Radial distribution (PC averaged over 1 ps)\\
 \includegraphics[width=7.0cm]{md_10c_2333si_pc_vasp.ps}
 \end{minipage}
 \begin{minipage}{5.0cm}
 \includegraphics[width=6.0cm]{md_10c_2333si_pcc_vasp.ps}
 \end{minipage}
 Diffusion:
 (Video \href{../video/md_10c_2333si_vasp.avi}{$\rhd_{\text{local}}$ } $|$
 \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_10c_2333si_vasp.avi}{$\rhd_{\text{remote url}}$})
 \begin{itemize}
  \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of
        boundary crossings
  \item Agglomeration of C? (Video)
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf
  Density Functional Theory
 }

 Hohenberg-Kohn theorem

 \small
 

\end{slide}

\end{document}

