\usepackage{pstricks}
\usepackage{pst-node}
+\usepackage{slashbox}
+
%\usepackage{epic}
%\usepackage{eepic}
\usepackage{upgreek}
+\usepackage{miller}
+
\begin{document}
\extraslideheight{10in}
\vspace{08pt}
- June 2009
+ July 2009
\end{center}
\end{slide}
\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 / 5.390 \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
- Interstitial configurations
+ Silicon point defects\\
}
- Silicon:
+ \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 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}
+ $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
+ \end{minipage}
- <100> interstitial:
+
+\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 Lattice constant:
- \item Cohesive energy:
+ \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)
+ }
- Hexagonal interstitial:
+ \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 Lattice constant:
- \item Cohesive energy:
+ \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}\\[0.2cm]
+ {\scriptsize\underline{PC (Vasp)}}
+ \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
+ ($3\times 3\times 3$ Type 2):
+
+ \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
+ ($3\times 3\times 3$ Type 2):
+
+ \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]
+ \[
+ \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV}
+ \]
+
+ \vspace*{0.5cm}
+ {\large\bf
+ Intermediate configuration {\color{red}not found} by now!
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C in Si interstitial configurations (VASP)
+ }
+
+ Check of Kohn-Sham eigenvalues\\
+
+ \small
+
+ \begin{minipage}{6cm}
+ \hkl<1 0 0> interstitial\\
+ \end{minipage}
+ \begin{minipage}{6cm}
+ Saddle point configuration\\
+ \end{minipage}
+ \underline{$4\times 4\times 3$ Type 1 - fixed border atoms}\\
+ \begin{minipage}{6cm}
+385: 4.8567 - 2.00000\\
+386: 4.9510 - 2.00000\\
+387: 5.3437 - 0.00000\\
+388: 5.4930 - 0.00000
+ \end{minipage}
+ \begin{minipage}{6cm}
+385: 4.8694 - 2.00000\\
+386: {\color{red}4.9917} - 1.92603\\
+387: {\color{red}5.1181} - 0.07397\\
+388: 5.4541 - 0.00000
+ \end{minipage}\\[0.2cm]
+ \underline{$4\times 4\times 3$ Type 1 - no constraints}\\
+ \begin{minipage}{6cm}
+385: 4.8586 - 2.00000\\
+386: 4.9458 - 2.00000\\
+387: 5.3358 - 0.00000\\
+388: 5.4915 - 0.00000
+ \end{minipage}
+ \begin{minipage}{6cm}
+385: 4.8693 - 2.00000\\
+386: {\color{red}4.9879} - 1.92065\\
+387: {\color{red}5.1120} - 0.07935\\
+388: 5.4544 - 0.00000
+ \end{minipage}\\[0.2cm]
+ \underline{$3\times 3\times 3$ Type 2 - no constraints}\\
+ \begin{minipage}{6cm}
+433: 4.8054 - 2.00000\\
+434: 4.9027 - 2.00000\\
+435: 5.2543 - 0.00000\\
+436: 5.5718 - 0.00000
+ \end{minipage}
+ \begin{minipage}{6cm}
+433: 4.8160 - 2.00000\\
+434: {\color{green}5.0109} - 1.00264\\
+435: {\color{green}5.0111} - 0.99736\\
+436: 5.5364 - 0.00000
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Once again: C \hkl<1 0 0> interstitial migration (VASP)
+ }
+
+ Method:
+ \begin{itemize}
+ \item Start in fully relaxed (assumed) saddle point configuration
+ \item Move towards \hkl<1 0 0> configuration using updated values
+ for $\Delta x$, $\Delta y$ and $\Delta z$ (CRT)
+ \item \hkl<1 1 0> constraints applied, 1 Si atom fixed
+ \item $4\times 4\times 3$ Type 1 supercell
+ \end{itemize}
+
+ Results:
+
+ \begin{minipage}{6.2cm}
+ \includegraphics[width=6.0cm]{c_100_110sp-i_vasp.ps}
+ \end{minipage}
+ \begin{minipage}{6.2cm}
+ \includegraphics[width=6.0cm]{c_100_110sp-i_rc_vasp.ps}
+ \end{minipage}
+
+ Reaction coordinate:
+ $r_{i+1}=r_i+\sum_{\text{atoms j}} \left| r_{j,i+1}-r_{j,i} \right|$
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Investigation of the migration path along \hkl<1 1 0> (VASP)
+ }
+
+ \small
+
+ \underline{Minimum:}\\
+ \begin{minipage}{4cm}
+ \includegraphics[width=3.5cm]{c_100_mig_vasp/110_c-si_split.eps}
+ \end{minipage}
+ \begin{minipage}{8cm}
+ \begin{itemize}
+ \item Starting conf: 35 \% displacement results (1443)
+ \item \hkl<1 1 0> constraint disabled
+ \end{itemize}
+ \begin{center}
+ $\Downarrow$
+ \end{center}
+ \begin{itemize}
+ \item C-Si \hkl<1 1 0> split interstitial
+ \item Stable configuration
+ \item $E_{\text{f}}=4.13\text{ eV}$
+ \end{itemize}
+ \end{minipage}\\[0.1cm]
+
+ \underline{Maximum:}\\
+ \begin{minipage}{6cm}
+ \begin{center}
+ \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_01.eps}
+ \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_02.eps}\\
+ 20 \% $\rightarrow$ 25 \%\\
+ Breaking of Si-C bond
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \includegraphics[width=6.2cm]{c_100_110sp-i_upd_vasp.ps}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Displacing the \hkl<1 1 0> Si-C split along \hkl<1 -1 0> (VASP)
+ }
+
+ \small
+
+ $4\times 4\times 3$ Type 1 supercell
+
+ \underline{Structures:}
+
+ \begin{minipage}[t]{4.1cm}
+ \includegraphics[height=3.0cm]{c_100_mig_vasp/start.eps}\\
+ \hkl<0 0 -1> dumbbell\\
+ $E_{\text{f}}={\color{orange}3.2254}\text{ eV}$
+ \end{minipage}
+ \begin{minipage}[t]{4.1cm}
+ \includegraphics[height=3.0cm]{c_100_mig_vasp/110_c-si_split.eps}\\
+ Assumed \hkl<1 1 0> C-Si split\\
+ $E_{\text{f}}=4.1314\text{ eV}$
+ \end{minipage}
+ \begin{minipage}[t]{4.1cm}
+ \includegraphics[height=3.0cm]{c_100_mig_vasp/110_dis_0-10.eps}\\
+ First guess: \hkl<0 -1 0> dumbbell\\
+ {\color{red}but:} $E_{\text{f}}={\color{orange}2.8924}\text{ eV}$\\
+ Third bond missing!
+ \end{minipage}\\
+
+ \underline{Occupancies:}
+
+ \scriptsize
+
+ \begin{minipage}{4.1cm}
+385: 4.8586 - 2.00000\\
+386: 4.9458 - 2.00000\\
+387: 5.3358 - 0.00000\\
+388: 5.4915 - 0.00000
+\hfill
+ \end{minipage}
+ \begin{minipage}{4.1cm}
+385: 4.7790 - 2.00000\\
+386: 4.8797 - 1.99964\\
+387: 5.1321 - 0.00036\\
+388: 5.4711 - 0.00000
+\hfill
+ \end{minipage}
+ \begin{minipage}{4.1cm}
+385: 4.7670 - 2.00000\\
+386: 4.9190 - 2.00000\\
+387: 5.2886 - 0.00000\\
+388: 5.4849 - 0.00000
+\hfill
+ \end{minipage}\\
+
+\small
+
+ \begin{center}
+ {\color{red}? ! ? ! ? ! ? ! ?}
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> interstitial migration (VASP)
+ }
+
+ \small
+
+ \begin{minipage}{6.2cm}
+ \begin{itemize}
+ \item $3\times 3\times 3$ Type 2 supercell
+ \item \hkl<1 1 0> constraints applied
+ (\href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/sd_rot.patch}{Patch})
+ \item Move from \hkl<1 0 0> towards\\
+ bond centered configuration
+ \end{itemize}
+ \underline{Sd Rot usage (POSCAR):}
+\begin{verbatim}
+cubic diamond
+5.480
+ 3.0 0.0 0.0
+ 0.0 3.0 0.0
+ 0.0 0.0 3.0
+216 1
+Transformed selective dynamics
+45.0 0.0
+Direct
+ ...
+\end{verbatim}
+Only works in direct mode!\\
+$z,x'$-axis rotation: $45.0^{\circ}$, $0.0^{\circ}$
+ \end{minipage}
+ \begin{minipage}{6.2cm}
+ \includegraphics[width=5cm]{c_100_110sp-i_2333_vasp.ps}
+ \includegraphics[width=5cm]{c_100_110sp-i_2333_rc_vasp.ps}\\
+ {\color{red}One fixed Si atom not enough!}\\
+ Video: \href{../video/c_in_si_233_110mig_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_in_si_233_110mig_vasp.avi}{$\rhd_{\text{remote url}}$}\\
+ \end{minipage}
+
+ {\color{blue}
+ Next: Migration calculation in 2333 using CRT
+ (\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1> and \hkl<0 -1 0>)
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Defect configurations in $4\times 4\times 3$ Type 1 supercells revisited
+ }
+
+ \footnotesize
+
+ \begin{tabular}{l|p{2.5cm}|p{2.5cm}|p{4cm}|}
+ & \hkl<0 0 -1> interstitial
+ & local minimum\newline
+ \hkl<1 1 0> C-Si split
+ & intermediate configuration\newline
+ (bond centered conf)\\
+ \hline
+ default & $E_{\text{f}}=3.3254\text{ eV}$\newline
+ {\tiny
+ 386: 4.9458 - 2.00000\newline
+ 387: 5.3358 - 0.00000}
+ & $E_{\text{f}}=4.1314\text{ eV}$\newline
+ {\tiny
+ 386: 4.8797 - 1.99964\newline
+ 387: 5.1321 - 0.00036}
+ & $E_{\text{f}}=4.2434\text{ eV}$\newline
+ {\tiny
+ 386: 4.9879 - 1.92065\newline
+ 387: 5.1120 - 0.07935} \\
+ \hline
+ No symmetry & $E_{\text{f}}=3.3154\text{ eV}$\newline
+ {\tiny
+ 386: 4.9456 - 2.00000\newline
+ 387: 5.3366 - 0.00000}
+ & $E_{\text{f}}=4.1314\text{ eV}$\newline
+ {\tiny
+ 386: 4.8798 - 1.99961\newline
+ 387: 5.1307 - 0.00039}
+ & $E_{\text{f}}=4.2454\text{ eV}$\newline
+ {\tiny
+ 386: 4.9841 - 1.92147\newline
+ 387: 5.1085 - 0.07853} \\
+ \hline
+ $+$ spin polarized & $E_{\text{f}}=3.3154\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 386: 4.9449 - 1.00000\newline
+ 387: 5.3365 - 0.00000\newline%
+ }%
+ {\color{green}%
+ 386: 4.9449 - 1.00000\newline
+ 387: 5.3365 - 0.00000}}
+ & $E_{\text{f}}={\color{red}4.1314}\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 386: 4.8799 - 0.99980\newline
+ 387: 5.1307 - 0.00020\newline%
+ }%
+ {\color{green}%
+ 386: 4.8799 - 0.99980\newline
+ 387: 5.1306 - 0.00020}}
+ & $E_{\text{f}}=4.0254\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 387: 4.8581 - 1.00000\newline
+ 388: 5.4662 - 0.00000\newline%
+ }%
+ {\color{green}%
+ 385: 4.8620 - 1.00000\newline
+ 386: 5.2951 - 0.00000}} \\
+ \hline
+ $+$ spin difference 2 & $E_{\text{f}}=3.6394\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 387: 5.2704 - 0.99891\newline
+ 388: 5.4886 - 0.00095\newline
+ 389: 5.5094 - 0.00011\newline
+ 390: 5.5206 - 0.00003\newline%
+ }%
+ {\color{green}%
+ 385: 4.8565 - 0.98603\newline
+ 386: 5.0119 - 0.01397}}
+ & Relaxation into\newline
+ bond centered\newline
+ configuration\newline
+ $\rightarrow$
+ & $E_{\text{f}}=4.0254\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 387: 4.8578 - 1.00000\newline
+ 388: 5.4661 - 0.00000\newline%
+ }%
+ {\color{green}%
+ 385: 4.8618 - 1.00000\newline
+ 386: 5.2950 - 0.00000}} \\
+ \hline
+ \end{tabular}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Defect configurations in $3\times 3\times 3$ Type 2 supercells revisited\\
+ }
+
+ \footnotesize
+
+ \begin{tabular}{l|p{2.5cm}|p{2.5cm}|p{4cm}|}
+ & \hkl<0 0 -1> interstitial
+ & local minimum\newline
+ \hkl<1 1 0> C-Si split
+ & intermediate configuration\newline
+ (bond centered conf)\\
+ \hline
+ default & $E_{\text{f}}=3.15407\text{ eV}$\newline
+ {\tiny
+ 434: 4.9027 - 2.00000\newline
+ 435: 5.2543 - 0.00000}
+ & $E_{\text{f}}=??\text{ eV}$\newline
+ {\tiny
+ ??\newline
+ ??}
+ & $E_{\text{f}}=4.40907\text{ eV}$\newline
+ {\tiny
+ 434: 5.0109 - 1.00264\newline
+ 435: 5.0111 - 0.99736}\\
+ \hline
+ No symmetry & $E_{\text{f}}=3.16107\text{ eV}$\newline
+ {\tiny
+ 434: 4.9032 - 2.00000\newline
+ 435: 5.2547 - 0.00000}
+ & $E_{\text{f}}=??\text{ eV}$\newline
+ {\tiny
+ ??\newline
+ ??}
+ & $E_{\text{f}}=4.41507\text{ eV}$\newline
+ {\tiny
+ 434: 5.0113 - 1.00140\newline
+ 435: 5.0114 - 0.99860} \\
+ \hline
+ $+$ spin polarized & $E_{\text{f}}=3.16107\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 434: 4.9033 - 1.00000\newline
+ 435: 5.2544 - 0.00000\newline%
+ }%
+ {\color{green}%
+ 434: 4.9035 - 1.00000\newline
+ 435: 5.2550 - 0.00000}}
+ & $E_{\text{f}}=??\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ ??\newline
+ ??\newline%
+ }%
+ {\color{green}%
+ ??\newline
+ ??}}
+ & $E_{\text{f}}=4.10307\text{ eV}$\newline
+ {\tiny
+ {\color{blue}
+ 435: 4.8118 - 1.00000\newline
+ 436: 5.5360 - 0.00000\newline%
+ }%
+ {\color{green}%
+ 433: 4.8151 - 1.00000\newline
+ 434: 5.3475 - 0.00000}} \\
+ \hline
+ \end{tabular}
+
+ \normalsize
+
+ \vspace*{0.3cm}
+
+ {\color{blue}Tracer:}\\
+ Smearing of electrons over two or more (degenerated) energy levels\\
+ $\Rightarrow$ use spin polarized calculations!
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Bond centered configuration revisited ($3\times 3\times 3$ Type 2)
+ }
+
+ Spin polarized calculations
+
+ {\small
+ \begin{minipage}[t]{5.8cm}
+ \underline{Kohn-Sham eigenvalues}\\
+ \begin{minipage}{2.8cm}
+ Spin up:\\
+ 430: 4.2639 - 1\newline
+ 431: 4.7332 - 1\newline
+ 432: 4.7354 - 1\newline
+ 433: 4.7700 - 1\newline
+ 434: 4.8116 - 1\newline
+ 435: 4.8118 - 1\newline
+ 436: 5.5360 - 0\newline
+ 437: 5.5623 - 0
+ \end{minipage}
+ \begin{minipage}{2.8cm}
+ Spin down:\\
+ 430: 4.2682 - 1\newline
+ 431: 4.7738 - 1\newline
+ 432: 4.8150 - 1\newline
+ 433: 4.8151 - 1\newline
+ 434: 5.3475 - 0\newline
+ 435: 5.3476 - 0\newline
+ 436: 5.5455 - 0\newline
+ 437: 5.5652 - 0
+ \end{minipage}\\[0.3cm]
+ \begin{itemize}
+ \item linear Si-C-Si bond
+ \item Each Si has another 3 Si neighbours
+ \end{itemize}
+ \begin{center}
+ {\color{blue}Spin polarized calculations necessary!}\\[0.3cm]
+ \end{center}
+ {\scriptsize Charge density isosurface of
+ {\color{gray}spin up}, {\color{green}spin down} and
+ the {\color{blue}resulting spin up} electrons.\\
+ Two {\color{yellow} Si} atoms and one {\color{red}C}
+ atom are shown.
+ }
+ \end{minipage}
+ \begin{minipage}[t]{6.5cm}
+ \underline{MO diagram}\\
+ \begin{minipage}[t]{1.2cm}
+ {\color{red}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{red}\uparrow}$}
+ \underline{${\color{red}\uparrow}$}
+ \underline{${\color{red}\uparrow}$}
+ \underline{${\color{red}\uparrow}$}\\
+ sp$^3$
+ \end{minipage}
+ \begin{minipage}[t]{1.4cm}
+ \begin{center}
+ {\color{red}M}{\color{blue}O}\\[1.0cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.0cm}
+ \begin{center}
+ {\color{blue}C}\\
+ {\tiny sp}\\[0.2cm]
+ \underline{${\color{white}\uparrow\uparrow}$}
+ \underline{${\color{white}\uparrow\uparrow}$}\\
+ 2p\\[0.4cm]
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
+ sp
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.4cm}
+ \begin{center}
+ {\color{blue}M}{\color{green}O}\\[1.0cm]
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[t]{1.2cm}
+ \begin{flushright}
+ {\color{green}Si}\\
+ {\tiny sp$^3$}\\[0.8cm]
+ \underline{${\color{green}\uparrow}$}
+ \underline{${\color{green}\uparrow}$}
+ \underline{${\color{green}\uparrow}$}
+ \underline{${\color{green}\uparrow}$}\\
+ sp$^3$
+ \end{flushright}
+ \end{minipage}\\[0.4cm]
+ \begin{flushright}
+ \includegraphics[width=6cm]{c_100_mig_vasp/im_spin_diff.eps}
+ \end{flushright}
+ \end{minipage}
+ }
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ \hkl<0 0 -1> configuration revisited ($3\times 3\times 3$ Type 2)
+ }
+
+ Spin polarized calculations
+
+ {\small
+ \begin{minipage}[t]{5.8cm}
+ \underline{Kohn-Sham eigenvalues}\\
+ \begin{minipage}{2.8cm}
+ Spin up:\\
+ 430: 4.3317 - 1\newline
+ 431: 4.7418 - 1\newline
+ 432: 4.8014 - 1\newline
+ 433: 4.8060 - 1\newline
+ 434: 4.9033 - 1\newline
+ 435: 5.2544 - 0\newline
+ 436: 5.5723 - 0\newline
+ 437: 5.5848 - 0
+ \end{minipage}
+ \begin{minipage}{2.8cm}
+ Spin down:\\
+ 430: 4.3317 - 1\newline
+ 431: 4.7420 - 1\newline
+ 432: 4.8013 - 1\newline
+ 433: 4.8059 - 1\newline
+ 434: 4.9035 - 1\newline
+ 435: 5.2550 - 0\newline
+ 436: 5.5724 - 0\newline
+ 437: 5.5846 - 0
+ \end{minipage}
+ \end{minipage}
+ \begin{minipage}[t]{6.5cm}
+ \underline{MO diagram}\\
+ \begin{minipage}[t]{1.2cm}
+ {\color{red}Si}\\
+ {\tiny sp$^2$}\\[0.1cm]
+ \underline{${\color{white}\uparrow}$}\\
+ p\\[0.4cm]
+ \underline{${\color{red}\uparrow\downarrow}$}
+ \underline{${\color{red}\uparrow}{\color{white}\downarrow}$}
+ \underline{${\color{red}\uparrow}{\color{white}\downarrow}$}\\
+ sp$^2$
+ \end{minipage}
+ \begin{minipage}[t]{1.2cm}
+ \begin{flushright}
+ {\color{red}M}\\[1.0cm]
+ \underline{${\color{white}\uparrow}{\color{white}\downarrow}$}\\
+ $\sigma_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
+ $\sigma_{\text{b}}$
+ \end{flushright}
+ \end{minipage}
+ \begin{minipage}[t]{1.2cm}
+ \begin{flushleft}
+ {\color{blue}O}\\[0.4cm]
+ \underline{${\color{white}\uparrow}{\color{white}\downarrow}$}\\
+ $\pi_{\text{ab}}$\\[0.5cm]
+ \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
+ $\pi_{\text{b}}$
+ \end{flushleft}
+ \end{minipage}
+ \begin{minipage}[t]{2.0cm}
+ \begin{center}
+ {\color{blue}C}\\
+ {\tiny sp$^2$}\\[0.5cm]
+ \underline{${\color{white}\uparrow\uparrow}$}\\
+ p\\[0.4cm]
+ \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}
+ \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
+ sp$^2$
+ \end{center}
+ \end{minipage}
+ \end{minipage}
+ }
+
+ \vspace*{0.4cm}
+
+ \begin{itemize}
+ \item Si-C double bond
+ \item Si and C atom have another 2 Si neighbours
+ \end{itemize}
+
+ \begin{center}
+ {\color{blue}Spin polarized calculations {\color{red}not} necessary!}
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Kohn-Sham levels visualized
+ }
+
+ \begin{minipage}{6cm}
+ \underline{\hkl<0 0 -1> configuration}
+ \begin{center}
+ \includegraphics[height=8cm]{c_100_mig_vasp/100_ksl.ps}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \underline{Saddle point configuration}
+ \begin{center}
+ \includegraphics[height=8cm]{c_100_mig_vasp/im_ksl.ps}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Saddle point configuration check
+ }
+
+ Simulations:
+ \begin{itemize}
+ \item Displacing the C atom in the BC configuration
+ \begin{itemize}
+ \item in \hkl<1 -1 0> direction\\
+ $(0.1240, 0.1240, 0.0409) \rightarrow
+ (0.1340, 0.1140, 0.0409)$
+ \item in \hkl<1 0 0> direction\\
+ $(0.1240, 0.1240, 0.0409) \rightarrow
+ (0.1440, 0.1240, 0.0409)$
+ \end{itemize}
+ \item Full relaxation of the configuration
+ \end{itemize}
+
+ Results:
+ \begin{itemize}
+ \item Both displacement simulations relax to
+ the BC configuration
+ \item Obviously the second derivative with respect to the
+ migration direction is also positive
+ \end{itemize}
+
+ \begin{center}
+ $\Downarrow$\\
+ Bond centered configuration is a
+ {\color{blue}real local minimum}
+ and {\color{red}not} a saddle point configuration
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ New default parameter set\\[1cm]
+ }
+
+ Since some defect configurations need spin polarized calculations ...\\[1cm]
+
+ from now on the default parameter set\\
+ {\bf\color{blue}
+ $+$ no symmetry\\
+ $+$ spin polarized\\
+ }
+ \ldots is used!\\[1cm]
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ BC to \hkl<0 0 -1> migration
+ in the $3\times 3\times 3$ Type 2 supercell
+ }
+
+ \begin{minipage}{6cm}
+ Method:
+ \begin{itemize}
+ \item Starting configuration:\\
+ C bond centered
+ \item CRT towards \hkl<0 0 -1> configuration
+ \item Spin polarized calculations
+ \end{itemize}
+ Results:\\
+ Video \href{../video/c_im_00-1_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_im_00-1_vasp.avi}{$\rhd_{\text{remote url}}$}
+ \begin{itemize}
+ \item Still abrupt changes in configuration and energy
+ \item Migration barrier $>$ 1 eV
+ \end{itemize}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \includegraphics[width=6cm]{c_im_001_mig_vasp.ps}
+ \includegraphics[width=6cm]{c_im_001_mig_rc_vasp.ps}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ \hkl<0 0 -1> to \hkl<0 -1 0> migration
+ in the $3\times 3\times 3$ Type 2 supercell
+ }
+
+ \includegraphics[width=6cm]{c_00-1_0-10_mig_vasp.ps}
+ \includegraphics[width=6cm]{c_00-1_0-10_mig_dis_vasp.ps}
+
+ Calculations without spin:\\
+ Video \href{../video/c_00-1_0-10_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/c_00-1_0-10_vasp.avi}{$\rhd_{\text{remote url}}$} ... WAAAAH!!!
+ \begin{itemize}
+ \item Refined starting from 70\% due to
+ abrubt jumps in energy and configuration
+ \item Displacement from 80 to 85\% disastrous
+ \item Subsequent displacements too large
+ \end{itemize}
+
+ Waiting for spin polarized calculations before deciding what to do ...
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ C \hkl<1 0 0> migration - yet another method!
+ }
+
+ {\color{red}Problem:}
+
+ Abrubt changes in atomic configurations (and energy)
+ in consecutive steps.
+ In addition - sometimes - the final configuration is not obtained!
+
+ {\color{blue}New method:}
+
+ Displace {\color{red}all} atoms towards the final configuration
+ and apply corresponding constraints for each atom.
+
+ Usage:
+ (\href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/sd_rot_all-atoms.patch}{Patch})
+
+\footnotesize
+
+\begin{verbatim}
+cubic diamond
+ 5.48000000000000
+ 2.9909698580839312 0.0039546630279804 -0.0039658085666586
+ 0.0039548953566878 2.9909698596656376 -0.0039660323646892
+ -0.0039680658132861 -0.0039674231313905 2.9909994291263242
+ 216 1
+Transformed selective dynamics
+Direct
+ 0.994174 0.994174 -0.000408732 T F T 45 36.5145
+ 0.182792 0.182792 0.981597 T F T -135 -5.95043
+ ...
+ 0.119896 0.119896 0.0385525 T F T -135 21.8036
+\end{verbatim}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ BC to \hkl<0 0 -1> migration (all atoms CRT)
+ }
+
+ \includegraphics[width=6cm]{im_00-1_nosym_sp_fullct.ps}
+ \includegraphics[width=6cm]{im_00-1_nosym_sp_fullct_rc.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ \hkl<0 0 -1> to \hkl<0 -1 0> migration (all atoms CRT)
+ }
+
+ \includegraphics[width=6cm]{00-1_0-10_nosym_sp_fullct.ps}
+ \includegraphics[width=6cm]{00-1_0-10_nosym_sp_fullct_rc.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ \hkl<0 0 -1> to \hkl<0 -1 0> migration in place (all atoms CRT)
+ }
+
+ \includegraphics[width=6cm]{00-1_ip0-10_nosym_sp_fullct.ps}
+ \includegraphics[width=6cm]{00-1_ip0-10_nosym_sp_fullct_rc.ps}
+
+ in progress ...
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ TODO: introduce some Si self-interstitials and C interstitials before\\
+ BUT: Concentrate on 100 C interstitial combinations and 100 C + vacancy\\
+
+ Agglomeration of 100 defects energetically favorable?
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Silicon point defects
+ }
+
+ \begin{minipage}{3.2cm}
+ \underline{Vacancy}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.63\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{si_pd_vasp/vac_2333.eps}\\
+ \underline{\hkl<1 1 0> interstitial}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.39\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{si_pd_vasp/110_2333.eps}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{si_pd_vasp/vac_2333_ksl.ps}\\
+ {\scriptsize Vacancy}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{si_pd_vasp/110_2333_ksl.ps}
+ {\scriptsize \hkl<1 1 0> interstitial}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Silicon point defects
+ }
+
+ \begin{minipage}{3.1cm}
+ \underline{Hexagonal}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.42\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{si_pd_vasp/hex_2333.eps}\\
+ \underline{Tetrahedral}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.77\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{si_pd_vasp/tet_2333.eps}
+ \end{minipage}
+ \begin{minipage}{3.7cm}
+ \begin{center}
+ \includegraphics[height=8cm]{si_pd_vasp/hex_2333_ksl.ps}\\
+ {\scriptsize Hexagonal}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{3.7cm}
+ \begin{center}
+ \includegraphics[height=8cm]{si_pd_vasp/tet_2333_ksl.ps}
+ {\scriptsize Tetrahedral}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}[c]{0.1cm}
+ \hfill
+ \end{minipage}
+ \begin{minipage}[c]{1.9cm}
+{\tiny
+\underline{Energy - Occup.}\\
+5.5063 - 0.32840\\
+5.5064 - 0.32793\\
+5.5064 - 0.32764\\
+5.5777 - 0.00691\\
+5.5777 - 0.00691\\
+5.6031 - 0.00074\\
+5.6031 - 0.00074\\
+5.6035 - 0.00071\\
+5.6357 - 0.00002\\
+5.6453 - 0.00001\\
+5.6453 - 0.00001
+}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Silicon point defects
+ }
+
+ \begin{minipage}{3.1cm}
+ \underline{\hkl<1 0 0> interstitial}
+ \begin{itemize}
+ \item $E_{\text{f}}=4.41\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{si_pd_vasp/100_2333.eps}\\
+ \end{minipage}
+ \begin{minipage}{3.7cm}
+ \begin{center}
+ \includegraphics[height=8cm]{si_pd_vasp/100_2333_ksl.ps}\\
+ {\scriptsize \hkl<1 0 0> interstitial}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Carbon point defects in silicon
+ }
+
+ \begin{minipage}{3.2cm}
+ \underline{C substitutional}
+ \begin{itemize}
+ \item $E_{\text{f}}=1.39\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{c_pd_vasp/sub_2333.eps}\\
+ \underline{\hkl<1 0 0> interstitial}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.15\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{c_pd_vasp/100_2333.eps}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{c_pd_vasp/sub_2333_ksl.ps}\\
+ {\scriptsize C substitutional}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{c_pd_vasp/100_2333_ksl.ps}
+ {\scriptsize \hkl<1 0 0> interstitial}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Carbon point defects in silicon
+ }
+
+ \begin{minipage}{3.2cm}
+ \underline{C bond centered}
+ \begin{itemize}
+ \item $E_{\text{f}}=4.10\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{c_pd_vasp/bc_2333.eps}
+ \underline{\hkl<1 1 0> interstitial}
+ \begin{itemize}
+ \item $E_{\text{f}}=3.60\text{ eV}$
+ \end{itemize}
+ \includegraphics[width=3cm]{c_pd_vasp/110_2333.eps}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{c_pd_vasp/110_2333_ksl.ps}
+ {\scriptsize \hkl<1 1 0> interstitial}
+ \end{center}
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \begin{center}
+ \includegraphics[height=8cm]{c_pd_vasp/bc_2333_ksl.ps}
+ {\scriptsize C bond centered}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Carbon point defects in silicon
+ }
+
+ The hexagonal and tetrahedral C configurations both relax into the
+ \hkl<0 0 1> interstitial configuration!
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \begin{itemize}
+ \item Supercell: $3\times 3\times 3$ Type 2
+ \item Starting configuration: \hkl<0 0 -1> C-Si interstitial
+ ($E_{\text{f}}=3.15\text{ eV}$)
+ \item Energies: $E_{\text{f}}$ of the interstitial combinations in eV
+ \end{itemize}
+
+ \underline{Along \hkl<1 1 0>:}
+
+ \begin{tabular}{|l|p{2.0cm}|p{1.8cm}|p{1.8cm}|p{1.8cm}|}
+ \hline
+ {\scriptsize
+ \backslashbox{2nd interstitial}{Distance $[\frac{a}{4}]$}
+ }
+ & \hkl<1 1 -1> & \hkl<2 2 0> & \hkl<3 3 -1> & \hkl<4 4 0>\\
+ \hline
+ \hkl<0 0 -1> & 6.23\newline {\color{blue}6.23514}
+ & 4.65\newline {\color{blue}4.65014}
+ & 5.97\newline {\color{blue}5.97314}
+ & 6.45\newline {\color{blue}6.45714} \\
+ \hline
+ \hkl<0 0 1> & 6.64\newline {\color{blue}6.65114}
+ & 4.78\newline {\color{blue}4.78314}
+ & 6.53\newline {\color{blue}6.53614}
+ & 6.18\newline {\color{blue}6.18914} \\
+ \hline
+ \hkl<1 0 0>, \hkl<0 1 0> & 4.06\newline alkmene
+ & 4.93
+ & 5.72
+ & 6.00\\
+ \hline
+ \hkl<-1 0 0>, \hkl<0 -1 0> & 3.92 & 4.43 & 6.02 & 6.02 \\
+ \hline
+ Vacancy & 1.39 ($\rightarrow\text{ C}_{\text{S}}$)& 5.81 & 5.47 & 6.50 \\
+ \hline
+ \end{tabular}
+
+ Spin polarized and {\color{blue}non spin polarized} results
+
+\end{slide}
+
+\begin{slide}
+
+ \begin{minipage}{5cm}
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \scriptsize
+
+ Initial insterstital at: $\frac{1}{4}\hkl<1 1 1>$
+
+ Relative silicon neighbour positions:
+ \begin{enumerate}
+ \item The dumbbell Si
+ \item $\frac{1}{4}\hkl<1 1 -1>$, $\frac{1}{4}\hkl<-1 -1 -1>$
+ \item $\frac{1}{2}\hkl<1 0 -1>$, $\frac{1}{2}\hkl<0 1 -1>$,
+ $\frac{1}{2}\hkl<0 -1 -1>$, $\frac{1}{2}\hkl<-1 0 -1>$
+ \item $\frac{1}{4}\hkl<1 -1 1>$, $\frac{1}{4}\hkl<-1 1 1>$
+ \item $\frac{1}{4}\hkl<-1 1 -3>$, $\frac{1}{4}\hkl<1 -1 -3>$
+ \item $\frac{1}{2}\hkl<-1 -1 0>$, $\frac{1}{2}\hkl<1 1 0>$
+ \item $\frac{1}{2}\hkl<1 -1 0>$, $\frac{1}{2}\hkl<-1 1 0>$
+ \item $\frac{1}{4}\hkl<-1 3 -1>$, $\frac{1}{4}\hkl<1 -3 -1>$,
+ $\frac{1}{4}\hkl<3 -1 -1>$. $\frac{1}{4}\hkl<-3 1 -1>$
+ \item $\hkl<0 0 -1>$
+ \item $\frac{1}{2}\hkl<1 0 1>$, $\frac{1}{2}\hkl<0 1 1>$,
+ $\frac{1}{2}\hkl<0 -1 1>$, $\frac{1}{2}\hkl<-1 0 1>$
+ \item $\frac{1}{4}\hkl<-1 -3 1>$, $\frac{1}{4}\hkl<-3 -1 1>$,
+ $\frac{1}{4}\hkl<1 3 1>$, $\frac{1}{4}\hkl<3 1 1>$
+ \item $\frac{1}{4}\hkl<1 3 -3>$, $\frac{1}{4}\hkl<3 1 -3>$,
+ $\frac{1}{4}\hkl<-1 -3 -3>$, $\frac{1}{4}\hkl<-3 -1 -3>$
+ \item $\hkl<1 0 0>$, $\hkl<0 1 0>$, $\hkl<-1 0 0>$, $\hkl<0 -1 0>$
+ \item $\frac{1}{4}\hkl<1 1 3>$, $\frac{1}{4}\hkl<-1 -1 3>$
+ \item $\frac{1}{4}\hkl<3 3 -1>$, $\frac{1}{4}\hkl<-3 -3 -1>$
+ \item $\frac{1}{2}\hkl<1 1 -2>$, $\frac{1}{2}\hkl<-1 -1 -2>$,
+ \item $\frac{1}{2}\hkl<1 -1 -2>$, $\frac{1}{2}\hkl<-1 1 -2>$
+ \end{enumerate}
+ One of a kind\\
+ {\color{red}Two of a kind}\\
+ {\color{blue}Four of a kind}
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \includegraphics[width=8cm]{c_100_next_neighbours_02.eps}
+ \begin{center}
+ \includegraphics[width=5cm]{c_100_res_bonds_vasp.ps}
+ \end{center}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ Initial C \hkl<0 0 -1> insterstital at: $\frac{1}{4}\hkl<1 1 1>$
+
+ {\footnotesize
+ \begin{tabular}{|l|l|l|l|l|l|}
+ \hline
+ & 2 & 3 & 4 & 5 & 6 \\
+ \hline
+C \hkl<0 0 -1> & 6.23/-0.08 & 5.16/-1.15 & 6.23/-0.08 & 6.35/0.04 & 4.65/-1.66\\
+ \hline
+C \hkl<0 0 1> & 6.64/0.34 & 6.31/0.01 & 4.26/-2.05 & 6.57/0.26 & 4.78/-1.53 \\
+ \hline
+C \hkl<1 0 0> & 4.06/-2.25 & 6.13/-0.17 & 6.21/-0.10 & 6.03/-0.27 & 4.93/-1.38 \\
+ \hline
+C \hkl<-1 0 0> & \hkl<0 -1 0> & 4.41/-1.90 & 4.06/-2.25 & 6.19/-0.12 & 4.43/-1.88 \\
+ \hline
+C \hkl<0 1 0> & \hkl<1 0 0> & 5.95/-0.36 & \hkl<-1 0 0> & \hkl<-1 0 0> & \hkl<1 0 0> \\
+ \hline
+C \hkl<0 -1 0> & 3.92/-2.39 & 4.15/-2.16 & \hkl<1 0 0> & \hkl<1 0 0> & \hkl <-1 0 0> \\
+ \hline
+Vacancy & 1.39/-5.39 ($\rightarrow\text{ C}_{\text{S}}$) & 6.19/-0.59 & 3.65/-3.14 & 6.24/-0.54 & 6.50/-0.50 \\
+ \hline
+C$_{\text{sub}}$ & 4.80/0.26 & 4.03/-0.51 & 3.62/-0.93 & 4.39/-0.15 & 5.03/0.49 \\
+\hline
+ \end{tabular}\\[0.2cm]
+ }
+
+ \begin{minipage}{8cm}
+ Energies: $x/y$\\
+ $x$: Defect formation energy of the complex\\
+ $y$:
+ $E_{\text{f}}^{\text{defect combination}}-
+ E_{\text{f}}^{\text{isolated C \hkl<0 0 -1>}}-
+ E_{\text{f}}^{\text{isolated 2nd defect}}
+ $\\[0.3cm]
+ {\color{blue}
+ If $y<0$ $\rightarrow$ favored compared to far-off isolated defects
+ }
+ \end{minipage}
+ \begin{minipage}{4.5cm}
+ \includegraphics[width=5.0cm]{00-1dc/energy.ps}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: \hkl<0 0 -1>
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/00-1_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/00-1_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/00-1_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/00-1_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/00-1_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_00x.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: \hkl<0 0 1>
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/001_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/001_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/001_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/001_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/001_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_001.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: \hkl<1 0 0> or equivalent one
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_100.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+
+ Type of second defect: \hkl<-1 0 0> or equivalent one
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/0-10_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/-100_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/-100_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/-100_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/0-10_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_x00.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: \hkl<0 1 0> or equivalent one
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/010_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/-100_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/-100_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_010.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+
+ Type of second defect: \hkl<0 -1 0> or equivalent one
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/0-10_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/0-10_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/100_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/0-10_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_0x0.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: Vacancy
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/vac_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/vac_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/vac_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/vac_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/vac_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_vac.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+ Combination of defects
+ }
+
+ \small
+
+ {\color{blue}
+ For defect position 3 and 5 (image 2 and 4) the unit cell is translated by
+ $\frac{a}{2} \hkl<0 -1 -1>$
+ }
+
+ Type of second defect: C$_{\text{sub}}$
+
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/csub_1.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/csub_3.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/csub_4.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/csub_5.eps}
+ \end{minipage}
+ \begin{minipage}{2.5cm}
+ \includegraphics[width=2.5cm]{00-1dc/csub_6.eps}
+ \end{minipage}
+
+ \includegraphics[width=5.0cm]{00-1dc/energy_csub.ps}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Brainstorming: Point defects in Si (as grown and as implanted)
+ }
+
+ \small
+
+ Supercell size: $2$ -- $2000 \cdot 10^{-21}\text{ cm}^3$
+
+ \underline{After crystal growth}
+ \begin{itemize}
+ \item Si point defects at $450\, ^{\circ}\text{C}$
+ \begin{itemize}
+ \item Interstitials:
+ \item Vacancies:
+ \end{itemize}
+ \item C impurities: $10^{17}\text{ cm}^{-3}$\\
+ $\Rightarrow$ $10^{-4}$ -- $10^{-1}$ per sc
+ $\rightarrow$ neglected in simulations
+ \end{itemize}
+
+ \underline{After/during implantation}
+ \begin{itemize}
+ \item Si point defects\\
+ $E_{\text{d}}^{\text{av}}=35\text{ eV}$,
+ $D_{\text{imp}}=1\text{ -- }4 \cdot 10^{17}\text{ cm }^{-2}$,
+ $d_{\text{sc}}=3\text{ -- }30\cdot 4.38\text { \AA}$,
+ $A=(3\text{ -- }30\text{ \AA})^2$,\\
+ Amount of collisions with $\Delta E > E_{\text{d}}$
+ in depth region $[h,h+d_{\text{sc}}]$: $n=$ .. (SRIM)\\
+ $\Rightarrow N_{\text{FP}}=nAD$
+ \item C point defects
+ \begin{itemize}
+ \item Substitutional C: ...
+ \item Intesrtitial C: ...
+ \end{itemize}
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Reminder (just for me to keep in mind ...)
+ }
+
+ \small
+
+ \underline{Volume of the MD cell}
+ \begin{itemize}
+ \item $T=450, 900, 1400\text{ K}$ - (no melting, N\underline{V}T!)
+ \item $\alpha=2.0 \cdot 10^{-6}\text{ K}^{-1}$
+ \item $a = a_0(1+\alpha \Delta T)$
+ \item Plain Si$(T=0)$: $a_0=5.4575\text{ \AA}$
+ $\rightarrow a(900\text{ K})=5.4674\text{ \AA}$
+ \item C \hkl<1 0 0> in Si$(T=0)$: $a_0^{\text{avg}}=
+ \frac{1}{3}(a_0^x+a_0^y+a_0^z)=5.4605\text{ \AA}$
+ $\rightarrow a(900\text{ K})=5.4704{ \AA}$
+ \end{itemize}
+ Used in first 900 K simulations: 5.4705 \AA\\
+ BUT: Better use plain Si lattice constant! (only local distortions)\\
+ $\Rightarrow a(1400\text{ K})=5.4728\text{ \AA}$
+
+ \underline{Zero total momentum simulations}
+ \begin{itemize}
+ \item If C is randomly inserted there is a net total momentum
+ \item No correction in the temperature control routine of VASP?
+ \item Relax a Si:C configuration first
+ (at T=0, no volume relaxation, scaled volume)
+ \item Use this configuration as the MD initial configuration
+ \end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Molecular dynamics simulations (VASP)
+ }
+
+ 2 C atoms in $2\times 2\times 2$ Type 2 supercell at $450\,^{\circ}\text{C}$
+
+ \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 at $450\,^{\circ}\text{C}$
+
+ \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
+ Molecular dynamics simulations (VASP)
+ }
+
+ 1 C atom in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$\\\\
+
+ Video \href{../video/md_01c_2333si_900_vasp.avi}{$\rhd_{\text{local}}$ } $|$
+ \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_01c_2333si_900_vasp.avi}{$\rhd_{\text{remote url}}$}\\\\
+
+ \begin{itemize}
+ \item Inserted C becomes a \hkl<0 0 1> interstitial after a few femto-seconds
+ \item {\color{red}There is a non-zero total momentum!}
+ \item Migration of the C atom not observed
+ \item C \hkl<0 0 1> configuration persists
+ \end{itemize}
+
+ Problem: Thermostat doesn't do momentum correction
+
+ TODO: Start MD using relaxed (at zero temperature) initial configuration
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Molecular dynamics simulations (VASP)
+ }
+
+ 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $900\,^{\circ}\text{C}$
+
+ in progress ...
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Density Functional Theory
+ }
+
+ Hohenberg-Kohn theorem
+
+ \small
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ More theory ...
+ }
+
+ Transition state theory\\
+ ART,NEB ...
+
+ Group theory
+
+ \small
+
+\end{slide}
+
+\end{document}
\end{document}
projects / lectures / latex.git / blobdiff