begin trafo

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

diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex
index 502375b..2d7db85 100644 (file)
--- a/posic/talks/upb-ua-xc.tex
+++ b/posic/talks/upb-ua-xc.tex
@@ -219,7 +219,7 @@ POTIM = 0.1
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
   \item Tetrahedron method with Blöchl corrections for
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
   \item Tetrahedron method with Blöchl corrections for

-        the partial occupancies $f_{nk}$
+        the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$

   \item Supercell 3 (8 atoms, 4 primitive cells)
  \end{itemize}
  \vspace*{0.6cm}
   \item Supercell 3 (8 atoms, 4 primitive cells)
  \end{itemize}
  \vspace*{0.6cm}


@@ -270,7 +270,7 @@ POTIM = 0.1
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
   \item Tetrahedron method with Blöchl corrections for
   \item Calculation of cohesive energies for different lattice constants
   \item No ionic update
   \item Tetrahedron method with Blöchl corrections for

-        the partial occupancies $f_{nk}$
+        the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$

  \end{itemize}
  \vspace*{0.6cm}
  \begin{minipage}{6.5cm}
  \end{itemize}
  \vspace*{0.6cm}
  \begin{minipage}{6.5cm}


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

-  for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
+  for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
+ }
+ \vspace*{0.5cm}
+ {\footnotesize
+ Does this matter in structural optimizaton simulations?
+ \begin{itemize}
+  \item Derivative might be continuous
+  \item Similar lattice constants where derivative equals zero
+ \end{itemize}

  }
  \end{center}
  \end{minipage}
  }
  \end{center}
  \end{minipage}


@@ -348,25 +356,30 @@ POTIM = 0.1
          \item Spin polarized calculation
          \item Interpolation formula according to Vosko Wilk and Nusair
                for the correlation part of the exchange correlation functional
          \item Spin polarized calculation
          \item Interpolation formula according to Vosko Wilk and Nusair
                for the correlation part of the exchange correlation functional

-         \item Gaussian smearing for the partial occupancies $f_{nk}$
+         \item Gaussian smearing for the partial occupancies
+               $f(\{\epsilon_{n{\bf k}}\})$

                ($\sigma=0.05$)
          \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
          \item Supercell: one atom in cubic
                $10\times 10\times 10$ \AA$^3$ box
         \end{itemize}
         {\color{blue}
                ($\sigma=0.05$)
          \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001
          \item Supercell: one atom in cubic
                $10\times 10\times 10$ \AA$^3$ box
         \end{itemize}
         {\color{blue}

-        $E_{\textrm{free,sp}}(\textrm{Si},250\, \textrm{eV})=
+        $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=

          -0.70036911\,\textrm{eV}$
          -0.70036911\,\textrm{eV}$

+        }\\
+        {\color{blue}
+        $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
+         -0.70021403\,\textrm{eV}$

         },
         {\color{gray}
         },
         {\color{gray}

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

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

- \vspace*{0.3cm}
+ \vspace*{0.2cm}

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


@@ -379,6 +392,49 @@ POTIM = 0.1
 
 \end{slide}
 
 
 \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
 \begin{slide}
 
  {\large\bf


@@ -430,22 +486,6 @@ POTIM = 0.1
 
  \small
 
 
  \small
 

- Calculation of formation energy $E_{\textrm{f}}$
- \begin{itemize}
-  \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
-        cohesive energy per atom of the initial system
-  \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
-        cohesive energy per atom of the interstitial system
-  \item N: amount of atoms in the interstitial system
- \end{itemize}
- \vspace*{0.2cm}
- {\color{blue}
- \[
- \Rightarrow
- E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
-               -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
- \]
- }

  Influence of supercell size\\
  \begin{minipage}{8cm}
  \includegraphics[width=7.0cm]{si_self_int.ps}
  Influence of supercell size\\
  \begin{minipage}{8cm}
  \includegraphics[width=7.0cm]{si_self_int.ps}


@@ -454,8 +494,28 @@ POTIM = 0.1
  $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\

- $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$
+ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
+ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$
+ \end{minipage}
+
+ Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
+ \begin{minipage}{8cm}
+ \begin{itemize}
+  \item GGA and LDA
+  \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
+  \item 216 atom supercell
+  \item Gamma point only calculations
+ \end{itemize}
+ \end{minipage}
+ \begin{minipage}{5cm}
+ $E_{\textrm{f}}^{\textrm{110}}=3.31 / 2.88\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$

  \end{minipage}
  \end{minipage}

+ 

 
 \end{slide}
 
 
 \end{slide}
 


@@ -479,6 +539,7 @@ POTIM = 0.1
          \item hence also connected to choice of smearing method?
          \item constraints can only be applied to the lattice vectors!
         \end{itemize}
          \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}
 
   \item \ldots
  \end{itemize}
 


@@ -490,7 +551,550 @@ POTIM = 0.1
   Review (so far) ...\\
  }
 
   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 110 interstitital in Si ($1\times 1\times 1$ Type 2):
+ $0.026\,\%$\\
+ $\Downarrow$\\
+ {\bf\color{blue}
+  Real space projection used for 'large supercell' simulations}
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Review (so far) ...
+ }
+
+ Energy cut-off\\
+
+ \begin{center}
+
+ {\small
+ 3C-SiC equilibrium lattice constant and free energy\\ 
+ \includegraphics[width=7cm]{plain_sic_lc.ps}\\
+ $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
+ }
+
+ $\Downarrow$\\
+
+ {\bf\color{blue}
+  650 eV used as energy cut-off
+ }
+
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Not answered (so far) ...\\
+ }
+
+\vspace{1.5cm}
+
+ \LARGE
+ \bf
+ \color{blue}
+
+ \begin{center}
+ Continue\\
+ with\\
+ US LDA?
+ \end{center}
+
+\vspace{1.5cm}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Final parameter choice
+ }
+
+ \footnotesize
+
+ \underline{Param 1}\\
+ My first choice. Used for more accurate calculations.
+ \begin{itemize}
+  \item $6\times 6 \times 6$ Monkhorst k-point mesh
+  \item $E_{\text{cut-off}}=650\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+ \end{itemize}
+ \vspace*{0.2cm}
+ \underline{Param 2}\\
+ After talking to the pros!
+ \begin{itemize}
+  \item $\Gamma$-point only
+  \item $E_{\text{cut-off}}=xyz\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+  \item Real space projection (Auto, Medium) for 'large' simulations
+ \end{itemize}
+ \vspace*{0.2cm}
+ {\color{blue}
+  In both parameter sets the ultra soft pseudo potential method
+  as well as the projector augmented wave method is used with both,
+  the LDA and GGA exchange correlation potential!
+ }
+\end{slide}
+
+\begin{slide}
+
+ \footnotesize
+
+ {\large\bf
+  Properties of Si, C and SiC using the new parameters\\
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
+ Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
+ Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
+ \hline
+ Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
+ Expt. [eV] & -4.63 & -7.374 & -6.340 \\
+ Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
+ \hline
+ \end{tabular}\\
+
+ \begin{minipage}{10cm}
+ $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
+ \hline
+ Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
+ Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
+ \hline
+ Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
+ Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{3cm}
+ US PP, GGA\\[0.2cm]
+ \begin{tabular}{|l|l|}
+ \hline
+ c-Si & c-C \\
+ \hline
+ 5.455 & 3.567 \\
+ {\color{green}0.5} & {\color{green}0.01} \\
+ \hline
+ -4.591 & -7.703 \\
+ {\color{green}0.8} & {\color{orange}4.5} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Energy cut-off for $\Gamma$-point only caclulations
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
+ $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
+ $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
+ \small
+ \begin{minipage}{10cm}
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
+ Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
+ \hline
+ Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
+ Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{2cm}
+ {\LARGE
+  ${\color{green}\surd}$
+ }
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  C 100 interstitial migration along 110 in c-Si (Albe potential)
+ }
+
+ \small
+
+ \begin{minipage}[t]{4.2cm}
+ \underline{Starting configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/start.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.0cm}
+ \vspace*{0.8cm}
+ $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta z=0.325\text{ \AA}$\\
+ \end{minipage}
+ \begin{minipage}[t]{4.2cm}
+ \underline{{\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}110 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
+  C 100 interstitial migration along 110 in c-Si (Albe potential)
+ }
+
+ \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
+  C 100 interstitial migration along 110 in c-Si (Albe potential)\\
+ }
+
+ 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
+  C 100 interstitial migration along 110 in c-Si (Albe potential)
+ }
+
+ {\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
+  C 100 interstitial migration along 110 in c-Si (VASP)
+ }
+
+ \small
+
+ {\color{blue}Method:}
+ \begin{itemize}
+  \item Place interstitial carbon atom at the respective coordinates
+        into perfect c-Si
+  \item 110 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
+  C 100 interstitial migration along 110 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 110 direction
+  \item 110 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}
+ \includegraphics[width=7cm]{c_100_110mig_01_vasp.ps} 
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Again: C 100 interstitial migration
+ }
+
+ \small
+
+ {\color{blue}The applied methods:}
+ \begin{enumerate}
+  \item Method
+        \begin{itemize}
+          \item Start in relaxed 100 interstitial configuration
+          \item Displace C atom along 110 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 110 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}110 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
+  Again: C 100 interstitial migration
+ }
+
+ Defining the transformation for the Type 1 supercell (VASP)
+
+ \small
+
+ \begin{minipage}[t]{4.2cm}
+ \underline{Starting configuration}\\
+ \includegraphics[width=4cm]{c_100_mig_vasp/start.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.0cm}
+ \vspace*{0.8cm}
+ $\Delta x=\frac{1}{4}a_{\text{Si}}=1.368\text{ \AA}$\\
+ $\Delta y=\frac{1}{4}a_{\text{Si}}=1.368\text{ \AA}$\\
+ $\Delta z=0.888\text{ \AA}$\\
+ \end{minipage}
+ \begin{minipage}[t]{4.2cm}
+ \underline{{\bf Expected} final configuration}\\
+ \includegraphics[width=4cm]{c_100_mig_vasp/final.eps}\\
+ \end{minipage}\\
+ Angle of rotation about the 1-10 axis:
+ \[
+ \Theta=\arctan\frac{\Delta z}{\sqrt{2}\Delta x}=24.666^{\circ}
+ \]
+ Transformation of basis:
+ \[
+ T(\Theta)=\left(\begin{array}{ccc}
+ 1 & 0 & 0\\
+ 0 & \cos\Theta & -\sin\Theta \\
+ 0 & \sin\Theta & \cos\Theta
+ \end{array}\right)
+ \]
+ Transformation of atom coordinates: $T(-\Theta)$
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Density Functional Theory
+ }
+
+ Hohenberg-Kohn theorem
+
+ \small

  
 
 \end{slide}
  
 
 \end{slide}