X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Ftalks%2Fupb-ua-xc.tex;h=23aff0777d5909a7b080b423d63a0713daa26105;hp=4197c0ab1ecbc77e0621617e5486c72275814a7e;hb=d6b64d4be7e48f04395e2862f4a53e0f06198e26;hpb=071e7bdd84bc69d2dde1835c2693fc1375d2bfb2

diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex
index 4197c0a..23aff07 100644
--- a/posic/talks/upb-ua-xc.tex
+++ b/posic/talks/upb-ua-xc.tex
@@ -46,6 +46,8 @@
 
 \usepackage{upgreek}
 
+\usepackage{miller}
+
 \begin{document}
 
 \extraslideheight{10in}
@@ -392,6 +394,49 @@ POTIM = 0.1
 
 \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
@@ -443,32 +488,36 @@ POTIM = 0.1
 
  \small
 
- Calculation of formation energy $E_{\textrm{f}}$
- \begin{itemize}
-  \item $E_{\textrm{coh}}^{\textrm{initial conf}}$:
-        cohesive energy per atom of the initial system
-  \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$:
-        cohesive energy per atom of the interstitial system
-  \item N: amount of atoms in the interstitial system
- \end{itemize}
- \vspace*{0.2cm}
- {\color{blue}
- \[
- \Rightarrow
- E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}}
-               -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N
- \]
- }
  Influence of supercell size\\
  \begin{minipage}{8cm}
  \includegraphics[width=7.0cm]{si_self_int.ps}
  \end{minipage}
  \begin{minipage}{5cm}
- $E_{\textrm{f}}^{\textrm{110},\,{\color{red}32}\textrm{pc}}=3.38\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\hkl<1 1 0>,\,32\textrm{pc}}=3.38\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
  $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
  $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
- $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$
+ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\hkl<1 1 0>,\,54\textrm{pc}}=3.48\textrm{ eV}$
+ \end{minipage}
+
+ Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm]
+ \begin{minipage}{8cm}
+ \begin{itemize}
+  \item GGA and LDA
+  \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$
+  \item 216 atom supercell
+  \item Gamma point only calculations
+ \end{itemize}
+ \end{minipage}
+ \begin{minipage}{5cm}
+ $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$
  \end{minipage}
+ 
 
 \end{slide}
 
@@ -507,88 +556,944 @@ POTIM = 0.1
  Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
  and $k$-point mesh
 
+ \begin{minipage}{4.4cm}
+  \includegraphics[width=4.4cm]{sic_smear_k.ps}
+ \end{minipage}
+ \begin{minipage}{4.4cm}
+  \includegraphics[width=4.4cm]{c_smear_k.ps}
+ \end{minipage}
+ \begin{minipage}{4.3cm}
+  \includegraphics[width=4.4cm]{si_smear_k.ps}
+ \end{minipage}\\[0.3cm]
+ \begin{itemize}
+  \item Convergence reached at $6\times 6\times 6$ k-point mesh
+  \item No difference between Gauss ($\sigma=0.05$)
+        and tetrahedron smearing method!
+ \end{itemize}
+ \begin{center}
+ $\Downarrow$\\
+ {\color{blue}\bf
+   Gauss ($\sigma=0.05$) smearing
+   and $6\times 6\times 6$ Monkhorst $k$-point mesh used
+ }
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Review (so far) ...\\
+ }
+
+ \underline{Symmetry (in defect simulations)}
+
+ \begin{center}
+ {\color{red}No}
+ difference in $1\times 1\times 1$ Type 2 defect calculations\\
+ $\Downarrow$\\
+ Symmetry precission (SYMPREC) small enough\\
+ $\Downarrow$\\
+ {\bf\color{blue}Symmetry switched on}\\
+ \end{center}
+
+ \underline{Real space projection}
+
+ \begin{center}
+ Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2):
+ $0.025\,\%$\\
+ Error in position of the \hkl<1 1 0> interstitital in Si
+ ($1\times 1\times 1$ Type 2):
+ $0.026\,\%$\\
+ $\Downarrow$\\
+ {\bf\color{blue}
+  Real space projection used for 'large supercell' simulations}
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Review (so far) ...
+ }
+
+ Energy cut-off\\
+
+ \begin{center}
+
+ {\small
+ 3C-SiC equilibrium lattice constant and free energy\\ 
+ \includegraphics[width=7cm]{plain_sic_lc.ps}\\
+ $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
+ }
+
+ $\Downarrow$\\
+
+ {\bf\color{blue}
+  650 eV used as energy cut-off
+ }
+
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Not answered (so far) ...\\
+ }
+
+\vspace{1.5cm}
+
+ \LARGE
+ \bf
+ \color{blue}
+
+ \begin{center}
+ Continue\\
+ with\\
+ US LDA?
+ \end{center}
+
+\vspace{1.5cm}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Final parameter choice
+ }
+
+ \footnotesize
+
+ \underline{Param 1}\\
+ My first choice. Used for more accurate calculations.
+ \begin{itemize}
+  \item $6\times 6 \times 6$ Monkhorst k-point mesh
+  \item $E_{\text{cut-off}}=650\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+ \end{itemize}
+ \vspace*{0.2cm}
+ \underline{Param 2}\\
+ After talking to the pros!
+ \begin{itemize}
+  \item $\Gamma$-point only
+  \item $E_{\text{cut-off}}=xyz\text{ eV}$
+  \item Gaussian smearing ($\sigma=0.05$)
+  \item Use symmetry
+  \item Real space projection (Auto, Medium) for 'large' simulations
+ \end{itemize}
+ \vspace*{0.2cm}
+ {\color{blue}
+  In both parameter sets the ultra soft pseudo potential method
+  as well as the projector augmented wave method is used with both,
+  the LDA and GGA exchange correlation potential!
+ }
+\end{slide}
+
+\begin{slide}
+
+ \footnotesize
+
+ {\large\bf
+  Properties of Si, C and SiC using the new parameters\\
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
+ Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
+ Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
+ \hline
+ Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
+ Expt. [eV] & -4.63 & -7.374 & -6.340 \\
+ Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
+ \hline
+ \end{tabular}\\
+
+ \begin{minipage}{10cm}
+ $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
+ \hline
+ Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
+ Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
+ \hline
+ Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
+ Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{3cm}
+ US PP, GGA\\[0.2cm]
+ \begin{tabular}{|l|l|}
+ \hline
+ c-Si & c-C \\
+ \hline
+ 5.455 & 3.567 \\
+ {\color{green}0.5} & {\color{green}0.01} \\
+ \hline
+ -4.591 & -7.703 \\
+ {\color{green}0.8} & {\color{orange}4.5} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+  Energy cut-off for $\Gamma$-point only caclulations
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
+ $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
+ $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
+ \small
+ \begin{minipage}{10cm}
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+  & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
+ Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
+ \hline
+ Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
+ Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{2cm}
+ {\LARGE
+  ${\color{green}\surd}$
+ }
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf\boldmath
+  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0>
+  in c-Si (Albe)
+ }
+
+ \small
+
+ \begin{minipage}[t]{4.2cm}
+ \underline{Starting configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/start.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.0cm}
+ \vspace*{0.8cm}
+ $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta z=0.325\text{ \AA}$\\
+ \end{minipage}
+ \begin{minipage}[t]{4.2cm}
+ \underline{{\bf Expected} final configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/final.eps}\\
+ \end{minipage}
+ \begin{minipage}{6cm}
  \begin{itemize}
-  \item $1\times 1\times 1$ Type 0 simulations
+  \item Fix border atoms of the simulation cell
+  \item Constraints and displacement of the C atom:
         \begin{itemize}
-         \item No difference in tetrahedron method and Gauss smearing
-         \item ...
+         \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 $1\times 1\times 1$ Type 2 simulations
+  \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 Again, no difference in tetrahedron method and Gauss smearing
-         \item ...
+         \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}
 
- {\LARGE\bf\color{red}
- More simulations running ...
+ \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
-  Review (so far) ...\\
+ {\large\bf\boldmath
+  C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP)
  }
 
- Symmetry (in defect simulations)
+ \small
 
- {\LARGE\bf\color{red}
- Simulations running ...
+ {\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
-  Review (so far) ...\\
+ {\large\bf\boldmath
+  Again: C \hkl<1 0 0> interstitial migration (Albe)
  }
 
- Real space projection
+ 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
-  Review (so far) ...\\
+ {\large\bf\boldmath
+  Again: C \hkl<1 0 0> interstitial migration (VASP)
  }
 
- Energy cut-off
+ 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
-  Review (so far) ...\\
+ 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 $\Delta z$
+  \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)
  }
 
- Size and type of supercell
+ \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
+    \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
-  Not answered (so far) ...\\
+  Molecular dynamics simulations (VASP)
  }
 
-\vspace{1.5cm}
+ 2 C atoms in $2\times 2\times 2$ Type 2 supercell at $450\,^{\circ}\text{C}$
 
- \LARGE
- \bf
- \color{blue}
+ \small
 
+ \begin{minipage}{7.6cm}
+ Radial distribution\\
+ \includegraphics[width=7.6cm]{md_02c_2222si_pc.ps}
+ \end{minipage}
+ \begin{minipage}{5.0cm}
  \begin{center}
- Continue\\
- with\\
- US LDA?
+ 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}
 
-\vspace{1.5cm}
+\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
+  Density Functional Theory
+ }
+
+ Hohenberg-Kohn theorem
+
+ \small
+ 
 
 \end{slide}