a bit more ...

[lectures/latex.git] / posic / talks / seminar_2010.tex

diff --git a/posic/talks/seminar_2010.tex b/posic/talks/seminar_2010.tex
index ce2956b..75ceb62 100644 (file)
--- a/posic/talks/seminar_2010.tex
+++ b/posic/talks/seminar_2010.tex
@@ -1,6 +1,6 @@
 \pdfoutput=0
 \pdfoutput=0

-\documentclass[landscape,semhelv,draft]{seminar}
-%\documentclass[landscape,semhelv]{seminar}
+%\documentclass[landscape,semhelv,draft]{seminar}
+\documentclass[landscape,semhelv]{seminar}

 
 \usepackage{verbatim}
 \usepackage[greek,german]{babel}
 
 \usepackage{verbatim}
 \usepackage[greek,german]{babel}


@@ -154,14 +154,14 @@
 
 \end{pspicture}
 
 
 \end{pspicture}
 

-\begin{picture}(0,0)(-10,68)
+\begin{picture}(0,0)(-3,68)

 \includegraphics[width=2.6cm]{wide_band_gap.eps}
 \end{picture}
 \includegraphics[width=2.6cm]{wide_band_gap.eps}
 \end{picture}

-\begin{picture}(0,0)(-295,-165)
-\includegraphics[width=3cm]{sic_led.eps}
+\begin{picture}(0,0)(-285,-162)
+\includegraphics[width=3.38cm]{sic_led.eps}

 \end{picture}
 \end{picture}

-\begin{picture}(0,0)(-215,-165)
-\includegraphics[width=2.5cm]{6h-sic_3c-sic.eps}
+\begin{picture}(0,0)(-195,-162)
+\includegraphics[width=2.8cm]{6h-sic_3c-sic.eps}

 \end{picture}
 \begin{picture}(0,0)(-313,65)
 \includegraphics[width=2.2cm]{infineon_schottky.eps}
 \end{picture}
 \begin{picture}(0,0)(-313,65)
 \includegraphics[width=2.2cm]{infineon_schottky.eps}


@@ -169,6 +169,12 @@
 \begin{picture}(0,0)(-220,65)
 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
 \end{picture}
 \begin{picture}(0,0)(-220,65)
 \includegraphics[width=2.9cm]{sic_wechselrichter_ise.eps}
 \end{picture}

+\begin{picture}(0,0)(0,-160)
+\includegraphics[width=3.0cm]{sic_proton.eps}
+\end{picture}
+\begin{picture}(0,0)(-60,65)
+\includegraphics[width=3.4cm]{sic_switch.eps}
+\end{picture}

 
 \end{slide}
 
 
 \end{slide}
 


@@ -181,7 +187,7 @@
 }
 
  \begin{itemize}
 }
 
  \begin{itemize}

-  \item Polyteps and fabrication of silicon carbide
+  \item Polytyps and fabrication of silicon carbide

   \item Supposed precipitation mechanism of SiC in Si
   \item Utilized simulation techniques
         \begin{itemize}
   \item Supposed precipitation mechanism of SiC in Si
   \item Utilized simulation techniques
         \begin{itemize}


@@ -314,6 +320,19 @@ Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
   }
   \end{minipage}
  \end{picture}
   }
   \end{minipage}
  \end{picture}

+ \begin{picture}(0,0)(-230,-35)
+ \framebox{
+ {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
+ }
+ \end{picture}
+ \begin{picture}(0,0)(-230,-10)
+ \framebox{
+ \begin{minipage}{3cm}
+ {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
+                               less advanced}
+ \end{minipage}
+ }
+ \end{picture}

 
 \end{slide}
 
 
 \end{slide}
 


@@ -441,8 +460,14 @@ Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)
 \psellipse[linecolor=blue](11.5,5.8)(0.3,0.5)
 \rput{-20}{\psellipse[linecolor=blue](3.3,8.1)(0.3,0.5)}
 \psline[linewidth=4pt]{->}(4.0,2)(4.5,2)

+\rput(12.7,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+ $4a_{\text{Si}}=5a_{\text{SiC}}$
+ }}}
+\rput(12.2,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+\hkl(h k l) planes match
+ }}}

 \end{pspicture}
 \end{pspicture}

- 
+

 \end{slide}
 
 \begin{slide}
 \end{slide}
 
 \begin{slide}


@@ -529,7 +554,7 @@ V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
 n(r)=\sum_i^N|\Phi_i(r)|^2
 \]
   \item \underline{Self-consistent solution}\\
 n(r)=\sum_i^N|\Phi_i(r)|^2
 \]
   \item \underline{Self-consistent solution}\\

-$n(r)$ depends on $\Phi_i$, which depends on $V_{\text{eff}}$,
+$n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,

 which in turn depends on $n(r)$
   \item \underline{Variational principle}
         - minimize total energy with respect to $n(r)$
 which in turn depends on $n(r)$
   \item \underline{Variational principle}
         - minimize total energy with respect to $n(r)$


@@ -561,8 +586,10 @@ which in turn depends on $n(r)$
 \[
 \rightarrow
 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
 \[
 \rightarrow
 \text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}

+\qquad ({\color{blue}300\text{ eV}})

 \]
 \]

-  \item \underline{$k$-point sampling} - $\Gamma$-point only calculations
+  \item \underline{Brillouin zone sampling} -
+        {\color{blue}$\Gamma$-point only} calculations

   \item \underline{Pseudo potential} 
         - consider only the valence electrons
   \item \underline{Code} - VASP 4.6
   \item \underline{Pseudo potential} 
         - consider only the valence electrons
   \item \underline{Code} - VASP 4.6


@@ -578,6 +605,10 @@ which in turn depends on $n(r)$
   \item Structural optimization: Conjugate gradient method
  \end{itemize}
 
   \item Structural optimization: Conjugate gradient method
  \end{itemize}
 

+\begin{pspicture}(0,0)(0,0)
+\psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
+\end{pspicture}
+

 \end{slide}
 
 \begin{slide}
 \end{slide}
 
 \begin{slide}


@@ -1763,9 +1794,9 @@ Potential enhanced problem of slow phase space propagation
 Increased temperature simulations without TAD corrections\\
 (accelerated methods or higher time scales exclusively not sufficient)
 
 Increased temperature simulations without TAD corrections\\
 (accelerated methods or higher time scales exclusively not sufficient)
 

-\begin{picture}(0,0)(-262,-10)
-\frame{
-\begin{minipage}{4.3cm}
+\begin{picture}(0,0)(-260,-30)
+\framebox{
+\begin{minipage}{4.2cm}

 \tiny
 \begin{center}
 \vspace{0.03cm}
 \tiny
 \begin{center}
 \vspace{0.03cm}


@@ -1781,9 +1812,9 @@ Increased temperature simulations without TAD corrections\\
 }
 \end{picture}
 
 }
 \end{picture}
 

-\begin{picture}(0,0)(-305,-152)
-\frame{
-\begin{minipage}{2.6cm}
+\begin{picture}(0,0)(-305,-155)
+\framebox{
+\begin{minipage}{2.5cm}

 \tiny
 \begin{center}
 retain proper\\
 \tiny
 \begin{center}
 retain proper\\


@@ -1798,65 +1829,319 @@ thermodynmic sampling
 \begin{slide}
 
  {\large\bf
 \begin{slide}
 
  {\large\bf

-  Increased temperature simulations
+  Increased temperature simulations at low C concentration

  }
 
 \small
 
  }
 
 \small
 

-Low concentration simulation
-
+\begin{minipage}{6.5cm}
+\includegraphics[width=6.4cm]{tot_pc_thesis.ps}
+\end{minipage}
+\begin{minipage}{6.5cm}
+\includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
+\end{minipage}

 
 

+\begin{minipage}{6.5cm}
+\includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
+\end{minipage}
+\begin{minipage}{6.5cm}
+\scriptsize
+ \underline{Si-C bonds:}
+ \begin{itemize}
+  \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
+  \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
+ \end{itemize}
+ \underline{Si-Si bonds:}
+ {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
+ ($\rightarrow$ 0.325 nm)\\[0.1cm]
+ \underline{C-C bonds:}
+ \begin{itemize}
+  \item C-C next neighbour pairs reduced (mandatory)
+  \item Peak at 0.3 nm slightly shifted
+        \begin{itemize}
+         \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
+               $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
+               combinations (|)\\
+               $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
+               ($\downarrow$)
+         \item Range [|-$\downarrow$]:
+               {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
+               with nearby Si$_{\text{I}}$}
+        \end{itemize}
+ \end{itemize}
+\end{minipage}

 
 

+\begin{picture}(0,0)(-330,-74)
+\color{blue}
+\framebox{
+\begin{minipage}{1.6cm}
+\tiny
+\begin{center}
+stretched SiC\\[-0.1cm]
+in c-Si
+\end{center}
+\end{minipage}
+}
+\end{picture}

 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf
 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf

-  Increased temperature simulations
+  Increased temperature simulations at high C concentration

  }
 
  }
 

-\small
+\footnotesize
+
+\begin{minipage}{6.5cm}
+\includegraphics[width=6.4cm]{12_pc_thesis.ps}
+\end{minipage}
+\begin{minipage}{6.5cm}
+\includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
+\end{minipage}

 
 

-High concentration simulation
+\begin{center}
+Decreasing cut-off artifact\\
+High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
+$\Rightarrow$ hard to categorize
+\end{center}
+
+\vspace{0.1cm}

 
 

+\framebox{
+\begin{minipage}[t]{6.0cm}
+0.186 nm: Si-C pairs $\uparrow$\\
+(as expected in 3C-SiC)\\[0.2cm]
+0.282 nm: Si-C-C\\[0.2cm]
+$\approx$0.35 nm: C-Si-Si
+\end{minipage}
+}
+\begin{minipage}{0.2cm}
+\hfill
+\end{minipage}
+\framebox{
+\begin{minipage}[t]{6.0cm}
+0.15 nm: C-C pairs $\uparrow$\\
+(as expected in graphite/diamond)\\[0.2cm]
+0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
+0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
+\end{minipage}
+}

 
 

+\vspace{0.1cm}

 
 

+\begin{center}
+{\color{red}Amorphous} SiC-like phase remains\\
+Slightly sharper peaks
+$\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
+due to temperature\\[0.1cm]
+\framebox{
+\bf
+Continue with higher temperatures and longer time scales
+}
+\end{center}

 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf
 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf

-  Silicon carbide precipitation simulations
+  Valuation of a practicable temperature limit

  }
 
  \small
  }
 
  \small

+
+\vspace{0.1cm}
+
+\begin{center}
+\framebox{
+{\color{blue}
+Recrystallization is a hard task!
+$\Rightarrow$ Avoid melting!
+}
+}
+\end{center}

  
  

- 4. temperature limit
+\vspace{0.1cm}

 
 

+\footnotesize
+
+\begin{minipage}{7.5cm}
+\includegraphics[width=7cm]{fe_and_t.ps}
+\end{minipage}
+\begin{minipage}{5.5cm}
+\underline{Melting does not occur instantly after}\\
+\underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
+\begin{itemize}
+\item required transition enthalpy
+\item hysterisis behaviour
+\end{itemize}
+\underline{Heating up c-Si by 1 K/ps}
+\begin{itemize}
+\item transition occurs at $\approx$ 3125 K
+\item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
+      (literature: 50.2 kJ/mole)
+\end{itemize}
+\end{minipage}
+
+\vspace{0.1cm}
+
+\framebox{
+\begin{minipage}{4cm}
+Initially chosen temperatures:\\
+$1.0 - 1.2 \cdot T_{\text{m}}$
+\end{minipage}
+}
+\begin{minipage}{3cm}
+\begin{center}
+$\Longrightarrow$
+\end{center}
+\end{minipage}
+\framebox{
+\begin{minipage}{5cm}
+Introduced C (defects)\\
+$\rightarrow$ reduction of transition point\\
+$\rightarrow$ melting already at $T_{\text{m}}$
+\end{minipage}
+}
+
+\vspace{0.4cm}
+
+\begin{center}
+\framebox{
+{\color{blue}
+Maximum temperature used: $0.95\cdot T_{\text{m}}$
+}
+}
+\end{center}

 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf
 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf

-  Silicon carbide precipitation simulations
+  Long time scale simulations at maximum temperature

  }
 
  }
 

- \small
+\small
+
+\vspace{0.1cm}

  
  

- 5. final TODO
+\underline{Differences}
+\begin{itemize}
+ \item Temperature set to $0.95 \cdot T_{\text{m}}$
+ \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
+ \item Amount of C atoms: 6000 $\rightarrow$ 5500
+       $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
+ \item Simulation volume: 21 unit cells of c-Si in each direction
+\end{itemize}
+
+\footnotesize
+
+\vspace{0.3cm}
+
+\begin{minipage}[t]{4.5cm}
+\begin{center}
+\underline{Low C concentration, Si-C}
+\includegraphics[width=4.5cm]{c_in_si_95_v1_si-c.ps}\\
+Sharper peaks!
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{4.5cm}
+\begin{center}
+\underline{Low C concentration, C-C}
+\includegraphics[width=4.5cm]{c_in_si_95_v1_c-c.ps}\\
+Sharper peaks!\\
+No C agglomeration!
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{4cm}
+\begin{center}
+\underline{High C concentration}
+\includegraphics[width=4.5cm]{c_in_si_95_v2.ps}\\
+No significant changes
+\end{center}
+\end{minipage}
+
+\begin{center}
+\framebox{
+Long time scales and high temperatures most probably not sufficient enough!
+}
+\end{center}

 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf
 
 \end{slide}
 
 \begin{slide}
 
  {\large\bf

-  Silicon carbide precipitation simulations
+  Investigation of a silicon carbide precipitate in silicon

  }
 
  }
 

- \small
+ \footnotesize
+
+\vspace{0.2cm}
+
+\framebox{
+\scriptsize
+\begin{minipage}{5.3cm}
+\[
+\frac{8}{a_{\text{Si}}^3}(
+\underbrace{21^3 a_{\text{Si}}^3}_{=V}
+-\frac{4}{3}\pi x^3)+
+\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
+=21^3\cdot 8
+\]
+\[
+\Downarrow
+\]
+\[
+\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
+\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
+\]
+\[
+y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
+\]
+\end{minipage}
+}
+\begin{minipage}{0.3cm}
+\hfill
+\end{minipage}
+\begin{minipage}{7.0cm}
+\underline{Construction}
+\begin{itemize}
+ \item Simulation volume: 21$^3$ unit cells of c-Si
+ \item Spherical topotactically aligned precipitate\\
+       $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
+ \item Create c-Si but skipped inside sphere of radius $x$
+ \item Create 3C-SiC inside sphere of radius $x$\\
+       and lattice constant $y$
+ \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
+\end{itemize}
+\end{minipage}
+
+\vspace{0.3cm}
+
+\begin{minipage}{6.2cm}
+\includegraphics[width=6cm]{pc_0.ps}
+\end{minipage}
+\begin{minipage}{6.8cm}
+\underline{Results}
+\begin{itemize}
+ \item Slight increase of c-Si lattice constant!
+ \item C-C peaks (imply same distanced Si-Si peaks)
+       \begin{itemize}
+        \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
+        \item Bumps ({\color{green}$\downarrow$}):
+              4$^{\text{th}}$ and 6$^{\text{th}}$ NN
+       \end{itemize}
+ \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
+       $\rightarrow$ compressed precipitate
+ \item Interface tension:\\
+       20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
+       (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
+\end{itemize}
+\end{minipage}

 
 \end{slide}
 
 
 \end{slide}
 


@@ -1866,9 +2151,174 @@ High concentration simulation
   Investigation of a silicon carbide precipitate in silicon
  }
 
   Investigation of a silicon carbide precipitate in silicon
  }
 

+ \footnotesize
+
+\begin{minipage}{7cm}
+\underline{Appended annealing steps}
+\begin{itemize}
+ \item artificially constructed interface\\
+       $\rightarrow$ allow for rearrangement of interface atoms
+ \item check SiC stability
+\end{itemize}
+\underline{Temperature schedule}
+\begin{itemize}
+ \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
+       (75 K/ps)
+ \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
+       by 1 K/ps\\
+       $\rightarrow$ melting at around 2840 K
+       (\href{../video/sic_prec_120.avi}{$\rhd$})
+ \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
+       $\rightarrow$ no energetically more favorable struture
+\end{itemize}
+\end{minipage}
+\begin{minipage}{6cm}
+\includegraphics[width=6.7cm]{fe_and_t_sic.ps}
+\end{minipage}
+
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{sic_prec/melt_01.eps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{sic_prec/melt_02.eps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{sic_prec/melt_03.eps}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Summary / Conclusion / Outlook
+ }
+
+ \scriptsize
+
+\vspace{0.1cm}
+
+\framebox{
+\begin{minipage}{12.9cm}
+ \underline{Defects}
+ \begin{itemize}
+  \item Summary \& conclusion
+        \begin{itemize}
+         \item Point defects excellently / fairly well described
+               by QM / classical potential simulations
+         \item Identified migration path explaining
+               diffusion and reorientation experiments
+         \item Agglomeration of point defects energetically favorable
+         \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
+        \end{itemize}
+  \item Todo
+        \begin{itemize}
+         \item Discussions concerning interpretation of QM results (Paderborn)
+         \item Compare migration barrier of
+               \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell
+         \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \&
+                            C-Si \hkl<1 0 0> dumbbell (IBS)
+        \end{itemize}
+ \end{itemize}
+\end{minipage}
+}
+
+\vspace{0.2cm}
+
+\framebox{
+\begin{minipage}[t]{6.2cm}
+ \underline{Pecipitation simulations}
+ \begin{itemize}
+  \item Summary \& conclusion
+        \begin{itemize}
+         \item Low T
+               $\rightarrow$ C-Si \hkl<1 0 0> dumbbell\\
+               dominated structure
+         \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
+         \item High C concentration\\
+               $\rightarrow$ amorphous SiC like phase
+        \end{itemize}
+  \item Todo
+        \begin{itemize}
+         \item Accelerated method: self-guided MD
+         \item Activation relaxation technique
+         \item Constrainted transition path
+        \end{itemize}
+ \end{itemize}
+\end{minipage}
+}
+\framebox{
+\begin{minipage}[t]{6.2cm}
+ \underline{Constructed 3C-SiC precipitate}
+ \begin{itemize}
+  \item Summary \& conclusion
+        \begin{itemize}
+         \item Small / stable / compressed 3C-SiC\\
+               precipitate in slightly stretched\\
+               c-Si matrix
+         \item Interface tension matches experiemnts
+        \end{itemize}
+  \item Todo
+        \begin{itemize}
+         \item Try to improve interface
+         \item Precipitates of different size
+        \end{itemize}
+ \end{itemize}
+\end{minipage}
+}
+
+ \small
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Acknowledgements
+ }
+
+ \vspace{0.1cm}
+

  \small
 
  \small
 

+ Thanks to \ldots
+
+ \underline{Augsburg}
+ \begin{itemize}
+  \item Prof. B. Stritzker (accepting a simulator at EP \RM{4})
+  \item Ralf Utermann (EDV)
+ \end{itemize}
+ 
+ \underline{Helsinki}
+ \begin{itemize}
+  \item Prof. K. Nordlund (MD)
+ \end{itemize}

  
  

+ \underline{Munich}
+ \begin{itemize}
+  \item Bayerische Forschungsstiftung (financial support)
+ \end{itemize}
+ 
+ \underline{Paderborn}
+ \begin{itemize}
+  \item Prof. J. Lindner (SiC)
+  \item Prof. G. Schmidt (DFT + financial support)
+  \item Dr. E. Rauls (DFT + SiC)
+ \end{itemize}
+
+\vspace{0.2cm}
+
+\begin{center}
+\framebox{
+\bf Thank you for your attention!
+}
+\end{center}

 
 \end{slide}
 
 
 \end{slide}