From: hackbard <hackbard@hackdaworld.org>
Date: Mon, 14 May 2012 19:50:15 +0000 (+0200)
Subject: almostr finished
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=310e93282535b42f7ae2e3cd167776cc6249171a;p=lectures%2Flatex.git

almostr finished
---

diff --git a/posic/talks/emrs2012.tex b/posic/talks/emrs2012.tex
new file mode 100644
index 0000000..b437eb1
--- /dev/null
+++ b/posic/talks/emrs2012.tex
@@ -0,0 +1,2054 @@
+\pdfoutput=0
+%\documentclass[landscape,semhelv,draft]{seminar}
+\documentclass[landscape,semhelv]{seminar}
+
+\usepackage{verbatim}
+\usepackage[greek,german]{babel}
+\usepackage[latin1]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath}
+\usepackage{stmaryrd}
+\usepackage{latexsym}
+\usepackage{ae}
+
+\usepackage{calc}               % Simple computations with LaTeX variables
+\usepackage{caption}            % Improved captions
+\usepackage{fancybox}           % To have several backgrounds
+
+\usepackage{fancyhdr}           % Headers and footers definitions
+\usepackage{fancyvrb}           % Fancy verbatim environments
+\usepackage{pstricks}           % PSTricks with the standard color package
+
+\usepackage{pstricks}
+\usepackage{pst-node}
+\usepackage{pst-grad}
+
+%\usepackage{epic}
+%\usepackage{eepic}
+
+\usepackage{layout}
+
+\usepackage{graphicx}
+\graphicspath{{../img/}}
+
+\usepackage{miller}
+
+\usepackage[setpagesize=false]{hyperref}
+
+% units
+\usepackage{units}
+
+\usepackage{semcolor}
+\usepackage{semlayer}           % Seminar overlays
+\usepackage{slidesec}           % Seminar sections and list of slides
+
+\input{seminar.bug}             % Official bugs corrections
+\input{seminar.bg2}             % Unofficial bugs corrections
+
+\articlemag{1}
+
+\special{landscape}
+
+% font
+%\usepackage{cmbright}
+%\renewcommand{\familydefault}{\sfdefault}
+%\usepackage{mathptmx}
+
+\usepackage{upgreek}
+
+%\newrgbcolor{hred}{0.9 0.13 0.13}
+%\newrgbcolor{hblue}{0.13 0.13 0.9}
+\newrgbcolor{hred}{1.0 0.0 0.0}
+\newrgbcolor{hblue}{0.0 0.0 1.0}
+
+\begin{document}
+
+\extraslideheight{10in}
+\slideframe{plain}
+
+\pagestyle{empty}
+
+% specify width and height
+\slidewidth 26.3cm 
+\slideheight 19.9cm 
+
+% margin
+\def\slidetopmargin{-0.15cm}
+
+\newcommand{\ham}{\mathcal{H}}
+\newcommand{\pot}{\mathcal{V}}
+\newcommand{\foo}{\mathcal{U}}
+\newcommand{\vir}{\mathcal{W}}
+
+% itemize level ii
+\renewcommand\labelitemii{{\color{gray}$\bullet$}}
+
+% nice phi
+\renewcommand{\phi}{\varphi}
+
+% roman letters
+\newcommand{\RM}[1]{\MakeUppercase{\romannumeral #1{}}}
+
+% colors
+\newrgbcolor{si-yellow}{.6 .6 0}
+\newrgbcolor{hb}{0.75 0.77 0.89}
+\newrgbcolor{lbb}{0.75 0.8 0.88}
+\newrgbcolor{hlbb}{0.825 0.88 0.968}
+\newrgbcolor{lachs}{1.0 .93 .81}
+
+% head
+\newcommand{\headphd}{
+\begin{pspicture}(0,0)(0,0)
+\rput(6.0,0.2){\psframebox[fillstyle=gradient,gradbegin=hb,gradend=white,gradlines=1000,gradmidpoint=1,linestyle=none]{
+\begin{minipage}{14cm}
+\hfill
+\vspace{0.7cm}
+\end{minipage}
+}}
+\end{pspicture}
+}
+
+% shortcuts
+\newcommand{\si}{Si$_{\text{i}}${}}
+\newcommand{\ci}{C$_{\text{i}}${}}
+\newcommand{\cs}{C$_{\text{sub}}${}}
+\newcommand{\degc}[1]{\unit[#1]{$^{\circ}$C}{}}
+\newcommand{\distn}[1]{\unit[#1]{nm}{}}
+\newcommand{\dista}[1]{\unit[#1]{\AA}{}}
+\newcommand{\perc}[1]{\unit[#1]{\%}{}}
+
+% no vertical centering
+%\centerslidesfalse
+
+% layout check
+%\layout
+\ifnum1=0
+\begin{slide}
+\center
+{\Huge
+E\\
+F\\
+G\\
+A B C D E F G H G F E D C B A
+G\\
+F\\
+E\\
+}
+\end{slide}
+\fi
+
+% topic
+
+\begin{slide}
+
+ \small
+
+ \vspace{16pt}
+
+ {\LARGE\bf
+  %\hrule
+  %\vspace{5pt}
+  First-principles and empirical\\[0.2cm]
+  potential simulation study of intrinsic\\[0.2cm]
+  and carbon-related defects in silicon
+  %\vspace{10pt}
+  %\hrule
+ }
+
+ \vspace{30pt}
+
+ {\bf\small
+  \underline{F. Zirkelbach} $\color{gray}\bullet$ B. Stritzker\\
+ }
+ {\color{gray}
+  Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany
+ }\\[0.3cm]
+ {\bf\small
+  K. Nordlund\\
+ }
+ {\color{gray}
+  Department of Physics, University of Helsinki, 00014 Helsinki, Finland
+ }\\[0.3cm]
+ {\bf\small
+  W. G. Schmidt $\color{gray}\bullet$ E. Rauls $\color{gray}\bullet$
+  J. K. N. Lindner\\
+ }
+ {\color{gray}
+  Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany
+ }
+
+ \vspace{30pt}
+
+ {
+  E-MRS Spring Meeting, Strasbourg, 17.05.2012
+ }
+
+\end{slide}
+
+% no vertical centering
+\centerslidesfalse
+
+% skip for preparation
+\ifnum1=0
+
+% intro
+
+\begin{slide}
+
+\headphd
+{\large\bf
+  Motivation \& Outline
+}
+
+\vspace{0.1cm}
+
+{\bf
+ Ion beam synthesis (IBS) of epitaxial single crystalline 3C-SiC
+}
+
+\vspace{0.1cm}
+
+\begin{minipage}{7.0cm}
+\small
+\begin{itemize}
+ \item \underline{Implantation}\\[0.1cm]
+        Stoichiometric dose | \unit[180]{keV} | \degc{500}\\
+        $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
+        {\color{blue}precipitates}
+ \item \underline{Annealing}\\[0.1cm]
+       \unit[10]{h} at \degc{1250}\\
+       $\Rightarrow$ Homogeneous 3C-SiC layer
+\end{itemize}
+\begin{center}
+{\color{blue}
+\framebox{
+\begin{minipage}{4.5cm}
+ \color{black}
+ \centering
+ 3C-SiC precipitation\\
+ not yet fully understood
+\end{minipage}
+}
+}
+\end{center}
+\end{minipage}
+\begin{minipage}{5.0cm}
+\includegraphics[width=5.5cm]{ibs_3c-sic.eps}\\[-0.4cm]
+\begin{center}
+{\tiny
+ XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
+}
+\end{center}
+\end{minipage}\\[0.2cm]
+
+{\bf
+ Outline
+}
+
+\begin{itemize}
+ \item Assumed SiC precipitation mechanisms / Controversy
+ \item Utilized simulation techniques
+ \item C and Si self-interstitial point defects in silicon
+ \item Silicon carbide precipitation simulations
+\end{itemize}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+  Supposed precipitation mechanism of SiC in Si
+}
+
+ \scriptsize
+
+ \vspace{0.1cm}
+
+ \framebox{
+ \begin{minipage}{3.6cm}
+ \begin{center}
+ Si \& SiC lattice structure\\[0.1cm]
+ \includegraphics[width=2.3cm]{sic_unit_cell.eps}
+ \end{center}
+{\tiny
+ \begin{minipage}{1.7cm}
+\underline{Silicon}\\
+{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
+$a=\unit[5.429]{\\A}$\\
+$\rho^*_{\text{Si}}=\unit[100]{\%}$
+ \end{minipage}
+ \begin{minipage}{1.7cm}
+\underline{Silicon carbide}\\
+{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
+$a=\unit[4.359]{\\A}$\\
+$\rho^*_{\text{Si}}=\unit[97]{\%}$
+ \end{minipage}
+}
+ \end{minipage}
+ }
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{tem_c-si-db.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{tem_3c-sic.eps}
+ \end{center}
+ \end{minipage}
+
+ \vspace{0.1cm}
+
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ C-Si dimers (dumbbells)\\[-0.1cm]
+ on Si lattice sites
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ Agglomeration of C-Si dumbbells\\[-0.1cm]
+ $\Rightarrow$ dark contrasts
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ Precipitation of 3C-SiC in Si\\[-0.1cm]
+ $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
+ \& release of Si self-interstitials
+ \end{center}
+ \end{minipage}
+
+ \vspace{0.1cm}
+
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
+ \end{center}
+ \end{minipage}
+
+\begin{pspicture}(0,0)(0,0)
+\psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
+\psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
+\rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
+\psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
+\rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+ $4a_{\text{Si}}=5a_{\text{SiC}}$
+ }}}
+\rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+\hkl(h k l) planes match
+ }}}
+\rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+r = \unit[2--4]{nm}
+ }}}
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Supposed precipitation mechanism of SiC in Si
+}
+
+ \scriptsize
+
+ \vspace{0.1cm}
+
+ \framebox{
+ \begin{minipage}{3.6cm}
+ \begin{center}
+ Si \& SiC lattice structure\\[0.1cm]
+ \includegraphics[width=2.3cm]{sic_unit_cell.eps}
+ \end{center}
+{\tiny
+ \begin{minipage}{1.7cm}
+\underline{Silicon}\\
+{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
+$a=\unit[5.429]{\\A}$\\
+$\rho^*_{\text{Si}}=\unit[100]{\%}$
+ \end{minipage}
+ \begin{minipage}{1.7cm}
+\underline{Silicon carbide}\\
+{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
+$a=\unit[4.359]{\\A}$\\
+$\rho^*_{\text{Si}}=\unit[97]{\%}$
+ \end{minipage}
+}
+ \end{minipage}
+ }
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{tem_c-si-db.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{tem_3c-sic.eps}
+ \end{center}
+ \end{minipage}
+
+ \vspace{0.1cm}
+
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ C-Si dimers (dumbbells)\\[-0.1cm]
+ on Si interstitial sites
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ Agglomeration of C-Si dumbbells\\[-0.1cm]
+ $\Rightarrow$ dark contrasts
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ Precipitation of 3C-SiC in Si\\[-0.1cm]
+ $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
+ \& release of Si self-interstitials
+ \end{center}
+ \end{minipage}
+
+ \vspace{0.1cm}
+
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.1cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
+ \end{center}
+ \end{minipage}
+ \hspace{0.1cm}
+ \begin{minipage}{4.0cm}
+ \begin{center}
+ \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
+ \end{center}
+ \end{minipage}
+
+\begin{pspicture}(0,0)(0,0)
+\psline[linewidth=2pt]{->}(8.3,2)(8.8,2)
+\psellipse[linecolor=blue](11.1,6.0)(0.3,0.5)
+\rput{-20}{\psellipse[linecolor=blue](3.1,8.2)(0.3,0.5)}
+\psline[linewidth=2pt]{->}(3.9,2)(4.4,2)
+\rput(11.8,0.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+ $4a_{\text{Si}}=5a_{\text{SiC}}$
+ }}}
+\rput(11.5,8){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+\hkl(h k l) planes match
+ }}}
+\rput(8.5,6.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+r = \unit[2--4]{nm}
+ }}}
+% controversial view!
+\rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
+\begin{minipage}{14cm}
+\hfill
+\vspace{12cm}
+\end{minipage}
+}}
+\rput(6.5,5.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
+\begin{minipage}{10cm}
+\small
+\vspace*{0.2cm}
+\begin{center}
+{\color{gray}\bf Controversial findings}
+\end{center}
+\begin{itemize}
+\item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
+ \begin{itemize}
+  \item {\color{blue}Substitutionally} incorporated C on regular Si lattice sites
+  \item \si{} reacting with further C in cleared volume
+ \end{itemize}
+\item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
+ \begin{itemize}
+  \item Room temperature implantation $\rightarrow$ high C diffusion
+  \item Elevated temperature implantation $\rightarrow$ no C redistribution
+ \end{itemize}
+ $\Rightarrow$ mobile {\color{red}\ci} opposed to
+ stable {\color{blue}\cs{}} configurations
+\item Strained Si$_{1-y}$C$_y$/Si heterostructures
+      {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
+ \begin{itemize}
+  \item Initial {\color{blue}coherent} SiC structures (tensile strain)
+  \item Incoherent SiC nanocrystals (strain relaxation)
+ \end{itemize}
+\end{itemize}
+\vspace{0.1cm}
+\begin{center}
+{\Huge${\lightning}$} \hspace{0.3cm}
+{\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
+{\Huge${\lightning}$}
+\end{center}
+\vspace{0.2cm}
+\end{minipage}
+ }}}
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Utilized computational methods
+}
+
+\vspace{0.3cm}
+
+\small
+
+{\bf Molecular dynamics (MD)}\\[0.1cm]
+\scriptsize
+\begin{tabular}{| p{4.5cm} | p{7.5cm} |}
+\hline
+System of $N$ particles &
+$N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
+Phase space propagation &
+Velocity Verlet | timestep: \unit[1]{fs} \\
+Analytical interaction potential &
+Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
+(Erhart/Albe)
+$\displaystyle
+E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
+    \pot_{ij} = {\color{red}f_C(r_{ij})}
+    \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
+$\\
+Observables: time/ensemble averages &
+NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
+\hline
+\end{tabular}
+
+\small
+
+\vspace{0.3cm}
+
+{\bf Density functional theory (DFT)}
+
+\scriptsize
+
+\begin{minipage}[t]{6cm}
+\begin{itemize}
+ \item Hohenberg-Kohn theorem:\\
+       $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
+ \item Kohn-Sham approach:\\
+       Single-particle effective theory
+\end{itemize}
+\hrule
+\begin{itemize}
+\item Code: \textsc{vasp}
+\item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
+%$\displaystyle
+%\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
+%$\\
+%$\displaystyle
+%E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
+%$
+\item Ultrasoft pseudopotential
+\item Exchange \& correlation: GGA
+\item Brillouin zone sampling: $\Gamma$-point
+\item Supercell: $N=216\pm2$
+\end{itemize}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{pspicture}(0,0)(0,0)
+\pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
+\rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
+\left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
+$
+}}
+\rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
+n(r)=\sum_i^N|\Phi_i(r)|^2
+$
+}}
+\rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
+V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
+                 +V_{\text{XC}}[n(r)]
+$
+}}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
+
+\end{pspicture}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+ {\large\bf
+  Point defects \& defect migration
+ }
+
+ \small
+
+ \vspace{0.2cm}
+
+\begin{minipage}[b]{7.5cm}
+{\bf Defect structure}\\
+  \begin{pspicture}(0,0)(7,4.4)
+  \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+   \parbox{7cm}{
+   \begin{itemize}
+    \item Creation of c-Si simulation volume
+    \item Periodic boundary conditions
+    \item $T=0\text{ K}$, $p=0\text{ bar}$
+   \end{itemize}
+  }}}}
+\rput(3.5,1.3){\rnode{insert}{\psframebox{
+ \parbox{7cm}{
+  \begin{center}
+  Insertion of interstitial C/Si atoms
+  \end{center}
+  }}}}
+  \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
+   \parbox{7cm}{
+   \begin{center}
+   Relaxation / structural energy minimization
+   \end{center}
+  }}}}
+  \ncline[]{->}{init}{insert}
+  \ncline[]{->}{insert}{cool}
+ \end{pspicture}
+\end{minipage}
+\begin{minipage}[b]{4.5cm}
+\begin{center}
+\includegraphics[width=3.8cm]{unit_cell_e.eps}\\
+\end{center}
+\begin{minipage}{2.21cm}
+{\scriptsize
+{\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
+{\color{green}$\bullet$} Hexagonal\\[-0.1cm]
+{\color{yellow}$\bullet$} \hkl<1 0 0> DB
+}
+\end{minipage}
+\begin{minipage}{2.21cm}
+{\scriptsize
+{\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
+{\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
+{\color{black}$\bullet$} Vac. / Sub.
+}
+\end{minipage}
+\end{minipage}
+
+\vspace{0.3cm}
+
+\begin{minipage}[t]{6cm}
+{\bf Defect formation energy}\\
+\framebox{
+$E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.5cm]
+%Particle reservoir: Si \& SiC\\[0.2cm]
+{\bf Binding energy}\\
+\framebox{
+$
+E_{\text{b}}=
+E_{\text{f}}^{\text{comb}}-
+E_{\text{f}}^{1^{\text{st}}}-
+E_{\text{f}}^{2^{\text{nd}}}
+$
+}\\[0.1cm]
+\footnotesize
+$E_{\text{b}}<0$: energetically favorable configuration\\
+$E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
+\end{minipage}
+\begin{minipage}[t]{6cm}
+\vspace{1.4cm}
+{\bf Migration barrier}
+\footnotesize
+\begin{itemize}
+ \item Displace diffusing atom
+ \item Constrain relaxation of (diffusing) atoms
+ \item Record configurational energy
+\end{itemize}
+\begin{picture}(0,0)(-60,-33)
+\includegraphics[width=4.5cm]{crt_mod.eps}
+\end{picture}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+\footnotesize
+
+\headphd
+{\large\bf
+ C interstitial point defects in silicon\\
+}
+
+\begin{tabular}{l c c c c c c r}
+\hline
+ $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
+ {\color{black} \cs{} \& \si}\\
+\hline
+ \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
+ Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
+\hline
+\end{tabular}\\[0.1cm]
+
+\framebox{
+\begin{minipage}{2.8cm}
+\underline{Hexagonal} \hspace{2pt}
+\href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
+$E_{\text{f}}^*=9.05\text{ eV}$\\
+\includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{2.8cm}
+\underline{\hkl<1 0 0>}\\
+$E_{\text{f}}=3.88\text{ eV}$\\
+\includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
+\end{minipage}
+}
+\begin{minipage}{1.4cm}
+\hfill
+\end{minipage}
+\begin{minipage}{3.0cm}
+\begin{flushright}
+\underline{Tetrahedral}\\
+\includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
+\end{flushright}
+\end{minipage}
+
+\framebox{
+\begin{minipage}{2.8cm}
+\underline{Bond-centered}\\
+$E_{\text{f}}^*=5.59\text{ eV}$\\
+\includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{2.8cm}
+\underline{\hkl<1 1 0> dumbbell}\\
+$E_{\text{f}}=5.18\text{ eV}$\\
+\includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
+\end{minipage}
+}
+\begin{minipage}{1.4cm}
+\hfill
+\end{minipage}
+\begin{minipage}{3.0cm}
+\begin{flushright}
+\underline{Substitutional}\\
+\includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
+\end{flushright}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf\boldmath
+ C interstitial migration
+}
+
+\scriptsize
+
+\vspace{0.3cm}
+
+\begin{minipage}{6.8cm}
+{\bf\underline{First-principles}} $\quad$ \hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]\\
+\begin{minipage}{2.0cm}
+\includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
+\end{minipage}
+\begin{minipage}{0.2cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{2.0cm}
+\includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
+\end{minipage}
+\begin{minipage}{0.2cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{2.0cm}
+\includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
+\end{minipage}\\[0.1cm]
+$\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
+$\Rightarrow$ {\color{blue}Migration mechanism identified!}\\
+Note: Change in orientation
+\end{minipage}
+\begin{minipage}{5.4cm}
+\includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
+\end{minipage}\\[0.4cm]
+\begin{minipage}{6.8cm}
+{\bf\underline{Empirical potential}} $\quad$
+\hkl[0 0 -1] $\rightarrow$ \hkl[1 1 0] $\rightarrow$ \hkl[0 -1 0]\\
+\begin{itemize}
+ \item Transition involving \hkl[1 1 0] DB\\
+       (instability of BC configuration)
+ \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
+ \item 2.4 -- 3.4 times higher than ab initio result
+ \item After all: Change of the DB orientation
+\end{itemize}
+\vspace{0.1cm}
+\begin{center}
+{\color{red}Drastically overestimated diffusion barrier}
+\end{center}
+\end{minipage}
+\begin{minipage}{5.4cm}
+\includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf\boldmath
+ Defect combinations --- ab inito
+}
+
+\footnotesize
+
+\vspace{0.3cm}
+
+\begin{minipage}{9cm}
+{\bf
+ Summary of combinations}\\[0.1cm]
+{\scriptsize
+\begin{tabular}{l c c c c c c}
+\hline
+ $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
+ \hline
+ \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
+ \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
+ \hkl[0 -1 0] & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl[0 1 0] & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hkl[-1 0 0] & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
+ \hkl[1 0 0] & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
+ \hline
+ C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
+ Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
+\hline
+\end{tabular}
+}
+\end{minipage}
+\begin{minipage}{3cm}
+\includegraphics[width=3.5cm]{comb_pos.eps}
+\end{minipage}
+
+\vspace{0.5cm}
+
+{\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
+\begin{minipage}{6.1cm}
+\begin{itemize}
+ \item \ci{} agglomeration energetically favorable
+ \item Reduction of strain
+ \item Capture radius exceeding \unit[1]{nm}
+ \item Disappearance of attractive forces\\
+       between two lowest separations.
+\end{itemize}
+\begin{center}
+{\color{blue}\ci{} agglomeration / no C clustering}
+\end{center}
+\end{minipage}
+
+\begin{picture}(0,0)(-180,-40)
+\begin{minipage}{6.0cm}
+\scriptsize\centering
+Interaction along \hkl[1 1 0]\\
+\includegraphics[width=6.2cm]{db_along_110_cc.ps}
+\end{minipage}
+\end{picture}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Defect combinations of C-Si dimers and vacancies
+}
+\footnotesize
+
+\vspace{0.2cm}
+
+\begin{minipage}[b]{2.6cm}
+\begin{flushleft}
+\underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
+\includegraphics[width=2.5cm]{00-1dc/0-59.eps}
+\end{flushleft}
+\end{minipage}
+\begin{minipage}[b]{7cm}
+\hfill
+\end{minipage}
+\begin{minipage}[b]{2.6cm}
+\begin{flushright}
+\underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
+\includegraphics[width=2.5cm]{00-1dc/3-14.eps}
+\end{flushright}
+\end{minipage}\\[0.2cm]
+
+\begin{minipage}{6.5cm}
+\includegraphics[width=6.0cm]{059-539.ps}
+\end{minipage}
+\begin{minipage}{5.7cm}
+\includegraphics[width=6.0cm]{314-539.ps}
+\end{minipage}
+
+\begin{pspicture}(0,0)(0,0)
+\psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)
+
+\rput(6.3,7.0){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=gray]{
+\begin{minipage}{6.5cm}
+\begin{center}
+IBS: Impinging C creates V \& far away \si\\[0.3cm]
+Low migration barrier towards C$_{\text{sub}}$\\
+\&\\
+High barrier for reverse process\\[0.3cm]
+{\color{blue}
+High probability of stable C$_{\text{sub}}$ configuration
+}
+\end{center}
+\end{minipage}
+}}}
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Combinations of substitutional C and Si self-interstitials
+}
+
+\scriptsize
+
+\vspace{0.3cm}
+
+\begin{minipage}{6.2cm}
+\begin{center}
+{\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
+\begin{itemize}
+ \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
+ \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
+ \item Interaction drops quickly to zero\\
+       $\rightarrow$ low capture radius
+\end{itemize}
+\end{center}
+\end{minipage}
+\begin{minipage}{0.2cm}
+\hfill
+\end{minipage}
+\begin{minipage}{6.0cm}
+\begin{center}
+{\bf Transition from the ground state}
+\begin{itemize}
+ \item Low transition barrier
+ \item Barrier smaller than \ci{} migration barrier
+ \item Low \si{} migration barrier (\unit[0.67]{eV})\\
+       $\rightarrow$ Separation of \cs{} \& \si{} most probable
+\end{itemize}
+\end{center}
+\end{minipage}\\[0.3cm]
+
+\begin{minipage}{6.0cm}
+\includegraphics[width=6.0cm]{c_sub_si110.ps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+\hfill
+\end{minipage}
+\begin{minipage}{6.0cm}
+\begin{flushright}
+\includegraphics[width=6.0cm]{162-097.ps}
+\end{flushright}
+\end{minipage}
+
+\begin{pspicture}(0,0)(0,0)
+\psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
+\rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
+\begin{minipage}{8cm}
+\begin{center}
+\vspace{0.1cm}
+{\color{black}
+\cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
+IBS --- process far from equilibrium\\
+}
+\end{center}
+\end{minipage}
+}}}
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Combinations of substitutional C and Si self-interstitials
+}
+
+\scriptsize
+
+\vspace{0.3cm}
+
+\begin{minipage}{6.2cm}
+\begin{center}
+{\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
+\begin{itemize}
+ \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
+ \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
+ \item Interaction drops quickly to zero\\
+       $\rightarrow$ low capture radius
+\end{itemize}
+\end{center}
+\end{minipage}
+\begin{minipage}{0.2cm}
+\hfill
+\end{minipage}
+\begin{minipage}{6.0cm}
+\begin{center}
+{\bf Transition from the ground state}
+\begin{itemize}
+ \item Low transition barrier
+ \item Barrier smaller than \ci{} migration barrier
+ \item Low \si{} migration barrier (\unit[0.67]{eV})\\
+       $\rightarrow$ Separation of \cs{} \& \si{} most probable
+\end{itemize}
+\end{center}
+\end{minipage}\\[0.3cm]
+
+\begin{minipage}{6.0cm}
+\includegraphics[width=6.0cm]{c_sub_si110.ps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+\hfill
+\end{minipage}
+\begin{minipage}{6.0cm}
+\begin{flushright}
+\includegraphics[width=6.0cm]{162-097.ps}
+\end{flushright}
+\end{minipage}
+
+\begin{pspicture}(0,0)(0,0)
+\psline[linewidth=0.05cm,linecolor=gray](6.5,0)(6.5,7.5)
+\rput(6.5,-0.7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.05cm,linecolor=blue]{
+\begin{minipage}{8cm}
+\begin{center}
+\vspace{0.1cm}
+{\color{black}
+\cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
+IBS --- process far from equilibrium\\
+}
+\end{center}
+\end{minipage}
+}}}
+\end{pspicture}
+
+% md support
+\begin{pspicture}(0,0)(0,0)
+\rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
+\begin{minipage}{14cm}
+\hfill
+\vspace{14cm}
+\end{minipage}
+}}
+\rput(6.5,4.3){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
+\begin{minipage}{11cm}
+\begin{center}
+\vspace{0.2cm}
+\scriptsize
+Ab initio MD at \degc{900}\\[0.4cm]
+\begin{minipage}{5.4cm}
+\centering
+\includegraphics[width=4.3cm]{md01_bonds.eps}\\
+$t=\unit[2230]{fs}$
+\end{minipage}
+\begin{minipage}{5.4cm}
+\centering
+\includegraphics[width=4.3cm]{md02_bonds.eps}\\
+$t=\unit[2900]{fs}$
+\end{minipage}\\[0.5cm]
+{\color{blue}
+Contribution of entropy to structural formation\\[0.1cm]
+}
+\end{center}
+\end{minipage}
+}}}
+\end{pspicture}
+
+\end{slide}
+
+\fi
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Silicon carbide precipitation simulations
+}
+
+\small
+
+\vspace{0.2cm}
+
+{\bf Procedure}
+
+{\scriptsize
+ \begin{pspicture}(0,0)(12,6.5)
+  % nodes
+  \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+   \parbox{7cm}{
+   \begin{itemize}
+    \item Create c-Si volume
+    \item Periodc boundary conditions
+    \item Set requested $T$ and $p=0\text{ bar}$
+    \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
+   \end{itemize}
+  }}}}
+  \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
+   \parbox{7cm}{
+   Insertion of C atoms at constant T
+   \begin{itemize}
+    \item total simulation volume {\pnode{in1}}
+    \item volume of minimal SiC precipitate size {\pnode{in2}}
+    %\item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
+    \item volume containing Si atoms to form a minimal {\pnode{in3}}\\
+          precipitate
+   \end{itemize} 
+  }}}}
+  \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
+   \parbox{7.0cm}{
+   Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
+  }}}}
+  \ncline[]{->}{init}{insert}
+  \ncline[]{->}{insert}{cool}
+  \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
+  \rput(7.6,6){\footnotesize $V_1$}
+  \psframe[fillstyle=solid,fillcolor=lightgray](8.7,2)(11.6,5)
+  \rput(8.9,4.85){\tiny $V_2$}
+  \psframe[fillstyle=solid,fillcolor=gray](8.95,2.25)(11.35,4.75)
+  \rput(9.25,4.45){\footnotesize $V_3$}
+  \rput(7.9,3.2){\pnode{ins1}}
+  \rput(8.92,2.8){\pnode{ins2}}
+  \rput(10.8,2.4){\pnode{ins3}}
+  \ncline[]{->}{in1}{ins1}
+  \ncline[]{->}{in2}{ins2}
+  \ncline[]{->}{in3}{ins3}
+ \end{pspicture}
+}
+
+\vspace{-0.5cm}
+
+{\bf Note}
+
+\footnotesize
+
+\begin{minipage}{5.7cm}
+\begin{itemize}
+ \item Amount of C atoms: 6000\\
+       ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
+ \item Simulation volume: $31^3$ Si unit cells\\
+       (238328 Si atoms)
+\end{itemize}
+\end{minipage}
+\begin{minipage}{0.3cm}
+\hfill
+\end{minipage}
+\framebox{
+\begin{minipage}{6.0cm}
+Restricted to classical potential caclulations\\
+$\rightarrow$ Low C diffusion / overestimated barrier\\
+$\rightarrow$ Consider $V_2$ and $V_3$
+%\begin{itemize}
+% \item $V_2$ and $V_3$ considered due to expected low C diffusion
+%\end{itemize}
+\end{minipage}
+}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Silicon carbide precipitation simulations
+}
+
+\small
+
+\begin{minipage}{6.3cm}
+\hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
+\hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
+\hfill
+\end{minipage} 
+\begin{minipage}{6.1cm}
+\scriptsize
+\underline{Temperature as used in IBS (\degc{450})}\\[0.2cm]
+\ci{} \hkl<1 0 0> dumbbell dominated structure\\
+\begin{pspicture}(0,0)(6.0,1.0)
+\rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
+\begin{minipage}{5cm}
+\vspace{0.1cm}
+\centering
+{\color{blue}Formation of \ci{} DBs}\\
+{\color{red}No agllomeration / precipitation}
+\end{minipage}
+}}
+\end{pspicture}\\[0.1cm]
+Limitations:
+\begin{itemize}
+ \item Time scale problem of MD\\
+       $\Rightarrow$ slow phase space propagation
+ \item Short range potential\\
+       $\Rightarrow$ overestimated diffusion barrier
+\end{itemize}
+\vspace{0.6cm}
+\underline{Increased temperatures}\\[0.2cm]
+\cs{} dominated structure\\
+\begin{pspicture}(0,0)(6.0,1.0)
+\rput(2.75,0.4){\psframebox[linewidth=0.05cm,linecolor=black]{
+\begin{minipage}{5cm}
+\vspace{0.1cm}
+\centering
+Si-{\color{blue}C$_{\text{sub}}$}-Si along \hkl<1 1 0>\\
+{\color{blue}\cs}-Si-{\color{blue}\cs} \& nearby \si
+\end{minipage}
+}}
+\end{pspicture}\\[0.1cm]
+Conclusions:
+\begin{itemize}
+ \item Stretched coherent SiC structures\\
+       $\Rightarrow$ \cs{} involved in precipitation mechanism
+ \item High T $\leftrightarrow$ non-equilibrium IBS conditions
+\end{itemize}
+\vspace{0.3cm}
+
+\end{minipage} 
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Summary and Conclusions
+}
+
+Summary
+
+\begin{itemize}
+ \item First-principles investigation of defect combinations
+       and mobilities in Si
+ \item Empirical potential MD simulations on SiC prcipitation in Si
+\end{itemize}
+
+
+% conclusions
+\rput(6.5,-4.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
+\begin{minipage}{9cm}
+\vspace{0.2cm}
+\small
+\begin{center}
+{\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
+{\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
+\end{center}
+\begin{itemize}
+\item Stretched coherent SiC structures directly observed\\
+\psframebox[linecolor=blue,linewidth=0.05cm]{
+\begin{minipage}{7cm}
+\centering
+\cs{} involved in the precipitation mechanism\\
+\end{minipage}
+}
+\item Emission of \si{} serves several needs:
+      \begin{itemize}
+       \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
+       \item Building block for surrounding Si host \& further SiC
+       \item Strain compensation \ldots\\
+             \ldots Si/SiC interface\\
+             \ldots within stretched coherent SiC structure
+      \end{itemize}
+\item Explains annealing behavior of high/low T C implantations
+      \begin{itemize}
+       \item Low T: highly mobile {\color{red}\ci}
+       \item High T: stable configurations of {\color{blue}\cs}
+      \end{itemize}
+\psframebox[linecolor=blue,linewidth=0.05cm]{
+\begin{minipage}{7cm}
+\centering
+High T $\leftrightarrow$ IBS conditions far from equilibrium\\
+\end{minipage}
+}
+\end{itemize}
+\end{minipage}
+\vspace{0.2cm}
+}}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf
+ Acknowledgements
+}
+
+ \vspace{0.1cm}
+
+ \small
+
+ Thanks to \ldots
+
+\begin{minipage}[t]{6cm}
+ \underline{Augsburg}
+ \begin{itemize}
+  \item Prof. B. Stritzker
+ \end{itemize}
+ 
+ \underline{Helsinki}
+ \begin{itemize}
+  \item Prof. K. Nordlund
+ \end{itemize}
+
+ \underline{Munich}
+ \begin{itemize}
+  \item Bayerische Forschungsstiftung
+ \end{itemize}
+\end{minipage}
+\begin{minipage}[t]{6cm}
+ \underline{Paderborn}
+ \begin{itemize}
+  \item Prof. J. Lindner
+  \item Prof. G. Schmidt
+  \item Dr. E. Rauls
+ \end{itemize}
+\end{minipage}
+
+\vspace{2.5cm}
+
+\begin{center}
+\framebox{
+\LARGE\bf Thank you for your attention!
+}
+\end{center}
+
+\end{slide}
+
+
+
+
+
+
+
+\ifnum1=0
+
+\begin{slide}
+
+\headphd
+ {\large\bf
+  Polytypes of SiC\\[0.6cm]
+ }
+
+\vspace{0.6cm}
+
+\includegraphics[width=3.8cm]{cubic_hex.eps}\\
+\begin{minipage}{1.9cm}
+{\tiny cubic (twist)}
+\end{minipage}
+\begin{minipage}{2.9cm}
+{\tiny hexagonal (no twist)}
+\end{minipage}
+
+\begin{picture}(0,0)(-150,0)
+ \includegraphics[width=7cm]{polytypes.eps}
+\end{picture}
+
+\vspace{0.6cm}
+
+\footnotesize
+
+\begin{tabular}{l c c c c c c}
+\hline
+ & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
+\hline
+Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
+Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
+Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
+Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
+Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
+Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
+Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
+\hline
+\end{tabular}
+
+\begin{pspicture}(0,0)(0,0)
+\psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
+\end{pspicture}
+\begin{pspicture}(0,0)(0,0)
+\psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
+\end{pspicture}
+\begin{pspicture}(0,0)(0,0)
+\psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+\footnotesize
+
+\headphd
+{\large\bf
+ Si self-interstitial point defects in silicon\\[0.1cm]
+}
+
+\begin{center}
+\begin{tabular}{l c c c c c}
+\hline
+ $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
+\hline
+ \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
+ Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
+\hline
+\end{tabular}\\[0.4cm]
+\end{center}
+
+\begin{minipage}{3cm}
+\begin{center}
+\underline{Vacancy}\\
+\includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{3cm}
+\begin{center}
+\underline{\hkl<1 1 0> DB}\\
+\includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{3cm}
+\begin{center}
+\underline{\hkl<1 0 0> DB}\\
+\includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{3cm}
+\begin{center}
+\underline{Tetrahedral}\\
+\includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
+\end{center}
+\end{minipage}\\
+
+\underline{Hexagonal} \hspace{2pt}
+\href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
+\framebox{
+\begin{minipage}{2.7cm}
+$E_{\text{f}}^*=4.48\text{ eV}$\\
+\includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
+\end{minipage}
+\begin{minipage}{0.4cm}
+\begin{center}
+$\Rightarrow$
+\end{center}
+\end{minipage}
+\begin{minipage}{2.7cm}
+$E_{\text{f}}=3.96\text{ eV}$\\
+\includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
+\end{minipage}
+}
+\begin{minipage}{5.5cm}
+\begin{center}
+{\tiny nearly T $\rightarrow$ T}\\
+\end{center}
+\includegraphics[width=6.0cm]{nhex_tet.ps}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+\headphd
+{\large\bf\boldmath
+ C-Si dimer \& bond-centered interstitial configuration
+}
+
+\footnotesize
+
+\vspace{0.1cm}
+
+\begin{minipage}[t]{4.1cm}
+{\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
+\begin{minipage}{2.0cm}
+\begin{center}
+\underline{Erhart/Albe}
+\includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
+\end{center}
+\end{minipage}
+\begin{minipage}{2.0cm}
+\begin{center}
+\underline{\textsc{vasp}}
+\includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
+\end{center}
+\end{minipage}\\[0.2cm]
+Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
+$\Rightarrow$ $sp$ hybridization\\[0.1cm]
+Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
+$\Rightarrow$ $sp^2$ hybridization
+\begin{center}
+\includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
+{\tiny Charge density isosurface}
+\end{center}
+\end{minipage}
+\begin{minipage}{0.2cm}
+\hfill
+\end{minipage}
+\begin{minipage}[t]{8.1cm}
+\begin{flushright}
+{\bf Bond-centered interstitial}\\[0.1cm]
+\begin{minipage}{4.4cm}
+%\scriptsize
+\begin{itemize}
+ \item Linear Si-C-Si bond
+ \item Si: one C \& 3 Si neighbours
+ \item Spin polarized calculations
+ \item No saddle point!\\
+       Real local minimum!
+\end{itemize}
+\end{minipage}
+\begin{minipage}{2.7cm}
+%\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
+\vspace{0.2cm}
+\includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
+\end{minipage}
+
+\framebox{
+ \tiny
+ \begin{minipage}[t]{6.5cm}
+  \begin{minipage}[t]{1.2cm}
+  {\color{red}Si}\\
+  {\tiny sp$^3$}\\[0.8cm]
+  \underline{${\color{black}\uparrow}$}
+  \underline{${\color{black}\uparrow}$}
+  \underline{${\color{black}\uparrow}$}
+  \underline{${\color{red}\uparrow}$}\\
+  sp$^3$
+  \end{minipage}
+  \begin{minipage}[t]{1.4cm}
+  \begin{center}
+  {\color{red}M}{\color{blue}O}\\[0.8cm]
+  \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}\\[0.8cm]
+  \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{${\color{black}\uparrow}$}
+  \underline{${\color{black}\uparrow}$}
+  \underline{${\color{black}\uparrow}$}\\
+  sp$^3$
+  \end{flushright}
+  \end{minipage}
+ \end{minipage}
+}\\[0.4cm]
+
+%\framebox{
+\begin{minipage}{3.0cm}
+%\scriptsize
+\underline{Charge density}\\
+{\color{gray}$\bullet$} Spin up\\
+{\color{green}$\bullet$} Spin down\\
+{\color{blue}$\bullet$} Resulting spin up\\
+{\color{yellow}$\bullet$} Si atoms\\
+{\color{red}$\bullet$} C atom
+\end{minipage}
+\begin{minipage}{3.6cm}
+\includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
+\end{minipage}
+%}
+
+\end{flushright}
+
+\end{minipage}
+\begin{pspicture}(0,0)(0,0)
+\psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
+\end{pspicture}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  Increased temperature simulations at high C concentration
+ }
+
+\footnotesize
+
+\begin{minipage}{6.0cm}
+\includegraphics[width=6.4cm]{12_pc_thesis.ps}
+\end{minipage}
+\begin{minipage}{6.0cm}
+\includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
+\end{minipage}
+
+\vspace{0.1cm}
+
+\scriptsize
+
+\framebox{
+\begin{minipage}[t]{5.5cm}
+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}{0.1cm}
+\hfill
+\end{minipage}
+\framebox{
+\begin{minipage}[t]{5.9cm}
+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}
+}
+
+\begin{itemize}
+\item Decreasing cut-off artifact
+\item {\color{red}Amorphous} SiC-like phase remains
+\item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
+\item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
+\end{itemize}
+
+\begin{center}
+{\color{blue}
+\framebox{
+{\color{black}
+High C \& small $V$ \& short $t$
+$\Rightarrow$
+}
+\begin{minipage}{4cm}
+\begin{center}
+Slow structural evolution due to strong C-C bonds
+\end{center}
+\end{minipage}
+{\color{black}
+$\Leftarrow$
+High C \& low T implants
+}
+}
+}
+\end{center}
+
+\end{slide}
+
+
+
+\begin{slide}
+
+ {\large\bf
+  Valuation of a practicable temperature limit
+ }
+
+ \small
+
+\vspace{0.1cm}
+
+\begin{center}
+\framebox{
+{\color{blue}
+Recrystallization is a hard task!
+$\Rightarrow$ Avoid melting!
+}
+}
+\end{center}
+ 
+\vspace{0.1cm}
+
+\footnotesize
+
+\begin{minipage}{6.4cm}
+\includegraphics[width=6.4cm]{fe_and_t.ps}
+\end{minipage}
+\begin{minipage}{5.7cm}
+\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}{2cm}
+\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
+  Long time scale simulations at maximum temperature
+ }
+
+\small
+
+\vspace{0.1cm}
+ 
+\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]{4.3cm}
+\begin{center}
+\underline{Low C concentration, Si-C}
+\includegraphics[width=4.3cm]{c_in_si_95_v1_si-c.ps}\\
+Sharper peaks!
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{4.3cm}
+\begin{center}
+\underline{Low C concentration, C-C}
+\includegraphics[width=4.3cm]{c_in_si_95_v1_c-c.ps}\\
+Sharper peaks!\\
+No C agglomeration!
+\end{center}
+\end{minipage}
+\begin{minipage}[t]{3.4cm}
+\begin{center}
+\underline{High C concentration}
+\includegraphics[width=4.3cm]{c_in_si_95_v2.ps}\\
+No significant changes\\
+iC-Si-Si $\uparrow$\\
+C-Si-C $\downarrow$
+\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
+  Investigation of a silicon carbide precipitate in silicon
+ }
+
+ \scriptsize
+
+\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}{0.1cm}
+\hfill
+\end{minipage}
+\begin{minipage}{6.3cm}
+\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}{6.0cm}
+\includegraphics[width=6cm]{pc_0.ps}
+\end{minipage}
+\begin{minipage}{6.1cm}
+\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}
+
+\begin{slide}
+
+ {\large\bf
+  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}{5cm}
+\includegraphics[width=5.5cm]{fe_and_t_sic.ps}
+\end{minipage}
+
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{sic_prec/melt_01.eps}
+\end{minipage}
+\begin{minipage}{0.2cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{4cm}
+\includegraphics[width=4cm]{sic_prec/melt_02.eps}
+\end{minipage}
+\begin{minipage}{0.2cm}
+$\rightarrow$
+\end{minipage}
+\begin{minipage}{3.7cm}
+\includegraphics[width=4cm]{sic_prec/melt_03.eps}
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  DFT parameters
+ }
+
+\scriptsize
+
+\vspace{0.1cm}
+
+Equilibrium lattice constants and cohesive energies
+
+\begin{tabular}{l r c c c c c}
+\hline
+\hline
+ & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\
+\hline
+Si (dia) & $a$ [\AA] & 5.389 & 5.455 & - & - & 5.429 \\
+         & $\Delta_a$ [\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\
+       & $E_{\text{coh}}$ [eV] & -5.277 & -4.591 & - & - & -4.63 \\
+       & $\Delta_E$ [\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\
+\hline
+C (dia) & $a$ [\AA] & 3.527 & 3.567 & - & - & 3.567 \\
+         & $\Delta_a$ [\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\
+       & $E_{\text{coh}}$ [eV] & -8.812 & -7.703 & - & - & -7.374 \\
+       & $\Delta_E$ [\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\
+\hline
+3C-SiC & $a$ [\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\
+         & $\Delta_a$ [\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\
+       & $E_{\text{coh}}$ [eV] & -7.318 & -6.426 &  -7.371 & -6.491 & -6.340 \\
+       & $\Delta_E$ [\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\
+\hline
+\hline
+\end{tabular}
+
+\vspace{0.3cm}
+
+\begin{minipage}{7cm}
+\begin{center}
+\begin{tabular}{l c c c}
+\hline
+\hline
+ & Si (dia) & C (dia) & 3C-SiC \\
+\hline
+$a$ [\AA] & 5.458 & 3.562 & 4.365 \\
+$\Delta_a$ [\%] & 0.5 & 0.1 & 0.1 \\
+\hline
+$E_{\text{coh}}$ [eV] & -4.577 & -7.695 & -6.419 \\
+$\Delta_E$ [\%] & 1.1 & 4.4 & 1.2 \\
+\hline
+\hline
+\end{tabular}
+\end{center}
+\end{minipage}
+\begin{minipage}{5cm}
+$\leftarrow$ entire parameter set
+\end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+  DFT parameters\\
+ }
+
+\footnotesize
+
+\begin{minipage}{6cm}
+\begin{center}
+\includegraphics[width=6cm]{sic_32pc_gamma_cutoff_lc.ps}
+\end{center}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{center}
+Lattice constants with respect to the PW cut-off energy
+\end{center}
+\end{minipage}
+
+\begin{minipage}{6cm}
+\begin{center}
+\includegraphics[width=6cm]{si_self_int_thesis.ps}
+\end{center}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{center}
+Defect formation energy with respect to the size of the supercell\\[0.1cm]
+\end{center}
+
+\end{minipage}
+
+\end{slide}
+
+\fi
+
+\end{document}
+