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\begin{document}

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% topic

\begin{slide}
\begin{center}

 \vspace{16pt}

 {\LARGE\bf
  Molecular dynamics simulation study\\
  of the silicon carbide precipitation process
 }

 \vspace{24pt}

 \textsc{\small \underline{F. Zirkelbach}$^1$, J. K. N. Lindner$^1$,
         K. Nordlund$^2$, B. Stritzker$^1$}\\

 \vspace{32pt}

 \begin{minipage}{2.0cm}
  \begin{center}
  \includegraphics[height=1.6cm]{uni-logo.eps}
  \end{center}
 \end{minipage}
 \begin{minipage}{8.0cm}
  \begin{center}
   {\footnotesize
    $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
         Universit"at Augsburg, Universit"atsstr. 1,\\
         D-86135 Augsburg, Germany
   }
  \end{center}
 \end{minipage}
 \begin{minipage}{2.3cm}
  \begin{center}
  \includegraphics[height=1.5cm]{Lehrstuhl-Logo.eps}
  \end{center}
 \end{minipage}

 \vspace{16pt}

 \begin{minipage}{4.0cm}
  \begin{center}
  \includegraphics[height=1.6cm]{logo_eng.eps}
  \end{center}
 \end{minipage}
 \begin{minipage}{8.0cm}
  \begin{center}
  {\footnotesize
   $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
   University of Helsinki, Pietari Kalmink. 2,\\
   00014 Helsinki, Finland
  }
  \end{center}
 \end{minipage}
\end{center}
\end{slide}

% contents

\begin{slide}

 \begin{center}
 {\bf
  Molecular dynamics simulation study\\
  of the silicon carbide precipitation process
 }
 \end{center}

 \vspace{16pt}

 {\large\bf
  Outline
 }

 \vspace{16pt}

 \begin{itemize}
  \item Motivation / Introduction
  \item Molecular dynamics simulation details
        \begin{itemize}
	 \item Integrator, potential, ensemble control
	 \item Simulation sequence
	\end{itemize}
  \item Simulation results
        \begin{itemize}
         \item Interstitials in silicon
         \item SiC-precipitation experiments
	\end{itemize}
  \item Conclusion / Outlook
 \end{itemize}
\end{slide}

% start of contents

\begin{slide}

 {\large\bf
  Motivation / Introduction
 }

 \vspace{16pt}

 Reasons for investigating C in Si:

 \begin{itemize}
  \item 3C-SiC wide band gap semiconductor formation
  \item Strained Si (no precipitation wanted!)
 \end{itemize}

 \vspace{16pt}

 Si / 3C-SiC facts:

 \begin{minipage}{8cm}
 \begin{itemize}
  \item Unit cell:
        \begin{itemize}
         \item {\color{yellow}fcc} $+$
         \item {\color{gray}fcc shifted $1/4$ of volume diagonal}
	\end{itemize}
  \item Lattice constants: $4a_{Si}\approx5a_{SiC}$
  \item Silicon density: 
        \[
        \frac{n_{SiC}}{n_{Si}}=
	\frac{4/a_{SiC}^3}{8/a_{Si}^3}=
        \frac{5^3}{2\cdot4^3}={\color{cyan}97,66}\,\%
        \]
 \end{itemize}
 \end{minipage}
 \hspace{8pt}
 \begin{minipage}{4cm}
 \includegraphics[width=4cm]{sic_unit_cell.eps}
 \end{minipage}

\end{slide}

 \small
\begin{slide}

 {\large\bf
  Motivation / Introduction
 }

 \small
 \vspace{6pt}

 Supposed conversion mechanism of heavily carbon doped Si into SiC:

 \vspace{8pt}

 \begin{minipage}{3.8cm}
 \includegraphics[width=3.7cm]{sic_prec_seq_01.eps}
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 \includegraphics[width=3.7cm]{sic_prec_seq_02.eps}
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 \includegraphics[width=3.7cm]{sic_prec_seq_03.eps}
 \end{minipage}

 \vspace{8pt}

 \begin{minipage}{3.8cm}
 Formation of C-Si dumbbells on regular c-Si lattice sites
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 Agglomeration into large clusters (embryos)\\
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 Precipitation of 3C-SiC + Creation of interstitials\\
 \end{minipage}

 \vspace{12pt}

 Experimentally observed:
 \begin{itemize}
  \item Minimal diameter of precipitation: 4 - 5 nm
  \item (hkl)-planes identical for Si and SiC
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \vspace{12pt}

 MD basics:
 \begin{itemize}
  \item Microscopic description of N particle system
  \item Analytical interaction potential
  \item Hamilton's equations of motion as propagation rule\\
        in 6N-dimensional phase space
  \item Observables obtained by time average
 \end{itemize}

 \vspace{12pt}

 Application details:
 \begin{itemize}
  \item Integrator: Velocity Verlet, timestep: $1\, fs$
  \item Ensemble control: NVT, Berendsen thermostat, $\tau=100.0$
  \item Potential: Tersoff-like bond order potential\\
        \[
	E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
	\pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
	\]
	\begin{center}
        {\scriptsize P. Erhart und K. Albe. Phys. Rev. B 71 (2005) 035211}
	\end{center}
 \end{itemize}

 \begin{picture}(0,0)(-240,-70)
  \includegraphics[width=5cm]{tersoff_angle.eps} 
 \end{picture}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \vspace{20pt}

 Interstitial experiments:

 \vspace{12pt}

 \begin{itemize}
  \item Initial configuration: $9\times9\times9$ unit cells Si
  \item Periodic boundary conditions
  \item $T=0 \, K$
  \item Insertion of Si / C atom at
        \begin{itemize}
         \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
         \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
         \item $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
	       $\rightarrow$ {\color{yellow}110 dumbbell}
	 \item random positions (critical distance check)
	\end{itemize}
  \item Relaxation time: $2\, ps$
  \item Optional heating-up 
 \end{itemize}

 \begin{picture}(0,0)(-210,-45)
  \includegraphics[width=6cm]{unit_cell.eps}
 \end{picture}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \small

 SiC precipitation experiments:

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   \begin{itemize}
    \item Initial configuration: $31\times31\times31$ unit cells Si
    \item Periodic boundary conditions
    \item $T=450\, ^{\circ}C$
    \item Equilibration of $E_{kin}$ and $E_{pot}$ for $600\, fs$
   \end{itemize}
  }}}}
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                             $T=450\pm 1\, ^{\circ}C$}}}
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                              \parbox{3cm}{
                              Insertion of 10 atoms\\
                              at random positions}}}}
  \rput(2,3.5){\rnode{adj1}{\psframebox[fillstyle=solid,fillcolor=red]{
                            \parbox{3.5cm}{
                            Adjusting temperature\\
                            for another $100\, fs$}}}}
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                          $N_{atoms}=6000$}}}
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                           $T=T_{set}$}}}
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  \rput(2,1){\rnode{adj2}{\psframebox[fillstyle=solid,fillcolor=cyan]{
                          \parbox{3.5cm}{
                          Adjusting temperature\\
                          for another $50\, fs$}}}}
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                         $T_{set}=20\, ^{\circ}C$}}}
  \rput(10,0){\rnode{end}{\psframebox{End}}}
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\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

 Si self-interstitial experiments:

 {\footnotesize
 {\bf Note:}
 \begin{itemize}
  \item $r_{cutoff}^{Si-Si}=2.96>\frac{5.43}{2}$
  \item Bond length near $r_{cutoff} \Rightarrow$ small bond strength
 \end{itemize}
 }

 \vspace{8pt}

 \small

 \begin{minipage}[t]{4.0cm}
 \underline{Tetrahedral}
 \begin{itemize}
  \item $E_f=3.41\, eV$
  \item essentialy tetrahedral\\
        bonds
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \underline{110 dumbbell}
 \begin{itemize}
  \item $E_f=4.39\, eV$
  \item essentially 4 bonds
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \underline{Hexagonal}
 \begin{itemize}
  \item $E_f^{\star}\approx4.48\, eV$
  \item unstable!
 \end{itemize}
 \end{minipage}

 \vspace{8pt}

 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
 \begin{center}
 \href{../video/si_self_int_hexa.avi}{$\rhd$}
 \end{center}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

 \vspace{8pt}

 Si self-interstitial \underline{random insertion} experiments:

 \small

 \vspace{8pt}

 \begin{minipage}[t]{4.0cm}
 \begin{itemize}
  \item $E_f=3.97\, eV$
  \item 3 identical weak bonds
  \item displaced in volume\\ diagonal
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \begin{itemize}
  \item $E_f=3.75\, eV$
  \item 4 identical weak bonds
  \item displaced in plane\\ diagonal
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \begin{itemize}
  \item $E_f=3.56\, eV$
  \item single weak bond
  \item displaced along\\ $x$-direction
  \item closest to tetrahedral\\ configuration
 \end{itemize}
 \end{minipage}

 \vspace{8pt}

 \begin{minipage}{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
 \end{minipage}
 \begin{minipage}{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
 \end{minipage}
 \begin{minipage}{4.3cm}
 \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
 \end{minipage}

 \vspace{8pt}

 \begin{center}
  {\footnotesize
   {\bf Note:} Displacements relative to tetrahedral configuration
  }
 \end{center}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

 \vspace{8pt}

 Carbon interstitial experiments:

 \vspace{12pt}

 \small

 \begin{minipage}[t]{4.0cm}
 \underline{Tetrahedral}
 \begin{itemize}
  \item $E_f=2.67\, eV$
  \item tetrahedral bond
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \underline{110 dumbbell}
 \begin{itemize}
  \item $E_f=1.76\, eV$
  \item C forms 3 bonds
 \end{itemize}
 \end{minipage}
 \hspace{0.3cm}
 \begin{minipage}[t]{4.0cm}
 \underline{Hexagonal}
 \begin{itemize}
  \item $E_f^{\star}\approx5.6\, eV$
  \item unstable!
 \end{itemize}
 \end{minipage}

 \vspace{8pt}

 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
 \begin{center}
 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}
 \end{center}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

 \vspace{8pt}

 Carbon \underline{random insertion} experiments:

 \vspace{8pt}

 High probability for dumbbell configurations!\\
 Others:\\
 \includegraphics[width=3.0cm]{c_in_si_int_001db_0.eps}
 \includegraphics[width=3.0cm]{c_in_si_int_rand_162_0.eps}
 \includegraphics[width=3.0cm]{c_in_si_int_rand_239_0.eps}
 \includegraphics[width=3.0cm]{c_in_si_int_rand_341_0.eps}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

 SiC-precipitation experiments:

\end{slide}

\begin{slide}

 {\large\bf
  Conclusion / Outlook
 }

\end{slide}

\end{document}

