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\begin{document}

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\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}

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  \begin{center}
   {\footnotesize
    $^1$ Experimentalphysik IV, Institut f"ur Physik,\\
         Universit"at Augsburg, Universit"atsstr. 1,\\
         D-86135 Augsburg, Germany
   }
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  \begin{center}
  {\footnotesize
   $^2$ Accelerator Laboratory, Department of Physical Sciences,\\
   University of Helsinki, Pietari Kalmink. 2,\\
   00014 Helsinki, Finland
  }
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\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 Results gained by simulation
        \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
 }

 \small
 \vspace{6pt}

 Supposed mechanism of the conversion 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}

 \begin{center}
 \[5a_{SiC}=4a_{Si} \quad \Rightarrow \quad
   \frac{n_{SiC}}{n_{Si}}=\frac{\frac{4}{a_{SiC}^3}}{\frac{8}{a_{Si}^3}}=
                          \frac{5^3}{2\cdot4^3}=97,66\%
 \]
 \end{center}

 Experimentally observed minimal diameter of precipitation: 4 - 5 nm

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 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-dimemnsional phase space
  \item Observables obtained by time average
 \end{itemize}

 \vspace{4pt}

 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}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \vspace{16pt}

 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: $2\, ps$
 \end{itemize}

 \begin{picture}(0,0)(-210,-65)
  \includegraphics[width=6cm]{unit_cell.eps}
 \end{picture}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \small

 SiC precipitation experiments:
 \begin{itemize}
  \item Initial configuration: $31\times31\times31$ unit cells Si
  \item Periodic boundary conditions
  \item $T=450\, ^{\circ}C$
  \item Steady state time: $600\, fs$
  \item C insertion steps:
        \begin{itemize}
	 \item If $T=450\pm 1\, ^{\circ}C$:\\
	       Insertion of 10 atoms at random positions within $V_{ins}$
	 \item Otherwise: Annealing for another $100\, fs$
	\end{itemize}
  \item Annealing: ($T_a: 450\rightarrow 20 \, ^{\circ}C$)
        \begin{itemize}
	 \item If $T=T_a$: Decrease $T_a$ by $1\, ^{\circ}C$
	 \item Otherwise: Annealing for another $50\, fs$
	\end{itemize}
 \end{itemize}

 Szenarios:
 \begin{enumerate}
  \item $V_{ins}$: total simulation volume $V$
  \item $V_{ins}$: $12\times12\times12$ SiC unit cells
                   ($\sim$ volume of minimal SiC precipitation)
  \item $V_{ins}$: $9\times9\times9$ SiC unit cells
                   ($\sim$ volume of necessary amount of Si)
 \end{enumerate}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

\end{slide}

\begin{slide}

 {\large\bf
  Results
 }

\end{slide}

\begin{slide}

 {\large\bf
  Conclusion / Outlook
 }

\end{slide}


\end{document}

