\documentclass[portrait,a0b,final]{a0poster}
\usepackage{epsf,psfig,pstricks,multicol,pst-grad,pst-node,color}
\usepackage{graphicx,amsmath,amssymb}
\graphicspath{{../img/}}
\usepackage[english,german]{babel}

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\selectlanguage{english}

\renewcommand\labelitemii{{\color{black}$\bullet$}}

\begin{document}

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\renewcommand{\columnfrac}{.31}

% potential
\newcommand{\pot}{\mathcal{V}}

% header
\vspace{-18cm}
\begin{header}
     \centerline{{\Huge \bfseries Molecular dynamics simulation
                                  of defect formation and precipitation}}
     \vspace*{0.5cm}
     \centerline{{\Huge \bfseries in heavily carbon doped silicon}}
     \vspace*{1cm}
     \centerline{\huge\textsc {\underline{F.~Zirkelbach}$^1$,
                               J.~K.~N.~Lindner$^1$,
                               K.~Nordlund$^2$, B.~Stritzker$^1$}}
     \vspace*{1cm}
     \begin{center}
       \begin{minipage}{.065\textwidth}
         \includegraphics[height=5.5cm]{uni-logo.eps}
       \end{minipage}
       \begin{minipage}{.57\textwidth}
         \centerline{\Large $^1$ Experimentalphysik IV, Institut f\"ur Physik,
                            Universit\"at Augsburg,}
         \centerline{\Large Universit\"atsstr. 1,  D-86135 Augsburg, Germany}
       \end{minipage}
       \begin{minipage} {.065\textwidth}
          \includegraphics[height=5cm]{Lehrstuhl-Logo.eps}
       \end{minipage}
     \end{center}
     \begin{center}
       \begin{minipage}{.20\textwidth}
         \includegraphics[height=5.5cm]{logo_eng.eps}
       \end{minipage}
       \begin{minipage}{.50\textwidth}
         \centerline{\Large $^2$ Accelerator Laboratory,
                            Department of Physical Sciences,
                            University of Helsinki,}
         \centerline{\Large Pietari Kalmink. 2, 00014 Helsinki, Finland}
       \end{minipage}
     \end{center}
\end{header}

\begin{poster}

%\vspace{-6cm}
\begin{pcolumn}
  \begin{pbox}
    \section*{Motivation}
    {\bf Reasons for understanding the 3C-SiC precipitation process}
    \begin{itemize}
      \item Significant technological progress
            in 3C-SiC wide band gap semiconductor thin film formation [1].
      \item New perspectives for processes relying upon prevention of
            precipitation, e.g. fabrication of strained pseudomorphic
	    $\text{Si}_{1-y}\text{C}_y$ heterostructures [2].
    \end{itemize}
    {\tiny
     [1] J. H. Edgar, J. Mater. Res. 7 (1992) 235.}\\
    {\tiny
     [2] J. W. Strane, S. R. Lee, H. J. Stein, S. T. Picraux,
         J. K. Watanabe, J. W. Mayer, J. Appl. Phys. 79 (1996) 637.}
  \end{pbox}
  \begin{pbox}
    \section*{Crystalline silicon and cubic silicon carbide}
    {\bf Lattice types and unit cells:}
    \begin{itemize}
      \item Crystalline silicon (c-Si) has diamond structure\\
            $\Rightarrow {\color{si-yellow}\bullet}$ and
            ${\color{gray}\bullet}$ are Si atoms
      \item Cubic silicon carbide (3C-SiC) has zincblende structure\\
            $\Rightarrow {\color{si-yellow}\bullet}$ are Si atoms,
            ${\color{gray}\bullet}$ are C atoms
    \end{itemize}
    \begin{minipage}{15cm}
    {\bf Lattice constants:}
    \[
    4a_{\text{c-Si}}\approx5a_{\text{3C-SiC}}
    \]
    {\bf Silicon density:}
    \[
    \frac{n_{\text{3C-SiC}}}{n_{\text{c-Si}}}=97,66\,\%
    \]
    \end{minipage}
    \begin{minipage}{10cm}
      \includegraphics[width=10cm]{sic_unit_cell.eps}
    \end{minipage}
  \end{pbox}
  \begin{pbox}
    \section*{Supposed Si to 3C-SiC conversion}
    {\bf Schematic of the conversion mechanism}\\\\
    \begin{minipage}{7.8cm}
    \includegraphics[width=7.7cm]{sic_prec_seq_01.eps}
    \end{minipage}
    \hspace{0.6cm}
    \begin{minipage}{7.8cm}
    \includegraphics[width=7.7cm]{sic_prec_seq_02.eps}
    \end{minipage}
    \hspace{0.6cm}
    \begin{minipage}{7.8cm}
    \includegraphics[width=7.7cm]{sic_prec_seq_03.eps}
    \end{minipage}
    \vspace{1cm}
    \begin{enumerate}
      \item Formation of C-Si dumbbells on regular c-Si lattice sites
      \item Agglomeration into large clusters (embryos)
      \item Precipitation of 3C-SiC + Creation of interstitials
    \end{enumerate}
    \vspace{1cm}
    {\bf Experimental observations} [3]
    \begin{itemize}
      \item Minimal diameter of precipitation: 2 - 4 nm
      \item Equal orientation of c-Si and 3C-SiC (hkl)-planes
    \end{itemize}
    {\tiny
     [3] J. K. N. Lindner, Appl. Phys. A 77 (2003) 27.
    }
  \end{pbox}
  \begin{pbox}
    \section*{Simulation details}
    {\bf MD basics:}
    \begin{itemize}
      \item Microscopic description of N particles
      \item Analytical interaction potential
      \item Propagation rule in 6N-dim. phase space:
            Hamilton's equations of motion
      \item Observables obtained by time or ensemble averages
    \end{itemize}
    {\bf Application details:}\\[0.5cm]
    \begin{minipage}{17cm}
    \begin{itemize}
      \item Integrator: Velocity Verlet, timestep: 1 fs
      \item Ensemble: isothermal-isobaric NPT [4]
            \begin{itemize}
	      \item Berendsen thermostat:
                    $\tau_{\text{T}}=100\text{ fs}$
	      \item Brendsen barostat:\\
                    $\tau_{\text{P}}=100\text{ fs}$,
		    $\beta^{-1}=100\text{ GPa}$
            \end{itemize}
      \item Potential: Tersoff-like bond order potential [5]
      \[
      E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
      \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
      \]
    \end{itemize}
    \end{minipage}
    \begin{minipage}{9cm}
      \includegraphics[width=9cm]{tersoff_angle.eps}
    \end{minipage}\\[1cm]
    {\tiny
     [4] L. Verlet, Phys. Rev. 159 (1967) 98.}\\
    {\tiny
     [5] P. Erhart and K. Albe, Phys. Rev. B 71 (2005) 35211.}
  \end{pbox}

\end{pcolumn}
\begin{pcolumn}

  \begin{pbox}
    \section*{Interstitial configurations}
    {\bf Simulation sequence:}\\

\begin{minipage}{15cm}
{\small
 \begin{pspicture}(0,0)(14,14)
  \rput(7,12.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
   \parbox{14cm}{
   \begin{itemize}
    \item Initial configuration: $9\times9\times9$ unit cells Si
    \item Periodic boundary conditions
    \item $T=0\text{ K}$, $p=0\text{ bar}$
   \end{itemize}
  }}}}
\rput(7,6){\rnode{insert}{\psframebox{
 \parbox{14cm}{
  Insertion of C / Si atom:
  \begin{itemize}
   \item $(0,0,0)$ $\rightarrow$ {\color{red}tetrahedral}
         (${\color{red}\triangleleft}$)
   \item $(-1/8,-1/8,1/8)$ $\rightarrow$ {\color{green}hexagonal}
         (${\color{green}\triangleright}$)
   \item $(-1/8,-1/8,-1/4)$, $(-3/8,-3/8,-1/4)$\\
         $\rightarrow$ {\color{magenta}110 dumbbell}
         (${\color{magenta}\Box}$,$\circ$)
   \item random positions (critical distance check)
  \end{itemize}
  }}}}
  \rput(7,1.5){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
   \parbox{7cm}{
   Relaxation time: 2 ps
  }}}}
  \ncline[]{->}{init}{insert}
  \ncline[]{->}{insert}{cool}
 \end{pspicture}
}
\end{minipage}
\begin{minipage}{10cm}
  \includegraphics[width=11cm]{unit_cell_s.eps}
\end{minipage}

    {\bf Si self-interstitial results:}\\

{\small
 \begin{minipage}[t]{8.5cm}
 \underline{Tetrahedral}\\
 $E_f=3.41$ eV\\
 \includegraphics[width=8cm]{si_self_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{8.5cm}
 \underline{110 dumbbell}\\
 $E_f=4.39$ eV\\
 \includegraphics[width=8cm]{si_self_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{8.5cm}
 \underline{Hexagonal}\\
 $E_f^{\star}\approx4.48$ eV (unstable!)\\
 \includegraphics[width=8cm]{si_self_int_hexa_0.eps}
 \end{minipage}\\[1cm]

 \underline{Random insertion}\\

 \begin{minipage}{8.5cm}
 $E_f=3.97$ eV\\
 \includegraphics[width=8cm]{si_self_int_rand_397_0.eps}
 \end{minipage}
 \begin{minipage}{8.5cm}
 $E_f=3.75$ eV\\
 \includegraphics[width=8cm]{si_self_int_rand_375_0.eps}
 \end{minipage}
 \begin{minipage}{8.5cm}
 $E_f=3.56$ eV\\
 \includegraphics[width=8cm]{si_self_int_rand_356_0.eps}
 \end{minipage}\\[1cm]
}

    {\bf C in Si interstitial results:}\\

{\small
 \begin{minipage}[t]{8.5cm}
 \underline{Tetrahedral}\\
 $E_f=2.67$ eV\\
 \includegraphics[width=8cm]{c_in_si_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{8.5cm}
 \underline{110 dumbbell}\\
 $E_f=1.76$ eV\\
 \includegraphics[width=8cm]{c_in_si_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{8.5cm}
 \underline{Hexagonal}\\
 $E_f^{\star}\approx5.6$ eV (unstable!)\\
 \includegraphics[width=8cm]{c_in_si_int_hexa_0.eps}
 \end{minipage}\\[1cm]
}
\begin{minipage}{17cm}
\underline{\flq100\frq{} dumbbell configuration}
\begin{itemize}
  \item $E_f=0.47$ eV
  \item Very often observed
  \item Most energetically favorable configuration
  \item Experimental evidence [6]
\end{itemize}
\end{minipage}
\begin{minipage}{8cm}
\includegraphics[width=8cm]{c_in_si_int_001db_0.eps}
\end{minipage}\\[1cm]
\begin{center}
\includegraphics[width=24cm]{100-c-si-db_s.eps}
\end{center}
{\tiny
 [6] G. D. Watkins and K. L. Brower, Phys. Rev. Lett. 36 (1976) 1329.}

  \end{pbox}

\end{pcolumn}
\begin{pcolumn}

  \begin{pbox}
    \section*{High C concentration simulations}

    {\bf Simulation sequence:}\\

{\small
 \begin{pspicture}(0,0)(30,13)
  % nodes
  \rput(7.5,11){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=green]{
   \parbox{15cm}{
   \begin{itemize}
    \item Initial configuration: $31\times31\times31$ unit cells Si
    \item Periodic boundary conditions
    \item $T=450\, ^{\circ}\textrm{C}$, $p=0\text{ bar}$
    \item Equilibration of $E_{kin}$ and $E_{pot}$
   \end{itemize}
  }}}}
  \rput(7.5,5){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=red]{
   \parbox{15cm}{
   Insertion of 6000 carbon atoms at constant\\
   temperature into:
   \begin{itemize}
    \item Total simulation volume $V_1$
    \item Volume of minimal 3C-SiC precipitation $V_2$
    \item Volume of necessary amount of Si $V_3$
   \end{itemize} 
  }}}}
  \rput(7.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=cyan]{
   \parbox{8cm}{
   Cooling down to $20\, ^{\circ}\textrm{C}$
  }}}}
  \ncline[]{->}{init}{insert}
  \ncline[]{->}{insert}{cool}
  \psframe[fillstyle=solid,fillcolor=white](16,2.6)(26,12.6)
  \psframe[fillstyle=solid,fillcolor=lightgray](18,4.6)(24,10.6)
  \psframe[fillstyle=solid,fillcolor=gray](18.5,5.1)(23.5,10.1)
  \rput(9,5.4){\pnode{in1}}
  \rput(15,5.4){\pnode{in-1}}
  \rput(17,7.2){\pnode{ins1}}
  \rput(14,4.2){\pnode{in2}}
  \rput(15,4.2){\pnode{in-2}}
  \rput(18.25,6.88){\pnode{ins2}}
  \rput(12,3.0){\pnode{in3}}
  \rput(15,3.0){\pnode{in-3}}
  \rput(21,7.6){\pnode{ins3}}
  \ncline[linewidth=0.05]{->}{in-1}{ins1}
  \ncline[linewidth=0.05]{->}{in-2}{ins2}
  \ncline[linewidth=0.05]{->}{in-3}{ins3}
  \ncline[linewidth=0.05]{-}{in1}{in-1}
  \ncline[linewidth=0.05]{-}{in2}{in-2}
  \ncline[linewidth=0.05]{-}{in3}{in-3}
 \end{pspicture}
}
    {\bf Results:}\\
    Si-C and C-C pair correlation function:\\
    \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-c_c-c.eps}
    \begin{center}
    {\tiny
     {\bf Dashed vertical lines:} Further calculated C-Si distances 
     in the \flq100\frq{} C-Si dumbbell interstitial configuration}\\[0.5cm]
    \end{center}
    Si-Si pair correlation function:\\
    \hspace*{1.3cm} \includegraphics[width=22cm]{pc_si-si.eps}\\
    {\bf Interpretation:}
    {\small
    \begin{itemize}
      \item C-C peak at 0.15 nm similar to next neighbour distance of graphite
            or diamond\\
	    $\Rightarrow$ Formation of strong C-C bonds
	                  (almost only for high C concentrations)
      \item C-C peak at 0.31 nm equals C-C distance in 3C-SiC\\
            (due to concatenated, differently oriented
	     \flq100\frq{} dumbbell interstitials)
      \item Si-Si shows non-zero g(r) values around 0.31 nm
            and a decrease at regular distances\\
            (no clear peak,
	     interval of enhanced g(r) corresponds to C-C peak width)
      \item Si-C peak at 0.19 nm similar to next neighbour distance in 3C-SiC
      \item Low C concentration (i.e. $V_1$):
            The \flq100\frq{} dumbbell configuration
            \begin{itemize}
              \item is identified to stretch the Si-Si next neighbour distance
	            to 0.3 nm
	      \item is identified to contribute to the Si-C peak at 0.19 nm
	      \item explains further C-Si peaks (dashed vertical lines)
	    \end{itemize}
            $\Rightarrow$ C atoms are first elements arranged at distances
	                  expected for 3C-SiC\\
            $\Rightarrow$ C atoms pull the Si atoms into the right
	                  configuration at a later stage
      \item High C concentration (i.e. $V_2$ and $V_3$):
            \begin{itemize}
	      \item High amount of damage introduced into the system
	      \item Short range order observed but almost no long range order
	    \end{itemize}
	    $\Rightarrow$ Start of amorphous SiC-like phase formation\\
	    $\Rightarrow$ Higher temperatures required for proper SiC formation
    \end{itemize}
    }

  \end{pbox}
  %\vspace{-0.5cm}
  \begin{pbox}
    \section*{Conclusion}
    \begin{itemize}
      \item \flq100\frq{} C-Si dumbbell interstitial configuration is observed
            to be the energetically most favorable configuration
      \item For low C concentrations C atoms introduced as differently
            oriented C-Si dumbbells in c-Si are properly arranged
	    for 3C-SiC formation
      \item For high C concentrations an amorphous SiC-like phase is observed
            which suggests higher temperature simulation runs for proper
	    3C-SiC formation
    \end{itemize}
  \end{pbox}

\end{pcolumn}
\end{poster}
\end{document}

