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\begin{document}

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% specify width and height
\slidewidth 27.7cm 
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% shift it into visual area properly
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\newcommand{\ham}{\mathcal{H}}
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% itemize level ii
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% topic

\begin{slide}
\begin{center}

 \vspace{16pt}

 {\LARGE\bf
  Molekulardynamische Untersuchung\\
  zum SiC-Ausscheidungsvorgang
 }

 \vspace{48pt}

 \textsc{F. Zirkelbach}

 \vspace{48pt}

 Lehrstuhlseminar

 \vspace{08pt}

 13. November 2008

\end{center}
\end{slide}

% contents

\begin{slide}

{\large\bf
 Gliederung
}

 \begin{itemize}
  \item Motivation
  \item SiC-Ausscheidungsvorgang
  \item Simulation
        \begin{itemize}
         \item Details der MD-Simulation
         \item Zwischengitter-Konfigurationen
         \item Simulationen zum Ausscheidungsvorgang
         \item SiC-Ausscheidungen in Si
        \end{itemize}
  \item Zusammenfassung und Ausblick
 \end{itemize}

\end{slide}

% start of contents

\begin{slide}

 {\large\bf
  Motivation
 }

 {\small

 Eigenschaften von SiC:

 \begin{itemize}
  \item gro"se Bandl"ucke (3C: 2.39 eV, 4H: 3.28 eV, 6H: 3.03 eV)
  \item hohe mechanische Stabilit"at
  \item gute Ladungstr"agermobilit"at
  \item sp"ate S"attigung der Elektronen-Driftgeschwindigkeit
  \item chemisch inerte Substanz
  \item hohe thermische Leitf"ahigkeit und Stabilit"at
  \item geringer Neutroneneinfangquerschnitt
  \item strahlungsresistent
 \end{itemize}

 Anwendungen:

 \begin{itemize}
  \item Hochfrequenz-, Hochtemperatur und Hochleistungsbauelemente
  \item blaue LEDs
  \item Kandidat f"ur Tr"ager und W"ande in Fusionsreaktoren
  \item Luft- und Raumfahrtindistrie, Milit"ar
  \item kohlenfaserverst"arkte SiC-Verbundkeramik
 \end{itemize}

 }

 \begin{picture}(0,0)(-275,-150)
  \includegraphics[width=4cm]{sic_inverter_ise.eps} 
 \end{picture}
 
 \begin{picture}(0,0)(-275,-20)
  \includegraphics[width=4cm]{cc_sic_brake_dlr.eps} 
 \end{picture}
 
\end{slide}

\begin{slide}

 {\large\bf
  Motivation
 }

 Problem:

 However, in order to become economically viable, several critical materials and processing issues still need to be solved. The most serious issue is the immature state of the crystal growth technology, where increases in wafer size and quality are urgently needed.

 Und andersrum:

 Modifikation der Bandl"ucke und Spannungen in Heterostrukturen

 Kein SiC-Ausscheidungsvorgang erw"unscht!

 {\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{slide}

\end{document}

\begin{slide}

 {\large\bf
  Crystalline silicon and cubic silicon carbide
 }

 \vspace{8pt}

 {\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}
 \vspace{8pt}
 \begin{minipage}{8cm}
 {\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}{5cm}
   \includegraphics[width=5cm]{sic_unit_cell.eps}         
 \end{minipage}

\end{slide}

 \small
\begin{slide}

 {\large\bf
  Supposed Si to 3C-SiC conversion
 }

 \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}

 \begin{minipage}{7cm}
 Experimentally observed [3]:
 \begin{itemize}
  \item Minimal diameter of precipitation: 4 - 5 nm
  \item Equal orientation of Si and SiC (hkl)-planes
 \end{itemize}
 \end{minipage}
 \begin{minipage}{6cm}
 \vspace{32pt}
 \hspace{16pt}
  {\tiny [3] J. K. N. Lindner, Appl. Phys. A 77 (2003) 27.}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation details
 }

 \small

 {\bf 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 or ensemble averages
 \end{itemize}
 {\bf Application details:}
 \begin{itemize}
  \item Integrator: Velocity Verlet, timestep: $1\text{ fs}$
  \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}
 {\tiny
  [4] L. Verlet, Phys. Rev. 159 (1967) 98.}\\
 {\tiny
  [5] P. Erhart and K. Albe, Phys. Rev. B 71 (2005) 35211.}

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

\end{slide}

\begin{slide}

 {\large\bf
  Simulation sequence
 }

 \vspace{8pt}

 Interstitial configurations:

 \vspace{8pt}

 \begin{pspicture}(0,0)(7,8)
  \rput(3.5,7){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
   \parbox{7cm}{
   \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(3.5,3.5){\rnode{insert}{\psframebox{
 \parbox{7cm}{
  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 $(-1/8,-1/8,-1/4)$, $(-1/4,-1/4,-1/4)$\\
         $\rightarrow$ {\color{magenta}110 dumbbell}
	 (${\color{magenta}\Box}$,$\circ$)
   \item random positions (critical distance check)
  \end{itemize}
  }}}}
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   \parbox{3.5cm}{
   Relaxation time: $2\, ps$
  }}}}
  \ncline[]{->}{init}{insert}
  \ncline[]{->}{insert}{cool}
 \end{pspicture}

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

\end{slide}

\begin{slide}

 {\large\bf
  Results
 } - Si self-interstitial runs

 \small

 \begin{minipage}[t]{4.3cm}
 \underline{Tetrahedral}\\
 $E_f=3.41$ eV\\
 \includegraphics[width=3.8cm]{si_self_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \underline{110 dumbbell}\\
 $E_f=4.39$ eV\\
 \includegraphics[width=3.8cm]{si_self_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \underline{Hexagonal} \hspace{4pt}
 \href{../video/si_self_int_hexa.avi}{$\rhd$}\\
 $E_f^{\star}\approx4.48$ eV (unstable!)\\
 \includegraphics[width=3.8cm]{si_self_int_hexa_0.eps}
 \end{minipage}

 \underline{Random insertion}

 \begin{minipage}{4.3cm}
 $E_f=3.97$ eV\\
 \includegraphics[width=3.8cm]{si_self_int_rand_397_0.eps}
 \end{minipage}
 \begin{minipage}{4.3cm}
 $E_f=3.75$ eV\\
 \includegraphics[width=3.8cm]{si_self_int_rand_375_0.eps}
 \end{minipage}
 \begin{minipage}{4.3cm}
 $E_f=3.56$ eV\\
 \includegraphics[width=3.8cm]{si_self_int_rand_356_0.eps}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 } - Carbon interstitial runs

 \small

 \begin{minipage}[t]{4.3cm}
 \underline{Tetrahedral}\\
 $E_f=2.67$ eV\\
 \includegraphics[width=3.8cm]{c_in_si_int_tetra_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \underline{110 dumbbell}\\
 $E_f=1.76$ eV\\
 \includegraphics[width=3.8cm]{c_in_si_int_dumbbell_0.eps}
 \end{minipage}
 \begin{minipage}[t]{4.3cm}
 \underline{Hexagonal} \hspace{4pt}
 \href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
 $E_f^{\star}\approx5.6$ eV (unstable!)\\
 \includegraphics[width=3.8cm]{c_in_si_int_hexa_0.eps}
 \end{minipage}

 \underline{Random insertion}

 \footnotesize

\begin{minipage}[t]{3.3cm}
   $E_f=0.47$ eV\\
   \includegraphics[width=3.3cm]{c_in_si_int_001db_0.eps}
   \begin{picture}(0,0)(-15,-3)
    100 dumbbell
   \end{picture}
\end{minipage}
\begin{minipage}[t]{3.3cm}
   $E_f=1.62$ eV\\
   \includegraphics[width=3.2cm]{c_in_si_int_rand_162_0.eps}
\end{minipage}
\begin{minipage}[t]{3.3cm}
   $E_f=2.39$ eV\\
   \includegraphics[width=3.1cm]{c_in_si_int_rand_239_0.eps}
\end{minipage}
\begin{minipage}[t]{3.0cm}
   $E_f=3.41$ eV\\
   \includegraphics[width=3.3cm]{c_in_si_int_rand_341_0.eps}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 } - <100> dumbbell configuration

 \vspace{8pt}

 \small

 \begin{minipage}{4cm}
 \begin{itemize}
  \item $E_f=0.47$ eV
  \item Very often observed
  \item Most energetically\\
        favorable configuration
  \item Experimental\\
        evidence [6]
 \end{itemize}
 \vspace{24pt}
 {\tiny
  [6] G. D. Watkins and K. L. Brower,\\
      Phys. Rev. Lett. 36 (1976) 1329.
 }
 \end{minipage}
 \begin{minipage}{8cm}
 \includegraphics[width=9cm]{100-c-si-db_s.eps}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Simulation sequence
 }

 \small

 \vspace{8pt}

 SiC precipitation simulations:

 \vspace{8pt}

 \begin{pspicture}(0,0)(12,8)
  % nodes
  \rput(3.5,6.5){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
   \parbox{7cm}{
   \begin{itemize}
    \item Initial configuration: $31\times31\times31$ unit cells Si
    \item Periodic boundary conditions
    \item $T=450\, ^{\circ}\text{C}$, $p=0\text{ bar}$
    \item Equilibration of $E_{kin}$ and $E_{pot}$
   \end{itemize}
  }}}}
  \rput(3.5,3.2){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
   \parbox{7cm}{
   Insertion of 6000 carbon atoms at constant\\
   temperature into:
   \begin{itemize}
    \item Total simulation volume {\pnode{in1}}
    \item Volume of minimal SiC precipitation {\pnode{in2}}
    \item Volume of necessary amount of Si {\pnode{in3}}
   \end{itemize} 
  }}}}
  \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
   \parbox{3.5cm}{
   Cooling down to $20\, ^{\circ}C$
  }}}}
  \ncline[]{->}{init}{insert}
  \ncline[]{->}{insert}{cool}
  \psframe[fillstyle=solid,fillcolor=white](7.5,1.8)(13.5,7.8)
  \psframe[fillstyle=solid,fillcolor=lightgray](9,3.3)(12,6.3)
  \psframe[fillstyle=solid,fillcolor=gray](9.25,3.55)(11.75,6.05)
  \rput(7.9,4.8){\pnode{ins1}}
  \rput(9.22,4.4){\pnode{ins2}}
  \rput(10.5,4.8){\pnode{ins3}}
  \ncline[]{->}{in1}{ins1}
  \ncline[]{->}{in2}{ins2}
  \ncline[]{->}{in3}{ins3}
 \end{pspicture}

\end{slide}

\begin{slide}

 {\large\bf
  Results
 } - SiC precipitation runs


 \includegraphics[width=6.3cm]{pc_si-c_c-c.eps}
 \includegraphics[width=6.3cm]{pc_si-si.eps}

 \begin{minipage}[t]{6.3cm}
 \tiny
    \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 Si-C peak at 0.19 nm similar to next neighbour distance in 3C-SiC
      \item C-C peak at 0.31 nm equals C-C distance in 3C-SiC\\
            (due to concatenated, differently oriented
             <100> dumbbell interstitials)
      \item Si-Si shows non-zero g(r) values around 0.31 nm like in 3C-SiC\\
            and a decrease at regular distances\\
            (no clear peak,
             interval of enhanced g(r) corresponds to C-C peak width)
    \end{itemize}
 \end{minipage}
 \begin{minipage}[t]{6.3cm}
 \tiny
   \begin{itemize}
      \item Low C concentration (i.e. $V_1$):
            The <100> 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{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Very first results of the SiC precipitation runs
 }

 \begin{minipage}[t]{6.9cm}
  \includegraphics[width=6.3cm]{../plot/sic_pc.ps}
  \includegraphics[width=6.3cm]{../plot/foo_end.ps}
  \hspace{12pt}
 \end{minipage}
 \begin{minipage}[c]{5.5cm}
  \includegraphics[width=6.0cm]{sic_si-c-n.eps}
 \end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Summary / Outlook
 }

\vspace{24pt}

\begin{itemize}
 \item Importance of understanding the SiC precipitation mechanism
 \item Interstitial configurations in silicon using the Albe potential
 \item Indication of SiC precipitation
\end{itemize}

\vspace{24pt}

\begin{itemize}
 \item Displacement and stress calculations
 \item Refinement of simulation sequence to create 3C-SiC
 \item Analyzing self-designed Si/SiC interface
\end{itemize}

\end{slide}

\end{document}

