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\begin{document}

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% specify width and height
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\newcommand{\ham}{\mathcal{H}}
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% itemize level ii
\renewcommand\labelitemii{{\color{gray}$\bullet$}}

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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 Optoelektronik (blaue LEDs), Sensoren
  \item Kandidat f"ur Tr"ager und W"ande in Fusionsreaktoren
  \item Luft- und Raumfahrtindustrie, Milit"ar
  \item kohlenfaserverst"arkte SiC-Verbundkeramik
 \end{itemize}

 }

 \begin{picture}(0,0)(-280,-150)
  %\includegraphics[width=4cm]{sic_inverter_ise.eps} 
 \end{picture}
 
 \begin{picture}(0,0)(-280,-20)
  %\includegraphics[width=4cm]{cc_sic_brake_dlr.eps} 
 \end{picture}
 
\end{slide}

\begin{slide}

 {\large\bf
  Motivation
 }

 \vspace{8pt}

 3C-SiC (\foreignlanguage{greek}{b}-SiC) /
 6H-SiC (\foreignlanguage{greek}{a}-SiC)
 \begin{itemize}
  \item h"ohere Ladungstr"agerbeweglichkeit in \foreignlanguage{greek}{b}-SiC
  \item Micropipes (Offene Kerne von Schraubenversetzungen) in c-Richtung
        bei \foreignlanguage{greek}{a}-SiC
  \item Herstellung gro"sfl"achiger einkristalliner 3C-SiC Filme
        im Anfangsstudium
 \end{itemize}

 \vspace{16pt}

 \begin{center}
  {\color{red}
  Genaues Verst"andnis des 3C-SiC-Ausscheidungsvorganges\\
  }
  $\Downarrow$\\ 
  signifikanter technologischen Fortschritt in 3C-SiC D"unnschichtherstellung
 \end{center}

 \vspace{16pt}

 Vermeidung von SiC-Ausscheidungen in
 $\text{Si}_{\text{1-y}}\text{C}_{\text{y}}$ Legierungen

 \begin{itemize}
  \item Ma"sschneidern der elektronischen Eigenschaften von Si
  \item gestreckte Heterostrukturen
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf
  Motivation bzw. SiC-Ausscheidungsvorgang
 }

 \vspace{64pt}

 Noch was zur Herstellung rein ...

\end{slide}

\begin{slide}

 {\large\bf
  SiC-Ausscheidungsvorgang
 }

 \vspace{8pt}

 {\bf Kristallstruktur und Einheitszelle:}
 \begin{itemize}
   \item kristallines Silizium (c-Si): Diamantstruktur\\
         ${\color{si-yellow}\bullet}$, ${\color{gray}\bullet}$
         $\leftarrow$ Si-Atome
   \item kubisches SiC (3C-SiC): Zinkblende-Struktur\\
         ${\color{si-yellow}\bullet} \leftarrow$ Si-Atome\\
         ${\color{gray}\bullet} \leftarrow$ C-Atome
 \end{itemize}
 \vspace{8pt}
 \begin{minipage}{8cm}
 {\bf Gitterkonstanten:}
 \[
 4a_{\text{c-Si}}\approx5a_{\text{3C-SiC}}
 \]
 {\bf Siliziumdichten:}
 \[
 \frac{n_{\text{3C-SiC}}}{n_{\text{c-Si}}}=97,66\,\%
 \]
 \end{minipage}
 \begin{minipage}{5cm}
   \includegraphics[width=5cm]{sic_unit_cell.eps}         
 \end{minipage}

\end{slide}

 \small
\begin{slide}

 {\large\bf
  SiC-Ausscheidungsvorgang
 }

 \small
 \vspace{6pt}

 Vermuteter SiC-Ausscheidungsvorgang in Si:

 \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}
 Bildung von C-Si Dumbbells auf regul"aren c-Si Gitterpl"atzen
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 Anh"aufung hin zu gro"sen Clustern (Embryos)\\
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 Ausscheidung von 3C-SiC + Erzeugung von Si-Zwischengitteratomen\\
 \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}

\end{document}

\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}

