[lectures/latex.git] / posic / talks / defense.tex

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% shortcuts
\newcommand{\si}{Si$_{\text{i}}${}}
\newcommand{\ci}{C$_{\text{i}}${}}
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\begin{slide}
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{\Huge
E\\
F\\
G\\
A B C D E F G H G F E D C B A
G\\
F\\
E\\
}
\end{slide}
\fi

% topic

\begin{slide}
\begin{center}

 \vspace{16pt}

 {\Large\bf
  \hrule
  \vspace{5pt}
  Atomistic simulation study on silicon carbide\\[0.2cm]
  precipitation in silicon\\
  \vspace{10pt}
  \hrule
 }

 \vspace{60pt}

 \textsc{Frank Zirkelbach}

 \vspace{60pt}

 Defense of doctor's thesis

 \vspace{08pt}

 Augsburg, 10.01.2012

\end{center}
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% intro

% motivation / properties / applications of silicon carbide

\begin{slide}

\vspace*{1.8cm}

\small

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 \rput[lt](0,4.6){\color{gray}PROPERTIES}

 \rput[lt](0.3,4){wide band gap}
 \rput[lt](0.3,3.5){high electric breakdown field}
 \rput[lt](0.3,3){good electron mobility}
 \rput[lt](0.3,2.5){high electron saturation drift velocity}
 \rput[lt](0.3,2){high thermal conductivity}

 \rput[lt](0.3,1.5){hard and mechanically stable}
 \rput[lt](0.3,1){chemically inert}

 \rput[lt](0.3,0.5){radiation hardness}

 \rput[rt](12.7,4.6){\color{gray}APPLICATIONS}

 \rput[rt](12.5,3.85){high-temperature, high power}
 \rput[rt](12.5,3.5){and high-frequency}
 \rput[rt](12.5,3.15){electronic and optoelectronic devices}

 \rput[rt](12.5,2.35){material suitable for extreme conditions}
 \rput[rt](12.5,2){microelectromechanical systems}
 \rput[rt](12.5,1.65){abrasives, cutting tools, heating elements}

 \rput[rt](12.5,0.85){first wall reactor material, detectors}
 \rput[rt](12.5,0.5){and electronic devices for space}

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%%%%
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% fabrication

\begin{slide}

\headphd
{\large\bf
 IBS of epitaxial single crystalline 3C-SiC
}

\footnotesize

\vspace{0.2cm}

\begin{center}
\begin{itemize}
 \item \underline{Implantation step 1}\\[0.1cm]
        Almost stoichiometric dose | \unit[180]{keV} | \degc{500}\\
        $\Rightarrow$ Epitaxial {\color{blue}3C-SiC} layer \&
        {\color{blue}precipitates}
 \item \underline{Implantation step 2}\\[0.1cm]
        Low remaining amount of dose | \unit[180]{keV} | \degc{250}\\
        $\Rightarrow$
        Destruction/Amorphization of precipitates at layer interface
 \item \underline{Annealing}\\[0.1cm]
       \unit[10]{h} at \degc{1250}\\
       $\Rightarrow$ Homogeneous 3C-SiC layer with sharp interfaces
\end{itemize}
\end{center}

\begin{minipage}{6.9cm}
\includegraphics[width=7cm]{ibs_3c-sic.eps}\\[-0.4cm]
\begin{center}
{\tiny
 XTEM: single crystalline 3C-SiC in Si\hkl(1 0 0)
}
\end{center}
\end{minipage}
\begin{minipage}{5cm}
\begin{center}
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 {\color{blue}
  3C-SiC precipitation\\
  not yet fully understood
 }
 \end{center}
% \vspace*{0.1cm}
% \renewcommand\labelitemi{$\Rightarrow$}
% Details of the SiC precipitation
% \begin{itemize}
%  \item significant technological progress\\
%        in SiC thin film formation
%  \item perspectives for processes relying\\
%        upon prevention of SiC precipitation
% \end{itemize}
\end{minipage}
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\begin{slide}

\headphd
{\large\bf
  Supposed precipitation mechanism of SiC in Si
}

 \scriptsize

 \vspace{0.1cm}

 \framebox{
 \begin{minipage}{3.6cm}
 \begin{center}
 Si \& SiC lattice structure\\[0.1cm]
 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
 \end{center}
{\tiny
 \begin{minipage}{1.7cm}
\underline{Silicon}\\
{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
$a=\unit[5.429]{\\A}$\\
$\rho^*_{\text{Si}}=\unit[100]{\%}$
 \end{minipage}
 \begin{minipage}{1.7cm}
\underline{Silicon carbide}\\
{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
$a=\unit[4.359]{\\A}$\\
$\rho^*_{\text{Si}}=\unit[97]{\%}$
 \end{minipage}
}
 \end{minipage}
 }
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 \end{center}
 \end{minipage}

 \vspace{0.1cm}

 \begin{minipage}{4.0cm}
 \begin{center}
 C-Si dimers (dumbbells)\\[-0.1cm]
 on Si lattice sites
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 Agglomeration of C-Si dumbbells\\[-0.1cm]
 $\Rightarrow$ dark contrasts
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 Precipitation of 3C-SiC in Si\\[-0.1cm]
 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 \& release of Si self-interstitials
 \end{center}
 \end{minipage}

 \vspace{0.1cm}

 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 \end{center}
 \end{minipage}

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\end{slide}

\begin{slide}

\headphd
{\large\bf
 Supposed precipitation mechanism of SiC in Si
}

 \scriptsize

 \vspace{0.1cm}

 \framebox{
 \begin{minipage}{3.6cm}
 \begin{center}
 Si \& SiC lattice structure\\[0.1cm]
 \includegraphics[width=2.3cm]{sic_unit_cell.eps}
 \end{center}
{\tiny
 \begin{minipage}{1.7cm}
\underline{Silicon}\\
{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} Si\\
$a=\unit[5.429]{\\A}$\\
$\rho^*_{\text{Si}}=\unit[100]{\%}$
 \end{minipage}
 \begin{minipage}{1.7cm}
\underline{Silicon carbide}\\
{\color{yellow}$\bullet$} Si | {\color{gray}$\bullet$} C\\
$a=\unit[4.359]{\\A}$\\
$\rho^*_{\text{Si}}=\unit[97]{\%}$
 \end{minipage}
}
 \end{minipage}
 }
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 \end{center}
 \end{minipage}

 \vspace{0.1cm}

 \begin{minipage}{4.0cm}
 \begin{center}
 C-Si dimers (dumbbells)\\[-0.1cm]
 on Si interstitial sites
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 Agglomeration of C-Si dumbbells\\[-0.1cm]
 $\Rightarrow$ dark contrasts
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 Precipitation of 3C-SiC in Si\\[-0.1cm]
 $\Rightarrow$ Moir\'e fringes\\[-0.1cm]
 \& release of Si self-interstitials
 \end{center}
 \end{minipage}

 \vspace{0.1cm}

 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.1cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 \end{center}
 \end{minipage}
 \hspace{0.1cm}
 \begin{minipage}{4.0cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 \end{center}
 \end{minipage}

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% controversial view!
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\small
\vspace*{0.2cm}
\begin{center}
{\color{gray}\bf Controversial findings}
\end{center}
\begin{itemize}
\item High-temperature implantation {\tiny\color{gray}/Nejim~et~al./}
 \begin{itemize}
  \item {\color{blue}Substitutionally} incorporated C on regular Si lattice sites
  \item \si{} reacting with further C in cleared volume
 \end{itemize}
\item Annealing behavior {\tiny\color{gray}/Serre~et~al./}
 \begin{itemize}
  \item Room temperature implantation $\rightarrow$ high C diffusion
  \item Elevated temperature implantation $\rightarrow$ no C redistribution
 \end{itemize}
 $\Rightarrow$ mobile {\color{red}\ci} opposed to
 stable {\color{blue}\cs{}} configurations
\item Strained Si$_{1-y}$C$_y$/Si heterostructures
      {\tiny\color{gray}/Strane~et~al./Guedj~et~al./}
 \begin{itemize}
  \item Initial {\color{blue}coherent} SiC structures (tensile strain)
  \item Incoherent SiC nanocrystals (strain relaxation)
 \end{itemize}
\end{itemize}
\vspace{0.1cm}
\begin{center}
{\Huge${\lightning}$} \hspace{0.3cm}
{\color{blue}\cs{}} --- vs --- {\color{red}\ci} \hspace{0.3cm}
{\Huge${\lightning}$}
\end{center}
\vspace{0.2cm}
\end{minipage}
 }}}
\end{pspicture}

\end{slide}

\begin{slide}

% contents

\headphd
{\large\bf
 Outline
}

 \begin{itemize}
  {\color{gray}
  \item Introduction / Motivation
  \item Assumed SiC precipitation mechanisms / Controversy
  }
  \item Utilized simulation techniques
        \begin{itemize}
         \item Molecular dynamics (MD) simulations
         \item Density functional theory (DFT) calculations
        \end{itemize}
  \item Simulation results
        \begin{itemize}
         \item C and Si self-interstitial point defects in silicon
         \item Silicon carbide precipitation simulations
        \end{itemize}
  \item Summary / Conclusion
 \end{itemize}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Utilized computational methods
}

\vspace{0.3cm}

\small

{\bf Molecular dynamics (MD)}\\[0.1cm]
\scriptsize
\begin{tabular}{| p{4.5cm} | p{7.5cm} |}
\hline
System of $N$ particles &
$N=5832\pm 1$ (Defects), $N=238328+6000$ (Precipitation)\\
Phase space propagation &
Velocity Verlet | timestep: \unit[1]{fs} \\
Analytical interaction potential &
Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential
(Erhart/Albe)
$\displaystyle
E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
    \pot_{ij} = {\color{red}f_C(r_{ij})}
    \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
$\\
Observables: time/ensemble averages &
NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\
\hline
\end{tabular}

\small

\vspace{0.3cm}

{\bf Density functional theory (DFT)}

\scriptsize

\begin{minipage}[t]{6cm}
\begin{itemize}
 \item Hohenberg-Kohn theorem:\\
       $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
 \item Kohn-Sham approach:\\
       Single-particle effective theory
\end{itemize}
\hrule
\begin{itemize}
\item Code: \textsc{vasp}
\item Plane wave basis set | $E_{\text{cut}}=\unit[300]{eV}$
%$\displaystyle
%\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
%$\\
%$\displaystyle
%E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
%$
\item Ultrasoft pseudopotential
\item Exchange \& correlation: GGA
\item Brillouin zone sampling: $\Gamma$-point
\item Supercell: $N=216\pm2$
\end{itemize}
\end{minipage}
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\end{slide}

\begin{slide}

\headphd
 {\large\bf
  Point defects \& defect migration
 }

 \small

 \vspace{0.2cm}

\begin{minipage}[b]{7.5cm}
{\bf Defect structure}\\
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   \begin{itemize}
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    \item $T=0\text{ K}$, $p=0\text{ bar}$
   \end{itemize}
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\begin{minipage}[b]{4.5cm}
\begin{center}
\includegraphics[width=3.8cm]{unit_cell_e.eps}\\
\end{center}
\begin{minipage}{2.21cm}
{\scriptsize
{\color{red}$\bullet$} Tetrahedral\\[-0.1cm]
{\color{green}$\bullet$} Hexagonal\\[-0.1cm]
{\color{yellow}$\bullet$} \hkl<1 0 0> DB
}
\end{minipage}
\begin{minipage}{2.21cm}
{\scriptsize
{\color{magenta}$\bullet$} \hkl<1 1 0> DB\\[-0.1cm]
{\color{cyan}$\bullet$} Bond-centered\\[-0.1cm]
{\color{black}$\bullet$} Vac. / Sub.
}
\end{minipage}
\end{minipage}

\vspace{0.3cm}

\begin{minipage}[t]{6cm}
{\bf Defect formation energy}\\
\framebox{
$E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.5cm]
%Particle reservoir: Si \& SiC\\[0.2cm]
{\bf Binding energy}\\
\framebox{
$
E_{\text{b}}=
E_{\text{f}}^{\text{comb}}-
E_{\text{f}}^{1^{\text{st}}}-
E_{\text{f}}^{2^{\text{nd}}}
$
}\\[0.1cm]
\footnotesize
$E_{\text{b}}<0$: energetically favorable configuration\\
$E_{\text{b}}\rightarrow 0$: non-interacting, isolated defects\\
\end{minipage}
\begin{minipage}[t]{6cm}
\vspace{1.4cm}
{\bf Migration barrier}
\footnotesize
\begin{itemize}
 \item Displace diffusing atom
 \item Constrain relaxation of (diffusing) atoms
 \item Record configurational energy
\end{itemize}
\begin{picture}(0,0)(-60,-33)
\includegraphics[width=4.5cm]{crt_mod.eps}
\end{picture}
\end{minipage}

\end{slide}

\begin{slide}

\footnotesize

\headphd
{\large\bf
 C interstitial point defects in silicon\\
}

\begin{tabular}{l c c c c c c r}
\hline
 $E_{\text{f}}$ [eV] & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B &
 {\color{black} \cs{} \& \si}\\
\hline
 \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
 Erhart/Albe & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
\hline
\end{tabular}\\[0.1cm]

\framebox{
\begin{minipage}{2.8cm}
\underline{Hexagonal} \hspace{2pt}
\href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
$E_{\text{f}}^*=9.05\text{ eV}$\\
\includegraphics[width=2.8cm]{c_pd_albe/hex_bonds.eps}
\end{minipage}
\begin{minipage}{0.4cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{2.8cm}
\underline{\hkl<1 0 0>}\\
$E_{\text{f}}=3.88\text{ eV}$\\
\includegraphics[width=2.8cm]{c_pd_albe/100_bonds.eps}
\end{minipage}
}
\begin{minipage}{1.4cm}
\hfill
\end{minipage}
\begin{minipage}{3.0cm}
\begin{flushright}
\underline{Tetrahedral}\\
\includegraphics[width=3.0cm]{c_pd_albe/tet_bonds.eps}
\end{flushright}
\end{minipage}

\framebox{
\begin{minipage}{2.8cm}
\underline{Bond-centered}\\
$E_{\text{f}}^*=5.59\text{ eV}$\\
\includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}
\end{minipage}
\begin{minipage}{0.4cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{2.8cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=5.18\text{ eV}$\\
\includegraphics[width=2.8cm]{c_pd_albe/110_bonds.eps}
\end{minipage}
}
\begin{minipage}{1.4cm}
\hfill
\end{minipage}
\begin{minipage}{3.0cm}
\begin{flushright}
\underline{Substitutional}\\
\includegraphics[width=3.0cm]{c_pd_albe/sub_bonds.eps}
\end{flushright}
\end{minipage}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 C interstitial migration --- ab initio
}

\scriptsize

\vspace{0.3cm}

\begin{minipage}{6.8cm}
\framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 0 1]}\\
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/bc_2333.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/100_next_2333.eps}
\end{minipage}\\[0.1cm]
Symmetry:\\
$\Rightarrow$ Sufficient to consider \hkl[00-1] to BC transition\\
$\Rightarrow$ Migration barrier to reach BC | $\Delta E=\unit[1.2]{eV}$
\end{minipage}
\begin{minipage}{5.4cm}
\includegraphics[width=6.0cm]{im_00-1_nosym_sp_fullct_thesis_vasp_s.ps}
%\end{minipage}\\[0.2cm]
\end{minipage}\\[0.4cm]
%\hrule
%
\begin{minipage}{6.8cm}
\framebox{\hkl[0 0 -1] $\rightarrow$ \hkl[0 -1 0]}\\
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/00-1-0-10_2333.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.0cm}
\includegraphics[width=2.0cm]{c_pd_vasp/0-10_2333.eps}
\end{minipage}\\[0.1cm]
$\Delta E=\unit[0.9]{eV}$ | Experimental values: \unit[0.70--0.87]{eV}\\
$\Rightarrow$ {\color{red}Migration mechanism identified!}\\
Note: Change in orientation
\end{minipage}
\begin{minipage}{5.4cm}
\includegraphics[width=6.0cm]{00-1_0-10_vasp_s.ps}
\end{minipage}\\[0.1cm]
%
%\begin{center}
%Reorientation pathway composed of two consecutive processes of the above type
%\end{center}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 C interstitial migration --- analytical potential
}

\scriptsize

\vspace{0.3cm}

\begin{minipage}[t]{6.0cm}
{\bf\boldmath BC to \hkl[0 0 -1] transition}\\[0.2cm]
\includegraphics[width=6.0cm]{bc_00-1_albe_s.ps}\\
\begin{itemize}
 \item Lowermost migration barrier
 \item $\Delta E \approx \unit[2.2]{eV}$
 \item 2.4 times higher than ab initio result
 \item Different pathway
\end{itemize}
\end{minipage}
\begin{minipage}[t]{0.2cm}
\hfill
\end{minipage}
\begin{minipage}[t]{6.0cm}
{\bf\boldmath Transition involving a \hkl<1 1 0> configuration}
\vspace{0.1cm}
\begin{itemize}
 \item Bond-centered configuration unstable\\
       $\rightarrow$ \ci{} \hkl<1 1 0> dumbbell
 \item Minimum of the \hkl[0 0 -1] to \hkl[0 -1 0] transition\\
       $\rightarrow$ \ci{} \hkl<1 1 0> DB
\end{itemize}
\vspace{0.1cm}
\includegraphics[width=6.0cm]{00-1_110_0-10_mig_albe.ps}
\begin{itemize}
 \item $\Delta E \approx \unit[2.2]{eV} \text{ \& } \unit[0.9]{eV}$
 \item 2.4 -- 3.4 times higher than ab initio result
 \item After all: Change of the DB orientation
\end{itemize}
\end{minipage}

\vspace{0.5cm}
\begin{center}
{\color{red}\bf Drastically overestimated diffusion barrier}
\end{center}

\begin{pspicture}(0,0)(0,0)
\psline[linewidth=0.05cm,linecolor=gray](6.1,1.0)(6.1,9.3)
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Defect combinations --- ab inito
}

\footnotesize

\vspace{0.3cm}

\begin{minipage}{9cm}
{\bf
 Summary of combinations}\\[0.1cm]
{\scriptsize
\begin{tabular}{l c c c c c c}
\hline
 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
 \hline
 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
 \hkl[0 -1 0] & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
 \hkl[0 1 0] & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
 \hkl[-1 0 0] & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
 \hkl[1 0 0] & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
 \hline
 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\hline
\end{tabular}
}
\vspace{0.2cm}
\begin{center}
{\color{blue}
 $E_{\text{b}}$ explainable by stress compensation / increase
}
\end{center}
\end{minipage}
\begin{minipage}{3cm}
\includegraphics[width=3.5cm]{comb_pos.eps}
\end{minipage}

\vspace{0.2cm}

{\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
\begin{minipage}[t]{3.2cm}
\underline{\hkl[1 0 0] at position 1}\\[0.1cm]
\includegraphics[width=2.8cm]{00-1dc/2-25.eps}
\end{minipage}
\begin{minipage}[t]{3.0cm}
\underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
\includegraphics[width=2.8cm]{00-1dc/2-39.eps}
\end{minipage}
\begin{minipage}[t]{6.1cm}
\vspace{0.7cm}
\begin{itemize}
 \item \ci{} agglomeration energetically favorable
 \item Most favorable: C clustering\\
       {\color{red}However \ldots}\\
        \ldots high migration barrier ($>4\,\text{eV}$)\\
        \ldots entropy:
        $4\times{\color{cyan}[-2.25]}$ versus
        $2\times{\color{orange}[-2.39]}$
\end{itemize}
\begin{center}
{\color{blue}\ci{} agglomeration / no C clustering}
\end{center}
\end{minipage}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Defect combinations --- ab inito
}

\footnotesize

\vspace{0.3cm}

\begin{minipage}{9cm}
{\bf
 Summary of combinations}\\[0.1cm]
{\scriptsize
\begin{tabular}{l c c c c c c}
\hline
 $E_{\text{b}}$ [eV] & 1 & 2 & 3 & 4 & 5 & R\\
 \hline
 \hkl[0 0 -1] & {\color{red}-0.08} & -1.15 & {\color{red}-0.08} & 0.04 & -1.66 & -0.19\\
 \hkl[0 0 1] & 0.34 & 0.004 & -2.05 & 0.26 & -1.53 & -0.19\\
 \hkl[0 -1 0] & {\color{orange}-2.39} & -0.17 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
 \hkl[0 1 0] & {\color{cyan}-2.25} & -1.90 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
 \hkl[-1 0 0] & {\color{orange}-2.39} & -0.36 & {\color{cyan}-2.25} & {\color{purple}-0.12} & {\color{magenta}-1.88} & {\color{gray}-0.05}\\
 \hkl[1 0 0] & {\color{cyan}-2.25} & -2.16 & {\color{green}-0.10} & {\color{blue}-0.27} & {\color{violet}-1.38} & {\color{yellow}-0.06}\\
 \hline
 C$_{\text{sub}}$ & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
 Vacancy & -5.39 ($\rightarrow$ C$_{\text{sub}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\hline
\end{tabular}
}
\vspace{0.2cm}
\begin{center}
{\color{blue}
 $E_{\text{b}}$ explainable by stress compensation / increase
}
\end{center}
\end{minipage}
\begin{minipage}{3cm}
\includegraphics[width=3.5cm]{comb_pos.eps}
\end{minipage}

\vspace{0.2cm}

{\bf\boldmath Combinations of \hkl<1 0 0>-type interstitials}\\[0.2cm]
\begin{minipage}[t]{3.2cm}
\underline{\hkl[1 0 0] at position 1}\\[0.1cm]
\includegraphics[width=2.8cm]{00-1dc/2-25.eps}
\end{minipage}
\begin{minipage}[t]{3.0cm}
\underline{\hkl[0 -1 0] at position 1}\\[0.1cm]
\includegraphics[width=2.8cm]{00-1dc/2-39.eps}
\end{minipage}
\begin{minipage}[t]{6.1cm}
\vspace{0.7cm}
\begin{itemize}
 \item \ci{} agglomeration energetically favorable
 \item Most favorable: C clustering\\
       {\color{red}However \ldots}\\
        \ldots high migration barrier ($>4\,\text{eV}$)\\
        \ldots entropy:
        $4\times{\color{cyan}[-2.25]}$ versus
        $2\times{\color{orange}[-2.39]}$
\end{itemize}
\begin{center}
{\color{blue}\ci{} agglomeration / no C clustering}
\end{center}
\end{minipage}

% insert graph ...
\begin{pspicture}(0,0)(0,0)
\rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
\begin{minipage}{14cm}
\hfill
\vspace{12cm}
\end{minipage}
}}
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\begin{minipage}{8cm}
\begin{center}
\vspace{0.2cm}
\scriptsize
Interaction along \hkl[1 1 0]
\includegraphics[width=7cm]{db_along_110_cc.ps}
\end{center}
\end{minipage}
}}}
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Defect combinations of C-Si dimers and vacancies
}
\footnotesize

\vspace{0.2cm}

\begin{minipage}[b]{2.6cm}
\begin{flushleft}
\underline{V at 2: $E_{\text{b}}=-0.59\text{ eV}$}\\[0.1cm]
\includegraphics[width=2.5cm]{00-1dc/0-59.eps}
\end{flushleft}
\end{minipage}
\begin{minipage}[b]{7cm}
\hfill
\end{minipage}
\begin{minipage}[b]{2.6cm}
\begin{flushright}
\underline{V at 3, $E_{\text{b}}=-3.14\text{ eV}$}\\[0.1cm]
\includegraphics[width=2.5cm]{00-1dc/3-14.eps}
\end{flushright}
\end{minipage}\\[0.2cm]

\begin{minipage}{6.5cm}
\includegraphics[width=6.0cm]{059-539.ps}
\end{minipage}
\begin{minipage}{5.7cm}
\includegraphics[width=6.0cm]{314-539.ps}
\end{minipage}

\begin{pspicture}(0,0)(0,0)
\psline[linewidth=0.05cm,linecolor=gray](6.3,9.0)(6.3,0)

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\begin{minipage}{6.5cm}
\begin{center}
IBS: Impinging C creates V \& far away \si\\[0.3cm]
Low migration barrier towards C$_{\text{sub}}$\\
\&\\
High barrier for reverse process\\[0.3cm]
{\color{blue}
High probability of stable C$_{\text{sub}}$ configuration
}
\end{center}
\end{minipage}
}}}
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Combinations of substitutional C and Si self-interstitials
}

\scriptsize

\vspace{0.3cm}

\begin{minipage}{6.2cm}
\begin{center}
{\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
\begin{itemize}
 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
 \item Interaction drops quickly to zero\\
       $\rightarrow$ low capture radius
\end{itemize}
\end{center}
\end{minipage}
\begin{minipage}{0.2cm}
\hfill
\end{minipage}
\begin{minipage}{6.0cm}
\begin{center}
{\bf Transition from the ground state}
\begin{itemize}
 \item Low transition barrier
 \item Barrier smaller than \ci{} migration barrier
 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
       $\rightarrow$ Separation of \cs{} \& \si{} most probable
\end{itemize}
\end{center}
\end{minipage}\\[0.3cm]

\begin{minipage}{6.0cm}
\includegraphics[width=6.0cm]{c_sub_si110.ps}
\end{minipage}
\begin{minipage}{0.4cm}
\hfill
\end{minipage}
\begin{minipage}{6.0cm}
\begin{flushright}
\includegraphics[width=6.0cm]{162-097.ps}
\end{flushright}
\end{minipage}

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\begin{minipage}{8cm}
\begin{center}
\vspace{0.1cm}
{\color{black}
\cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
IBS --- process far from equilibrium\\
}
\end{center}
\end{minipage}
}}}
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Combinations of substitutional C and Si self-interstitials
}

\scriptsize

\vspace{0.3cm}

\begin{minipage}{6.2cm}
\begin{center}
{\bf\boldmath C$_{\text{sub}}$ - \si{} \hkl<1 1 0> interaction}
\begin{itemize}
 \item Most favorable: \cs{} along \hkl<1 1 0> chain of \si{}
 \item Less favorable than ground-state \ci{} \hkl<1 0 0> DB
 \item Interaction drops quickly to zero\\
       $\rightarrow$ low capture radius
\end{itemize}
\end{center}
\end{minipage}
\begin{minipage}{0.2cm}
\hfill
\end{minipage}
\begin{minipage}{6.0cm}
\begin{center}
{\bf Transition from the ground state}
\begin{itemize}
 \item Low transition barrier
 \item Barrier smaller than \ci{} migration barrier
 \item Low \si{} migration barrier (\unit[0.67]{eV})\\
       $\rightarrow$ Separation of \cs{} \& \si{} most probable
\end{itemize}
\end{center}
\end{minipage}\\[0.3cm]

\begin{minipage}{6.0cm}
\includegraphics[width=6.0cm]{c_sub_si110.ps}
\end{minipage}
\begin{minipage}{0.4cm}
\hfill
\end{minipage}
\begin{minipage}{6.0cm}
\begin{flushright}
\includegraphics[width=6.0cm]{162-097.ps}
\end{flushright}
\end{minipage}

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\begin{minipage}{8cm}
\begin{center}
\vspace{0.1cm}
{\color{black}
\cs{} \& \si{} instead of thermodynamic ground state\\[0.1cm]
IBS --- process far from equilibrium\\
}
\end{center}
\end{minipage}
}}}
\end{pspicture}

% md support
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\begin{minipage}{14cm}
\hfill
\vspace{14cm}
\end{minipage}
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\begin{center}
\vspace{0.2cm}
\scriptsize
Ab initio MD at \degc{900}\\[0.4cm]
\begin{minipage}{5.4cm}
\centering
\includegraphics[width=4.3cm]{md01_bonds.eps}\\
$t=\unit[2230]{fs}$
\end{minipage}
\begin{minipage}{5.4cm}
\centering
\includegraphics[width=4.3cm]{md02_bonds.eps}\\
$t=\unit[2900]{fs}$
\end{minipage}\\[0.5cm]
{\color{blue}
Contribution of entropy to structural formation\\[0.1cm]
}
\end{center}
\end{minipage}
}}}
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Silicon carbide precipitation simulations
}

\small

\vspace{0.2cm}

{\bf Procedure}

{\scriptsize
 \begin{pspicture}(0,0)(12,6.5)
  % nodes
  \rput(3.5,5.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
   \parbox{7cm}{
   \begin{itemize}
    \item Create c-Si volume
    \item Periodc boundary conditions
    \item Set requested $T$ and $p=0\text{ bar}$
    \item Equilibration of $E_{\text{kin}}$ and $E_{\text{pot}}$
   \end{itemize}
  }}}}
  \rput(3.5,2.7){\rnode{insert}{\psframebox[fillstyle=solid,fillcolor=lachs]{
   \parbox{7cm}{
   Insertion of C atoms at constant T
   \begin{itemize}
    \item total simulation volume {\pnode{in1}}
    \item volume of minimal SiC precipitate size {\pnode{in2}}
    %\item volume consisting of Si atoms to form a minimal {\pnode{in3}}\\
    \item volume containing Si atoms to form a minimal {\pnode{in3}}\\
          precipitate
   \end{itemize} 
  }}}}
  \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
   \parbox{7.0cm}{
   Run for 100 ps followed by cooling down to $20\, ^{\circ}\textrm{C}$
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  \ncline[]{->}{insert}{cool}
  \psframe[fillstyle=solid,fillcolor=white](7.3,0.7)(12.8,6.3)
  \rput(7.6,6){\footnotesize $V_1$}
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  \rput(8.9,4.85){\tiny $V_2$}
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  \rput(8.92,2.8){\pnode{ins2}}
  \rput(10.8,2.4){\pnode{ins3}}
  \ncline[]{->}{in1}{ins1}
  \ncline[]{->}{in2}{ins2}
  \ncline[]{->}{in3}{ins3}
 \end{pspicture}
}

\vspace{-0.5cm}

{\bf Note}

\footnotesize

\begin{minipage}{5.7cm}
\begin{itemize}
 \item Amount of C atoms: 6000\\
       ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: \unit[2--4]{nm})
 \item Simulation volume: $31^3$ Si unit cells\\
       (238328 Si atoms)
\end{itemize}
\end{minipage}
\begin{minipage}{0.3cm}
\hfill
\end{minipage}
\framebox{
\begin{minipage}{6.0cm}
Restricted to classical potential caclulations\\
$\rightarrow$ Low C diffusion / overestimated barrier\\
$\rightarrow$ Consider $V_2$ and $V_3$
%\begin{itemize}
% \item $V_2$ and $V_3$ considered due to expected low C diffusion
%\end{itemize}
\end{minipage}
}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Silicon carbide precipitation simulations at \degc{450} as in IBS
}

\small

\begin{minipage}{6.3cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
\hfill
\end{minipage} 
\begin{minipage}{6.1cm}
\scriptsize
\underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
\ci{} \hkl<1 0 0> dumbbell dominated structure
\begin{itemize}
 \item Si-C bumbs around \unit[0.19]{nm}
 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
       concatenated differently oriented \ci{} DBs
 \item Si-Si NN distance stretched to \unit[0.3]{nm}
\end{itemize}
\begin{pspicture}(0,0)(6.0,1.0)
\rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{6cm}
\centering
Formation of \ci{} dumbbells\\
C atoms separated as expected in 3C-SiC
\end{minipage}
}}
\end{pspicture}\\[0.1cm]
\underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
\begin{itemize}
\item High amount of strongly bound C-C bonds
\item Increased defect \& damage density\\
      $\rightarrow$ Arrangements hard to categorize and trace
\item Only short range order observable
\end{itemize}
\begin{pspicture}(0,0)(6.0,0.8)
\rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{6cm}
\centering
Amorphous SiC-like phase
\end{minipage}
}}
\end{pspicture}\\[0.3cm]
\begin{pspicture}(0,0)(6.0,2.0)
\rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=white]{
\begin{minipage}{6cm}
\hfill
\vspace{2.5cm}
\end{minipage}
}}
\end{pspicture}
\end{minipage} 

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Silicon carbide precipitation simulations at \degc{450} as in IBS
}

\small

\begin{minipage}{6.3cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-c.ps}\\
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{sic_prec_450_si-si_c-c.ps}
\hfill
\end{minipage} 
\begin{minipage}{6.1cm}
\scriptsize
\underline{Low C concentration --- {\color{red}$V_1$}}\\[0.1cm]
\ci{} \hkl<1 0 0> dumbbell dominated structure
\begin{itemize}
 \item Si-C bumbs around \unit[0.19]{nm}
 \item C-C peak at \unit[0.31]{nm} (expected in 3C-SiC):\\
       concatenated differently oriented \ci{} DBs
 \item Si-Si NN distance stretched to \unit[0.3]{nm}
\end{itemize}
\begin{pspicture}(0,0)(6.0,1.0)
\rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{6cm}
\centering
Formation of \ci{} dumbbells\\
C atoms separated as expected in 3C-SiC
\end{minipage}
}}
\end{pspicture}\\[0.1cm]
\underline{High C concentration --- {\color{green}$V_2$}/{\color{blue}$V_3$}}
\begin{itemize}
\item High amount of strongly bound C-C bonds
\item Increased defect \& damage density\\
      $\rightarrow$ Arrangements hard to categorize and trace
\item Only short range order observable
\end{itemize}
\begin{pspicture}(0,0)(6.0,0.8)
\rput(3.2,0.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{6cm}
\centering
Amorphous SiC-like phase
\end{minipage}
}}
\end{pspicture}\\[0.3cm]
\begin{pspicture}(0,0)(6.0,2.0)
\rput(3.2,1.0){\psframebox[linewidth=0.05cm,linecolor=black]{
\begin{minipage}{6cm}
\vspace{0.1cm}
\centering
{\bf\color{red}Formation of 3C-SiC fails to appear}\\[0.3cm]
\begin{minipage}{0.8cm}
{\bf\boldmath $V_1$:}
\end{minipage}
\begin{minipage}{5.1cm}
Formation of \ci{} indeed occurs\\
Agllomeration not observed
\end{minipage}\\[0.3cm]
\begin{minipage}{0.8cm}
{\bf\boldmath $V_{2,3}$:}
\end{minipage}
\begin{minipage}{5.1cm}
Amorphous SiC-like structure\\
(not expected at \degc{450})\\[0.05cm]
No rearrangement/transition into 3C-SiC
\end{minipage}\\[0.1cm]
\end{minipage}
}}
\end{pspicture}
\end{minipage} 

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Limitations of MD and short range potentials
}

\small

\vspace{0.2cm}

{\bf Time scale problem of MD}\\[0.2cm]
Minimize integration error \& precise thermodynamic sampling\\
$\Rightarrow$ $\Delta t \ll \left( \max{\omega} \right)^{-1}$,
              $\omega$: vibrational mode\\
$\Rightarrow$ {\color{red}\underline{Slow}} phase space propagation\\[0.2cm]
Several local minima separated by large energy barriers\\
$\Rightarrow$ Transition event corresponds to a multiple
              of vibrational periods\\
$\Rightarrow$ Phase transition consists of {\color{red}\underline{many}}
              infrequent transition events\\[0.2cm]
{\color{blue}Accelerated methods:}
\underline{Temperature accelerated} MD (TAD), self-guided MD \ldots

\vspace{0.2cm}

{\bf Limitations related to the short range potential}\\[0.2cm]
Cut-off function limits interaction to next neighbours\\
$\Rightarrow$ Overestimated diffusion barrier (factor: 2.4--3.4)

\vspace{1.4cm}

{\bf Approach to the (twofold) problem}\\[0.2cm]
Increased temperature simulations without TAD corrections\\
Accelerated methods or higher time scales exclusively not sufficient!

\begin{pspicture}(0,0)(0,0)
\rput(4.0,2.8){\psframebox[linewidth=0.07cm,linecolor=red]{
\begin{minipage}{7.5cm}
\centering
\vspace{0.05cm}
Potential enhanced slow phase space propagation
\end{minipage}
}}
\rput(11.3,7.5){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{2.7cm}
\tiny
\centering
retain proper\\
thermodynamic sampling
\end{minipage}
}}
\psline[linewidth=0.03cm,linecolor=blue]{<-}(11.3,7.0)(11.0,5.7)
\rput(10.85,2.6){\psframebox[linewidth=0.03cm,linecolor=blue]{
\begin{minipage}{3.6cm}
\tiny
\centering
\underline{IBS}\\[0.1cm]
3C-SiC also observed for higher T\\[0.1cm]
Higher T inside sample\\[0.1cm]
Structural evolution vs.\\
equilibrium properties
\end{minipage}
}}
\psline[linewidth=0.03cm,linecolor=blue]{->}(10.85,1.75)(9.0,1.0)
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Increased temperature simulations --- $V_1$
}

\small

\begin{minipage}{6.2cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
\hfill
\end{minipage}
\begin{minipage}{6.2cm}
\includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
\end{minipage}

\begin{minipage}{6.2cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
\hfill
\end{minipage}
\begin{minipage}{6.3cm}
\scriptsize
 \underline{Si-C bonds:}
 \begin{itemize}
  \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
  \item Structural change: \ci{} \hkl<1 0 0> DB $\rightarrow$
        {\color{blue}\cs{}}
 \end{itemize}
 \underline{Si-Si bonds:}
 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
 ($\rightarrow$ 0.325 nm)\\[0.1cm]
 \underline{C-C bonds:}
 \begin{itemize}
  \item C-C next neighbour pairs reduced (mandatory)
  \item Peak at 0.3 nm slightly shifted\\[0.05cm]
        $\searrow$ \ci{} combinations (dashed arrows)\\
        $\nearrow$ \ci{} \hkl<1 0 0> \& {\color{blue}\cs{} combinations} (|)\\
        $\nearrow$ \ci{} pure \cs{} combinations ($\downarrow$)\\[0.05cm]
        Range [|-$\downarrow$]: {\color{blue}\cs{} \& \cs{} with nearby \si}
 \end{itemize}
\end{minipage}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 Increased temperature simulations --- $V_1$
}

\small

\begin{minipage}{6.2cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc_thesis.ps}
\hfill
\end{minipage}
\begin{minipage}{6.2cm}
\includegraphics[width=6.5cm]{tot_pc3_thesis.ps}
\end{minipage}

\begin{minipage}{6.2cm}
\hspace*{-0.4cm}\includegraphics[width=6.5cm]{tot_pc2_thesis.ps}
\hfill
\end{minipage}
\begin{minipage}{6.3cm}
\scriptsize
 \underline{Si-C bonds:}
 \begin{itemize}
  \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
  \item Structural change: \ci{} \hkl<1 0 0> DB $\rightarrow$
        {\color{blue}\cs{}}
 \end{itemize}
 \underline{Si-Si bonds:}
 {\color{blue}Si-C$_{\text{sub}}$-Si} along \hkl<1 1 0>
 ($\rightarrow$ 0.325 nm)\\[0.1cm]
 \underline{C-C bonds:}
 \begin{itemize}
  \item C-C next neighbour pairs reduced (mandatory)
  \item Peak at 0.3 nm slightly shifted\\[0.05cm]
        $\searrow$ \ci{} combinations (dashed arrows)\\
        $\nearrow$ \ci{} \hkl<1 0 0> \& {\color{blue}\cs{} combinations} (|)\\
        $\nearrow$ \ci{} pure \cs{} combinations ($\downarrow$)\\[0.05cm]
        Range [|-$\downarrow$]: {\color{blue}\cs{} \& \cs{} with nearby \si}
 \end{itemize}
\end{minipage}

% conclusions
\begin{pspicture}(0,0)(0,0)
\rput(6.5,5.0){\psframebox[fillstyle=solid,opacity=0.5,fillcolor=black]{
\begin{minipage}{14cm}
\hfill
\vspace{14cm}
\end{minipage}
}}
\rput(6.5,5.0){\psframebox[fillstyle=solid,fillcolor=white,linewidth=0.1cm]{
\begin{minipage}{9cm}
\vspace{0.2cm}
\small
\begin{center}
{\color{gray}\bf Conclusions on SiC precipitation}\\[0.1cm]
{\Huge$\lightning$} {\color{red}\ci{}} --- vs --- {\color{blue}\cs{}} {\Huge$\lightning$}\\
\end{center}
\begin{itemize}
\item Stretched coherent SiC structures directly observed\\
\psframebox[linecolor=blue,linewidth=0.05cm]{
\begin{minipage}{7cm}
\centering
\cs{} involved in the precipitation mechanism\\
\end{minipage}
}
\item Emission of \si{} serves several needs:
      \begin{itemize}
       \item Vehicle to rearrange \cs --- [\cs{} \& \si{} $\leftrightarrow$ \ci]
       \item Building block for surrounding Si host \& further SiC
       \item Strain compensation \ldots\\
             \ldots Si/SiC interface\\
             \ldots within stretched coherent SiC structure
      \end{itemize}
\item Explains annealing behavior of high/low T C implantations
      \begin{itemize}
       \item Low T: highly mobile {\color{red}\ci}
       \item High T: stable configurations of {\color{blue}\cs}
      \end{itemize}
\psframebox[linecolor=blue,linewidth=0.05cm]{
\begin{minipage}{7cm}
\centering
High T $\leftrightarrow$ IBS conditions far from equilibrium\\
\end{minipage}
}
\end{itemize}
\end{minipage}
\vspace{0.2cm}
}}
\end{pspicture}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Summary and Conclusions
}

\footnotesize

\vspace{0.1cm}

\framebox{
\begin{minipage}{12.3cm}
 \underline{Defects}
 \begin{itemize}
   \item DFT / EA
        \begin{itemize}
         \item Point defects excellently / fairly well described
               by DFT / EA
         \item Identified \ci{} migration path
         \item EA drastically overestimates the diffusion barrier
        \end{itemize}
   \item Combinations of defects (DFT)
         \begin{itemize}
          \item Agglomeration of point defects energetically favorable
          \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
          \item \ci{} \hkl<1 0 0> $\leftrightarrow$ \cs{} \& \si{} \hkl<1 1 0>\\
                Low barrier (\unit[0.77]{eV}) \& low capture radius
        \end{itemize}
 \end{itemize}
\end{minipage}
}

\framebox{
\begin{minipage}[t]{12.3cm}
 \underline{Pecipitation simulations}
 \begin{itemize}
  \item Problem of potential enhanced slow phase space propagation
  \item High T necessary to simulate IBS conditions (far from equilibrium)
  \item Low T $\rightarrow$ C-Si \hkl<1 0 0> dumbbell dominated structure
  \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
        / Structures of stretched SiC\\
        $\Rightarrow$
        \cs{} involved in the precipitation process at elevated temperatures
  \item \si{}: vehicle to form \cs{} \& supply of Si \& stress compensation
        (stretched SiC, interface)
 \end{itemize}
\end{minipage}
}

\begin{center}
{\color{blue}\bf
\framebox{IBS: 3C-SiC precipitation occurs by successive agglomeration of \cs{}}
}
\end{center}

\end{slide}

\begin{slide}

\headphd
{\large\bf
 Acknowledgements
}

 \vspace{0.1cm}

 \small

 Thanks to \ldots

\begin{minipage}[t]{6cm}
 \underline{Augsburg}
 \begin{itemize}
  \item Prof. B. Stritzker
  \item Ralf Utermann
  \item EP \RM{4}
 \end{itemize}
 
 \underline{Helsinki}
 \begin{itemize}
  \item Prof. K. Nordlund
 \end{itemize}
 
 \underline{Munich}
 \begin{itemize}
  \item Bayerische Forschungsstiftung
 \end{itemize}
 
 \underline{Paderborn}
 \begin{itemize}
  \item Prof. J. Lindner
  \item Prof. G. Schmidt
  \item Dr. E. Rauls
 \end{itemize}
\end{minipage}
\begin{minipage}[t]{6cm}
\underline{Referees}
 \begin{itemize}
  \item PD V. Eyert
  \item Prof. F. Haider
 \end{itemize}
\end{minipage}

\vspace{0.5cm}
\begin{center}
\framebox{
\Large\bf Thank you for your attention!
}
\end{center}

\end{slide}







\begin{slide}

\headphd
 {\large\bf
  Polytypes of SiC\\[0.6cm]
 }

\vspace{0.6cm}

\includegraphics[width=3.8cm]{cubic_hex.eps}\\
\begin{minipage}{1.9cm}
{\tiny cubic (twist)}
\end{minipage}
\begin{minipage}{2.9cm}
{\tiny hexagonal (no twist)}
\end{minipage}

\begin{picture}(0,0)(-150,0)
 \includegraphics[width=7cm]{polytypes.eps}
\end{picture}

\vspace{0.6cm}

\footnotesize

\begin{tabular}{l c c c c c c}
\hline
 & 3C-SiC & 4H-SiC & 6H-SiC & Si & GaN & Diamond\\
\hline
Hardness [Mohs] & \multicolumn{3}{c}{------ 9.6 ------}& 6.5 & - & 10 \\
Band gap [eV] & 2.36 & 3.23 & 3.03 & 1.12 & 3.39 & 5.5 \\
Break down field [$10^6$ V/cm] & 4 & 3 & 3.2 & 0.6 & 5 & 10 \\
Saturation drift velocity [$10^7$ cm/s] & 2.5 & 2.0 & 2.0 & 1 & 2.7 & 2.7 \\
Electron mobility [cm$^2$/Vs] & 800 & 900 & 400 & 1100 & 900 & 2200 \\
Hole mobility [cm$^2$/Vs] & 320 & 120 & 90 & 420 & 150 & 1600 \\
Thermal conductivity [W/cmK] & 5.0 & 4.9 & 4.9 & 1.5 & 1.3 & 22 \\
\hline
\end{tabular}

\begin{pspicture}(0,0)(0,0)
\psellipse[linecolor=green](5.7,2.05)(0.4,0.50)
\end{pspicture}
\begin{pspicture}(0,0)(0,0)
\psellipse[linecolor=green](5.6,0.89)(0.4,0.20)
\end{pspicture}
\begin{pspicture}(0,0)(0,0)
\psellipse[linecolor=red](10.45,0.42)(0.4,0.20)
\end{pspicture}

\end{slide}

\begin{slide}

\footnotesize

\headphd
{\large\bf
 Si self-interstitial point defects in silicon\\[0.1cm]
}

\begin{center}
\begin{tabular}{l c c c c c}
\hline
 $E_{\text{f}}$ [eV] & \hkl<1 1 0> DB & H & T & \hkl<1 0 0> DB & V \\
\hline
 \textsc{vasp} & \underline{3.39} & 3.42 & 3.77 & 4.41 & 3.63 \\
 Erhart/Albe & 4.39 & 4.48$^*$ & \underline{3.40} & 5.42 & 3.13 \\
\hline
\end{tabular}\\[0.4cm]
\end{center}

\begin{minipage}{3cm}
\begin{center}
\underline{Vacancy}\\
\includegraphics[width=2.8cm]{si_pd_albe/vac.eps}
\end{center}
\end{minipage}
\begin{minipage}{3cm}
\begin{center}
\underline{\hkl<1 1 0> DB}\\
\includegraphics[width=2.8cm]{si_pd_albe/110_bonds.eps}
\end{center}
\end{minipage}
\begin{minipage}{3cm}
\begin{center}
\underline{\hkl<1 0 0> DB}\\
\includegraphics[width=2.8cm]{si_pd_albe/100_bonds.eps}
\end{center}
\end{minipage}
\begin{minipage}{3cm}
\begin{center}
\underline{Tetrahedral}\\
\includegraphics[width=2.8cm]{si_pd_albe/tet_bonds.eps}
\end{center}
\end{minipage}\\

\underline{Hexagonal} \hspace{2pt}
\href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
\framebox{
\begin{minipage}{2.7cm}
$E_{\text{f}}^*=4.48\text{ eV}$\\
\includegraphics[width=2.7cm]{si_pd_albe/hex_a_bonds.eps}
\end{minipage}
\begin{minipage}{0.4cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{2.7cm}
$E_{\text{f}}=3.96\text{ eV}$\\
\includegraphics[width=2.8cm]{si_pd_albe/hex_bonds.eps}
\end{minipage}
}
\begin{minipage}{5.5cm}
\begin{center}
{\tiny nearly T $\rightarrow$ T}\\
\end{center}
\includegraphics[width=6.0cm]{nhex_tet.ps}
\end{minipage}

\end{slide}

\begin{slide}

\headphd
{\large\bf\boldmath
 C-Si dimer \& bond-centered interstitial configuration
}

\footnotesize

\vspace{0.1cm}

\begin{minipage}[t]{4.1cm}
{\bf\boldmath C \hkl<1 0 0> DB interstitial}\\[0.1cm]
\begin{minipage}{2.0cm}
\begin{center}
\underline{Erhart/Albe}
\includegraphics[width=2.0cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\begin{minipage}{2.0cm}
\begin{center}
\underline{\textsc{vasp}}
\includegraphics[width=2.0cm]{c_pd_vasp/100_cmp.eps}
\end{center}
\end{minipage}\\[0.2cm]
Si-C-Si bond angle $\rightarrow$ \unit[180]{$^{\circ}$}\\
$\Rightarrow$ $sp$ hybridization\\[0.1cm]
Si-Si-Si bond angle $\rightarrow$ \unit[120]{$^{\circ}$}\\
$\Rightarrow$ $sp^2$ hybridization
\begin{center}
\includegraphics[width=3.4cm]{c_pd_vasp/eden.eps}\\[-0.1cm]
{\tiny Charge density isosurface}
\end{center}
\end{minipage}
\begin{minipage}{0.2cm}
\hfill
\end{minipage}
\begin{minipage}[t]{8.1cm}
\begin{flushright}
{\bf Bond-centered interstitial}\\[0.1cm]
\begin{minipage}{4.4cm}
%\scriptsize
\begin{itemize}
 \item Linear Si-C-Si bond
 \item Si: one C \& 3 Si neighbours
 \item Spin polarized calculations
 \item No saddle point!\\
       Real local minimum!
\end{itemize}
\end{minipage}
\begin{minipage}{2.7cm}
%\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
\vspace{0.2cm}
\includegraphics[width=2.8cm]{c_pd_albe/bc_bonds.eps}\\
\end{minipage}

\framebox{
 \tiny
 \begin{minipage}[t]{6.5cm}
  \begin{minipage}[t]{1.2cm}
  {\color{red}Si}\\
  {\tiny sp$^3$}\\[0.8cm]
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{red}\uparrow}$}\\
  sp$^3$
  \end{minipage}
  \begin{minipage}[t]{1.4cm}
  \begin{center}
  {\color{red}M}{\color{blue}O}\\[0.8cm]
  \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
  $\sigma_{\text{ab}}$\\[0.5cm]
  \underline{${\color{red}\uparrow}{\color{blue}\downarrow}$}\\
  $\sigma_{\text{b}}$
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.0cm}
  \begin{center}
  {\color{blue}C}\\
  {\tiny sp}\\[0.2cm]
  \underline{${\color{white}\uparrow\uparrow}$}
  \underline{${\color{white}\uparrow\uparrow}$}\\
  2p\\[0.4cm]
  \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}
  \underline{${\color{blue}\uparrow}{\color{blue}\downarrow}$}\\
  sp
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.4cm}
  \begin{center}
  {\color{blue}M}{\color{green}O}\\[0.8cm]
  \underline{${\color{blue}\uparrow}{\color{white}\downarrow}$}\\
  $\sigma_{\text{ab}}$\\[0.5cm]
  \underline{${\color{green}\uparrow}{\color{blue}\downarrow}$}\\
  $\sigma_{\text{b}}$
  \end{center}
  \end{minipage}
  \begin{minipage}[t]{1.2cm}
  \begin{flushright}
  {\color{green}Si}\\
  {\tiny sp$^3$}\\[0.8cm]
  \underline{${\color{green}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}
  \underline{${\color{black}\uparrow}$}\\
  sp$^3$
  \end{flushright}
  \end{minipage}
 \end{minipage}
}\\[0.4cm]

%\framebox{
\begin{minipage}{3.0cm}
%\scriptsize
\underline{Charge density}\\
{\color{gray}$\bullet$} Spin up\\
{\color{green}$\bullet$} Spin down\\
{\color{blue}$\bullet$} Resulting spin up\\
{\color{yellow}$\bullet$} Si atoms\\
{\color{red}$\bullet$} C atom
\end{minipage}
\begin{minipage}{3.6cm}
\includegraphics[width=3.8cm]{c_100_mig_vasp/im_spin_diff.eps}
\end{minipage}
%}

\end{flushright}

\end{minipage}
\begin{pspicture}(0,0)(0,0)
\psline[linecolor=gray,linewidth=0.05cm](-7.8,-8.7)(-7.8,0)
\end{pspicture}

\end{slide}

\begin{slide}

 {\large\bf
  Increased temperature simulations at high C concentration
 }

\footnotesize

\begin{minipage}{6.0cm}
\includegraphics[width=6.4cm]{12_pc_thesis.ps}
\end{minipage}
\begin{minipage}{6.0cm}
\includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
\end{minipage}

\vspace{0.1cm}

\scriptsize

\framebox{
\begin{minipage}[t]{5.5cm}
0.186 nm: Si-C pairs $\uparrow$\\
(as expected in 3C-SiC)\\[0.2cm]
0.282 nm: Si-C-C\\[0.2cm]
$\approx$0.35 nm: C-Si-Si
\end{minipage}
}
\begin{minipage}{0.1cm}
\hfill
\end{minipage}
\framebox{
\begin{minipage}[t]{5.9cm}
0.15 nm: C-C pairs $\uparrow$\\
(as expected in graphite/diamond)\\[0.2cm]
0.252 nm: C-C-C (2$^{\text{nd}}$ NN for diamond)\\[0.2cm]
0.31 nm: shifted towards 0.317 nm $\rightarrow$ C-Si-C
\end{minipage}
}

\begin{itemize}
\item Decreasing cut-off artifact
\item {\color{red}Amorphous} SiC-like phase remains
\item High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
\item Slightly sharper peaks $\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics} due to temperature
\end{itemize}

\begin{center}
{\color{blue}
\framebox{
{\color{black}
High C \& small $V$ \& short $t$
$\Rightarrow$
}
\begin{minipage}{4cm}
\begin{center}
Slow structural evolution due to strong C-C bonds
\end{center}
\end{minipage}
{\color{black}
$\Leftarrow$
High C \& low T implants
}
}
}
\end{center}

\end{slide}



\begin{slide}

 {\large\bf
  Valuation of a practicable temperature limit
 }

 \small

\vspace{0.1cm}

\begin{center}
\framebox{
{\color{blue}
Recrystallization is a hard task!
$\Rightarrow$ Avoid melting!
}
}
\end{center}
 
\vspace{0.1cm}

\footnotesize

\begin{minipage}{6.4cm}
\includegraphics[width=6.4cm]{fe_and_t.ps}
\end{minipage}
\begin{minipage}{5.7cm}
\underline{Melting does not occur instantly after}\\
\underline{exceeding the melting point $T_{\text{m}}=2450\text{ K}$}
\begin{itemize}
\item required transition enthalpy
\item hysterisis behaviour
\end{itemize}
\underline{Heating up c-Si by 1 K/ps}
\begin{itemize}
\item transition occurs at $\approx$ 3125 K
\item $\Delta E=0.58\text{ eV/atom}=55.7\text{ kJ/mole}$\\
      (literature: 50.2 kJ/mole)
\end{itemize}
\end{minipage}

\vspace{0.1cm}

\framebox{
\begin{minipage}{4cm}
Initially chosen temperatures:\\
$1.0 - 1.2 \cdot T_{\text{m}}$
\end{minipage}
}
\begin{minipage}{2cm}
\begin{center}
$\Longrightarrow$
\end{center}
\end{minipage}
\framebox{
\begin{minipage}{5cm}
Introduced C (defects)\\
$\rightarrow$ reduction of transition point\\
$\rightarrow$ melting already at $T_{\text{m}}$
\end{minipage}
}

\vspace{0.4cm}

\begin{center}
\framebox{
{\color{blue}
Maximum temperature used: $0.95\cdot T_{\text{m}}$
}
}
\end{center}

\end{slide}

\begin{slide}

 {\large\bf
  Long time scale simulations at maximum temperature
 }

\small

\vspace{0.1cm}
 
\underline{Differences}
\begin{itemize}
 \item Temperature set to $0.95 \cdot T_{\text{m}}$
 \item Cubic insertion volume $\Rightarrow$ spherical insertion volume
 \item Amount of C atoms: 6000 $\rightarrow$ 5500
       $\Leftrightarrow r_{\text{prec}}=0.3\text{ nm}$
 \item Simulation volume: 21 unit cells of c-Si in each direction
\end{itemize}

\footnotesize

\vspace{0.3cm}

\begin{minipage}[t]{4.3cm}
\begin{center}
\underline{Low C concentration, Si-C}
\includegraphics[width=4.3cm]{c_in_si_95_v1_si-c.ps}\\
Sharper peaks!
\end{center}
\end{minipage}
\begin{minipage}[t]{4.3cm}
\begin{center}
\underline{Low C concentration, C-C}
\includegraphics[width=4.3cm]{c_in_si_95_v1_c-c.ps}\\
Sharper peaks!\\
No C agglomeration!
\end{center}
\end{minipage}
\begin{minipage}[t]{3.4cm}
\begin{center}
\underline{High C concentration}
\includegraphics[width=4.3cm]{c_in_si_95_v2.ps}\\
No significant changes\\
iC-Si-Si $\uparrow$\\
C-Si-C $\downarrow$
\end{center}
\end{minipage}

\begin{center}
\framebox{
Long time scales and high temperatures most probably not sufficient enough!
}
\end{center}

\end{slide}

\begin{slide}

 {\large\bf
  Investigation of a silicon carbide precipitate in silicon
 }

 \scriptsize

\vspace{0.2cm}

\framebox{
\scriptsize
\begin{minipage}{5.3cm}
\[
\frac{8}{a_{\text{Si}}^3}(
\underbrace{21^3 a_{\text{Si}}^3}_{=V}
-\frac{4}{3}\pi x^3)+
\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
=21^3\cdot 8
\]
\[
\Downarrow
\]
\[
\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
\]
\[
y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
\]
\end{minipage}
}
\begin{minipage}{0.1cm}
\hfill
\end{minipage}
\begin{minipage}{6.3cm}
\underline{Construction}
\begin{itemize}
 \item Simulation volume: 21$^3$ unit cells of c-Si
 \item Spherical topotactically aligned precipitate\\
       $r=3.0\text{ nm}$ $\Leftrightarrow$ $\approx$ 5500 C atoms
 \item Create c-Si but skipped inside sphere\\
       of radius $x$
 \item Create 3C-SiC inside sphere of radius $x$\\
       and lattice constant $y$
 \item Strong coupling to heat bath ($T=20\,^{\circ}\mathrm{C}$)
\end{itemize}
\end{minipage}

\vspace{0.3cm}

\begin{minipage}{6.0cm}
\includegraphics[width=6cm]{pc_0.ps}
\end{minipage}
\begin{minipage}{6.1cm}
\underline{Results}
\begin{itemize}
 \item Slight increase of c-Si lattice constant!
 \item C-C peaks\\
       (imply same distanced Si-Si peaks)
       \begin{itemize}
        \item New peak at 0.307 nm: 2$^{\text{nd}}$ NN in 3C-SiC
        \item Bumps ({\color{green}$\downarrow$}):
              4$^{\text{th}}$ and 6$^{\text{th}}$ NN
       \end{itemize}
 \item 3C-SiC lattice constant: 4.34 \AA (bulk: 4.36 \AA)\\
       $\rightarrow$ compressed precipitate
 \item Interface tension:\\
       20.15 eV/nm$^2$ or $3.23 \times 10^{-4}$ J/cm$^2$\\
       (literature: $2 - 8 \times 10^{-4}$ J/cm$^2$)
\end{itemize}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Investigation of a silicon carbide precipitate in silicon
 }

 \footnotesize

\begin{minipage}{7cm}
\underline{Appended annealing steps}
\begin{itemize}
 \item artificially constructed interface\\
       $\rightarrow$ allow for rearrangement of interface atoms
 \item check SiC stability
\end{itemize}
\underline{Temperature schedule}
\begin{itemize}
 \item rapidly heat up structure up to $2050\,^{\circ}\mathrm{C}$\\
       (75 K/ps)
 \item slow heating up to $1.2\cdot T_{\text{m}}=2940\text{ K}$
       by 1 K/ps\\
       $\rightarrow$ melting at around 2840 K
       (\href{../video/sic_prec_120.avi}{$\rhd$})
 \item cooling down structure at 100 \% $T_{\text{m}}$ (1 K/ps)\\
       $\rightarrow$ no energetically more favorable struture
\end{itemize}
\end{minipage}
\begin{minipage}{5cm}
\includegraphics[width=5.5cm]{fe_and_t_sic.ps}
\end{minipage}

\begin{minipage}{4cm}
\includegraphics[width=4cm]{sic_prec/melt_01.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4cm}
\includegraphics[width=4cm]{sic_prec/melt_02.eps}
\end{minipage}
\begin{minipage}{0.2cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{3.7cm}
\includegraphics[width=4cm]{sic_prec/melt_03.eps}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  DFT parameters
 }

\scriptsize

\vspace{0.1cm}

Equilibrium lattice constants and cohesive energies

\begin{tabular}{l r c c c c c}
\hline
\hline
 & & USPP, LDA & USPP, GGA & PAW, LDA & PAW, GGA & Exp. \\
\hline
Si (dia) & $a$ [\AA] & 5.389 & 5.455 & - & - & 5.429 \\
         & $\Delta_a$ [\%] & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - & - & - \\
       & $E_{\text{coh}}$ [eV] & -5.277 & -4.591 & - & - & -4.63 \\
       & $\Delta_E$ [\%] & \unit[{\color{red}14.0}]{\%} & \unit[{\color{green}0.8}]{\%} & - & - & - \\
\hline
C (dia) & $a$ [\AA] & 3.527 & 3.567 & - & - & 3.567 \\
         & $\Delta_a$ [\%] & \unit[{\color{green}1.1}]{\%} & \unit[{\color{green}0.01}]{\%} & - & - & - \\
       & $E_{\text{coh}}$ [eV] & -8.812 & -7.703 & - & - & -7.374 \\
       & $\Delta_E$ [\%] & \unit[{\color{red}19.5}]{\%} & \unit[{\color{orange}4.5}]{\%} & - & - & - \\
\hline
3C-SiC & $a$ [\AA] & 4.319 & 4.370 & 4.330 & 4.379 & 4.359 \\
         & $\Delta_a$ [\%] & \unit[{\color{green}0.9}]{\%} & \unit[{\color{green}0.3}]{\%} & \unit[{\color{green}0.7}]{\%} & \unit[{\color{green}0.5}]{\%} & - \\
       & $E_{\text{coh}}$ [eV] & -7.318 & -6.426 &  -7.371 & -6.491 & -6.340 \\
       & $\Delta_E$ [\%] & \unit[{\color{red}15.4}]{\%} & \unit[{\color{green}1.4}]{\%} & \unit[{\color{red}16.3}]{\%} & \unit[{\color{orange}2.4}]{\%} & - \\
\hline
\hline
\end{tabular}

\vspace{0.3cm}

\begin{minipage}{7cm}
\begin{center}
\begin{tabular}{l c c c}
\hline
\hline
 & Si (dia) & C (dia) & 3C-SiC \\
\hline
$a$ [\AA] & 5.458 & 3.562 & 4.365 \\
$\Delta_a$ [\%] & 0.5 & 0.1 & 0.1 \\
\hline
$E_{\text{coh}}$ [eV] & -4.577 & -7.695 & -6.419 \\
$\Delta_E$ [\%] & 1.1 & 4.4 & 1.2 \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\begin{minipage}{5cm}
$\leftarrow$ entire parameter set
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  DFT parameters\\
 }

\footnotesize

\begin{minipage}{6cm}
\begin{center}
\includegraphics[width=6cm]{sic_32pc_gamma_cutoff_lc.ps}
\end{center}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
Lattice constants with respect to the PW cut-off energy
\end{center}
\end{minipage}

\begin{minipage}{6cm}
\begin{center}
\includegraphics[width=6cm]{si_self_int_thesis.ps}
\end{center}
\end{minipage}
\begin{minipage}{6cm}
\begin{center}
Defect formation energy with respect to the size of the supercell\\[0.1cm]
\end{center}

\end{minipage}

\end{slide}

\end{document}