basically finished point dfect chapter

[lectures/latex.git] / posic / talks / seminar_2010.tex

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% topic

\begin{slide}
\begin{center}

 \vspace{16pt}

 {\LARGE\bf
  Atomistic simulation study of the silicon carbide precipitation
  in silicon
 }

 \vspace{48pt}

 \textsc{F. Zirkelbach}

 \vspace{48pt}

 Lehrstuhlseminar

 \vspace{08pt}

 17. Juni 2010

\end{center}
\end{slide}

% motivation / properties / applications of silicon carbide
\begin{slide}

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 \rput[lt](0.2,4.6){\color{gray}PROPERTIES}

 \rput[lt](0.5,4){wide band gap}
 \rput[lt](0.5,3.5){high electric breakdown field}
 \rput[lt](0.5,3){good electron mobility}
 \rput[lt](0.5,2.5){high electron saturation drift velocity}
 \rput[lt](0.5,2){high thermal conductivity}

 \rput[lt](0.5,1.5){hard and mechanically stable}
 \rput[lt](0.5,1){chemically inert}

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

 \rput[rt](13.3,4.6){\color{gray}APPLICATIONS}

 \rput[rt](13,3.85){high-temperature, high power}
 \rput[rt](13,3.5){and high-frequency}
 \rput[rt](13,3.15){electronic and optoelectronic devices}

 \rput[rt](13,2.35){material suitable for extreme conditions}
 \rput[rt](13,2){microelectromechanical systems}
 \rput[rt](13,1.65){abrasives, cutting tools, heating elements}

 \rput[rt](13,0.85){first wall reactor material, detectors}
 \rput[rt](13,0.5){and electronic devices for space}

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% contents

\begin{slide}

{\large\bf
 Outline
}

 \begin{itemize}
  \item Polytyps and fabrication of silicon carbide
  \item Supposed precipitation mechanism of SiC in Si
  \item Utilized simulation techniques
        \begin{itemize}
         \item Molecular dynamics (MD) simulations
         \item Density functional theory (DFT) calculations
        \end{itemize}
  \item C and Si self-interstitial point defects in silicon
  \item Silicon carbide precipitation simulations
  \item Investigation of a silicon carbide precipitate in silicon
  \item Summary / Conclusion / Outlook
 \end{itemize}

\end{slide}

% start of contents

\begin{slide}

 {\large\bf
  Polytypes of SiC
 }

 \vspace{4cm}

 \small

\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}

{\tiny
 Values for $T=300$ K
}

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 {\tiny cubic (twist)}
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 {\tiny hexagonal (no twist)}
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\begin{slide}

 {\large\bf
  Fabrication of silicon carbide
 }

 \small
 
 \vspace{4pt}

 SiC - \emph{Born from the stars, perfected on earth.}
 
 \vspace{4pt}

 Conventional thin film SiC growth:
 \begin{itemize}
  \item \underline{Sublimation growth using the modified Lely method}
        \begin{itemize}
         \item SiC single-crystalline seed at $T=1800 \, ^{\circ} \text{C}$
         \item Surrounded by polycrystalline SiC in a graphite crucible\\
               at $T=2100-2400 \, ^{\circ} \text{C}$
         \item Deposition of supersaturated vapor on cooler seed crystal
        \end{itemize}
  \item \underline{Homoepitaxial growth using CVD}
        \begin{itemize}
         \item Step-controlled epitaxy on off-oriented 6H-SiC substrates
         \item C$_3$H$_8$/SiH$_4$/H$_2$ at $1100-1500 \, ^{\circ} \text{C}$
         \item Angle, temperature $\rightarrow$ 3C/6H/4H-SiC
         \item High quality but limited in size of substrates
        \end{itemize}
  \item \underline{Heteroepitaxial growth of 3C-SiC on Si using CVD/MBE}
        \begin{itemize}
         \item Two steps: carbonization and growth
         \item $T=650-1050 \, ^{\circ} \text{C}$
         \item Quality and size not yet sufficient
        \end{itemize}
 \end{itemize}

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  \begin{minipage}{5cm}
  {\tiny
   NASA: 6H-SiC and 3C-SiC LED\\[-7pt]
   on 6H-SiC substrate
  }
  \end{minipage}
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  {\tiny
   1. Lid\\[-7pt]
   2. Heating\\[-7pt]
   3. Source\\[-7pt]
   4. Crucible\\[-7pt]
   5. Insulation\\[-7pt]
   6. Seed crystal
  }
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 {\footnotesize\color{blue}\bf Hex: micropipes along c-axis}
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 \begin{minipage}{3cm}
 {\footnotesize\color{blue}\bf 3C-SiC fabrication\\
                               less advanced}
 \end{minipage}
 }
 \end{picture}

\end{slide}

\begin{slide}

 {\large\bf
  Fabrication of silicon carbide
 }

 \small

 Alternative approach:
 Ion beam synthesis (IBS) of burried 3C-SiC layers in Si\hkl(1 0 0)
 \begin{itemize}
  \item \underline{Implantation step 1}\\
        180 keV C$^+$, $D=7.9\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=500\,^{\circ}\mathrm{C}$\\
        $\Rightarrow$ box-like distribution of equally sized
                       and epitactically oriented SiC precipitates
                       
  \item \underline{Implantation step 2}\\
        180 keV C$^+$, $D=0.6\times 10^{17}$ cm$^{-2}$, $T_{\text{i}}=250\,^{\circ}\mathrm{C}$\\
        $\Rightarrow$ destruction of SiC nanocrystals
                      in growing amorphous interface layers
  \item \underline{Annealing}\\
        $T=1250\,^{\circ}\mathrm{C}$, $t=10\,\text{h}$\\
        $\Rightarrow$ homogeneous, stoichiometric SiC layer
                      with sharp interfaces
 \end{itemize}

 \begin{minipage}{6.3cm}
 \includegraphics[width=6cm]{ibs_3c-sic.eps}\\[-0.2cm]
 {\tiny
  XTEM micrograph of single crystalline 3C-SiC in Si\hkl(1 0 0)
 }
 \end{minipage}
\framebox{
 \begin{minipage}{6.3cm}
 \begin{center}
 {\color{blue}
  Precipitation mechanism not yet fully understood!
 }
 \renewcommand\labelitemi{$\Rightarrow$}
 \small
 \underline{Understanding 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{center}
 \end{minipage}
}
 
\end{slide}

\begin{slide}

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

 \scriptsize

 \vspace{0.1cm}

 \begin{minipage}{3.8cm}
 Si \& SiC lattice structure\\[0.2cm]
 \includegraphics[width=3.5cm]{sic_unit_cell.eps}\\[-0.3cm]
 \hrule
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_c-si-db.eps}
 \end{center}
 \end{minipage}
 \hspace{0.6cm}
 \begin{minipage}{3.8cm}
 \begin{center}
 \includegraphics[width=3.3cm]{tem_3c-sic.eps}
 \end{center}
 \end{minipage}

 \begin{minipage}{4cm}
 \begin{center}
 C-Si dimers (dumbbells)\\[-0.1cm]
 on Si interstitial sites
 \end{center}
 \end{minipage}
 \hspace{0.2cm}
 \begin{minipage}{4.2cm}
 \begin{center}
 Agglomeration of C-Si dumbbells\\[-0.1cm]
 $\Rightarrow$ dark contrasts
 \end{center}
 \end{minipage}
 \hspace{0.2cm}
 \begin{minipage}{4cm}
 \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}

 \begin{minipage}{3.8cm}
 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_01.eps}
 \end{center}
 \end{minipage}
 \hspace{0.6cm}
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 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_02.eps}
 \end{center}
 \end{minipage}
 \hspace{0.6cm}
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 \begin{center}
 \includegraphics[width=3.3cm]{sic_prec_seq_03.eps}
 \end{center}
 \end{minipage}

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\begin{slide}

 {\large\bf
  Molecular dynamics (MD) simulations
 }

 \vspace{12pt}

 \small

 {\bf MD basics:}
 \begin{itemize}
  \item Microscopic description of N particle system
  \item Analytical interaction potential
  \item Numerical integration using Newtons equation of motion\\
        as a propagation rule in 6N-dimensional phase space
  \item Observables obtained by time and/or ensemble averages
 \end{itemize}
 {\bf Details of the simulation:}
 \begin{itemize}
  \item Integration: Velocity Verlet, timestep: $1\text{ fs}$
  \item Ensemble: NpT (isothermal-isobaric)
        \begin{itemize}
         \item Berendsen thermostat:
               $\tau_{\text{T}}=100\text{ fs}$
         \item Berendsen barostat:\\
               $\tau_{\text{P}}=100\text{ fs}$,
               $\beta^{-1}=100\text{ GPa}$
        \end{itemize}
  \item Erhart/Albe potential: Tersoff-like bond order potential
  \vspace*{12pt}
        \[
        E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
        \pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
        \]
 \end{itemize}

 \begin{picture}(0,0)(-230,-30)
  \includegraphics[width=5cm]{tersoff_angle.eps} 
 \end{picture}
 
\end{slide}

\begin{slide}

 {\large\bf
  Density functional theory (DFT) calculations
 }

 \small

 Basic ingredients necessary for DFT

 \begin{itemize}
  \item \underline{Hohenberg-Kohn theorem} - ground state density $n_0(r)$ ...
        \begin{itemize}
         \item ... uniquely determines the ground state potential
               / wavefunctions
         \item ... minimizes the systems total energy
        \end{itemize}
  \item \underline{Born-Oppenheimer}
        - $N$ moving electrons in an external potential of static nuclei
\[
H\Psi = \left[-\sum_i^N \frac{\hbar^2}{2m}\nabla_i^2
              +\sum_i^N V_{\text{ext}}(r_i)
              +\sum_{i<j}^N V_{e-e}(r_i,r_j)\right]\Psi=E\Psi
\]
  \item \underline{Effective potential}
        - averaged electrostatic potential \& exchange and correlation
\[
V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
                 +V_{\text{XC}}[n(r)]
\]
  \item \underline{Kohn-Sham system}
        - Schr\"odinger equation of N non-interacting particles
\[
\left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
=\epsilon_i\Phi_i(r)
\quad
\Rightarrow
\quad
n(r)=\sum_i^N|\Phi_i(r)|^2
\]
  \item \underline{Self-consistent solution}\\
$n(r)$ depends on $\Phi_i$, which depend on $V_{\text{eff}}$,
which in turn depends on $n(r)$
  \item \underline{Variational principle}
        - minimize total energy with respect to $n(r)$
 \end{itemize}

\end{slide}

\begin{slide}

 {\large\bf
  Density functional theory (DFT) calculations
 }

 \small

 \vspace*{0.2cm}

 Details of applied DFT calculations in this work

 \begin{itemize}
  \item \underline{Exchange correlation functional}
        - approximations for the inhomogeneous electron gas
        \begin{itemize}
         \item LDA: $E_{\text{XC}}^{\text{LDA}}[n]=\int \epsilon_{\text{XC}}(n)n(r)d^3r$
         \item GGA: $E_{\text{XC}}^{\text{GGA}}[n]=\int \epsilon_{\text{XC}}(n,\nabla n)n(r)d^3r$
        \end{itemize}
  \item \underline{Plane wave basis set}
        - approximation of the wavefunction $\Phi_i$ by plane waves $\phi_j$
\[
\rightarrow
\text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
\qquad ({\color{blue}300\text{ eV}})
\]
  \item \underline{Brillouin zone sampling} -
        {\color{blue}$\Gamma$-point only} calculations
  \item \underline{Pseudo potential} 
        - consider only the valence electrons
  \item \underline{Code} - VASP 4.6
 \end{itemize}

 \vspace*{0.2cm}

 MD and structural optimization

 \begin{itemize}
  \item MD integration: Gear predictor corrector algorithm
  \item Pressure control: Parrinello-Rahman pressure control
  \item Structural optimization: Conjugate gradient method
 \end{itemize}

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\begin{slide}

 {\large\bf
  C and Si self-interstitial point defects in silicon
 }

 \small

 \vspace*{0.3cm}

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 \begin{tabular}{l c c}
 \hline
 & size [unit cells] & \# atoms\\
\hline
VASP & $3\times 3\times 3$ & $216\pm 1$ \\
Erhart/Albe & $9\times 9\times 9$ & $5832\pm 1$\\
\hline
 \end{tabular}
\end{minipage}
\begin{minipage}{4cm}
{\color{red}$\bullet$} Tetrahedral\\
{\color{green}$\bullet$} Hexagonal\\
{\color{yellow}$\bullet$} \hkl<1 0 0> dumbbell\\
{\color{magenta}$\bullet$} \hkl<1 1 0> dumbbell\\
{\color{cyan}$\bullet$} Bond-centered\\
{\color{black}$\bullet$} Vacancy / Substitutional
\end{minipage}

\end{slide}

\begin{slide}

 \footnotesize

\begin{minipage}{9.5cm}

 {\large\bf
  Si self-interstitial point defects in silicon\\
 }

\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
 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}\\[0.2cm]

\begin{minipage}{4.7cm}
\includegraphics[width=4.7cm]{e_kin_si_hex.ps}
\end{minipage}
\begin{minipage}{4.7cm}
\begin{center}
{\tiny nearly T $\rightarrow$ T}\\
\end{center}
\includegraphics[width=4.7cm]{nhex_tet.ps}
\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.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.eps}
\end{minipage}
}
\begin{minipage}{2.9cm}
\begin{flushright}
\underline{Vacancy}\\
\includegraphics[width=3.0cm]{si_pd_albe/vac.eps}
\end{flushright}
\end{minipage}

\end{minipage}
\begin{minipage}{3.5cm}

\begin{flushright}
\underline{\hkl<1 1 0> dumbbell}\\
\includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
\underline{Tetrahedral}\\
\includegraphics[width=3.0cm]{si_pd_albe/tet.eps}\\
\underline{\hkl<1 0 0> dumbbell}\\
\includegraphics[width=3.0cm]{si_pd_albe/100.eps}
\end{flushright}

\end{minipage}

\end{slide}

\begin{slide}

\footnotesize

 {\large\bf
  C interstitial point defects in silicon\\[-0.1cm]
 }

\begin{tabular}{l c c c c c c}
\hline
 $E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\hline
 VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 \\
 Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & 0.75 & 5.59$^*$ \\
\hline
\end{tabular}\\[0.1cm]

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\begin{minipage}{2.7cm}
\underline{Hexagonal} \hspace{2pt}
\href{../video/c_in_si_int_hexa.avi}{$\rhd$}\\
$E_{\text{f}}^*=9.05\text{ eV}$\\
\includegraphics[width=2.7cm]{c_pd_albe/hex.eps}
\end{minipage}
\begin{minipage}{0.4cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{2.7cm}
\underline{\hkl<1 0 0>}\\
$E_{\text{f}}=3.88\text{ eV}$\\
\includegraphics[width=2.7cm]{c_pd_albe/100.eps}
\end{minipage}
}
\begin{minipage}{2cm}
\hfill
\end{minipage}
\begin{minipage}{3cm}
\begin{flushright}
\underline{Tetrahedral}\\
\includegraphics[width=3.0cm]{c_pd_albe/tet.eps}
\end{flushright}
\end{minipage}

\framebox{
\begin{minipage}{2.7cm}
\underline{Bond-centered}\\
$E_{\text{f}}^*=5.59\text{ eV}$\\
\includegraphics[width=2.7cm]{c_pd_albe/bc.eps}
\end{minipage}
\begin{minipage}{0.4cm}
\begin{center}
$\Rightarrow$
\end{center}
\end{minipage}
\begin{minipage}{2.7cm}
\underline{\hkl<1 1 0> dumbbell}\\
$E_{\text{f}}=5.18\text{ eV}$\\
\includegraphics[width=2.7cm]{c_pd_albe/110.eps}
\end{minipage}
}
\begin{minipage}{2cm}
\hfill
\end{minipage}
\begin{minipage}{3cm}
\begin{flushright}
\underline{Substitutional}\\
\includegraphics[width=3.0cm]{c_pd_albe/sub.eps}
\end{flushright}
\end{minipage}

\end{slide}

\begin{slide}

\footnotesize

 {\large\bf\boldmath
  C \hkl<1 0 0> dumbbell interstitial configuration\\
 }

{\tiny
\begin{tabular}{l c c c c c c c c}
\hline
 Distances [nm] & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
\hline
Erhart/Albe & 0.175 & 0.329 & 0.186 & 0.226 & 0.300 & 0.343 & 0.423 & 0.425 \\
VASP & 0.174 & 0.341 & 0.182 & 0.229 & 0.286 & 0.347 & 0.422 & 0.417 \\
\hline
\end{tabular}\\[0.2cm]
\begin{tabular}{l c c c c }
\hline
 Angles [$^{\circ}$] & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
\hline
Erhart/Albe & 140.2 & 109.9 & 134.4 & 112.8 \\
VASP & 130.7 & 114.4 & 146.0 & 107.0 \\
\hline
\end{tabular}\\[0.2cm]
\begin{tabular}{l c c c}
\hline
 Displacements [nm]& $a$ & $b$ & $|a|+|b|$ \\
\hline
Erhart/Albe & 0.084 & -0.091 & 0.175 \\
VASP & 0.109 & -0.065 & 0.174 \\
\hline
\end{tabular}\\[0.6cm]
}

\begin{minipage}{3.0cm}
\begin{center}
\underline{Erhart/Albe}
\includegraphics[width=3.0cm]{c_pd_albe/100_cmp.eps}
\end{center}
\end{minipage}
\begin{minipage}{3.0cm}
\begin{center}
\underline{VASP}
\includegraphics[width=3.0cm]{c_pd_vasp/100_cmp.eps}
\end{center}
\end{minipage}\\

\begin{picture}(0,0)(-185,10)
\includegraphics[width=6.8cm]{100-c-si-db_cmp.eps}
\end{picture}
\begin{picture}(0,0)(-280,-150)
\includegraphics[width=3.3cm]{c_pd_vasp/eden.eps}
\end{picture}

\begin{pspicture}(0,0)(0,0)
\psellipse[linecolor=green](5.18,5.92)(0.5,0.3)
\psellipse[linecolor=red](3.45,5.92)(1.0,0.4)
\psellipse[linecolor=blue](2.7,6.92)(0.9,0.2)
\psellipse[linecolor=blue](4.65,6.92)(0.9,0.2)
\end{pspicture}

\end{slide}

\begin{slide}

\small

\begin{minipage}{8.5cm}

 {\large\bf
  Bond-centered interstitial configuration\\[-0.1cm]
 }

\begin{minipage}{3.0cm}
\includegraphics[width=2.8cm]{c_pd_vasp/bc_2333.eps}\\
\end{minipage}
\begin{minipage}{5.2cm}
\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}

\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}
}\\[0.1cm]

\framebox{
\begin{minipage}{4.5cm}
\includegraphics[width=4cm]{c_100_mig_vasp/im_spin_diff.eps}
\end{minipage}
\begin{minipage}{3.5cm}
{\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}
}

\end{minipage}
\begin{minipage}{4.2cm}
\begin{flushright}
\includegraphics[width=4.3cm]{c_pd_vasp/bc_2333_ksl.ps}\\
{\color{green}$\Box$} {\tiny unoccupied}\\
{\color{red}$\bullet$} {\tiny occupied}
\end{flushright}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Migration of the C \hkl<1 0 0> dumbbell interstitial
 }

\scriptsize

 {\small Investigated pathways}

\begin{minipage}{8.5cm}
\begin{minipage}{8.3cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 0 1>}\\
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/bc_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/100_next_2333.eps}
\end{minipage}
\end{minipage}\\
\begin{minipage}{8.3cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0>}\\
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/00-1-0-10_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/0-10_2333.eps}
\end{minipage}
\end{minipage}\\
\begin{minipage}{8.3cm}
\underline{\hkl<0 0 -1> $\rightarrow$ \hkl<0 -1 0> (in place)}\\
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/100_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/00-1_ip0-10_2333.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{2.4cm}
\includegraphics[width=2.4cm]{c_pd_vasp/0-10_ip_2333.eps}
\end{minipage}
\end{minipage}
\end{minipage}
\framebox{
\begin{minipage}{4.2cm}
 {\small Constrained relaxation\\
         technique (CRT) method}\\
\includegraphics[width=4cm]{crt_orig.eps}
\begin{itemize}
 \item Constrain diffusing atom
 \item Static constraints 
\end{itemize}
\vspace*{0.3cm}
 {\small Modifications}\\
\includegraphics[width=4cm]{crt_mod.eps}
\begin{itemize}
 \item Constrain all atoms
 \item Update individual\\
       constraints
\end{itemize}
\end{minipage}
}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Migration of the C \hkl<1 0 0> dumbbell interstitial
 }

\scriptsize

\framebox{
\begin{minipage}{5.9cm}
\begin{flushleft}
\includegraphics[width=5.8cm]{im_00-1_nosym_sp_fullct_thesis.ps}\\[0.45cm]
\end{flushleft}
\begin{center}
\begin{picture}(0,0)(60,0)
\includegraphics[width=1cm]{vasp_mig/00-1.eps}
\end{picture}
\begin{picture}(0,0)(-5,0)
\includegraphics[width=1cm]{vasp_mig/bc_00-1_sp.eps}
\end{picture}
\begin{picture}(0,0)(-55,0)
\includegraphics[width=1cm]{vasp_mig/bc.eps}
\end{picture}
\begin{picture}(0,0)(12.5,10)
\includegraphics[width=1cm]{110_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,0)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace*{0.35cm}
\end{minipage}
}
\begin{minipage}{0.3cm}
\hfill
\end{minipage}
\framebox{
\begin{minipage}{5.9cm}
\begin{flushright}
\includegraphics[width=5.9cm]{vasp_mig/00-1_0-10_nosym_sp_fullct.ps}\\[0.5cm]
\end{flushright}
\begin{center}
\begin{picture}(0,0)(60,0)
\includegraphics[width=1cm]{vasp_mig/00-1_a.eps}
\end{picture}
\begin{picture}(0,0)(5,0)
\includegraphics[width=1cm]{vasp_mig/00-1_0-10_sp.eps}
\end{picture}
\begin{picture}(0,0)(-55,0)
\includegraphics[width=1cm]{vasp_mig/0-10.eps}
\end{picture}
\begin{picture}(0,0)(12.5,10)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,0)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace*{0.3cm}
\end{minipage}\\
}

\vspace*{0.05cm}

\framebox{
\begin{minipage}{5.9cm}
\begin{flushleft}
\includegraphics[width=5.9cm]{vasp_mig/00-1_ip0-10_nosym_sp_fullct.ps}\\[0.6cm]
\end{flushleft}
\begin{center}
\begin{picture}(0,0)(60,0)
\includegraphics[width=0.9cm]{vasp_mig/00-1_b.eps}
\end{picture}
\begin{picture}(0,0)(10,0)
\includegraphics[width=0.9cm]{vasp_mig/00-1_ip0-10_sp.eps}
\end{picture}
\begin{picture}(0,0)(-60,0)
\includegraphics[width=0.9cm]{vasp_mig/0-10_b.eps}
\end{picture}
\begin{picture}(0,0)(12.5,10)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,0)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace*{0.3cm}
\end{minipage}
}
\begin{minipage}{0.3cm}
\hfill
\end{minipage}
\begin{minipage}{6.5cm}
VASP results
\begin{itemize}
 \item Energetically most favorable path
       \begin{itemize}
        \item Path 2
        \item Activation energy: $\approx$ 0.9 eV 
        \item Experimental values: 0.73 ... 0.87 eV
       \end{itemize}
       $\Rightarrow$ {\color{blue}Diffusion} path identified!
 \item Reorientation (path 3)
       \begin{itemize}
        \item More likely composed of two consecutive steps of type 2
        \item Experimental values: 0.77 ... 0.88 eV
       \end{itemize}
       $\Rightarrow$ {\color{blue}Reorientation} transition identified!
\end{itemize}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Migration of the C \hkl<1 0 0> dumbbell interstitial
 }

\scriptsize

\begin{minipage}{6.5cm}

\framebox{
\begin{minipage}{5.9cm}
\begin{flushleft}
\includegraphics[width=5.9cm]{bc_00-1.ps}\\[2.35cm]
\end{flushleft}
\begin{center}
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=red,fillstyle=none](-2.8,1.35)(3.3,2.7)
\end{pspicture}
\begin{picture}(0,0)(60,-50)
\includegraphics[width=1cm]{albe_mig/bc_00-1_red_00.eps}
\end{picture}
\begin{picture}(0,0)(5,-50)
\includegraphics[width=1cm]{albe_mig/bc_00-1_red_01.eps}
\end{picture}
\begin{picture}(0,0)(-55,-50)
\includegraphics[width=1cm]{albe_mig/bc_00-1_red_02.eps}
\end{picture}
\begin{picture}(0,0)(12.5,-40)
\includegraphics[width=1cm]{110_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,-45)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}\\
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=blue,fillstyle=none](-2.8,0)(3.3,1.6)
\end{pspicture}
\begin{picture}(0,0)(60,-15)
\includegraphics[width=0.9cm]{albe_mig/bc_00-1_01.eps}
\end{picture}
\begin{picture}(0,0)(35,-15)
\includegraphics[width=0.9cm]{albe_mig/bc_00-1_02.eps}
\end{picture}
\begin{picture}(0,0)(-5,-15)
\includegraphics[width=0.9cm]{albe_mig/bc_00-1_03.eps}
\end{picture}
\begin{picture}(0,0)(-55,-15)
\includegraphics[width=0.9cm]{albe_mig/bc_00-1_04.eps}
\end{picture}
\begin{picture}(0,0)(12.5,-5)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,-15)
\includegraphics[height=0.9cm]{010_arrow.eps}
\end{picture}
\end{center}
\end{minipage}
}\\[0.1cm]

\begin{minipage}{5.9cm}
Erhart/Albe results
\begin{itemize}
 \item Lowest activation energy: $\approx$ 2.2 eV
 \item 2.4 times higher than VASP
 \item Different pathway
 \item Transition minima ($\rightarrow$ \hkl<1 1 0> dumbbell)
\end{itemize}
\end{minipage}

\end{minipage}
\begin{minipage}{6.5cm}

\framebox{
\begin{minipage}{5.9cm}
\begin{flushright}
\includegraphics[width=5.9cm]{00-1_0-10.ps}\\[0.75cm]
\end{flushright}
\begin{center}
\begin{pspicture}(0,0)(0,0)
\psframe[linecolor=red,fillstyle=none](-2.8,-0.25)(3.3,1.1)
\end{pspicture}
\begin{picture}(0,0)(60,-5)
\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_00.eps}
\end{picture}
\begin{picture}(0,0)(0,-5)
\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_min.eps}
\end{picture}
\begin{picture}(0,0)(-55,-5)
\includegraphics[width=0.9cm]{albe_mig/00-1_0-10_red_03.eps}
\end{picture}
\begin{picture}(0,0)(12.5,5)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(90,0)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace{0.2cm}
\end{minipage}
}\\[0.2cm]

\framebox{
\begin{minipage}{5.9cm}
\includegraphics[width=5.9cm]{00-1_ip0-10.ps}
\end{minipage}
}

\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Migrations involving the C \hkl<1 1 0> dumbbell interstitial
 }

\small

\vspace*{0.1cm}

VASP

\begin{minipage}{6.0cm}
\includegraphics[width=6cm]{vasp_mig/110_mig_vasp.ps}
\end{minipage}
\begin{minipage}{7cm}
\underline{Alternative pathway and energies [eV]}\\[0.1cm]
\hkl<0 -1 0> $\stackrel{0.7}{{\color{red}\longrightarrow}}$
\hkl<1 1 0> $\stackrel{0.95}{{\color{blue}\longrightarrow}}$
BC $\stackrel{0.25}{\longrightarrow}$ \hkl<0 0 -1>\\[0.3cm]
Composed of three single transitions\\[0.3cm]
Activation energy of second transition slightly\\
higher than direct transition (path 2)\\[0.3cm]
$\Rightarrow$ very unlikely to happen
\end{minipage}\\[0.2cm]

Erhart/Albe

\begin{minipage}{6.0cm}
\includegraphics[width=6cm]{110_mig.ps}
\end{minipage}
\begin{minipage}{7cm}
\underline{Alternative pathway and energies [eV]}\\[0.1cm]
\hkl<0 0 -1> $\stackrel{2.2}{{\color{green}\longrightarrow}}$
\hkl<1 1 0> $\stackrel{0.9}{{\color{red}\longrightarrow}}$
\hkl<0 0 -1>\\[0.3cm]
Composed of two single transitions\\[0.3cm]
Compared to direct transition: (2.2 eV \& 0.5 eV)\\[0.3cm]
$\Rightarrow$ more readily constituting a probable transition
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Combinations with a C-Si \hkl<1 0 0>-type interstitial
 }

\small

\vspace*{0.1cm}

Binding energy: 
$
E_{\text{b}}=
E_{\text{f}}^{\text{defect combination}}-
E_{\text{f}}^{\text{C \hkl<0 0 -1> dumbbell}}-
E_{\text{f}}^{\text{2nd defect}}
$

\vspace*{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 substitutional (C$_{\text{S}}$) & 0.26 & -0.51 & -0.93 & -0.15 & 0.49 & -0.05\\
 Vacancy & -5.39 ($\rightarrow$ C$_{\text{S}}$) & -0.59 & -3.14 & -0.54 & -0.50 & -0.31\\
\hline
\end{tabular}
}

\vspace*{0.3cm}

\footnotesize

\begin{minipage}[t]{3.8cm}
\underline{\hkl<1 0 0> at position 1}\\[0.1cm]
\includegraphics[width=3.5cm]{00-1dc/2-25.eps}
\end{minipage}
\begin{minipage}[t]{3.5cm}
\underline{\hkl<0 -1 0> at position 1}\\[0.1cm]
\includegraphics[width=3.2cm]{00-1dc/2-39.eps}
\end{minipage}
\begin{minipage}[t]{5.5cm}
\begin{itemize}
 \item Restricted to VASP simulations
 \item $E_{\text{b}}=0$ for isolated non-interacting defects
 \item $E_{\text{b}} \rightarrow 0$ for increasing distance (R)
 \item Stress compensation / increase
 \item Most favorable: C clustering
 \item Unfavored: antiparallel orientations
 \item Indication of energetically favored\\
       agglomeration
\end{itemize}
\end{minipage}

\begin{picture}(0,0)(-295,-130)
\includegraphics[width=3.5cm]{comb_pos.eps}
\end{picture}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Combinations of C-Si \hkl<1 0 0>-type interstitials
 }

\small

\vspace*{0.1cm}

Energetically most favorable combinations along \hkl<1 1 0>

\vspace*{0.1cm}

{\scriptsize
\begin{tabular}{l c c c c c c}
\hline
 & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
$E_{\text{b}}$ [eV] & -2.39 & -1.88 & -0.59 & -0.31 & -0.24 & -0.21 \\
C-C distance [\AA] & 1.4 & 4.6 & 6.5 & 8.6 & 10.5 & 10.8 \\
Type & \hkl<-1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0> & \hkl<1 0 0>, \hkl<0 -1 0>\\
\hline
\end{tabular}
}

\vspace*{0.3cm}

\begin{minipage}{7.0cm}
\includegraphics[width=7cm]{db_along_110_cc.ps}
\end{minipage}
\begin{minipage}{6.0cm}
\begin{center}
{\color{blue}
 Interaction proportional to reciprocal cube of C-C distance
}\\[0.2cm]
 Saturation in the immediate vicinity
\end{center}
\end{minipage}

\vspace{0.2cm}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Combinations of substitutional C and \hkl<1 1 0> Si self-interstitials
 }

 \scriptsize

\begin{center}
\begin{minipage}{3.2cm}
\includegraphics[width=3cm]{sub_110_combo.eps}
\end{minipage}
\begin{minipage}{7.8cm}
\begin{tabular}{l c c c c c c}
\hline
C$_{\text{sub}}$ & \hkl<1 1 0> & \hkl<-1 1 0> & \hkl<0 1 1> & \hkl<0 -1 1> &
                   \hkl<1 0 1> & \hkl<-1 0 1> \\
\hline
1 & \RM{1} & \RM{3} & \RM{3} & \RM{1} & \RM{3} & \RM{1} \\
2 & \RM{2} & A & A & \RM{2} & C & \RM{5} \\
3 & \RM{3} & \RM{1} & \RM{3} & \RM{1} & \RM{1} & \RM{3} \\
4 & \RM{4} & B & D & E & E & D \\
5 & \RM{5} & C & A & \RM{2} & A & \RM{2} \\
\hline
\end{tabular}
\end{minipage}
\end{center}

\begin{center}
\begin{tabular}{l c c c c c c c c c c}
\hline
Conf & \RM{1} & \RM{2} & \RM{3} & \RM{4} & \RM{5} & A & B & C & D & E \\
\hline
$E_{\text{f}}$ [eV]& 4.37 & 5.26 & 5.57 & 5.37 & 5.12 & 5.10 & 5.32 & 5.28 & 5.39 & 5.32 \\
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\
$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\
\hline
\end{tabular}
\end{center}

\begin{minipage}{6.0cm}
\includegraphics[width=5.8cm]{c_sub_si110.ps}
\end{minipage}
\begin{minipage}{7cm}
\small
\begin{itemize}
 \item IBS: C may displace Si\\
       $\Rightarrow$ C$_{\text{sub}}$ + \hkl<1 1 0> Si self-interstitial
 \item Assumption:\\
       \hkl<1 1 0>-type $\rightarrow$ favored combination
 \renewcommand\labelitemi{$\Rightarrow$}
 \item Less favorable than C-Si \hkl<1 0 0> dumbbell\\
       ($E_{\text{f}}=3.88\text{ eV}$)
 \item Interaction drops quickly to zero\\
       (low interaction capture radius)
\end{itemize}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Migration in C-Si \hkl<1 0 0> and vacancy combinations
 }

 \footnotesize

\vspace{0.1cm}

\begin{minipage}[t]{3cm}
\underline{Pos 2, $E_{\text{b}}=-0.59\text{ eV}$}\\
\includegraphics[width=2.8cm]{00-1dc/0-59.eps}
\end{minipage}
\begin{minipage}[t]{7cm}
\vspace{0.2cm}
\begin{center}
 Low activation energies\\
 High activation energies for reverse processes\\
 $\Downarrow$\\
 {\color{blue}C$_{\text{sub}}$ very stable}\\
\vspace*{0.1cm}
 \hrule
\vspace*{0.1cm}
 Without nearby \hkl<1 1 0> Si self-interstitial (IBS)\\
 $\Downarrow$\\
 {\color{blue}Formation of SiC by successive substitution by C}

\end{center}
\end{minipage}
\begin{minipage}[t]{3cm}
\underline{Pos 3, $E_{\text{b}}=-3.14\text{ eV}$}\\
\includegraphics[width=2.8cm]{00-1dc/3-14.eps}
\end{minipage}


\framebox{
\begin{minipage}{5.9cm}
\includegraphics[width=5.9cm]{vasp_mig/comb_mig_3-2_vac_fullct.ps}\\[0.6cm]
\begin{center}
\begin{picture}(0,0)(70,0)
\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_init.eps}
\end{picture}
\begin{picture}(0,0)(30,0)
\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_03.eps}
\end{picture}
\begin{picture}(0,0)(-10,0)
\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_seq_06.eps}
\end{picture}
\begin{picture}(0,0)(-48,0)
\includegraphics[width=1.4cm]{vasp_mig/comb_2-1_final.eps}
\end{picture}
\begin{picture}(0,0)(12.5,5)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(97,-10)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace{0.1cm}
\end{minipage}
}
\begin{minipage}{0.3cm}
\hfill
\end{minipage}
\framebox{
\begin{minipage}{5.9cm}
\includegraphics[width=5.9cm]{vasp_mig/comb_mig_4-2_vac_fullct.ps}\\[0.1cm]
\begin{center}
\begin{picture}(0,0)(60,0)
\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_init.eps}
\end{picture}
\begin{picture}(0,0)(25,0)
\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_03.eps}
\end{picture}
\begin{picture}(0,0)(-20,0)
\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_seq_07.eps}
\end{picture}
\begin{picture}(0,0)(-55,0)
\includegraphics[width=0.9cm]{vasp_mig/comb_3-1_final.eps}
\end{picture}
\begin{picture}(0,0)(12.5,5)
\includegraphics[width=1cm]{100_arrow.eps}
\end{picture}
\begin{picture}(0,0)(95,0)
\includegraphics[height=0.9cm]{001_arrow.eps}
\end{picture}
\end{center}
\vspace{0.1cm}
\end{minipage}
}

\end{slide}

\begin{slide}

 {\large\bf
  Conclusion of defect / migration / combined defect simulations
 }

 \small

\vspace*{0.1cm}

Defect structures
\begin{itemize}
 \item Accurately described by quantum-mechanical simulations
 \item Less correct description by classical potential simulations
\end{itemize}
\vspace*{0.2cm}
\begin{itemize}
 \item Consistent with solubility data of C in Si
 \item \hkl<1 0 0> C-Si dumbbell interstitial ground state configuration
 \item Consistent with reorientation and diffusion experiments
 \item C migration pathway in Si identified
\end{itemize} 

\vspace*{0.2cm}

Concerning the precipitation mechanism
\begin{itemize}
 \item Agglomeration of C-Si dumbbells energetically favorable
 \item C-Si indeed favored compared to
       C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
 \item Possible low interaction capture radius of
       C$_{\text{sub}}$ \& \hkl<1 1 0> Si self-interstitial
 \item In absence of nearby \hkl<1 1 0> Si self-interstitial:
       C-Si \hkl<1 0 0> + Vacancy $\rightarrow$ C$_{\text{sub}}$ (SiC)
\end{itemize} 

\vspace*{0.1cm}
\begin{center}
{\color{blue}Some results point to a different precipitation mechanism!}
\end{center}

\end{slide}

\begin{slide}

 {\large\bf
  Silicon carbide precipitation simulations
 }

 \small

{\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 {\pnode{in2}}
    \item volume consisting of 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 $20\, ^{\circ}\textrm{C}$
  }}}}
  \ncline[]{->}{init}{insert}
  \ncline[]{->}{insert}{cool}
  \psframe[fillstyle=solid,fillcolor=white](7.5,0.7)(13.5,6.3)
  \rput(7.8,6){\footnotesize $V_1$}
  \psframe[fillstyle=solid,fillcolor=lightgray](9,2)(12,5)
  \rput(9.2,4.85){\tiny $V_2$}
  \psframe[fillstyle=solid,fillcolor=gray](9.25,2.25)(11.75,4.75)
  \rput(9.55,4.45){\footnotesize $V_3$}
  \rput(7.9,3.2){\pnode{ins1}}
  \rput(9.22,2.8){\pnode{ins2}}
  \rput(11.0,2.4){\pnode{ins3}}
  \ncline[]{->}{in1}{ins1}
  \ncline[]{->}{in2}{ins2}
  \ncline[]{->}{in3}{ins3}
 \end{pspicture}
}

\begin{itemize}
 \item Restricted to classical potential simulations
 \item $V_2$ and $V_3$ considered due to low diffusion
 \item Amount of C atoms: 6000
       ($r_{\text{prec}}\approx 3.1\text{ nm}$, IBS: 2 ... 4 nm)
 \item Simulation volume: $31\times 31\times 31$ unit cells
       (238328 Si atoms)
\end{itemize}

\end{slide}

\begin{slide}

 {\large\bf\boldmath
  Silicon carbide precipitation simulations at $450\,^{\circ}\mathrm{C}$ as in IBS
 }

 \small

\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{sic_prec_450_si-si_c-c.ps}
\end{minipage} 
\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{sic_prec_450_energy.ps}
\end{minipage} 

\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{sic_prec_450_si-c.ps}
\end{minipage} 
\begin{minipage}{6.5cm}
\scriptsize
\underline{Low C concentration ($V_1$)}\\
\hkl<1 0 0> C-Si dumbbell dominated structure
\begin{itemize}
 \item Si-C bumbs around 0.19 nm
 \item C-C peak at 0.31 nm (as expected in 3C-SiC):\\
       concatenated dumbbells of various orientation
 \item Si-Si NN distance stretched to 0.3 nm
\end{itemize}
{\color{blue}$\Rightarrow$ C atoms in proper 3C-SiC distance first}\\
\underline{High C concentration ($V_2$, $V_3$)}\\
High amount of strongly bound C-C bonds\\
Defect density $\uparrow$ $\Rightarrow$ considerable amount of damage\\
Only short range order observable\\
{\color{blue}$\Rightarrow$ amorphous SiC-like phase}
\end{minipage} 

\end{slide}

\begin{slide}

 {\large\bf
  Limitations of molecular dynamics and short range potentials
 }

\footnotesize

\vspace{0.2cm}

\underline{Time scale problem of MD}\\[0.2cm]
Minimize integration error\\
$\Rightarrow$ discretization considerably smaller than
              reciprocal of fastest vibrational mode\\[0.1cm]
Order of fastest vibrational mode: $10^{13} - 10^{14}\text{ Hz}$\\
$\Rightarrow$ suitable choice of time step:
              $\tau=1\text{ fs}=10^{-15}\text{ s}$\\
$\Rightarrow$ {\color{red}\underline{slow}} phase space propagation\\[0.1cm]
Several local minima in energy surface separated by large energy barriers\\
$\Rightarrow$ transition event corresponds to a multiple
              of vibrational periods\\
$\Rightarrow$ phase transition made up of {\color{red}\underline{many}}
              infrequent transition events\\[0.1cm]
{\color{blue}Accelerated methods:}
\underline{Temperature accelerated} MD (TAD), self-guided MD \ldots

\vspace{0.3cm}

\underline{Limitations related to the short range potential}\\[0.2cm]
Cut-off function pushing forces and energies to zero between 1$^{\text{st}}$
and 2$^{\text{nd}}$ next neighbours\\
$\Rightarrow$ overestimated unphysical high forces of next neighbours

\vspace{0.3cm}

\framebox{
\color{red}
Potential enhanced problem of slow phase space propagation
}

\vspace{0.3cm}

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

\begin{picture}(0,0)(-260,-30)
\framebox{
\begin{minipage}{4.2cm}
\tiny
\begin{center}
\vspace{0.03cm}
\underline{IBS}
\end{center}
\begin{itemize}
\item 3C-SiC also observed for higher T
\item higher T inside sample
\item structural evolution vs.\\
      equilibrium properties
\end{itemize}
\end{minipage}
}
\end{picture}

\begin{picture}(0,0)(-305,-155)
\framebox{
\begin{minipage}{2.5cm}
\tiny
\begin{center}
retain proper\\
thermodynmic sampling
\end{center}
\end{minipage}
}
\end{picture}

\end{slide}

\begin{slide}

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

\small

\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{tot_pc_thesis.ps}
\end{minipage}
\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{tot_pc3_thesis.ps}
\end{minipage}

\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{tot_pc2_thesis.ps}
\end{minipage}
\begin{minipage}{6.5cm}
\scriptsize
 \underline{Si-C bonds:}
 \begin{itemize}
  \item Vanishing cut-off artifact (above $1650\,^{\circ}\mathrm{C}$)
  \item Structural change: C-Si \hkl<1 0 0> $\rightarrow$ C$_{\text{sub}}$
 \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
        \begin{itemize}
         \item C-Si \hkl<1 0 0> combinations (dashed arrows)\\
               $\rightarrow$ C-Si \hkl<1 0 0> \& C$_{\text{sub}}$
               combinations (|)\\
               $\rightarrow$ pure {\color{blue}C$_{\text{sub}}$ combinations}
               ($\downarrow$)
         \item Range [|-$\downarrow$]:
               {\color{blue}C$_{\text{sub}}$ \& C$_{\text{sub}}$
               with nearby Si$_{\text{I}}$}
        \end{itemize}
 \end{itemize}
\end{minipage}

\begin{picture}(0,0)(-330,-74)
\color{blue}
\framebox{
\begin{minipage}{1.6cm}
\tiny
\begin{center}
stretched SiC\\[-0.1cm]
in c-Si
\end{center}
\end{minipage}
}
\end{picture}

\end{slide}

\begin{slide}

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

\footnotesize

\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{12_pc_thesis.ps}
\end{minipage}
\begin{minipage}{6.5cm}
\includegraphics[width=6.4cm]{12_pc_c_thesis.ps}
\end{minipage}

\begin{center}
Decreasing cut-off artifact\\
High amount of {\color{red}damage} \& alignement to c-Si host matrix lost
$\Rightarrow$ hard to categorize
\end{center}

\vspace{0.1cm}

\framebox{
\begin{minipage}[t]{6.0cm}
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}{0.2cm}
\hfill
\end{minipage}
\framebox{
\begin{minipage}[t]{6.0cm}
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}
}

\vspace{0.1cm}

\begin{center}
{\color{red}Amorphous} SiC-like phase remains\\
Slightly sharper peaks
$\Rightarrow$ indicate slight {\color{blue}acceleration of dynamics}
due to temperature\\[0.1cm]
\framebox{
\bf
Continue with higher temperatures and longer time scales
}
\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}{7.5cm}
\includegraphics[width=7cm]{fe_and_t.ps}
\end{minipage}
\begin{minipage}{5.5cm}
\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}{3cm}
\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]{4.5cm}
\begin{center}
\underline{Low C concentration, Si-C}
\includegraphics[width=4.5cm]{c_in_si_95_v1_si-c.ps}\\
Sharper peaks!
\end{center}
\end{minipage}
\begin{minipage}[t]{4.5cm}
\begin{center}
\underline{Low C concentration, C-C}
\includegraphics[width=4.5cm]{c_in_si_95_v1_c-c.ps}\\
Sharper peaks!\\
No C agglomeration!
\end{center}
\end{minipage}
\begin{minipage}[t]{4cm}
\begin{center}
\underline{High C concentration}
\includegraphics[width=4.5cm]{c_in_si_95_v2.ps}\\
No significant changes
\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
 }

 \footnotesize

\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}{0.3cm}
\hfill
\end{minipage}
\begin{minipage}{7.0cm}
\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}{6.2cm}
\includegraphics[width=6cm]{pc_0.ps}
\end{minipage}
\begin{minipage}{6.8cm}
\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}{6cm}
\includegraphics[width=6.7cm]{fe_and_t_sic.ps}
\end{minipage}

\begin{minipage}{4cm}
\includegraphics[width=4cm]{sic_prec/melt_01.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4cm}
\includegraphics[width=4cm]{sic_prec/melt_02.eps}
\end{minipage}
\begin{minipage}{0.4cm}
$\rightarrow$
\end{minipage}
\begin{minipage}{4cm}
\includegraphics[width=4cm]{sic_prec/melt_03.eps}
\end{minipage}

\end{slide}

\begin{slide}

 {\large\bf
  Summary / Conclusion / Outlook
 }

 \scriptsize

\vspace{0.1cm}

\framebox{
\begin{minipage}{12.9cm}
 \underline{Defects}
 \begin{itemize}
  \item Summary \& conclusion
        \begin{itemize}
         \item Point defects excellently / fairly well described
               by QM / classical potential simulations
         \item Identified migration path explaining
               diffusion and reorientation experiments
         \item Agglomeration of point defects energetically favorable
         \item C$_{\text{sub}}$ favored conditions (conceivable in IBS)
        \end{itemize}
  \item Todo
        \begin{itemize}
         \item Discussions concerning interpretation of QM results (Paderborn)
         \item Compare migration barrier of
               \hkl<1 1 0> Si and C-Si \hkl<1 0 0> dumbbell
         \item Combination: Vacancy \& \hkl<1 1 0> Si self-interstitial \&
                            C-Si \hkl<1 0 0> dumbbell (IBS)
        \end{itemize}
 \end{itemize}
\end{minipage}
}

\vspace{0.2cm}

\framebox{
\begin{minipage}[t]{6.2cm}
 \underline{Pecipitation simulations}
 \begin{itemize}
  \item Summary \& conclusion
        \begin{itemize}
         \item Low T
               $\rightarrow$ C-Si \hkl<1 0 0> dumbbell\\
               dominated structure
         \item High T $\rightarrow$ C$_{\text{sub}}$ dominated structure
         \item High C concentration\\
               $\rightarrow$ amorphous SiC like phase
        \end{itemize}
  \item Todo
        \begin{itemize}
         \item Accelerated method: self-guided MD
         \item Activation relaxation technique
         \item Constrainted transition path
        \end{itemize}
 \end{itemize}
\end{minipage}
}
\framebox{
\begin{minipage}[t]{6.2cm}
 \underline{Constructed 3C-SiC precipitate}
 \begin{itemize}
  \item Summary \& conclusion
        \begin{itemize}
         \item Small / stable / compressed 3C-SiC\\
               precipitate in slightly stretched\\
               c-Si matrix
         \item Interface tension matches experiemnts
        \end{itemize}
  \item Todo
        \begin{itemize}
         \item Try to improve interface
         \item Precipitates of different size
        \end{itemize}
 \end{itemize}
\end{minipage}
}

 \small

\end{slide}

\begin{slide}

 {\large\bf
  Acknowledgements
 }

 \vspace{0.1cm}

 \small

 Thanks to \ldots

 \underline{Augsburg}
 \begin{itemize}
  \item Prof. B. Stritzker (accepting a simulator at EP \RM{4})
  \item Ralf Utermann (EDV)
 \end{itemize}
 
 \underline{Helsinki}
 \begin{itemize}
  \item Prof. K. Nordlund (MD)
 \end{itemize}
 
 \underline{Munich}
 \begin{itemize}
  \item Bayerische Forschungsstiftung (financial support)
 \end{itemize}
 
 \underline{Paderborn}
 \begin{itemize}
  \item Prof. J. Lindner (SiC)
  \item Prof. G. Schmidt (DFT + financial support)
  \item Dr. E. Rauls (DFT + SiC)
 \end{itemize}

\vspace{0.2cm}

\begin{center}
\framebox{
\bf Thank you for your attention!
}
\end{center}

\end{slide}

\end{document}