\begin{slide}
{\large\bf
- Basics of molecular dynamics (MD) simulations
+ Molecular dynamics (MD) simulations
}
\vspace{12pt}
$\tau_{\text{P}}=100\text{ fs}$,
$\beta^{-1}=100\text{ GPa}$
\end{itemize}
- \item Potential: Tersoff-like bond order potential
+ \item Erhart/Albe potential: Tersoff-like bond order potential
\vspace*{12pt}
\[
E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
\begin{slide}
{\large\bf
- Basics of density functional theory (DFT) calculations
+ Density functional theory (DFT) calculations
}
\small
- Ingredients
+ Basic ingredients necessary for DFT
+
\begin{itemize}
- \item Hohenberg-Kohn (HK) theorem
+ \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\\
+ - $N$ 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}
- - replace electrostatic potential by an average over e$^-$ positions\\
+ - 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
\[
-V_{\text{eff}}=...
+\left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) \right] \Phi_i(r)
+=\epsilon_i\Phi_i(r)
+\quad
+\Rightarrow
+\quad
+n(r)=\sum_i^N|\Phi_i(r)|^2
\]
- \item Exchange correlation (EC) LDA / GGA
- \item Self-consistent solution
- \item Plane wave basis set
- \item Pseudo potential
+ \item \underline{Self-consistent solution}\\
+$n(r)$ depends on $\Phi_i$, which depends 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 $\phi_j$
+\[
+\rightarrow
+\text{Fourier series: } \Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r), \quad E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}
+\]
+ \item \underline{$k$-point sampling} - $\Gamma$-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}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ C and Si self-interstitial point defects in silicon
+ }
+
+ \small
+
+ \vspace*{0.3cm}
+
+\begin{minipage}{8cm}
+Procedure:\\[0.3cm]
+ \begin{pspicture}(0,0)(7,5)
+ \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+ \parbox{7cm}{
+ \begin{itemize}
+ \item Creation of c-Si simulation volume
+ \item Periodic boundary conditions
+ \item $T=0\text{ K}$, $p=0\text{ bar}$
+ \end{itemize}
+ }}}}
+\rput(3.5,2.1){\rnode{insert}{\psframebox{
+ \parbox{7cm}{
+ \begin{center}
+ Insertion of interstitial C/Si atoms
+ \end{center}
+ }}}}
+ \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
+ \parbox{7cm}{
+ \begin{center}
+ Relaxation / structural energy minimization
+ \end{center}
+ }}}}
+ \ncline[]{->}{init}{insert}
+ \ncline[]{->}{insert}{cool}
+ \end{pspicture}
+\end{minipage}
+\begin{minipage}{5cm}
+ \includegraphics[width=5cm]{unit_cell_e.eps}\\
+\end{minipage}
+
+\begin{minipage}{9cm}
+ \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]
+
+\framebox{
+\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 & $r(1C)$ & $r(2C)$ & $r(3C)$ & $r(12)$ & $r(13)$ & $r(34)$ & $r(23)$ & $r(25)$ \\
+\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 & $\theta_1$ & $\theta_2$ & $\theta_3$ & $\theta_4$ \\
+\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 & $a$ & $b$ & $|a|+|b|$ \\
+\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}
+
+\end{slide}
+
+\begin{slide}
+
+\footnotesize
+
+ {\large\bf
+ Bond-centered interstitial configuration\\
+ }
+
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Silicon carbide precipitation simulations
+ }
+
+ \small
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Investigation of a silicon carbide precipitate in silicon
+ }
+
+ \small
+
+\end{slide}
\end{document}
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