\end{slide}
-% continue here
-\fi
-
\begin{slide}
\headphd
Utilized computational methods
}
- \vspace{0.1cm}
+\vspace{0.3cm}
- \small
+\small
-{\bf Molecular dynamics (MD):}\\
+{\bf Molecular dynamics (MD)}\\[0.1cm]
\scriptsize
-\begin{tabular}{l r}
+\begin{tabular}{| p{4.5cm} | p{7.5cm} |}
\hline
-Basics & Details\\
-\hline
-Microscopic description of N particle system & \\
-Analytical interaction potential & Tersoff-like bond order potential (Erhart/Albe) \\
-Numerical integration using Newtons equation of motion as a propagation rule in 6N-dimensional phase space & Velocity Verlet | timestep: \unit[1]{fs} \\
-Observables obtained by time and/or ensemble averages & 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}\\
+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}
- \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}
- \[
- E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad
- \pot_{ij} = {\color{red}f_C(r_{ij})}
- \left[ f_R(r_{ij}) + {\color{blue}b_{ij}} f_A(r_{ij}) \right]
- \]
- \end{itemize}
-
- \begin{picture}(0,0)(-230,-30)
- \includegraphics[width=5cm]{tersoff_angle.eps}
- \end{picture}
-
-\end{slide}
-
-\end{document}
-\ifnum1=0
-
-\begin{slide}
+\small
- {\large\bf
- Density functional theory (DFT) calculations
- }
+\vspace{0.3cm}
- \small
+{\bf Density functional theory (DFT)}
- Basic ingredients necessary for DFT
+\scriptsize
- \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
-\[
+\begin{minipage}[t]{6cm}
+\begin{itemize}
+ \item Hohenberg-Kohn theorem:\\
+ $\Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)]$, $E_0=E[n_0]$
+ \item Kohn-Sham approach:\\
+ Single-particle effective theory
+\end{itemize}
+\hrule
+\begin{itemize}
+\item Code: \textsc{vasp}
+\item Plane wave basis set
+%$\displaystyle
+%\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_{i,k+G} \exp{\left(i(k+G)r\right)}
+%$\\
+%$\displaystyle
+%E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV}
+%$
+\item Ultrasoft pseudopotential
+\item Exchange \& correlation: GGA
+\item Brillouin zone sampling: $\Gamma$-point
+\item Supercell: $N=216\pm2$
+\end{itemize}
+\end{minipage}
+\begin{minipage}{6cm}
+\begin{pspicture}(0,0)(0,0)
+\pscircle[fillcolor=yellow,fillstyle=solid,linestyle=none](3.5,-2.0){2.5}
+\rput(2.7,-0.7){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
+\left[ -\frac{\hbar^2}{2m}\nabla^2 + V_{\text{eff}}(r) - \epsilon_i \right] \Phi_i(r) = 0
+$
+}}
+\rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
+n(r)=\sum_i^N|\Phi_i(r)|^2
+$
+}}
+\rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{
+$\displaystyle
V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r'
+V_{\text{XC}}[n(r)]
-\]
- \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}
+$
+}}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165}
+\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310}
-\begin{pspicture}(0,0)(0,0)
-\psellipse[linecolor=blue](1.5,6.75)(0.5,0.3)
\end{pspicture}
+\end{minipage}
\end{slide}
\begin{slide}
+\headphd
{\large\bf
- C and Si self-interstitial point defects in silicon
+ Point defects \& defect migration
}
\small
- \vspace*{0.3cm}
+ \vspace{0.2cm}
-\begin{minipage}{8cm}
-Procedure:\\[0.3cm]
- \begin{pspicture}(0,0)(7,5)
- \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
+\begin{minipage}[b]{7.5cm}
+{\bf Defect structure}\\
+ \begin{pspicture}(0,0)(7,4.4)
+ \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{
\parbox{7cm}{
\begin{itemize}
\item Creation of c-Si simulation volume
\item $T=0\text{ K}$, $p=0\text{ bar}$
\end{itemize}
}}}}
-\rput(3.5,2.1){\rnode{insert}{\psframebox{
+\rput(3.5,1.3){\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]{
+ \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{
\parbox{7cm}{
\begin{center}
Relaxation / structural energy minimization
\ncline[]{->}{insert}{cool}
\end{pspicture}
\end{minipage}
-\begin{minipage}{5cm}
- \includegraphics[width=5cm]{unit_cell_e.eps}\\
+\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}
-\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}
+\vspace{0.2cm}
+
+\begin{minipage}[b]{6cm}
+{\bf Defect formation energy}\\
+\framebox{
+$E_{\text{f}}=E-\sum_i N_i\mu_i$}\\[0.1cm]
+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}{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
+\begin{minipage}[b]{6cm}
+{\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}
+% continue here
+\fi
+
\begin{slide}
\footnotesize
\begin{minipage}{9.5cm}
+\headphd
{\large\bf
- Si self-interstitial point defects in silicon\\
+ Si self-interstitial point defects in silicon\\[0.1cm]
}
\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 \\
+ \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}\\[0.2cm]
+\end{tabular}\\[0.3cm]
\begin{minipage}{4.7cm}
\includegraphics[width=4.7cm]{e_kin_si_hex.ps}
{\tiny nearly T $\rightarrow$ T}\\
\end{center}
\includegraphics[width=4.7cm]{nhex_tet.ps}
-\end{minipage}\\
+\end{minipage}\\[0.1cm]
\underline{Hexagonal} \hspace{2pt}
\href{../video/si_self_int_hexa.avi}{$\rhd$}\\[0.1cm]
\end{minipage}
\end{minipage}
-\begin{minipage}{3.5cm}
+\begin{minipage}{2.5cm}
\begin{flushright}
+\vspace*{0.2cm}
\underline{\hkl<1 1 0> dumbbell}\\
\includegraphics[width=3.0cm]{si_pd_albe/110.eps}\\
\underline{Tetrahedral}\\
\footnotesize
+\headphd
{\large\bf
C interstitial point defects in silicon\\[-0.1cm]
}
+{\scriptsize
\begin{tabular}{l c c c c c c r}
\hline
$E_{\text{f}}$ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B & \cs{} \& \si\\
\hline
- VASP & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
+ \textsc{vasp} & unstable & unstable & \underline{3.72} & 4.16 & 1.95 & 4.66 & {\color{green}4.17}\\
Erhart/Albe MD & 6.09 & 9.05$^*$ & \underline{3.88} & 5.18 & {\color{red}0.75} & 5.59$^*$ & {\color{green}4.43} \\
\hline
-\end{tabular}\\[0.1cm]
+\end{tabular}
+}\\[0.1cm]
\framebox{
\begin{minipage}{2.7cm}
\end{slide}
+\end{document}
+\ifnum1=0
+
\begin{slide}
\footnotesize
\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}
projects / lectures / latex.git / blobdiff