author hackbard Mon, 7 Nov 2011 14:24:37 +0000 (15:24 +0100) committer hackbard Mon, 7 Nov 2011 14:24:37 +0000 (15:24 +0100)

index c05db6d..e4cdd48 100644 (file)
@@ -991,9 +991,6 @@ r = \unit[2--4]{nm}

\end{slide}

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-\fi
-
\begin{slide}

@@ -1001,21 +998,18 @@ r = \unit[2--4]{nm}
Utilized computational methods
}

-\vspace{0.2cm}
+\vspace{0.3cm}

\small

-{\bf Molecular dynamics (MD)}\\
+{\bf Molecular dynamics (MD)}\$0.1cm] \scriptsize -\begin{tabular}{p{4.5cm} p{7.5cm}} -Basics & Details\\ +\begin{tabular}{| p{4.5cm} | p{7.5cm} |} \hline System of N particles & N=5832\pm 1 (Defects), N=238328+6000 (Precipitation)\\ -\hline Phase space propagation & Velocity Verlet | timestep: \unit[1]{fs} \\ -\hline Analytical interaction potential & Tersoff-like {\color{red}short-range}, {\color{blue}bond order} potential (Erhart/Albe) @@ -1024,7 +1018,6 @@ 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] \\ -\hline Observables: time/ensemble averages & NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\ \hline NpT (isothermal-isobaric) | Berendsen thermostat/barostat\\ \small -\vspace{0.1cm} +\vspace{0.3cm} {\bf Density functional theory (DFT)} \scriptsize \begin{minipage}[t]{6cm} -\underline{Basics} \begin{itemize} - \item \Psi_0(r_1,r_2,\ldots,r_N)=\Psi[n_0(r)], E_0=E[n_0] - \item Single-particle effective theory -% \item Born-Oppenheimer approximation:\\ -% Decouple electronic \& ionic motion -% \item Hohenberg-Kohn theorem:\\ -% n_0(r) \stackrel{\text{uniquely}}{\rightarrow} -% V_0 / H / \Phi_i / \underline{E_0} + \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} -\underline{Details} +\hrule \begin{itemize} \item Code: \textsc{vasp} -\item Plane wave basis set \{\phi_j\}\\[0.1cm] -\displaystyle -\Phi_i=\sum_{|G+k|<G_{\text{cut}}} c_j^i \phi_j(r) -\\ -\displaystyle -E_{\text{cut}}=\frac{\hbar^2}{2m}G^2_{\text{cut}}=\unit[300]{eV} - +\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}[t]{6cm} - -\[ +\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 -$
-$+ +}} +\rput(5.2,-2.0){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{ +\displaystyle n(r)=\sum_i^N|\Phi_i(r)|^2 -$
-$-V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r' - +V_{\text{XC}}[n(r)] -$
-
-\end{minipage}
-
-\end{slide}
-
-\end{document}
-\ifnum1=0
-
-\begin{slide}
-
- \small
- {\large\bf
-  Density functional theory (DFT) calculations
- }
-
- 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
-$+ +}} +\rput(3.0,-4.5){\psframebox[fillstyle=solid,opacity=0.8,fillcolor=white]{ +\displaystyle V_{\text{eff}}(r)=V_{\text{ext}}(r)+\int\frac{e^2 n(r')}{|r-r'|}d^3r' +V_{\text{XC}}[n(r)] -$
-  \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 $\phi_j$
-  \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}
+$+}} +\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{130}{15} +\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{230}{165} +\psarcn[linewidth=0.07cm,linestyle=dashed]{->}(3.5,-2.0){2.5}{345}{310} -\begin{pspicture}(0,0)(0,0) -\psellipse[linecolor=blue](1.5,6.75)(0.5,0.3) \end{pspicture} +\end{minipage} \end{slide} \begin{slide} +\headphd {\large\bf - C and Si self-interstitial point defects in silicon + Point defects \& defect migration } \small - \vspace*{0.3cm} + \vspace{0.2cm} -\begin{minipage}{8cm} -Procedure:\\[0.3cm] - \begin{pspicture}(0,0)(7,5) - \rput(3.5,4){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ +\begin{minipage}[b]{7.5cm} +{\bf Defect structure}\\ + \begin{pspicture}(0,0)(7,4.4) + \rput(3.5,3.2){\rnode{init}{\psframebox[fillstyle=solid,fillcolor=hb]{ \parbox{7cm}{ \begin{itemize} \item Creation of c-Si simulation volume @@ -1197,13 +1104,13 @@ Procedure:\\[0.3cm] \item$T=0\text{ K}$,$p=0\text{ bar}$\end{itemize} }}}} -\rput(3.5,2.1){\rnode{insert}{\psframebox{ +\rput(3.5,1.3){\rnode{insert}{\psframebox{ \parbox{7cm}{ \begin{center} Insertion of interstitial C/Si atoms \end{center} }}}} - \rput(3.5,1){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{ + \rput(3.5,0.2){\rnode{cool}{\psframebox[fillstyle=solid,fillcolor=lbb]{ \parbox{7cm}{ \begin{center} Relaxation / structural energy minimization @@ -1213,49 +1120,83 @@ Procedure:\\[0.3cm] \ncline[]{->}{insert}{cool} \end{pspicture} \end{minipage} -\begin{minipage}{5cm} - \includegraphics[width=5cm]{unit_cell_e.eps}\\ +\begin{minipage}[b]{4.5cm} +\begin{center} +\includegraphics[width=3.8cm]{unit_cell_e.eps}\\ +\end{center} +\begin{minipage}{2.21cm} +{\scriptsize +{\color{red}$\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} @@ -1265,7 +1206,7 @@ Erhart/Albe &$9\times 9\times 9$&$5832\pm 1$\\ {\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] @@ -1292,9 +1233,10 @@$E_{\text{f}}=3.96\text{ eV}$\\ \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}\\ @@ -1311,18 +1253,21 @@$E_{\text{f}}=3.96\text{ eV}$\\ \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} @@ -1381,6 +1326,9 @@$E_{\text{f}}=5.18\text{ eV}$\\ \end{slide} +\end{document} +\ifnum1=0 + \begin{slide} \footnotesize @@ -1614,25 +1562,6 @@$\rightarrow\$
\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}