From: hackbard <hackbard@hackdaworld.org>
Date: Mon, 7 Nov 2011 14:24:37 +0000 (+0100)
Subject: started point defects
X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=ab8104bce0b6ea93877f9a5102a9fa933c4bf17f;p=lectures%2Flatex.git

started point defects
---

diff --git a/posic/talks/mpi_app.tex b/posic/talks/mpi_app.tex
index c05db6d..e4cdd48 100644
--- a/posic/talks/mpi_app.tex
+++ b/posic/talks/mpi_app.tex
@@ -991,9 +991,6 @@ r = \unit[2--4]{nm}
 
 \end{slide}
 
-% continue here
-\fi
-
 \begin{slide}
 
 \headphd
@@ -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
@@ -1032,164 +1025,78 @@ 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}