X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=posic%2Ftalks%2Fupb-ua-xc.tex;h=23aff0777d5909a7b080b423d63a0713daa26105;hp=617ec4f2107d0f9c2b7e8f738b8711106c8eb7e1;hb=d6b64d4be7e48f04395e2862f4a53e0f06198e26;hpb=5affbb4badd4957136391742e083f6fbdbfc663c diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex index 617ec4f..23aff07 100644 --- a/posic/talks/upb-ua-xc.tex +++ b/posic/talks/upb-ua-xc.tex @@ -46,6 +46,8 @@ \usepackage{upgreek} +\usepackage{miller} + \begin{document} \extraslideheight{10in} @@ -491,14 +493,14 @@ POTIM = 0.1 \includegraphics[width=7.0cm]{si_self_int.ps} \end{minipage} \begin{minipage}{5cm} - $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\ + $E_{\textrm{f}}^{\hkl<1 1 0>,\,32\textrm{pc}}=3.38\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\ - $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$ + $E_{\textrm{f}}^{\hkl<1 1 0>,\,54\textrm{pc}}=3.48\textrm{ eV}$ \end{minipage} Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm] @@ -511,7 +513,7 @@ POTIM = 0.1 \end{itemize} \end{minipage} \begin{minipage}{5cm} - $E_{\textrm{f}}^{\textrm{110}}=3.31 / 2.88\textrm{ eV}$\\ + $E_{\textrm{f}}^{\hkl<1 1 0>}=3.31 / 2.88\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$ \end{minipage} @@ -600,7 +602,8 @@ POTIM = 0.1 \begin{center} Error in lattice constant of plain Si ($1\times 1\times 1$ Type 2): $0.025\,\%$\\ - Error in position of the 110 interstitital in Si ($1\times 1\times 1$ Type 2): + Error in position of the \hkl<1 1 0> interstitital in Si + ($1\times 1\times 1$ Type 2): $0.026\,\%$\\ $\Downarrow$\\ {\bf\color{blue} @@ -747,7 +750,7 @@ POTIM = 0.1 \begin{slide} - {\large\bf + {\large\bf\boldmath Energy cut-off for $\Gamma$-point only caclulations } @@ -780,8 +783,9 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (Albe potential) + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> + in c-Si (Albe) } \small @@ -805,7 +809,7 @@ POTIM = 0.1 \item Fix border atoms of the simulation cell \item Constraints and displacement of the C atom: \begin{itemize} - \item along {\color{green}110 direction}\\ + \item along {\color{green}\hkl<1 1 0> direction}\\ displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$} \item C atom {\color{red}entirely fixed in position}\\ displaced by @@ -826,8 +830,9 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (Albe potential) + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> + in c-Si (Albe) } \footnotesize @@ -885,8 +890,9 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (Albe potential)\\ + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> + in c-Si (Albe)\\ } Displacement step size decreased to @@ -918,8 +924,9 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (Albe potential) + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> + in c-Si (Albe) } {\color{blue}New approach:}\\ @@ -954,8 +961,8 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (VASP) + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP) } \small @@ -964,7 +971,7 @@ POTIM = 0.1 \begin{itemize} \item Place interstitial carbon atom at the respective coordinates into perfect c-Si - \item 110 direction fixed for the C atom + \item \hkl<1 1 0> direction fixed for the C atom \item $4\times 4\times 3$ Type 1, $198+1$ atoms \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed \end{itemize} @@ -986,8 +993,8 @@ POTIM = 0.1 \begin{slide} - {\large\bf - C 100 interstitial migration along 110 in c-Si (VASP) + {\large\bf\boldmath + C \hkl<1 0 0> interstitial migration along \hkl<1 1 0> in c-Si (VASP) } \small @@ -995,8 +1002,8 @@ POTIM = 0.1 {\color{blue}Method:} \begin{itemize} \item Continue with atomic positions of the last run - \item Displace the C atom in 110 direction - \item 110 direction fixed for the C atom + \item Displace the C atom in \hkl<1 1 0> direction + \item \hkl<1 1 0> direction fixed for the C atom \item $4\times 4\times 3$ Type 1, $198+1$ atoms \item Atoms with $x=0$ or $y=0$ or $z=0$ fixed \end{itemize} @@ -1009,8 +1016,8 @@ POTIM = 0.1 \begin{slide} - {\large\bf - Again: C 100 interstitial migration + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration } \small @@ -1019,8 +1026,8 @@ POTIM = 0.1 \begin{enumerate} \item Method \begin{itemize} - \item Start in relaxed 100 interstitial configuration - \item Displace C atom along 110 direction + \item Start in relaxed \hkl<1 0 0> interstitial configuration + \item Displace C atom along \hkl<1 1 0> direction \item Relaxation (Berendsen thermostat) \item Continue with configuration of the last run \end{itemize} @@ -1034,7 +1041,7 @@ POTIM = 0.1 {\color{blue}In both methods:} \begin{itemize} \item Fixed border atoms - \item Applied 110 constraint for the C atom + \item Applied \hkl<1 1 0> constraint for the C atom \end{itemize} {\color{red}Pitfalls} and {\color{green}refinements}: \begin{itemize} @@ -1042,7 +1049,7 @@ POTIM = 0.1 Relaxation of stress not possible\\ $\Rightarrow$ {\color{green}Fix only one Si atom} (the one furthermost to the defect) - \item {\color{red}110 constraint not sufficient}\\ + \item {\color{red}\hkl<1 1 0> constraint not sufficient}\\ $\Rightarrow$ {\color{green}Apply 11x constraint} (connecting line of initial and final C positions) \end{itemize} @@ -1051,11 +1058,11 @@ POTIM = 0.1 \begin{slide} - {\large\bf - Again: C 100 interstitial migration (Albe) + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration (Albe) } - Constraint applied by modyfing the Velocity Verlet algorithm + Constraint applied by modifying the Velocity Verlet algorithm {\color{blue}Results:} (Video \href{../video/c_in_si_fmig_albe.avi}{$\rhd_{\text{local}}$ } $|$ @@ -1080,8 +1087,8 @@ POTIM = 0.1 \begin{slide} - {\large\bf - Again: C 100 interstitial migration (VASP) + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration (VASP) } Transformation for the Type 2 supercell @@ -1137,8 +1144,8 @@ POTIM = 0.1 \begin{slide} - {\large\bf - Again: C 100 interstitial migration\\ + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration\\ } {\color{blue}Reminder:}\\ @@ -1164,16 +1171,18 @@ POTIM = 0.1 \begin{slide} - {\large\bf - The C 100 defect configuration + {\large\bf\boldmath + The C \hkl<1 0 0> defect configuration } Needed so often for input configurations ...\\[0.8cm] - \begin{minipage}{7.7cm} - \includegraphics[width=7cm]{100-c-si-db_light.eps} - \hfill + \begin{minipage}{7.0cm} + \includegraphics[width=6.5cm]{100-c-si-db_light.eps}\\ + Qualitative {\color{red}and} quantitative {\color{red}difference}! \end{minipage} - \begin{minipage}{4.5cm} + \begin{minipage}{5.5cm} + \scriptsize + \begin{center} \begin{tabular}{|l|l|l|} \hline & a & b \\ @@ -1186,39 +1195,292 @@ POTIM = 0.1 fractional & 0.1547 & 0.1676 \\ in \AA & 0.84 & 0.91 \\ \hline - \end{tabular} + \end{tabular}\\[0.2cm] + {\scriptsize\underline{PC (Vasp)}} + \includegraphics[width=6.1cm]{c_100_pc_vasp.ps} + \end{center} \end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration (VASP) + } + + $\hkl<0 0 -1> \rightarrow \hkl<0 0 1>$ migration + ($3\times 3\times 3$ Type 2): + + \small + + \begin{minipage}[t]{4.1cm} + \underline{Starting configuration}\\ + \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps} \begin{center} - Qualitative {\color{red}and} quantitative {\color{red}difference}! + $E_{\text{f}}=3.15 \text{ eV}$ \end{center} + \end{minipage} + \begin{minipage}[t]{4.1cm} + \underline{Intermediate configuration}\\ + \includegraphics[height=3.2cm]{c_100_mig_vasp/00-1_001_im.eps} + \begin{center} + $E_{\text{f}}=4.41 \text{ eV}$ + \end{center} + \end{minipage} + \begin{minipage}[t]{4.1cm} + \underline{Final configuration}\\ + \includegraphics[height=3.2cm]{c_100_mig_vasp/final.eps} + \begin{center} + $E_{\text{f}}=3.17 \text{ eV}$ + \end{center} + \end{minipage}\\[0.4cm] + \[ + \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = 1.26 \text{ eV} + \] + Unexpected \& ({\color{red}more} or {\color{orange}less}) fatal: + \begin{itemize} + \renewcommand\labelitemi{{\color{orange}$\bullet$}} + \item Difference in formation energy (0.02 eV) + of the initial and final configuration + \renewcommand\labelitemi{{\color{red}$\bullet$}} + \item Huge discrepancy (0.3 - 0.4 eV) to the migration barrier + of Type 1 (198+1 atoms) calculations + \renewcommand\labelitemi{{\color{black}$\bullet$}} + \end{itemize} + \end{slide} \begin{slide} - {\large\bf - Again: C 100 interstitial migration + {\large\bf\boldmath + Again: C \hkl<1 0 0> interstitial migration (VASP) } + $\hkl<0 0 -1> \rightarrow \hkl<0 -1 0>$ migration + ($3\times 3\times 3$ Type 2): + \small - \underline{$00-1 \rightarrow$ bond centered $\rightarrow 001$} + \begin{minipage}[t]{4.1cm} + \underline{Starting configuration}\\ + \includegraphics[height=3.2cm]{c_100_mig_vasp/start.eps} + \begin{center} + $E_{\text{f}}=3.154 \text{ eV}$ + \end{center} + \end{minipage} + \begin{minipage}[t]{4.1cm} + \underline{Intermediate configuration}\\ + in progress ... + \begin{center} + $E_{\text{f}}=?.?? \text{ eV}$ + \end{center} + \end{minipage} + \begin{minipage}[t]{4.1cm} + \underline{Final configuration}\\ + \includegraphics[height=3.2cm]{c_100_mig_vasp/0-10.eps} + \begin{center} + $E_{\text{f}}=3.157 \text{ eV}$ + \end{center} + \end{minipage}\\[0.4cm] + \[ + \Rightarrow \Delta E_{\text{f}} = E_{\text{mig}} = ?.?? \text{ eV} + \] - $\Delta E_{\text{coh}}$\\ - $\Delta E_{\text{f}}$ + \vspace*{0.5cm} + {\large\bf + Intermediate configuration {\color{red}not found} by now! + } \end{slide} +\begin{slide} + + {\large\bf\boldmath + C in Si interstitial configurations (VASP) + } + + Check of Kohn-Sham eigenvalues\\ + + \small + + \begin{minipage}{6cm} + \hkl<1 0 0> interstitial\\ + \end{minipage} + \begin{minipage}{6cm} + Saddle point configuration\\ + \end{minipage} + \underline{$4\times 4\times 3$ Type 1 - fixed border atoms}\\ + \begin{minipage}{6cm} +385: 4.8567 - 2.00000\\ +386: 4.9510 - 2.00000\\ +387: 5.3437 - 0.00000\\ +388: 5.4930 - 0.00000 + \end{minipage} + \begin{minipage}{6cm} +385: 4.8694 - 2.00000\\ +386: {\color{red}4.9917} - 1.92603\\ +387: {\color{red}5.1181} - 0.07397\\ +388: 5.4541 - 0.00000 + \end{minipage}\\[0.2cm] + \underline{$4\times 4\times 3$ Type 1 - no constraints}\\ + \begin{minipage}{6cm} +385: 4.8586 - 2.00000\\ +386: 4.9458 - 2.00000\\ +387: 5.3358 - 0.00000\\ +388: 5.4915 - 0.00000 + \end{minipage} + \begin{minipage}{6cm} +385: 4.8693 - 2.00000\\ +386: {\color{red}4.9879} - 1.92065\\ +387: {\color{red}5.1120} - 0.07935\\ +388: 5.4544 - 0.00000 + \end{minipage}\\[0.2cm] + \underline{$3\times 3\times 3$ Type 2 - no constraints}\\ + \begin{minipage}{6cm} +433: 4.8054 - 2.00000\\ +434: 4.9027 - 2.00000\\ +435: 5.2543 - 0.00000\\ +436: 5.5718 - 0.00000 + \end{minipage} + \begin{minipage}{6cm} +433: 4.8160 - 2.00000\\ +434: {\color{green}5.0109} - 1.00264\\ +435: {\color{green}5.0111} - 0.99736\\ +436: 5.5364 - 0.00000 + \end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Once again: C \hkl<1 0 0> interstitial migration (VASP) + } + + Method: + \begin{itemize} + \item Start in fully relaxed (assumed) saddle point configuration + \item Move towards \hkl<1 0 0> configuration using updated values + for $\Delta x$, $\Delta y$ and $\Delta z$ + \item \hkl<1 1 0> constraints applied, 1 Si atom fixed + \item $4\times 4\times 3$ Type 1 supercell + \end{itemize} + + Results: + + \begin{minipage}{6.2cm} + \includegraphics[width=6.0cm]{c_100_110sp-i_vasp.ps} + \end{minipage} + \begin{minipage}{6.2cm} + \includegraphics[width=6.0cm]{c_100_110sp-i_rc_vasp.ps} + \end{minipage} + + Reaction coordinate: + $r_{i+1}=r_i+\sum_{\text{atoms j}} \left| r_{j,i+1}-r_{j,i} \right|$ + +\end{slide} + +\begin{slide} + + {\large\bf\boldmath + Investigation of the migration path along \hkl<1 1 0> (VASP) + } + + \small + + \underline{Minimum:}\\ + \begin{minipage}{4cm} + \includegraphics[width=3.5cm]{c_100_mig_vasp/110_c-si_split.eps} + \end{minipage} + \begin{minipage}{8cm} + \begin{itemize} + \item Starting conf: 35 \% displacement results + \item \hkl<1 1 0> constraint disabled + \end{itemize} + \begin{center} + $\Downarrow$ + \end{center} + \begin{itemize} + \item C-Si \hkl<1 1 0> split interstitial + \item Stable configuration + \item $E_{\text{f}}=4.13\text{ eV}$ + \end{itemize} + \end{minipage}\\[0.1cm] + + \underline{Maximum:}\\ + \begin{minipage}{6cm} + \begin{center} + \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_01.eps} + \includegraphics[width=2.3cm]{c_100_mig_vasp/100-110_02.eps}\\ + 20 \% $\rightarrow$ 25 \%\\ + Breaking of Si-C bond + \end{center} + \end{minipage} + \begin{minipage}{6cm} + \includegraphics[width=6.2cm]{c_100_110sp-i_upd_vasp.ps} + \end{minipage} + +\end{slide} + \begin{slide} {\large\bf Molecular dynamics simulations (VASP) } - + 2 C atoms in $2\times 2\times 2$ Type 2 supercell at $450\,^{\circ}\text{C}$ + \small - + + \begin{minipage}{7.6cm} + Radial distribution\\ + \includegraphics[width=7.6cm]{md_02c_2222si_pc.ps} + \end{minipage} + \begin{minipage}{5.0cm} + \begin{center} + PC average from\\ + $t_1=50$ ps to $t_2=50.93$ ps + \end{center} + \end{minipage} + Diffusion: + \begin{itemize} + \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of + boundary crossings + \item No jumps recognized in the + Video \href{../video/md_02c_2222si_vasp.avi}{$\rhd_{\text{local}}$ } $|$ + \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_02c_2222si_vasp.avi}{$\rhd_{\text{remote url}}$} + \end{itemize} + +\end{slide} + +\begin{slide} + + {\large\bf + Molecular dynamics simulations (VASP) + } + + 10 C atoms in $3\times 3\times 3$ Type 2 supercell at $450\,^{\circ}\text{C}$ + + \small + + \begin{minipage}{7.2cm} + Radial distribution (PC averaged over 1 ps)\\ + \includegraphics[width=7.0cm]{md_10c_2333si_pc_vasp.ps} + \end{minipage} + \begin{minipage}{5.0cm} + \includegraphics[width=6.0cm]{md_10c_2333si_pcc_vasp.ps} + \end{minipage} + Diffusion: + (Video \href{../video/md_10c_2333si_vasp.avi}{$\rhd_{\text{local}}$ } $|$ + \href{http://www.physik.uni-augsburg.de/~zirkelfr/download/posic/md_10c_2333si_vasp.avi}{$\rhd_{\text{remote url}}$}) + \begin{itemize} + \item $<(x(t)-x(0))^2>$ hard to determine due to missing info of + boundary crossings + \item Agglomeration of C? (Video) + \end{itemize} \end{slide}