From: hackbard Date: Wed, 22 Jul 2009 15:40:43 +0000 (+0200) Subject: new params X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=162206abf206f18e6d23e452b95a903bd2aafae1 new params --- diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex index 4197c0a..0419252 100644 --- a/posic/talks/upb-ua-xc.tex +++ b/posic/talks/upb-ua-xc.tex @@ -392,6 +392,49 @@ POTIM = 0.1 \end{slide} +\begin{slide} + + {\large\bf + Calculation of the defect formation energy\\ + } + + \small + + {\color{blue}Method 1} (single species) + \begin{itemize} + \item $E_{\textrm{coh}}^{\textrm{initial conf}}$: + cohesive energy per atom of the initial system + \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$: + cohesive energy per atom of the interstitial system + \item N: amount of atoms in the interstitial system + \end{itemize} + \vspace*{0.2cm} + {\color{blue} + \[ + \Rightarrow + E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}} + -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N + \] + }\\[0.4cm] + {\color{magenta}Method 2} (two and more species) + \begin{itemize} + \item $E$: energy of the interstitial system + (with respect to the ground state of the free atoms!) + \item $N_{\text{Si}}$, $N_{\text{C}}$: + amount of Si and C atoms + \item $\mu_{\text{Si}}$, $\mu_{\text{C}}$: + chemical potential (cohesive energy) of Si and C + \end{itemize} + \vspace*{0.2cm} + {\color{magenta} + \[ + \Rightarrow + E_{\textrm{f}}=E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}} + \] + } + +\end{slide} + \begin{slide} {\large\bf @@ -443,33 +486,37 @@ POTIM = 0.1 \small - Calculation of formation energy $E_{\textrm{f}}$ - \begin{itemize} - \item $E_{\textrm{coh}}^{\textrm{initial conf}}$: - cohesive energy per atom of the initial system - \item $E_{\textrm{coh}}^{\textrm{interstitial conf}}$: - cohesive energy per atom of the interstitial system - \item N: amount of atoms in the interstitial system - \end{itemize} - \vspace*{0.2cm} - {\color{blue} - \[ - \Rightarrow - E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}} - -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N - \] - } Influence of supercell size\\ \begin{minipage}{8cm} \includegraphics[width=7.0cm]{si_self_int.ps} \end{minipage} \begin{minipage}{5cm} - $E_{\textrm{f}}^{\textrm{110},\,{\color{red}32}\textrm{pc}}=3.38\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\ - $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$ + $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$ \end{minipage} + Comparison with literature (PRL 88 235501 (2002)):\\[0.2cm] + \begin{minipage}{8cm} + \begin{itemize} + \item GGA and LDA + \item $E_{\text{cut-off}}=35 / 25\text{ Ry}=476 / 340\text{ eV}$ + \item 216 atom supercell + \item Gamma point only calculations + \end{itemize} + \end{minipage} + \begin{minipage}{5cm} + $E_{\textrm{f}}^{\textrm{110}}=3.31 / 2.88\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{hex}}=3.31 / 2.87\textrm{ eV}$\\ + $E_{\textrm{f}}^{\textrm{vac}}=3.17 / 3.56\textrm{ eV}$ + \end{minipage} + + \end{slide} \begin{slide} @@ -507,22 +554,27 @@ POTIM = 0.1 Smearing method for the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$ and $k$-point mesh + \begin{minipage}{4.4cm} + \includegraphics[width=4.4cm]{sic_smear_k.ps} + \end{minipage} + \begin{minipage}{4.4cm} + \includegraphics[width=4.4cm]{c_smear_k.ps} + \end{minipage} + \begin{minipage}{4.3cm} + \includegraphics[width=4.4cm]{si_smear_k.ps} + \end{minipage}\\[0.3cm] \begin{itemize} - \item $1\times 1\times 1$ Type 0 simulations - \begin{itemize} - \item No difference in tetrahedron method and Gauss smearing - \item ... - \end{itemize} - \item $1\times 1\times 1$ Type 2 simulations - \begin{itemize} - \item Again, no difference in tetrahedron method and Gauss smearing - \item ... - \end{itemize} + \item Convergence reached at $6\times 6\times 6$ k-point mesh + \item No difference between Gauss ($\sigma=0.05$) + and tetrahedron smearing method! \end{itemize} - - {\LARGE\bf\color{red} - More simulations running ... + \begin{center} + $\Downarrow$\\ + {\color{blue}\bf + Gauss ($\sigma=0.05$) smearing + and $6\times 6\times 6$ Monkhorst $k$-point mesh used } + \end{center} \end{slide} @@ -532,41 +584,54 @@ POTIM = 0.1 Review (so far) ...\\ } - Symmetry (in defect simulations) - - {\LARGE\bf\color{red} - Simulations running ... - } - -\end{slide} + \underline{Symmetry (in defect simulations)} -\begin{slide} + \begin{center} + {\color{red}No} + difference in $1\times 1\times 1$ Type 2 defect calculations\\ + $\Downarrow$\\ + Symmetry precission (SYMPREC) small enough\\ + $\Downarrow$\\ + {\bf\color{blue}Symmetry switched on}\\ + \end{center} - {\large\bf - Review (so far) ...\\ - } + \underline{Real space projection} - Real space projection + \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): + $0.026\,\%$\\ + $\Downarrow$\\ + {\bf\color{blue} + Real space projection used for 'large supercell' simulations} + \end{center} \end{slide} \begin{slide} {\large\bf - Review (so far) ...\\ + Review (so far) ... } - Energy cut-off + Energy cut-off\\ -\end{slide} + \begin{center} -\begin{slide} + {\small + 3C-SiC equilibrium lattice constant and free energy\\ + \includegraphics[width=7cm]{plain_sic_lc.ps}\\ + $\rightarrow$ Convergence reached at 650 eV\\[0.2cm] + } - {\large\bf - Review (so far) ...\\ + $\Downarrow$\\ + + {\bf\color{blue} + 650 eV used as energy cut-off } - Size and type of supercell + \end{center} \end{slide} @@ -590,6 +655,72 @@ POTIM = 0.1 \vspace{1.5cm} +\end{slide} + +\begin{slide} + + {\large\bf + Final parameter choice + } + + \footnotesize + + \underline{Param 1}\\ + My first choice. Used for more accurate calculations. + \begin{itemize} + \item $6\times 6 \times 6$ Monkhorst k-point mesh + \item $E_{\text{cut-off}}=650\text{ eV}$ + \item Gaussian smearing ($\sigma=0.05$) + \item Use symmetry + \end{itemize} + \vspace*{0.2cm} + \underline{Param 2}\\ + After talking to the pros! Used for 'large' simulations. + \begin{itemize} + \item $\Gamma$-point only + \item $E_{\text{cut-off}}=xyz\text{ eV}$ + \item Gaussian smearing ($\sigma=0.05$) + \item Use symmetry + \item Real space projection (Auto, Medium) + \end{itemize} + \vspace*{0.2cm} + {\color{blue} + In both parameter sets the ultra soft pseudo potential method + as well as the projector augmented wave method is used! + } +\end{slide} + +\begin{slide} + + {\large\bf + Properties of Si, C and SiC using the new parameters\\ + } + + $2\times 2\times 2$ Type 2 supercell, Param 1\\[0.2cm] + \begin{tabular}{|l|l|l|l|} + \hline + & c-Si & c-C (diamond) & 3C-SiC \\ + \hline + Lattice constant [\AA] & 5.389 & 3.527 & \\ + Expt. [\AA] & 5.429 & 3.567 & \\ + Error [\%] & {\color{green}0.7} & 1.1 & \\ + \hline + Cohesive energy [eV] & -4.674 & -8.812 & \\ + Expt. [eV] & -4.63 & -7.374 & \\ + Error [\%] & {\color{green}1.0} & {\color{red}19.5} & \\ + \hline + \end{tabular}\\ + +\end{slide} + +\begin{slide} + + {\large\bf + C interstitial in c-Si + } + + + \end{slide} \end{document}