X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Ftalks%2Fupb-ua-xc.tex;h=e95c0fb64ce6488fd76c19a89902aaad9c85d829;hb=fee777e5d2af71751eddd51826ede4945aea40e7;hp=a3d1048f9c5eadf272cfff136966e1de6a1c37fb;hpb=200984a06c22985aa9403bd9950934430d770906;p=lectures%2Flatex.git diff --git a/posic/talks/upb-ua-xc.tex b/posic/talks/upb-ua-xc.tex index a3d1048..e95c0fb 100644 --- a/posic/talks/upb-ua-xc.tex +++ b/posic/talks/upb-ua-xc.tex @@ -97,7 +97,7 @@ \vspace{08pt} - June 2009 + July 2009 \end{center} \end{slide} @@ -185,6 +185,7 @@ POTIM = 0.1 \item Supercell: $x_1=(2,0,0),\, x_2=(0,2,0),\, x_3=(0,0,2)$; 64 atoms (32 pc) \end{enumerate} + \begin{minipage}{6cm} Cohesive energy / Lattice constant: \begin{enumerate} \item $E_{\textrm{cut-off}}=150\, \textrm{eV}$: 5.955 eV / 5.378 \AA\\ @@ -197,32 +198,215 @@ POTIM = 0.1 $E_{\textrm{cut-off}}=300\, \textrm{eV}^{*}$: 5.975 eV / 5.390 \AA \item $E_{\textrm{cut-off}}=300\, \textrm{eV}$: 5.977 eV / 5.389 \AA \end{enumerate} + \end{minipage} + \begin{minipage}{7cm} + \includegraphics[width=7cm]{si_lc_and_ce.ps} + \end{minipage}\\[0.3cm] + {\scriptsize + $^*$special settings (p. 138, VASP manual): + spin polarization, no symmetry, ... + } \end{slide} \begin{slide} {\large\bf - Interstitial configurations + Silicon bulk properties + } + + \begin{itemize} + \item Calculation of cohesive energies for different lattice constants + \item No ionic update + \item tetrahedron method with Blöchl corrections for + the partial occupancies $f_{nk}$ + \item Supercell 3 (8 atoms, 4 primitive cells) + \end{itemize} + \vspace*{0.6cm} + \begin{minipage}{6.5cm} + \begin{center} + $E_{\textrm{cut-off}}=150$ eV\\ + \includegraphics[width=6.5cm]{si_lc_fit.ps} + \end{center} + \end{minipage} + \begin{minipage}{6.5cm} + \begin{center} + $E_{\textrm{cut-off}}=250$ eV\\ + \includegraphics[width=6.5cm]{si_lc_fit_250.ps} + \end{center} + \end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf + 3C-SiC bulk properties\\[0.2cm] } - Silicon: + \begin{minipage}{6.5cm} + \includegraphics[width=6.5cm]{sic_lc_and_ce2.ps} + \end{minipage} + \begin{minipage}{6.5cm} + \includegraphics[width=6.5cm]{sic_lc_and_ce.ps} + \end{minipage}\\[0.3cm] \begin{itemize} - \item Lattice constant: - \item Cohesive energy: 5.95 eV, 5.99 eV, 5.96 eV, 5.98 eV + \item Supercell 3 (4 primitive cells, 4+4 atoms) + \item Error in equilibrium lattice constant: {\color{green} $0.9\,\%$} + \item Error in cohesive energy: {\color{red} $31.6\,\%$} \end{itemize} - <100> interstitial: +\end{slide} + +\begin{slide} + + {\large\bf + 3C-SiC bulk properties\\[0.2cm] + } + + \small + \begin{itemize} - \item Lattice constant: - \item Cohesive energy: + \item Calculation of cohesive energies for different lattice constants + \item No ionic update + \item tetrahedron method with Blöchl corrections for + the partial occupancies $f_{nk}$ \end{itemize} + \vspace*{0.6cm} + \begin{minipage}{6.5cm} + \begin{center} + Supercell 3, $4\times 4\times 4$ k-points\\ + \includegraphics[width=6.5cm]{sic_lc_fit.ps} + \end{center} + \end{minipage} + \begin{minipage}{6.5cm} + \begin{center} + {\color{red} + Non-continuous energies\\ + for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$! + } + \end{center} + \end{minipage} + +\end{slide} + +\begin{slide} - Hexagonal interstitial: + {\large\bf + 3C-SiC bulk properties\\[0.2cm] + } + + \footnotesize + +\begin{picture}(0,0)(-188,80) + %Supercell 1, $3\times 3\times 3$ k-points\\ + \includegraphics[width=6.5cm]{sic_lc_fit_k3.ps} +\end{picture} + + \begin{minipage}{6.5cm} \begin{itemize} - \item Lattice constant: - \item Cohesive energy: + \item Supercell 1 simulations + \item Variation of k-points + \item Continuous energies for + $E_{\textrm{cut-off}} > 550\,\textrm{eV}$ + \item Critical $E_{\textrm{cut-off}}$ for + different k-points\\ + depending on supercell? \end{itemize} + \end{minipage}\\[1.0cm] + \begin{minipage}{6.5cm} + \begin{center} + \includegraphics[width=6.5cm]{sic_lc_fit_k5.ps} + \end{center} + \end{minipage} + \begin{minipage}{6.5cm} + \begin{center} + \includegraphics[width=6.5cm]{sic_lc_fit_k7.ps} + \end{center} + \end{minipage} + +\end{slide} + +\begin{slide} + + {\large\bf + Cohesive energies + } + + {\bf\color{red} From now on ...} + + {\small Energies used: free energy without entropy ($\sigma \rightarrow 0$)} + + \small + + \begin{itemize} + \item $E_{\textrm{free,sp}}$: + energy of spin polarized free atom + \begin{itemize} + \item $k$-points: Monkhorst $1\times 1\times 1$ + \item Symmetry switched off + \item Spin polarized calculation + \item Interpolation formula according to Vosko Wilk and Nusair + for the correlation part of the exchange correlation functional + \item Gaussian smearing for the partial occupancies $f_{nk}$ + ($\sigma=0.05$) + \item Magnetic mixing: AMIX = 0.2, BMIX = 0.0001 + \item Supercell: one atom in cubic + $10\times 10\times 10$ \AA$^3$ box + \end{itemize} + {\color{blue} + $E_{\textrm{free,sp}}(\textrm{Si},250\, \textrm{eV})= + -0.70036911\,\textrm{eV}$ + }, + {\color{gray} + $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})= + yyy\,\textrm{eV}$ + } + \item $E$: + energy (non-polarized) of system of interest composed of\\ + n atoms of type N, m atoms of type M, \ldots + \end{itemize} + \vspace*{0.3cm} + {\color{red} + \[ + \Rightarrow + E_{\textrm{coh}}=\frac{ + -\Big(E(N_nM_m\ldots)-nE_{\textrm{free,sp}}(N)-mE_{\textrm{free,sp}}(M) + -\ldots\Big)} + {n+m+\ldots} + \] + } + +\end{slide} + +\begin{slide} + + {\large\bf + Silicon point defects\\ + } + + \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} + \[ + \Rightarrow + E_{\textrm{f}}=\Big(E_{\textrm{coh}}^{\textrm{interstitial conf}} + -E_{\textrm{coh}}^{\textrm{initial conf}}\Big) N + \] + } + + \begin{center} + \includegraphics[width=7.0cm]{si_self_int.ps} + \end{center} \end{slide}