\item Calculation of cohesive energies for different lattice constants
\item No ionic update
\item Tetrahedron method with Blöchl corrections for
- the partial occupancies $f_{nk}$
+ the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
\item Supercell 3 (8 atoms, 4 primitive cells)
\end{itemize}
\vspace*{0.6cm}
\item Calculation of cohesive energies for different lattice constants
\item No ionic update
\item Tetrahedron method with Blöchl corrections for
- the partial occupancies $f_{nk}$
+ the partial occupancies $f(\{\epsilon_{n{\bf k}}\})$
\end{itemize}
\vspace*{0.6cm}
\begin{minipage}{6.5cm}
\begin{center}
{\color{red}
Non-continuous energies\\
- for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!
+ for $E_{\textrm{cut-off}}<1050\,\textrm{eV}$!\\
+ }
+ \vspace*{0.5cm}
+ {\footnotesize
+ Does this matter in structural optimizaton simulations?
+ \begin{itemize}
+ \item Derivative might be continuous
+ \item Similar lattice constants where derivative equals zero
+ \end{itemize}
}
\end{center}
\end{minipage}
\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 $f_{nk}$
+ \item Gaussian smearing for the partial occupancies
+ $f(\{\epsilon_{n{\bf k}}\})$
($\sigma=0.05$)
\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})=
+ $E_{\textrm{free,sp}}(\textrm{Si},{\color{green}250}\, \textrm{eV})=
-0.70036911\,\textrm{eV}$
+ }\\
+ {\color{blue}
+ $E_{\textrm{free,sp}}(\textrm{Si},{\color{red}650}\, \textrm{eV})=
+ -0.70021403\,\textrm{eV}$
},
{\color{gray}
- $E_{\textrm{free,sp}}(\textrm{C},xxx\, \textrm{eV})=
- yyy\,\textrm{eV}$
+ $E_{\textrm{free,sp}}(\textrm{C},{\color{red}650}\, \textrm{eV})=
+ -1.3535731\,\textrm{eV}$
}
\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}
+ \vspace*{0.2cm}
{\color{red}
\[
\Rightarrow
\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
\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}
$E_{\textrm{f}}^{\textrm{110},\,32\textrm{pc}}=3.38\textrm{ eV}$\\
$E_{\textrm{f}}^{\textrm{tet},\,32\textrm{pc}}=3.41\textrm{ eV}$\\
$E_{\textrm{f}}^{\textrm{hex},\,32\textrm{pc}}=3.42\textrm{ eV}$\\
- $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$
+ $E_{\textrm{f}}^{\textrm{vac},\,32\textrm{pc}}=3.51\textrm{ eV}$\\\\
+ $E_{\textrm{f}}^{\textrm{hex},\,54\textrm{pc}}=3.42\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{tet},\,54\textrm{pc}}=3.45\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{vac},\,54\textrm{pc}}=3.47\textrm{ eV}$\\
+ $E_{\textrm{f}}^{\textrm{110},\,54\textrm{pc}}=3.48\textrm{ eV}$
\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}
\item hence also connected to choice of smearing method?
\item constraints can only be applied to the lattice vectors!
\end{itemize}
+ \item Use of real space projection operators?
\item \ldots
\end{itemize}
Review (so far) ...\\
}
-
-
+ 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 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}
+ \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}
+
+\begin{slide}
+
+ {\large\bf
+ Review (so far) ...\\
+ }
+
+ \underline{Symmetry (in defect simulations)}
+
+ \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}
+
+ \underline{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) ...
+ }
+
+ Energy cut-off\\
+
+ \begin{center}
+
+ {\small
+ 3C-SiC equilibrium lattice constant and free energy\\
+ \includegraphics[width=7cm]{plain_sic_lc.ps}\\
+ $\rightarrow$ Convergence reached at 650 eV\\[0.2cm]
+ }
+
+ $\Downarrow$\\
+
+ {\bf\color{blue}
+ 650 eV used as energy cut-off
+ }
+
+ \end{center}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Not answered (so far) ...\\
+ }
+
+\vspace{1.5cm}
+
+ \LARGE
+ \bf
+ \color{blue}
+
+ \begin{center}
+ Continue\\
+ with\\
+ US LDA?
+ \end{center}
+
+\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!
+ \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) for 'large' simulations
+ \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 with both,
+ the LDA and GGA exchange correlation potential!
+ }
+\end{slide}
+
+\begin{slide}
+
+ \footnotesize
+
+ {\large\bf
+ Properties of Si, C and SiC using the new parameters\\
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 1, LDA, US PP\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+ & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.389 & 3.527 & 4.319 \\
+ Expt. [\AA] & 5.429 & 3.567 & 4.359 \\
+ Error [\%] & {\color{green}0.7} & {\color{green}1.1} & {\color{green}0.9} \\
+ \hline
+ Cohesive energy [eV] & -5.277 & -8.812 & -7.318 \\
+ Expt. [eV] & -4.63 & -7.374 & -6.340 \\
+ Error [\%] & {\color{red}14.0} & {\color{red}19.5} & {\color{red}15.4} \\
+ \hline
+ \end{tabular}\\
+
+ \begin{minipage}{10cm}
+ $2\times 2\times 2$ Type 2 supercell, 3C-SiC, Param 1\\[0.2cm]
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+ & {\color{magenta}US PP, GGA} & PAW, LDA & PAW, GGA \\
+ \hline
+ Lattice constant [\AA] & 4.370 & 4.330 & 4.379 \\
+ Error [\%] & {\color{green}0.3} & {\color{green}0.7} & {\color{green}0.5} \\
+ \hline
+ Cohesive energy [eV] & -6.426 & -7.371 & -6.491 \\
+ Error [\%] & {\color{green}1.4} & {\color{red}16.3} & {\color{green}2.4} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{3cm}
+ US PP, GGA\\[0.2cm]
+ \begin{tabular}{|l|l|}
+ \hline
+ c-Si & c-C \\
+ \hline
+ 5.455 & 3.567 \\
+ {\color{green}0.5} & {\color{green}0.01} \\
+ \hline
+ -4.591 & -7.703 \\
+ {\color{green}0.8} & {\color{orange}4.5} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ Energy cut-off for $\Gamma$-point only caclulations
+ }
+
+ $2\times 2\times 2$ Type 2 supercell, Param 2, US PP, LDA, 3C-SiC\\[0.2cm]
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff.ps}
+ \includegraphics[width=5.5cm]{sic_32pc_gamma_cutoff_lc.ps}\\
+ $\Rightarrow$ Use 300 eV as energy cut-off?\\[0.2cm]
+ $2\times 2\times 2$ Type 2 supercell, Param 2, 300 eV, US PP, GGA\\[0.2cm]
+ \small
+ \begin{minipage}{10cm}
+ \begin{tabular}{|l|l|l|l|}
+ \hline
+ & c-Si & c-C (diamond) & 3C-SiC \\
+ \hline
+ Lattice constant [\AA] & 5.470 & 3.569 & 4.364 \\
+ Error [\%] & {\color{green}0.8} & {\color{green}0.1} & {\color{green}0.1} \\
+ \hline
+ Cohesive energy [eV] & -4.488 & -7.612 & -6.359 \\
+ Error [\%] & {\color{orange}3.1} & {\color{orange}3.2} & {\color{green}0.3} \\
+ \hline
+ \end{tabular}
+ \end{minipage}
+ \begin{minipage}{2cm}
+ {\LARGE
+ ${\color{green}\surd}$
+ }
+ \end{minipage}
+
+\end{slide}
+
+\begin{slide}
+
+ {\large\bf
+ C 100 interstitial migration along 110 in c-Si (Albe potential)
+ }
+
+ \small
+
+ \begin{minipage}[t]{4.2cm}
+ \underline{Starting configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/start.eps}
+ \end{minipage}
+ \begin{minipage}[t]{4.0cm}
+ \vspace*{0.8cm}
+ $\Delta x=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta y=\frac{1}{4}a_{\text{Si}}=1.357\text{ \AA}$\\
+ $\Delta z=0.325\text{ \AA}$\\
+ \end{minipage}
+ \begin{minipage}[t]{4.2cm}
+ \underline{{\bf Expected} final configuration}\\
+ \includegraphics[width=4cm]{c_100_mig/final.eps}\\
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \begin{itemize}
+ \item Fix border atoms of the simulation cell
+ \item Constraints and displacement of the C atom:
+ \begin{itemize}
+ \item along {\color{green}110 direction}\\
+ displaced by {\color{green} $\frac{1}{10}(\Delta x,\Delta y)$}
+ \item C atom {\color{red}entirely fixed in position}\\
+ displaced by
+ {\color{red}$\frac{1}{10}(\Delta x,\Delta y,\Delta z)$}
+ \end{itemize}
+ \end{itemize}
+ {\bf\color{blue}Expected configuration not obtained!}
+ \end{minipage}
+ \begin{minipage}{0.5cm}
+ \hfill
+ \end{minipage}
+ \begin{minipage}{6cm}
+ \includegraphics[width=6.0cm]{c_100_110mig_01_albe.ps}
+ \end{minipage}
+
\end{slide}
projects / lectures / latex.git / blobdiff