From: hackbard Date: Thu, 14 Jun 2007 07:38:59 +0000 (+0000) Subject: tersoff pot, albe still missing ... X-Git-Url: https://hackdaworld.org/gitweb/?a=commitdiff_plain;h=1b9983243e764a394f2ba2d7d1aa5babece75218;p=lectures%2Flatex.git tersoff pot, albe still missing ... --- diff --git a/posic/talks/md_simulation_von_silizium.tex b/posic/talks/md_simulation_von_silizium.tex index b78dd11..6590157 100644 --- a/posic/talks/md_simulation_von_silizium.tex +++ b/posic/talks/md_simulation_von_silizium.tex @@ -565,29 +565,98 @@ Problemstellung: Finden der Nachbarn f"ur Wechselwirkung \end{itemize} \end{slide} +%\begin{slide} +%{\large\bf +% Thermodynamische Gr"o"sen +%} +%\begin{itemize} +% \item W"armekapazit"at +% \item Struktur Werte +% \item Diffusion +%\end{itemize} +%\end{slide} + \begin{slide} {\large\bf - Thermodynamische Gr"o"sen + Idee des Tersoff Potentials } + \begin{picture}(350,10) + \end{picture} \begin{itemize} - \item W"armekapazit"at - \item Struktur Werte - \item Diffusion + \item Potential f"ur kovalente Bindungen\\ + ($Si$: $sp^3$-Hybridisierung, 4 "au"sere Elektronen, + 4 gerichtete Bindungen, Winkel: $109,47 ^{\circ}$)\\ + $\Rightarrow$ Bindungsenergie von 3 Atomen $i,j,k$ + abh"angig von $r_{ij},r_{ik},r_{jk}$ {\color{red} und} + $\theta_{ijk},\theta_{ikj},\theta_{kij}$ + \item {\em\color{blue} bond order} Potential + im Gegensatz zu {\em explicit angular}\\ + \[ + \pot = \pot_R(r_{ij}) + {\color{blue} b_{ijk}} \pot_A(r_ij) + \] + \begin{picture}(350,10) + \end{picture} + \begin{itemize} + \item $b_{ijk}$: umgebungsabh"angiger Term + \item $b_{ijk}=const.$ $\Rightarrow$ Paarpotential + \item Schw"achung der Paarbindung je mehr Nachbarn vorhanden\\ + qualitative Motivation: Anzahl der Elektronenpaare pro Bindung + \item St"arke der Bindung monoton fallend mit Koordinationszahl\\ + steiler Abfall $\Rightarrow$ Dimer\\ + schwacher Abfall $\Rightarrow$ maximale Koordinationszahl + (hcp-Struktur) + \item Pseudopotentialtheorie: + \[ + b_{ijk} \sim Z^{-\delta} + \] + \begin{center} + {\scriptsize Abell et al. Phys. Rev. B 31 (1985) 6184.} + \end{center} + \end{itemize} \end{itemize} \end{slide} \begin{slide} {\large\bf - Tersoff -} + Form des Tersoff Potentials: +}\\ +Gesamtenergie: +\[ +E = \frac{1}{2} \sum_{i \neq j} \pot_{ij}, \quad +\pot_{ij} = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right] +\] +Repulsiver und attraktiver Beitrag: +\begin{eqnarray} +f_R(r_{ij}) &=& A_{ij} \exp(-\lambda_{ij} r_{ij}) \nonumber \\ +f_A(r_{ij}) &=& - B_{ij} \exp(-\mu_{ij} r_{ij}) \nonumber +\end{eqnarray} +Cut-Off Funktion: +\[ +f_C(r_{ij})=\left\{\begin{array}{ll} + 1, & r_{ij} < R_{ij} \\ + \frac{1}{2} + + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], + & R_{ij} < r_{ij} < S_{ij} \\ + 0, & r_{ij} > S_{ij} +\end{array} \right. +\] +{\em bond order} Term: +\begin{eqnarray} +b_{ij} &=& \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} +\nonumber \\ +\zeta_{ij} &=& \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) +\nonumber \\ +g(\theta_{ijk})&=&1+c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] +\nonumber +\end{eqnarray} \end{slide} -\begin{slide} -{\large\bf - EAM -} - -\end{slide} +%\begin{slide} +%{\large\bf +% EAM +%} +% +%\end{slide} \begin{slide} {\large\bf @@ -609,10 +678,10 @@ Gear Predictor Corrector & ${\color{red} \times}$ & GEAR-5 & $\bullet\bullet$ \\ {\bf Potential} & & & \\ Harmonischer Oszillator & ${\color{green} \surd}$ & & - \\ Lennard-Jones &$ {\color{green} \surd}$ & & - \\ -Tersoff/Albe & ${\color{green} \surd\surd}$ & & - \\ +Tersoff & ${\color{green} \surd}$ & & - \\ +Albe & ${\color{green} \surd}$ & & - \\ Tersoff/Albe (inkl. $\lambda^3$) & ${\color{red} \times\times}$ & & $\bullet\bullet\bullet$ \\ -EAM & ${\color{red} \times}$ & & $\bullet\bullet$ \\ {\bf Ensembles} & & & \\ {\em temperature scaling} & ${\color{green} \surd}$ & & - \\ {\em pressure scaling} & ${\color{green} \surd}$ & & - \\