X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=posic%2Ftalks%2Fmd_simulation_von_silizium.tex;h=6590157c7ca6bd7b0b6fc6b1b06ed4c434a4c823;hb=1b9983243e764a394f2ba2d7d1aa5babece75218;hp=47e144acc28ece1cdc369945cf31b6cc1a24ef62;hpb=74bdf88a71df101efc703ee94986c45418f56ad7;p=lectures%2Flatex.git diff --git a/posic/talks/md_simulation_von_silizium.tex b/posic/talks/md_simulation_von_silizium.tex index 47e144a..6590157 100644 --- a/posic/talks/md_simulation_von_silizium.tex +++ b/posic/talks/md_simulation_von_silizium.tex @@ -38,9 +38,12 @@ \def\slideleftmargin{5.1cm} \def\slidetopmargin{-0.6cm} +\def\slidetopmargin{-0.6cm} \newcommand{\ham}{\mathcal{H}} \newcommand{\pot}{\mathcal{V}} +\newcommand{\foo}{\mathcal{U}} +\newcommand{\vir}{\mathcal{W}} % topic @@ -468,7 +471,7 @@ Simulation von Oberfl"achen: \begin{minipage}{4cm} \begin{itemize} \item Zuf"alliges Hinzuf"ugen von Kohlenstoff\\ - (schaffrierter Bereich)\\ + (schraffierter Bereich)\\ $\Rightarrow$ Energie- und Impulszufuhr in die MD-Zelle \item $T$-Skalierung,\\ Kopplung ans W"armebad\\ (blauer Bereich)\\ @@ -521,43 +524,140 @@ Problemstellung: Finden der Nachbarn f"ur Wechselwirkung E = + = < \sum_i \frac{{\bf p}_i^2}{2m_i} > + \] \item Temperatur/Druck - \begin{eqnarray} - &=& k_BT \nonumber \\ - &=& k_BT \nonumber - \end{eqnarray} + \[ + = k_BT, \quad + = k_BT + \] \begin{center} - "Aquipartitionstheorem + {\em "Aquipartitionstheorem} \end{center} Temperatur: \[ - \sum_i {\bf p}_i \frac{{\bf p}_i}{m_i} = 3Nk_BT \quad - \Rightarrow \quad T=\frac{1}{3Nk_B} \sum_i \frac{{\bf p}_i^2}{m_i} + <\sum_i {\bf p}_i \frac{{\bf p}_i}{m_i}> = 3Nk_BT \quad + \Rightarrow \quad T=\frac{1}{3Nk_B} <\sum_i \frac{{\bf p}_i^2}{m_i}> \] Druck: \[ - \sum_i {\bf q}_i \nabla_{{\bf q}_i} \pot = 3Nk_BT \quad + <\sum_i {\bf q}_i \nabla_{{\bf q}_i} \foo> = 3Nk_BT \quad \stackrel{\textrm{kart. Koord.}}{\Rightarrow} \quad - - \sum_i {\bf r}_i \nabla_{{\bf r}_i} \pot = -3Nk_BT \quad + - \frac{1}{3} <\sum_i {\bf r}_i \nabla_{{\bf r}_i} \foo> = -Nk_BT + \] + \begin{center} + mit + \end{center} + \[ + - \nabla_{{\bf r}_i} \foo = {\bf f}_i^{tot} = {\bf f}_i^{ext} + {\bf f}_i^{int} + \] + \begin{center} + wobei + \end{center} + \[ + \frac{1}{3} \sum_i {\bf r}_i {\bf f}_i^{ext}=-pV, \quad + \frac{1}{3} \sum_i {\bf r}_i {\bf f}_i^{int}= + - \frac{1}{3} \sum_i {\bf r}_i \nabla_{{\bf r}_i} \pot = \vir + \] + \begin{center} + folgt + \end{center} + \[ + pV = Nk_BT + <\vir> \] - \item W"armekapazit"at - \item Struktur Werte - \item Diffusion \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 - Tersoff + Idee des Tersoff Potentials } + \begin{picture}(350,10) + \end{picture} +\begin{itemize} + \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 - EAM -} - + 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 Albe Reparametrisierung @@ -578,10 +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} & ${\color{green} \surd}$ & & - \\