From: hackbard Date: Wed, 11 Nov 2009 17:46:09 +0000 (+0100) Subject: next: start with si self-ints X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=af10a7cb7b83bf31dfbb61af01ec87b718f3fd0d;p=lectures%2Flatex.git next: start with si self-ints --- diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex index e763151..89bd382 100644 --- a/posic/thesis/defects.tex +++ b/posic/thesis/defects.tex @@ -15,17 +15,25 @@ The ions are relaxed by a conjugate gradient method. The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{}. Periodic boundary conditions in each direction are applied. -\begin{figure} +\begin{figure}[h] \begin{center} -\includegraphics[width=10cm]{unit_cell_e.eps} +\includegraphics[width=9cm]{unit_cell_e.eps} \end{center} -\caption{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}) and \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) interstitial configurations.} +\caption{Insertion positions for the tetrahedral ({\color{red}$\bullet$}), hexagonal ({\color{green}$\bullet$}), \hkl<1 0 0> dumbbell ({\color{yellow}$\bullet$}), \hkl<1 1 0> dumbbell ({\color{magenta}$\bullet$}) and bond-centered ({\color{cyan}$\bullet$}) interstitial configuration.} \label{fig:defects:ins_pos} \end{figure} -The interstitial atom positions are displayed in Fig. \ref{fig:defects:ins_pos}. -In seperated simulation runs the silicon or carbon atom is inserted at the tetrahedral $(0,0,0)$ ({\color{red}$\bullet$}), the hexagonal $(-1/8,-1/8,1/8)$ ({\color{green}$\bullet$}), the nearly \hkl<1 0 0> dumbbell $(-1/4,-1/4,-1/8)$ ({\color{yellow}$\bullet$}) and the nearly \hkl<1 1 0> dumbbell $(-1/8,-1/8,-1/4)$ ({\color{magenta}$\bullet$}) interstitial position. -For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid to high forces. +The interstitial atom positions are displayed in figure \ref{fig:defects:ins_pos}. +In seperated simulation runs the silicon or carbon atom is inserted at the +\begin{itemize} + \item tetrahedral, $\vec{p}=(0,0,0)$, ({\color{red}$\bullet$}) + \item hexagonal, $\vec{p}=(-1/8,-1/8,1/8)$, ({\color{green}$\bullet$}) + \item nearly \hkl<1 0 0> dumbbell, $\vec{p}=(-1/4,-1/4,-1/8)$, ({\color{yellow}$\bullet$}) + \item nearly \hkl<1 1 0> dumbbell, $\vec{p}=(-1/8,-1/8,-1/4)$, ({\color{magenta}$\bullet$}) + \item bond-centered, $\vec{p}=(-1/8,-1/8,-3/8)$, ({\color{cyan}$\bullet$}) +\end{itemize} +interstitial position. +For the dumbbell configurations the nearest silicon atom is displaced by $(0,0,-1/8)$ and $(-1/8,-1/8,0)$ respectively of the unit cell length to avoid too high forces. A vacancy or a substitutional atom is realized by removing one silicon atom and switching the type of one silicon atom respectively. From an energetic point of view the free energy of formation $E_{\text{f}}$ is suitable for the characterization of defect structures. @@ -33,13 +41,22 @@ For defect configurations consisting of a single atom species the formation ener \begin{equation} E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}} -E_{\text{coh}}^{\text{defect-free}}\right)N +\label{eq:defects:ef1} \end{equation} where $N$ and $E_{\text{coh}}^{\text{defect}}$ are the number of atoms and the cohesive energy per atom in the defect configuration and $E_{\text{coh}}^{\text{defect-free}}$ is the cohesive energy per atom of the defect-free structure. -Evtl Paper mit Ef rauskramen lenen schreiben ... -Defects consisting of two or more atom species ... +The formation energy of defects consisting of two or more atom species is defined as +\begin{equation} +E_{\text{f}}=E-N_1\mu_1-N_2\mu_2 - \ldots +\label{eq:defects:ef2} +\end{equation} +where $E$ is the free energy of the interstitial system and $N_i$ and $\mu_i$ are the amount of atoms and the chemical potential of species $i$. +The chemical potential is determined by the cohesive energy of the structure of the specific type in equilibrium at zero Kelvin. +For a defect configuration of a single species equation \ref{eq:defects:ef2} is equivalent to equation \ref{eq:defects:ef1}. \section{Silicon self-interstitials} + + \section{Carbon related point defects} \section[Migration of the carbon \hkl<1 0 0> interstitial]{\boldmath Migration of the carbon \hkl<1 0 0> interstitial}