From 91836b70fa4e7f4bc2834d0ef176c1e5b825759d Mon Sep 17 00:00:00 2001 From: hackbard Date: Mon, 12 Oct 2009 19:42:06 +0200 Subject: [PATCH] new structure ... --- posic/thesis/basics.tex | 131 +++++++++++++++++++----------------- posic/thesis/defects.tex | 10 +++ posic/thesis/simulation.tex | 52 ++++++-------- posic/thesis/thesis.tex | 4 ++ posic/thesis/title.tex | 6 +- 5 files changed, 109 insertions(+), 94 deletions(-) create mode 100644 posic/thesis/defects.tex diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index 6addfab..af46974 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -1,4 +1,4 @@ -\chapter{Basics} +\chapter{Basic principles of utilized simulation techniques} \begin{quotation} \dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{} @@ -75,63 +75,63 @@ A three body potential has to be included for these types of elements. In the following, relevant potentials for this work are discussed. -\subsubsection{The Lennard-Jones potential} - -The L-J potential is a realistic two body pair potential and is of the form -\begin{equation} -U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,} -\label{eq:lj-p} -\end{equation} -where $r$ denotes the distance between the two atoms. - -The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction. -It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way. - -The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds. -It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances. -Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way. - -The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data. -$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius. - -Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$) -\begin{equation} -\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big) -\label{eq:lj-d} -\end{equation} -one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$. -Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth. -The $i$th component of the force is given by -\begin{equation} -F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.} -\label{eq:lj-f} -\end{equation} - -\subsubsection{The Stillinger Weber potential} - -The Stillinger Weber potential \cite{stillinger_weber} \ldots - -\begin{equation} -U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) -\end{equation} - -\begin{equation} -U_2(r_{ij}) = \left\{ - \begin{array}{ll} - \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1} \Big] & r_{ij} < a \sigma \\ - 0 & r_{ij} \ge a \sigma - \end{array} \right. -\end{equation} - -\begin{equation} -U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) = -\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big] -\end{equation} - -\begin{equation} -h(r_{ij},r_{ik},\theta_{jik}) = -\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2 -\end{equation} +%\subsubsection{The Lennard-Jones potential} +% +%The L-J potential is a realistic two body pair potential and is of the form +%\begin{equation} +%U^{LJ}(r) = 4 \epsilon \Big[ \Big( \frac{\sigma}{r} \Big)^{12} - \Big( \frac{\sigma}{r} \Big)^6 \Big] \, \textrm{,} +%\label{eq:lj-p} +%\end{equation} +%where $r$ denotes the distance between the two atoms. +% +%The attractive tail for large separations $(\sim r^{-6})$ is essentially due to correlations between electron clouds surrounding the atoms. The attractive part is also known as {\em van der Waals} or {\em London} interaction. +%It can be derived classically by considering how two charged spheres induce dipol-dipol interactions into each other, or by considering the interaction between two oscillators in a quantum mechanical way. +% +%The repulsive term $(\sim r^{-12})$ captures the non-bonded overlap of the electron clouds. +%It does not have a true physical motivation, other than the exponent being larger than $6$ to get a steep rising repulsive potential wall at short distances. +%Chosing $12$ as the exponent of the repulsive term it is just the square of the attractive term which makes the potential evaluable in a very efficient way. +% +%The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data. +%$\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius. +% +%Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$) +%\begin{equation} +%\frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big) +%\label{eq:lj-d} +%\end{equation} +%one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$. +%Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth. +%The $i$th component of the force is given by +%\begin{equation} +%F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.} +%\label{eq:lj-f} +%\end{equation} +% +%\subsubsection{The Stillinger Weber potential} +% +%The Stillinger Weber potential \cite{stillinger_weber} \ldots +% +%\begin{equation} +%U = \sum_{i,j} U_2({\bf r}_i,{\bf r}_j) + \sum_{i,j,k} U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) +%\end{equation} +% +%\begin{equation} +%U_2(r_{ij}) = \left\{ +% \begin{array}{ll} +% \epsilon A \Big( B (r_{ij} / \sigma)^{-p} - 1\Big) \exp \Big[ (r_{ij} / \sigma - 1)^{-1} \Big] & r_{ij} < a \sigma \\ +% 0 & r_{ij} \ge a \sigma +% \end{array} \right. +%\end{equation} +% +%\begin{equation} +%U_3({\bf r}_i,{\bf r}_j,{\bf r}_k) = +%\epsilon \Big[ h(r_{ij},r_{ik},\theta_{jik}) + h(r_{ji},r_{jk},\theta_{ijk}) + h(r_{ki},r_{kj},\theta_{ikj}) \Big] +%\end{equation} +% +%\begin{equation} +%h(r_{ij},r_{ik},\theta_{jik}) = +%\lambda \exp \Big[ \gamma (r_{ij}/\sigma -a)^{-1} + \gamma (r_{ik}/\sigma - a)^{-1} \Big] \Big( \cos \theta_{jik} + \frac{1}{3} \Big)^2 +%\end{equation} \subsubsection{The Tersoff potential} @@ -192,10 +192,21 @@ F^i = - \nabla_{{\bf r}_i} E \textrm{ .} \end{equation} The details of the Tersoff potential derivative can be seen in appendix \ref{app:d_tersoff}. -And here comes why we use it. Advantages and disadvantages compared to other interaction potentials, maybe this is best at the very end of all potentials \ldots +\subsubsection{A reparametrized Tersoff-like bond order potential} -\subsubsection{The Brenner potential} +Erhart-Albe potential ... \subsection{Statistical ensembles} \label{subsection:statistical_ensembles} +\section{Denstiy functional theory} +\label{section:dft} + +\subsection{Born-Oppenheimer (adiabatic) approximation} + +\subsection{Hohenberg-Kohn theorem} + +\subsection{Exchange correlation} + +\subsection{Pseudopotentials} + diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex new file mode 100644 index 0000000..aba7078 --- /dev/null +++ b/posic/thesis/defects.tex @@ -0,0 +1,10 @@ +\chapter{Point defects in silicon} + +\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} + +\section{Combination of point defects} + diff --git a/posic/thesis/simulation.tex b/posic/thesis/simulation.tex index c28b982..6f836fa 100644 --- a/posic/thesis/simulation.tex +++ b/posic/thesis/simulation.tex @@ -1,31 +1,21 @@ -\chapter{Simulation} - -\section{Cohesive energies} - -\begin{figure}[!h] - \begin{center} - \includegraphics[width=10cm]{min_si.eps} - \includegraphics[width=10cm]{min_c.eps} - \includegraphics[width=10cm]{min_sic.eps} - \caption{Cohesive energies for different lattice constants of $Si$, $C$ and cubic $SiC$.} - \label{img:ec_vs_lc} - \end{center} -\end{figure} - - -\section{Silicon self-interstitials} - -\begin{itemize} - \item Tetrahedral: - \begin{itemize} - \item Cohesive energy: $3.405 \, eV$ - \end{itemize} - \item Hexagonal: - \begin{itemize} - \item Cohesive energy: $4.480 \, eV$ - \end{itemize} - \item 110 dumbbell: - \begin{itemize} - \item Cohesive energy: $4.392 \, eV$ - \end{itemize} -\end{itemize} +\chapter{Simulation parameters and test calculations} + +\section{Classical potential MD} + +\subsection{Tersoff vs. Erhart-Albe SiC potential} + +\subsection{Temperature and volume control} + +\section{DFT calculations / MD} + +\subsection{Used types of super cells} + +\subsection[$k$-point sampling]{\boldmath $k$-point sampling} + +\subsection{Energy cutoff} + +\subsection{Other parameters} + +Symmetry, spin, smearing method, real space projection, choice of ensemble and convergence criteria for electronic and ionic relaxation ... + + diff --git a/posic/thesis/thesis.tex b/posic/thesis/thesis.tex index 643c9c2..7a08668 100644 --- a/posic/thesis/thesis.tex +++ b/posic/thesis/thesis.tex @@ -19,6 +19,8 @@ %\usepackage{fancyhdr} %\pagestyle{fancy} +\usepackage{miller} + % (re)new commands \newcommand{\printimg}[5]{% \begin{figure}[#1]% @@ -53,10 +55,12 @@ \mainmatter{} \include{intro} +\include{sic} \include{basics} %\include{exp_findings} %\include{model} \include{simulation} +\include{defects} \include{results} \include{summary_outlook} diff --git a/posic/thesis/title.tex b/posic/thesis/title.tex index e4625a7..0c72325 100644 --- a/posic/thesis/title.tex +++ b/posic/thesis/title.tex @@ -5,7 +5,7 @@ \hrule \vspace{15pt} {\LARGE\bf - Simulation study of the precipitation process of silicon carbide in carbon doped silicon \\ + Atomistic simulation study of the silicon carbide precipitation in silicon\\ } \vspace{10pt} \hrule @@ -44,7 +44,7 @@ \vspace{60pt} {\large - Augsburg, im Oktober 2006 + Augsburg, im November 2009 } \end{center} @@ -56,7 +56,7 @@ \begin{tabular}{ll} Erstkorrektor: & Prof. Dr. Bernd Stritzker \\ Zweitkorrektor: & Prof. Dr. Kai Nordlund \\ -Tag der m"undlichen Pr"ufung: & 17. November 2006 \\ +Tag der m"undlichen Pr"ufung: & 17. November 2009 \\ \end{tabular} % switch back to english -- 2.20.1