From a59b8f7e1388e9f1289cacbfaa0ae9e898759ac4 Mon Sep 17 00:00:00 2001 From: hackbard Date: Tue, 10 Nov 2009 17:52:48 +0100 Subject: [PATCH] started defects chapter .. --- posic/thesis/defects.tex | 38 ++++++++++++++++++++++++++++++++++++++ posic/thesis/sic.tex | 3 ++- 2 files changed, 40 insertions(+), 1 deletion(-) diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex index aba7078..e763151 100644 --- a/posic/thesis/defects.tex +++ b/posic/thesis/defects.tex @@ -1,5 +1,43 @@ \chapter{Point defects in silicon} +Given the conversion mechnism of SiC in crystalline silicon introduced in \ref{section:assumed_prec} the understanding of carbon and silicon interstitial point defects in c-Si is of great interest. +Both types of defects are examined in the following both by classical potential as well as density functional theory calculations. + +In case of the classical potential calculations a simulation volume of nine silicon lattice constants in each direction is used. +Calculations are performed in an isothermal-isobaric NPT ensemble. +Coupling to the heat bath is achieved by the Berendsen thermostat with a time constant of 100 fs. +The temperature is set to zero Kelvin. +Pressure is controlled by a Berendsen barostat again using a time constant of 100 fs and a bulk modulus of 100 GPa for silicon. +To exclude surface effects periodic boundary conditions are applied. + +Due to the restrictions in computer time three silicon lattice constants in each direction are considered sufficiently large enough for DFT calculations. +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{center} +\includegraphics[width=10cm]{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.} +\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. +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. +For defect configurations consisting of a single atom species the formation energy is defined as +\begin{equation} +E_{\text{f}}=\left(E_{\text{coh}}^{\text{defect}} + -E_{\text{coh}}^{\text{defect-free}}\right)N +\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 ... + \section{Silicon self-interstitials} \section{Carbon related point defects} diff --git a/posic/thesis/sic.tex b/posic/thesis/sic.tex index 8d7ac2c..631f3b7 100644 --- a/posic/thesis/sic.tex +++ b/posic/thesis/sic.tex @@ -12,7 +12,7 @@ It is extremely rare and almost impossible to find in nature. \subsection{SiC polytypes} Each of the four sp$^3$ hybridized orbitals of the Si atom overlaps with one of the four sp$^3$ hybridized orbitals of the four surrounding C atoms and vice versa. -This results in fourfold coordinated covalent $\sigma$ bond of equal length and strength for each atom with its neighbours. +This results in fourfold coordinated covalent $\sigma$ bonds of equal length and strength for each atom with its neighbours. Although the local order of Si and C next neighbour atoms characterized by the tetrahedral bonding is the same, more than 250 different types of structures called polytypes of SiC exist \cite{fischer90}. The polytypes differ in the one-dimensional stacking sequence of identical, closed-packed SiC bilayers. @@ -22,6 +22,7 @@ The polytypes differ in the one-dimensional stacking sequence of identical, clos \section{Ion beam synthesis of cubic silicon carbide} \section{Assumed precipitation mechanism of cubic silicon carbide in silicon} +\label{section:assumed_prec} \section{Substoichiometric concentrations of carbon in crystalline silicon} -- 2.20.1