From 8b24d2f201fb3295f5a9b19c675271b8801f27d4 Mon Sep 17 00:00:00 2001 From: hackbard Date: Thu, 25 Mar 2010 18:46:36 +0100 Subject: [PATCH] started molecular dynamics chapter --- posic/thesis/defects.tex | 13 ++++++++ posic/thesis/md.tex | 69 ++++++++++++++++++++++++++++++++++++++++ posic/thesis/thesis.tex | 3 +- 3 files changed, 84 insertions(+), 1 deletion(-) create mode 100644 posic/thesis/md.tex diff --git a/posic/thesis/defects.tex b/posic/thesis/defects.tex index e0ddc83..5465ef8 100644 --- a/posic/thesis/defects.tex +++ b/posic/thesis/defects.tex @@ -1133,3 +1133,16 @@ Low migration barriers are necessary to obtain this configuration and in contras Thus, carbon interstitials and vacancies located close together are assumed to end up in such a configuration in which the carbon atom is tetrahedrally coordinated and bound to four silicon atoms as expected in silicon carbide. In contrast to the above, this would suggest a silicon carbide precipitation by succesive creation of substitutional carbon instead of the agglomeration of C-Si dumbbell interstitials followed by an abrupt precipitation. +{\color{red}Todo: +C atoms in 100 DB tightend in 110 direction, which is important for stress compensation in some combined constellations. +} +{\color{red}Todo: +Most of the combinations should only be thought of intermediate configurations, which need to be transformed in the later SiC precipitation process. +} +{\color{red}Todo: +Better structure, better language, better methodology! +} +{\color{red}Todo: +Fit of lennard-jones an other rep + attr potentials in 110 interaction data! +} + diff --git a/posic/thesis/md.tex b/posic/thesis/md.tex new file mode 100644 index 0000000..6910563 --- /dev/null +++ b/posic/thesis/md.tex @@ -0,0 +1,69 @@ +\chapter{Molecular dynamics simulations} + +The molecular dynamics (MD) technique is used to gain insight into the behavior of carbon existing in different concentrations in crystalline silicon on the microscopic level at finite temperatures. +Both, quantum-mechanical and classical potential molecular dynamics simulations are performed. +While quantum-mechanical calculations are restricted to a few hundreds of atoms only small volumes composed of three unit cells in each direction and small carbon concentrations are simulated using the VASP code. +Thus, investigations are restricted to the diffusion process of single carbon interstitials and the agglomeration of a few dumbbell interstitials in silicon. +Using classical potentials volume sizes up to 31 unit cells in each direction and high carbon concentrations are realizable. +Simulations targeting the formation of silicon carbide precipitates are, thus, attempted in classical potential calculations only. + +\section{Ab initio MD simulations} + +Molecular dynamics simulations of a single, two and ten carbon atoms in $3\times 3\times 3$ unit cells of crytsalline silicon are performed. + +\section{Classical potential MD simulations} + +\subsection{Initial simulations} + +In initial simulations aiming to reproduce a precipitation process simulation volumes of $31\times 31\times 31$ unit cells are utilized. +Periodic boundary conditions in each direction are applied. +The system temperature is set to $450\, ^{\circ}\mathrm{C}$, the temperature for which epitaxial growth of 3C-SiC films is achieved by ion beam synthesis (IBS). +After equilibration of the kinetic and potential energy carbon atoms are consecutively inserted. +The number of carbon atoms $N_{\text{Carbon}}$ necessary to form a spherical precipitate with radius $r$ is given by +\begin{equation} + N_{\text{Carbon}}=\frac{4}{3}\pi r^3 \cdot \frac{4}{a_{\text{SiC}}^3} + =\frac{16}{3} \pi \left( \frac{r}{a_{\text{SiC}}}\right)^3 +\label{eq:md:spheric_prec} +\end{equation} +with $a_{\text{SiC}}$ being the lattice constant of 3C-SiC. +A total amount of 6000 carbon atoms corresponds to a radius of approximately 3 nm, which is discovered to be the minimal size for precipitates in IBS experiments. +In separated simulations these 6000 carbon atoms are inserted in three regions of different volume ($V_1$, $V_2$, $V_3$) within the simulation cell. +For reasons of simplification these regions are rectangularly shaped. +$V_1$ is chosen to be the total simulation volume. +$V_2$ approximately corresponds to the volume of a minimal 3C-SiC precipitate. +$V_3$ is approximately the volume containing the necessary amount of silicon atoms to form such a precipitate, which is slightly smaller than $V_2$ due to the slightly lower silicon density of 3C-SiC compared to c-Si. +For rectangularly shaped precipitates with side length $L$ equation \eqref{eq:md:quadratic_prec} holds. +\begin{equation} + N_{\text{Carbon}} =4 \left( \frac{L}{a_{\text{SiC}}}\right)^3 +\label{eq:md:quadratic_prec} +\end{equation} +Table \ref{table:md:ins_vols} summarizes the side length of each of the three different insertion volumes determined by the equations mentioned above. +\begin{table} +\begin{center} +\begin{tabular}{l c c c} +\hline +\hline + & $V_1$ & $V_2$ & $V_3$ \\ +\hline +Side length [\AA] & 168.3 & 50.0 & 49.0 \\ +\hline +\hline +\end{tabular} +\end{center} +\caption{Side lengthes of the insertion volumes $V_1$, $V_2$ and $V_3$ used for the incoorperation of 6000 carbon atoms.} +\label{table:md:ins_vols} +\end{table} +The insertion is realized in a way to keep the system temperature constant. +In each of 600 insertion steps 10 carbon atoms are inserted at random positions within the respective region, which involves an increase in kinetic energy. +Thus, the simulation is continued without adding more carbon atoms until the system temperature is equal to the chosen temperature again, which is realized by the thermostat decoupling excessive energy. +Every inserted carbon atom must exhibit a distance greater or equal than 1.5 \AA{} to present neighboured atoms to prevent too high temperatures. +Once the total amount of carbon is inserted the simulation is continued for 100 ps followed by a cooling-down process until room temperature, that is $20\, ^{\circ}\mathrm{C}$ is reached. +Figure \ref{} displays a flow chart of the applied steps involved in the simulation sequence. + +The radial distribution function for Si-C and C-C distances is shown in figure \ref{}. + + +\subsection{Increased temperature simulations} + +\subsection{Simulations close to the silicon melting point} + diff --git a/posic/thesis/thesis.tex b/posic/thesis/thesis.tex index 48c2546..ca2a362 100644 --- a/posic/thesis/thesis.tex +++ b/posic/thesis/thesis.tex @@ -76,7 +76,8 @@ %\include{model} \include{simulation} \include{defects} -\include{results} +\include{md} +%\include{results} \include{summary_outlook} \appendix{} -- 2.20.1