Residing in the office of the undergraduates, I would like to thank all of the many fellow students I met during this time.
Interesting and not necessarily scientific discussions appeared on the daily agenda.
-I am likewise thankful to all the other members of the devision for the distinctive and pleasant working environment and the permanent willingness to discuss.
+It was a great time of lively exchange of ideas and excellent coffee that I will keep in good memory.
+I am likewise thankful to all the other members of the devision for the distinctive and pleasant working environment and the permanent willingness to discuss, which very often happened on fridays burning the midnight oil.
For discussions aside from natural science, I am grateful to the small but fine reading group formed on the fringes of the temporary student protests in 2009.
I would like to thank Meta Schnell, Berthold Arlt, Michael Lippok, Erika Rempel, Leo Sellinger, Michaela Strattner, Katja Teich and Matthias Link for the numerous sessions reviewing elements of the critical theory of society.
Dear Mali, you will always be in my mind, wherever life as a teacher may take you.
I would like to thank Stefanie Rajkay accompanying me during the last stages of this work.
-Although there cannot be a right life amidst wrongs \cite{adorno_mm}, I am glad for our joint effort to organize and try to anticipate a way of life that would be the actual right one in a free society.
+Although there cannot be a right life amidst wrongs\footnote[1]{Theodor W. Adorno, {\em Minima Moralia: Reflexionen aus dem besch\"adigten Leben}, Suhrkamp 1994, p. 19.}, I am glad for our joint effort trying to organize and anticipate a way of (private) life that would be the actual right one in a free society.
+Dear Stoffl, somehow, we will manage!
Last but not least, I would like to express my gratitude to my parents Wilfriede and Karl Zirkelbach, who contributed to a great extent to pursue a scientific career without major difficulties.
-My brother Till Zirkelbach is greatly acknowledged for backup of any kind of concerns.
+My brother Till Zirkelbach is greatly acknowledged for backup of any kind of concern.
\\
\\
Thanks!
+
In the following the simulation methods used within the scope of this study are introduced.
Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
-For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/download/posic/posic.tar.bz2}.
+For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential \cite{albe_sic_pot} an appropriate MD code called {\textsc posic}\footnote{{\textsc posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
The basic ideas of MD in general and the adopted techniques as implemented in {\textsc posic} in particular are outlined in section \ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix \ref{app:d_tersoff}.
An overview of the most important tools within the MD package is given in appendix \ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \\
f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} )
\end{eqnarray}
-The function $f_C$ is the a cutoff function to limit the range of interaction to nearest neighbors.
+The function $f_C$ is a cutoff function to limit the range of interaction to nearest neighbors.
It is designed to have a smooth transition of the potential at distances $R_{ij}$ and $S_{ij}$.
\begin{equation}
f_C(r_{ij}) = \left\{
\subsection{Verlet integration}
\label{subsection:integrate_algo}
-A numerical method to integrate Newton's equation of motion was presented by Verlet in 1967 \cite{verlet67}.
+A numerical method to integrate Newton's equations of motion was presented by Verlet in 1967 \cite{verlet67}.
The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionaly generates directly the velocities, is explained in the following.
Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
\begin{equation}
\chapter{Description of programs and tools}
\label{app:code}
-Programs and tools utilized within this study are introduced in the following.
+Some selected programs and tools utilized within this study are introduced in the following.
+The source code is available for download\footnote{http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
+The \textsc{vasp} utilities reside in the {\em vasp\_tools} subdirectory included within the \textsc{posic} source.
-\section{Contents of the \textsc{posic} program suite}
+\section[Contents of the {\normalfont\textsc{posic}} program suite]{Contents of the POSIC program suite}
-\begin{itemize}
- \item mdrun
- \item moldyn.\{c,h\}
- \item potentials/albe.\{c,h\}
- \item ...
- \item ...
-\end{itemize}
+\subsection{The molecular dynamics application}
-\section{\textsc{vasp} utilities}
+\paragraph{mdrun.\{c,h\}}
+constitutes the actual, executable molecular dynamics application program.
+\paragraph{config.default}
+is a sample configuration file that is parsed by the {\em mdrun} application.
+\paragraph{moldyn.\{c,h\}}
+includes all the molecular dynamics routines.
+\paragraph{potentials/albe.\{c,h\}}
+implements the energy and force evaluation of the potential.
+\paragraph{list/list.\{c,h\}}
+contains code for the management of linked lists.
+\paragraph{random/random.\{c,h\}}
+deals with random numbers and distributions.
+\paragraph{math/math.h}
+provides inlined mathematical functions.
+\paragraph{runmd, runmd\_rx200}
+script starting the {\em mdrun} application and postprocessing.
-\begin{itemize}
- \item mig\_fullct.sh, mig\_calc
- \item e\_form\_tersoff, e\_fc\_tersoff
- \item ...
- \item ...
- \item ...
-\end{itemize}
+\subsection{Postprocessing tools}
+
+\paragraph{calc\_delta\_e}
+determines defect formation energies using SiC as a particle reservoir.
+\paragraph{pair\_correlation\_calc.c}
+computes the radial dsitribution function.
+\paragraph{display\_atom\_data.c}
+displays atom specific information.
+\paragraph{bond\_analyze.c}
+counts the amount of C atoms that have four Si neighbors.
+\paragraph{bond\_analyze\_script}
+performs bond analysis on a large quantity of data.
+\paragraph{search\_bonds.c}
+prints out pairs of atoms featuring specific bond properties.
+\paragraph{visual\_atoms.c}
+creates a detailed atomic datafile.
+\paragraph{visualize}
+creates images of atomic configurations.
+\paragraph{parcasconv}
+converts \textsc{parcas} output to \textsc{posic} format.
+\paragraph{povconv}
+converts \textsc{posic} output to \textsc{parcas/rasmol} format.
+\paragraph{s2xyz.c}
+extracts (modified) {\em xyz} data from \textsc{posic} save files.
+\paragraph{ppm2avi}
+creates a movie out of atomic configuration images.
+
+\section[{\normalfont\textsc{vasp}} utilities]{VASP utilities}
+
+\subsection[Operating {\normalfont\textsc{vasp}}]{Operating VASP}
+
+\paragraph{create\_lattice.c}
+create the lattice in \textsc{vasp} POSCAR fomat.
+\paragraph{runvasp\_rx200}
+executing \textsc{vasp} on the Augsburg Linux Compute Cluster.
+\paragraph{sd\_rot\_all-atoms.patch}
+enables selected dynamics in a user-defined basis for every atom.
+\paragraph{mig\_fullct.sh}
+calculates a series of configurations within a migration path.
+
+\subsection{Postprocessing utilities}
+
+\paragraph{mig\_calc}
+prints out the configurational energies within a migration path.
+\paragraph{e\_coh}
+calculates the cohesive energy.
+\paragraph{e\_form\_tersoff}
+calculates defect formation energies using SiC as a particle reservoir.
+\paragraph{e\_fc}
+calculates the binding energy of a defect pair.
+\paragraph{get\_ks\_levels}
+creates the Kohn-Sham level diagram.
+\paragraph{visualize}
+creates images of atomic configurations.
-\ldots will be written out in full soon!
\begin{equation}
\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .}
\end{equation}
-In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are done.
+In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are written down.
- \section{Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
+ \section[Derivative of $V_{ij}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ij} & = & \nabla_{{\bf r}_i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber \\
& = & \Big[ \frac{\cos\theta_{ijk}}{r_{ij}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ij} + \Big[ \frac{\cos\theta_{ijk}}{r_{ik}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ik}
\end{eqnarray}
- \section{Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
+ \section[Derivative of $V_{ji}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{ji}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{ji} & = & \nabla_{{\bf r}_i} f_C(r_{ji}) \big[ f_R(r_{ji}) + b_{ji} f_A(r_{ji}) \big] + \nonumber \\
& = & \frac{1}{r_{ji} r_{jk}} {\bf r}_{jk} - \frac{\cos\theta_{jik}}{r_{ji}^2} {\bf r}_{ji}
\end{eqnarray}
- \section{Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
+ \section[Derivative of $V_{jk}$ with respect to ${r}_i$]{\boldmath Derivative of $V_{jk}$ with respect to ${\bf r}_i$}
\begin{eqnarray}
\nabla_{{\bf r}_i} V_{jk} & = & f_C(r_{jk}) f_A(r_{jk}) \nabla_{{\bf r}_i} b_{jk} \\
keeping in mind that all the necessary force contributions for atom $i$
are calculated and added in subsequent loops.
-\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
+\subsection[Derivative of $V_{ij}$ with respect to ${r}_j$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_j$}
\begin{eqnarray}
\nabla_{{\bf r}_j} V_{ij} & = &
\frac{\cos\theta_{ijk}}{r_{ij}^2}{\bf r}_{ij}
\end{eqnarray}
-\subsection{Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
+\subsection[Derivative of $V_{ij}$ with respect to ${r}_k$]{\boldmath Derivative of $V_{ij}$ with respect to ${\bf r}_k$}
The derivative of $V_{ij}$ with respect to ${\bf r}_k$ just consists of the
single term
These results support the above assumptions of an increased entropic contribution to structural formation involving C$_{\text{s}}$ to a greater extent.
% link to migration of \si{}!
+% todo - make it a subsection
The possibility for separated configurations of \cs{} and \si{} becomes even more likely if one of the constituents exhibits a low barrier of migration.
In this case, the \si{} is assumed to constitute the mobile defect compared to the stable \cs{} atom.
Thus, migration paths of \si{} are investigated in the following excursus.
\addcontentsline{toc}{chapter}{List of Figures}
\listoftables
\addcontentsline{toc}{chapter}{List of Tables}
+
In addition, sharper peaks in the radial distribution functions lead to the assumption of expeditious structural formation.
The increase in temperature leads to the occupation of new defect states, which is particularly evident but not limited to the low C concentration simulations.
+% todo - cut-off effect increases for non-equilibrium processes, thus, to mimic IBS increased temperatures are exceptionally necessary
The question remains, whether these states are only occupied due to the additional supply of kinetic energy and, thus, have to be considered unnatural for temperatures applied in IBS or whether the increase in temperature indeed enables infrequent transitions to occur faster, thus, leading to the intended acceleration of the dynamics and weakening of the unphysical quirks inherent to the potential.
As already pointed out in section~\ref{section:defects:noneq_process_01} and section~\ref{section:defects:noneq_process_02}, IBS is a non-equilibrium process, which might result in the formation of the thermodynamically less stable \cs{} and \si{} configuration.
Indeed, 3C-SiC is metastable and if not fabricated by IBS requires strong deviation from equilibrium and low temperatures to stabilize the 3C polytype.
\label{chapter:summary}
{\setlength{\parindent}{0pt}
-\paragraph{To summarize,}
+%\paragraph{To summarize,}
+{\bf To summarize},
in a short review of the C/Si compound and the fabrication of the technologically promising semiconductor SiC by IBS, two controversial assumptions of the precipitation mechanism of 3C-SiC in c-Si are elaborated.
}
These propose the precipitation of SiC by agglomeration of \ci{} DBs followed by a sudden formation of SiC and otherwise a formation by successive accumulation of \cs{} via intermediate stretched SiC structures, which are coherent to the Si lattice.
\si{} is often found in the direct surrounding.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
Indeed, utilizing increased temperatures is assumed to constitute a necessary condition to simulate IBS of 3C-SiC in c-Si.
-
-
+\\
+\\
% todo - sync with respective conclusion chapter
-
+%
% conclusions 2nd part
-\paragraph{Conclusions}
+%\paragraph{Conclusions}
+{\bf Conclusions}
concerning the SiC conversion mechanism are derived from results of both, first-principles and classical potential calculations.
Although classical potential MD calculations fail to directly simulate the precipitation of SiC, obtained results, on the one hand, reinforce previous findings of the first-principles investigations and, on the other hand, allow further conclusions on the SiC precipitation in Si.
The strained structure is found to be stable up to \degc{810}.
Coherent clustering followed by precipitation is suggested if these structures are annealed at higher temperatures.
%
-Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C}\cite{werner96} annealing temperature.
+Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C} \cite{werner96} annealing temperature.
Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples \cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
According to this, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
%
projects / lectures / latex.git / commitdiff