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/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}.
+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.
Thus, in order to obtain more accurate results quantum-mechanical calculations from first principles based on density functional theory (DFT) were performed.
The Vienna {\em ab initio} simulation package (\textsc{vasp})~\cite{kresse96} is used for this purpose.
-The relevant basics of DFT are described in section \ref{section:dft} while an overview of utilities mainly used to create input or parse output data of \textsc{vasp} is given in appendix \ref{app:code}.
+The relevant basics of DFT are described in section~\ref{section:dft} while an overview of utilities mainly used to create input or parse output data of \textsc{vasp} is given in appendix~\ref{app:code}.
The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the simulated system to be much smaller in size.
Thus, investigations based on DFT are restricted to single defects or combinations of two defects in a rather small Si supercell, their structural relaxation as well as some selected diffusion processes.
-Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section \ref{section:basics:defects}.
-The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section \ref{section:basics:migration} and modifications to the \textsc{vasp} code implementing this method are presented in appendix \ref{app:patch_vasp}.
+Next to the structure, defects can be characterized by the defect formation energy, a scalar indicating the costs necessary for the formation of the defect, which is explained in section~\ref{section:basics:defects}.
+The method used to investigate migration pathways to identify the prevalent diffusion mechanism is introduced in section~\ref{section:basics:migration} and modifications to the \textsc{vasp} code implementing this method are presented in appendix~\ref{app:patch_vasp}.
\section{Molecular dynamics simulations}
\label{section:md}
Three ingredients are required for a MD simulation:
\begin{enumerate}
\item A model for the interaction between system constituents is needed.
- Interaction potentials and their accuracy for describing certain systems of elements will be outlined in section \ref{subsection:interact_pot}.
+ Interaction potentials and their accuracy for describing certain systems of elements will be outlined in section~\ref{subsection:interact_pot}.
\item An integrator is needed, which propagates the particle positions and velocities from time $t$ to $t+\delta t$, realized by a finite difference scheme which moves trajectories discretely in time.
- This is explained in section \ref{subsection:integrate_algo}.
+ This is explained in section~\ref{subsection:integrate_algo}.
\item A statistical ensemble has to be chosen, which allows certain thermodynamic quantities to be controlled or to stay constant.
- This is discussed in section \ref{subsection:statistical_ensembles}.
+ This is discussed in section~\ref{subsection:statistical_ensembles}.
\end{enumerate}
These ingredients will be outlined in the following.
The discussion is restricted to methods employed within this study.
\begin{equation}
F^i = - \nabla_{{\bf r}_i} E \textrm{ .}
\end{equation}
-Details of the Tersoff potential derivative are presented in appendix \ref{app:d_tersoff}.
+Details of the Tersoff potential derivative are presented in appendix~\ref{app:d_tersoff}.
\subsubsection{Improved analytical bond order potential}
For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction~\cite{albe_sic_pot}.
The functional form is similar to the one proposed by Tersoff.
-Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
+Differences in the energy functional and the force evaluation routine are pointed out in appendix~\ref{app:d_tersoff}.
Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
The new parameter set for the C-C interaction yields improved dimer properties while at the same time delivers a description of the bulk phase similar to the Tersoff potential.
The potential succeeds in the description of the low as well as high coordinated structures.
Structures of maximum configurational energy do not necessarily constitute saddle point configurations, i.e.\ the method does not guarantee to find the true minimum energy path.
Whether a saddle point configuration and, thus, the minimum energy path is obtained by the CRT method, needs to be verified by calculating the respective vibrational modes.
-Modifications used to add the CRT feature to the \textsc{vasp} code and a short instruction on how to use it can be found in appendix \ref{app:patch_vasp}.
+Modifications used to add the CRT feature to the \textsc{vasp} code and a short instruction on how to use it can be found in appendix~\ref{app:patch_vasp}.
% todo - advantages of pw basis concenring hf forces
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