\chapter{Details of simulation parameters and test calculations}
\label{chapter:simulation}
-All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unti cells and periodic boundary conditions are imposed on the boundaries of that simulation cell.
+All calculations are carried out utilizing the supercell approach, which means that the simulation cell contains a multiple of unit cells and periodic boundary conditions are imposed on the boundaries of that simulation cell.
Strictly, these supercells become the unit cells, which, by a periodic sequence, compose the bulk material that is actually investigated by this approach.
Thus, importance need to be attached to the construction of the supercell.
Three basic types of supercells to compose the initial Si bulk lattice, which can be scaled by integers in the different directions, are considered.
-The basis vectors of the supercells are shown in Fig. \ref{fig:simulation:sc}.
+The basis vectors of the supercells are shown in Fig.~\ref{fig:simulation:sc}.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:simulation:sc1}\includegraphics[width=0.3\textwidth]{sc_type0.eps}}
\caption{Basis vectors of three basic types of supercells used to create the initial Si bulk lattice.}
\label{fig:simulation:sc}
\end{figure}
-Type 1 (Fig. \ref{fig:simulation:sc1}) constitutes the primitive cell.
-The basis is face-centered cubic (fcc) and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
+Type 1 (Fig.~\ref{fig:simulation:sc1}) constitutes the primitive cell.
+The basis is face-centered cubic and is given by $x_1=(0.5,0.5,0)$, $x_2=(0,0.5,0.5)$ and $x_3=(0.5,0,0.5)$.
Two atoms, one at $(0,0,0)$ and the other at $(0.25,0.25,0.25)$ with respect to the basis, generate the Si diamond primitive cell.
-Type 2 (Fig. \ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
+Type 2 (Fig.~\ref{fig:simulation:sc2}) covers two primitive cells with 4 atoms.
The basis is given by $x_1=(0.5,-0.5,0)$, $x_2=(0.5,0.5,0)$ and $x_3=(0,0,1)$.
-Type 3 (Fig. \ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig. \ref{fig:sic:unit_cell}.
+Type 3 (Fig.~\ref{fig:simulation:sc3}) contains 4 primitive cells with 8 atoms and corresponds to the unit cell shown in Fig.~\ref{fig:sic:unit_cell}.
The basis is simple cubic.
-In the following an overview of the different simulation procedures and respective parameters is presented.
-These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used and on what is going to be investigated.
+In the following, an overview of the different simulation procedures and respective parameters is presented.
+These procedures and parameters differ depending on whether classical potentials or {\em ab initio} methods are used as well as on the object of investigation.
\section{DFT calculations}
\label{section:simulation:dft_calc}
-The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package ({\textsc vasp}) \cite{kresse96}.
-The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
-The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in {\textsc vasp} \cite{vanderbilt90}.
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp})~\cite{kresse96}.
+The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91)~\cite{perdew86,perdew92}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp}~\cite{vanderbilt90}.
An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
Spin polarization has been fully accounted for.
-The electronic ground state is calculated by an interative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
+The electronic ground state is calculated by an iterative Davidson scheme~\cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
-The conjugate gradiant algorithm is used for ionic relaxation.
-The cell volume and shape is allowed to change using the pressure control algorithm of Parinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
-Due to restrictions by the {\textsc vasp} code, {\em ab initio} MD could only be performed at constant volume.
-In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a timestep of \unit[1]{fs}.
+The conjugate gradient algorithm is used for ionic relaxation.
+Migration paths are determined by the modified version of the CRT method as explained in section~\ref{section:basics:migration}.
+The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman~\cite{parrinello81} in order to realize constant pressure simulations.
+Due to restrictions by the \textsc{vasp} code, {\em ab initio} MD could only be performed at constant volume.
+In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
-% todo
-% All point defects are calculated for the neutral charge state.
+% todo - point defects are calculated for the neutral charge state.
Most of the parameter settings, as determined above, constitute a tradeoff regarding the tasks that need to be addressed.
-These parameters include the size of the supercell, cut-off energy and $k$ point mesh.
+These parameters include the size of the supercell, cut-off energy and $\vec{k}$-point mesh.
The choice of these parameters is considered to reflect a reasonable treatment with respect to both, computational efficiency and accuracy, as will be shown in the next sections.
Furthermore, criteria concerning the choice of the potential and the exchange-correlation (XC) functional are being outlined.
Finally, the utilized parameter set is tested by comparing the calculated values of the cohesive energy and the lattice constant to experimental data.
\caption{Defect formation energies of several defects in c-Si with respect to the size of the supercell.}
\label{fig:simulation:ef_ss}
\end{figure}
-To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated.
-An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh \cite{monkhorst76} is used.
-The results are displayed in Fig. \ref{fig:simulation:ef_ss}.
+To estimate a critical size, the formation energies of several intrinsic defects in Si with respect to the system size are calculated.
+An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $\vec{k}$-point mesh~\cite{monkhorst76} is used.
+The results are displayed in Fig.~\ref{fig:simulation:ef_ss}.
The formation energies converge fast with respect to the system size.
Thus, investigating supercells containing more than 56 primitive cells or $112\pm1$ atoms should be reasonably accurate.
Throughout this work sampling of the BZ is restricted to the $\Gamma$ point.
The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since $c_{i,q}=c_{i,-q}^*$, where the $c_{i,q}$ are the Fourier coefficients of the wave function.
-As discussed in section \ref{subsection:basics:bzs} this does not pose a severe limitation if the supercell is large enough.
-Indeed, it was shown \cite{dal_pino93} that already for calculations involving only 32 atoms energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point \cite{baldereschi73}, which constitutes a mean-value point in the BZ.
-Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $k$-point mesh.
+As discussed in section~\ref{subsection:basics:bzs}, this does not pose a severe limitation if the supercell is large enough.
+Indeed, it was shown~\cite{dal_pino93} that already for calculations involving only 32 atoms, energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point~\cite{baldereschi73}, which constitutes a mean-value point in the BZ.
+Thus, the calculations of the present study on supercells containing $108$ primitive cells can be considered sufficiently converged with respect to the $\vec{k}$-point mesh.
\subsection{Energy cut-off}
-To determine an appropriate cut-off energy of the plane-wave basis set a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA.
-% todo
-% mention that results are within lda
+To determine an appropriate cut-off energy of the plane-wave basis set, a $2\times2\times2$ supercell of type 3 containing $32$ Si and $32$ C atoms in the 3C-SiC structure is equilibrated for different cut-off energies in the LDA.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{sic_32pc_gamma_cutoff_lc.ps}
\caption{Lattice constants of 3C-SiC with respect to the cut-off energy used for the plane-wave basis set.}
\label{fig:simulation:lc_ce}
\end{figure}
-Fig. \ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
+Fig.~\ref{fig:simulation:lc_ce} shows the respective lattice constants of the relaxed 3C-SiC structure with respect to the cut-off energy.
As can be seen, convergence is reached already for low energies.
Obviously, an energy cut-off of \unit[300]{eV}, although the minimum acceptable, is sufficient for the plane-wave expansion.
\subsection{Potential and exchange-correlation functional}
-To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
-To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e. an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
-Next to the ultra-soft pseudopotentials \cite{vanderbilt90} {\textsc vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
-The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in {\textsc vasp}.
+To find the most suitable combination of potential and XC functional for the C/Si system, a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
+To exclude a possibly corrupting influence of the other parameters, highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $\vec{k}$-point mesh is used.
+Next to the ultra-soft pseudopotentials~\cite{vanderbilt90}, \textsc{vasp} offers the projector augmented-wave method (PAW)~\cite{bloechl94} to describe the ion-electron interaction.
+The two XC functionals included in the test are of the LDA~\cite{ceperley80,perdew81} and GGA~\cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
\begin{table}[t]
\begin{center}
\hline
\end{tabular}
\end{center}
-\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultr-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
+\caption[Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials and XC functionals.]{Equilibrium lattice constants and cohesive energies of fully relaxed structures of Si, C (diamond) and 3C-SiC for different potentials (ultra-soft PP and PAW) and XC functionals (LDA and GGA). Deviations of the respective values from experimental values are given. Values are in good (green), fair (orange) and poor (red) agreement.}
\label{table:simulation:potxc}
\end{table}
-Table \ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional.
+Table~\ref{table:simulation:potxc} shows the lattice constants and cohesive energies obtained for the fully relaxed structures with respect to the utilized potential and XC functional.
As expected, cohesive energies are poorly reproduced by the LDA whereas the equilibrium lattice constants are in good agreement.
Using GGA together with the ultra-soft pseudopotential yields improved lattice constants and, more importantly, a very nice agreement of the cohesive energies to the experimental data.
-The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e. overestimated binding energies.
+The 3C-SiC calculations employing the PAW method in conjunction with the LDA suffers from the general problem inherent to LDA, i.e.\ overestimated binding energies.
Thus, the PAW \& LDA combination is not pursued.
Since the lattice constant and cohesive energy of 3C-SiC calculated by the PAW method using the GGA are not improved compared to the ultra-soft pseudopotential calculations using the same XC functional, this concept is likewise stopped.
-To conclude, the combination of ultr-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
+To conclude, the combination of ultra-soft pseudopotentials and the GGA XC functional are considered the optimal choice for the present study.
\subsection{Lattice constants and cohesive energies}
\caption{Equilibrium lattice constants and cohesive energies of Si, C (diamond) and 3C-SiC using the entire parameter set as determined in the beginning of this section.}
\label{table:simulation:paramf}
\end{table}
-Table \ref{table:simulation:paramf} shows the respective results and deviations from experiment.
+Table~\ref{table:simulation:paramf} shows the respective results and deviations from experiment.
A nice agreement with experimental results is achieved.
-Clearly, a competent parameter set is found, which is capabale of describing the C/Si system by {\em ab initio} calculations.
+Clearly, a competent parameter set is found, which is capable of describing the C/Si system by {\em ab initio} calculations.
-% todo
-% rewrite dft chapter
-% ref for experimental values!
+% todo - ref for experimental values!
\section{Classical potential MD}
+\label{section:classpotmd}
+
+The classical potential MD method is much less computationally costly compared to the highly accurate quantum-mechanical method.
+Thus, the method is capable of performing structural optimizations on large systems and MD calculations may be used to model a system over long time scales.
+Defect structures are modeled in a cubic supercell (type 3) of nine Si lattice constants in each direction containing 5832 Si atoms.
+Reproducing the SiC precipitation was attempted in cubic c-Si supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
+A Tersoff-like bond order potential by Erhart and Albe (EA)~\cite{albe_sic_pot} is used to describe the atomic interaction.
+Constant pressure simulations are realized by the Berendsen barostat~\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
+The temperature is controlled by the Berendsen thermostat~\cite{berendsen84} with a time constant of \unit[100]{fs}.
+Integration of the equations of motion is realized by the velocity Verlet algorithm~\cite{verlet67} using a fixed time step of \unit[1]{fs}.
+For structural relaxation of defect structures, the same algorithm is utilized with the temperature set to zero Kelvin.
+This also applies for the relaxation of structures within the CRT calculations to find migration pathways.
+In the latter case the time constant of the Berendsen thermostat is set to \unit[1]{fs} in order to achieve direct velocity scaling, which corresponds to a steepest descent minimization driving the system into a local minimum, if the temperature is set to zero Kelvin.
+However, in some cases a time constant of \unit[100]{fs} turned out to result in lower barriers.
+Defect structures as well as the simulations modeling the SiC precipitation are performed in the isothermal-isobaric $NpT$ ensemble.
+
+In addition to the bond order formalism, the EA potential provides a set of parameters to describe the interaction in the C/Si system, as discussed in section~\ref{subsection:interact_pot}.
+There are basically no free parameters, which could be set by the user, and the properties of the potential and its parameters are well known and have been extensively tested by the authors~\cite{albe_sic_pot}.
+Therefore, test calculations are restricted to the time step used in the Verlet algorithm to integrate the equations of motion.
+Nevertheless, a further and rather uncommon test is carried out to roughly estimate the capabilities of the EA potential regarding the description of 3C-SiC precipitation in c-Si.
+
+\subsection{Time step}
+
+The quality of the integration algorithm and the occupied time step of \unit[1]{fs} is determined by the ability to conserve the total energy.
+Therefor, simulations of a $9\times9\times9$ 3C-SiC unit cell containing 5832 atoms in total are carried out in the $NVE$ ensemble.
+The calculations are performed for \unit[100]{ps} corresponding to $10^5$ integration steps and two different initial temperatures are considered, i.e.\ \unit[0]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}.
+\begin{figure}[t]
+\begin{center}
+\includegraphics[width=0.7\textwidth]{verlet_e.ps}
+\end{center}
+\caption{Evolution of the total energy of 3C-SiC in the $NVE$ ensemble for two different initial temperatures.}
+\label{fig:simulation:verlet_e}
+\end{figure}
+The evolution of the total energy is displayed in Fig.~\ref{fig:simulation:verlet_e}.
+Almost no shift in energy is observable for the simulation at \unit[0]{$^{\circ}$C}.
+Even for \unit[1000]{$^{\circ}$C} the shift is as small as \unit[0.04]{eV}, which is a quite acceptable error for $10^5$ integration steps.
+Thus, using a time step of \unit[1]{fs} is considered small enough.
-fast method, amoun tof atoms ...
-
-\subsection{Tersoff vs. Erhart-Albe SiC potential}
-
-\subsection{Temperature and volume control}
-
-
-\subsection{Test calculations}
-
-Give cohesive energies of Si, C (Dia) and (3C-)SiC and the respective lattice parameters ...
-
-\subsection{3C-SiC precipitate in crystalline silicon}
+\subsection{3C-SiC precipitate in c-Si}
\label{section:simulation:prec}
-A spherical 3C-SiC precipitate enclosed in a c-Si surrounding is constructed.
+Below, a spherical 3C-SiC precipitate enclosed in a c-Si surrounding is investigated by means of MD.
+On the one hand, these investigations are meant to draw conclusions on the capabilities of the potential for modeling the respective tasks in the C/Si system.
+Since, on the other hand, properties of the 3C-SiC precipitate, its surrounding as well as the interface can be obtained, the calculations could be considered to constitute a first investigation rather than a test of the capabilities of the potential.
- as it is expected from IBS experiments and from simulations that finally succeed in simulating the precipitation event.
-On the one hand this sheds light on characteristic values like the radial distribution function or the total amount of free energy for such a configuration that is aimed to be reproduced by simulation.
-On the other hand, assuming a correct alignment of the precipitate with the c-Si matrix, properties of such precipitates and the surrounding as well as the interface can be investiagted.
-Furthermore these investigations might establish the prediction of conditions necessary for the simulation of the precipitation process.
+\subsubsection{Interfacial energy}
To construct a spherical and topotactically aligned 3C-SiC precipitate in c-Si, the approach illustrated in the following is applied.
A total simulation volume $V$ consisting of 21 unit cells of c-Si in each direction is created.
-To obtain a minimal and stable precipitate 5500 carbon atoms are considered necessary.
-This corresponds to a spherical 3C-SiC precipitate with a radius of approximately 3 nm.
+To obtain a minimal and stable precipitate, 5500 carbon atoms are considered necessary according to experimental results as discussed in section~\ref{subsection:ibs} and~\ref{section:assumed_prec}.
+This corresponds to a spherical 3C-SiC precipitate with a radius of approximately \unit[3]{nm}.
The initial precipitate configuration is constructed in two steps.
-In the first step the surrounding silicon matrix is created.
-This is realized by just skipping the generation of silicon atoms inside a sphere of radius $x$, which is the first unknown variable.
-The silicon lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
+In the first step, the surrounding Si matrix is created.
+This is realized by just skipping the generation of Si atoms inside a sphere of radius $x$, which is the first unknown variable.
+The Si lattice constant $a_{\text{Si}}$ of the surrounding c-Si matrix is assumed to not alter dramatically and, thus, is used for the initial lattice creation.
In a second step 3C-SiC is created inside the empty sphere of radius $x$.
-The lattice constant $y$, the second unknown variable, is chosen in such a way, that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
-This is entirely described by the system of equations \eqref{eq:md:constr_sic_01}
+The lattice constant $y$, the second unknown variable, is chosen in such a way that the necessary amount of carbon is generated and that the total amount of silicon atoms corresponds to the usual amount contained in the simulation volume.
+This is entirely described by the equation
\begin{equation}
\frac{8}{a_{\text{Si}}^3}(
-\underbrace{21^3 a_{\text{Si}}^3}_{=V}
+V
-\frac{4}{3}\pi x^3)+
-\underbrace{\frac{4}{y^3}\frac{4}{3}\pi x^3}_{\stackrel{!}{=}5500}
+\frac{4}{y^3}\frac{4}{3}\pi x^3
=21^3\cdot 8
-\label{eq:md:constr_sic_01}
\text{ ,}
+\label{eq:simulation:constr_sic_01}
\end{equation}
-which can be simplified to read
+where the volume is given by $V=21^3 a_{\text{Si}}^3$ and the the additional condition $\frac{4}{y^3}\frac{4}{3}\pi x^3=5500$.
+This can be simplified to read
\begin{equation}
\frac{8}{a_{\text{Si}}^3}\frac{4}{3}\pi x^3=5500
\Rightarrow x = \left(\frac{5500 \cdot 3}{32 \pi} \right)^{1/3}a_{\text{Si}}
-\label{eq:md:constr_sic_02}
+\label{eq:simulation:constr_sic_02}
\end{equation}
and
\begin{equation}
%\Rightarrow
y=\left(\frac{1}{2} \right)^{1/3}a_{\text{Si}}
\text{ .}
-\label{eq:md:constr_sic_03}
+\label{eq:simulation:constr_sic_03}
\end{equation}
-By this means values of 2.973 nm and 4.309 \AA{} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
-Since the generation of atoms is a discrete process with regard to the size of the volume the expected amounts of atoms are not obtained.
-However, by applying these values the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in table \ref{table:md:sic_prec}.
-\begin{table}[!ht]
+By this means, values of \unit[2.973]{nm} and \unit[4.309]{\AA} are obtained for the initial precipitate radius and lattice constant of 3C-SiC.
+Since the generation of atoms is a discrete process with regard to the size of the volume, the expected amounts of atoms are not obtained.
+However, by applying these values, the final configuration varies only slightly from the expected one by five carbon and eleven silicon atoms, as can be seen in Table~\ref{table:simulation:sic_prec}.
+\begin{table}[t]
\begin{center}
\begin{tabular}{l c c c c}
\hline
\hline
\hline
\end{tabular}
-\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:md:constr_sic_02} and \eqref{eq:md:constr_sic_03} in the 3C-SiC precipitate construction approach.}
-\label{table:md:sic_prec}
+\caption{Comparison of the expected and obtained amounts of Si and C atoms by applying the values from equations \eqref{eq:simulation:constr_sic_02} and \eqref{eq:simulation:constr_sic_03} in the 3C-SiC precipitate construction approach.}
+\label{table:simulation:sic_prec}
\end{center}
\end{table}
-After the initial configuration is constructed some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
-Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be $20\,^{\circ}\mathrm{C}$.
-Once the main part of the excess energy is carried out previous settings for the Berendsen thermostat are restored and the system is relaxed for another 10 ps.
+After the initial configuration is constructed, some of the atoms located at the 3C-SiC/c-Si interface show small distances, which results in high repulsive forces acting on the atoms.
+Thus, the system is equilibrated using strong coupling to the heat bath, which is set to be \unit[20]{$^{\circ}$C}.
+Once the main part of the excess energy is carried out, previous settings for the Berendsen thermostat are restored and the system is relaxed for another \unit[10]{ps}.
-\begin{figure}[!ht]
+\begin{figure}[t]
\begin{center}
-\includegraphics[width=12cm]{pc_0.ps}
+\includegraphics[width=0.7\textwidth]{pc_0.ps}
\end{center}
-\caption[Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embeeded in c-Si at $20\,^{\circ}\mathrm{C}$. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
-\label{fig:md:pc_sic-prec}
+\caption[Radial distribution of a 3C-SiC precipitate embedded in c-Si at $20\,^{\circ}\mathrm{C}$.]{Radial distribution of a 3C-SiC precipitate embedded in c-Si at \unit[20]{$^{\circ}$C}. The Si-Si radial distribution of plain c-Si is plotted for comparison. Green arrows mark bumps in the Si-Si distribution of the precipitate configuration, which do not exist in plain c-Si.}
+\label{fig:simulation:pc_sic-prec}
\end{figure}
-Figure \ref{fig:md:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
-The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of 0.235 nm, which is the distance of next neighboured Si atoms in c-Si.
-Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly there is no change at all within observational accuracy.
+Fig.~\ref{fig:simulation:pc_sic-prec} shows the radial distribution of the obtained precipitate configuration.
+The Si-Si radial distribution for both, plain c-Si and the precipitate configuration show a maximum at a distance of \unit[0.235]{nm}, which is the distance of next neighbored Si atoms in c-Si.
+Although no significant change of the lattice constant of the surrounding c-Si matrix was assumed, surprisingly, there is no change at all within observational accuracy.
Looking closer at higher order Si-Si peaks might even allow the guess of a slight increase of the lattice constant compared to the plain c-Si structure.
-A new Si-Si peak arises at 0.307 nm, which is identical to the peak of the C-C distribution around that value.
-It corresponds to second next neighbours in 3C-SiC, which applies for Si as well as C pairs.
+A new Si-Si peak arises at \unit[0.307]{nm}, which is identical to the peak of the C-C distribution around that value.
+It corresponds to second next neighbors in 3C-SiC, which applies for Si as well as C pairs.
The bumps of the Si-Si distribution at higher distances marked by the green arrows can be explained in the same manner.
-They correspond to the fourth and sixth next neighbour distance in 3C-SiC.
+They correspond to the fourth and sixth next neighbor distance in 3C-SiC.
It is easily identifiable how these C-C peaks, which imply Si pairs at same distances inside the precipitate, contribute to the bumps observed in the Si-Si distribution.
-The Si-Si and C-C peak at 0.307 nm enables the determination of the lattic constant of the embedded 3C-SiC precipitate.
-A lattice constant of 4.34 \AA{} compared to 4.36 \AA{} for bulk 3C-SiC is obtained.
-This is in accordance with the peak of Si-C pairs at a distance of 0.188 nm.
+The Si-Si and C-C peak at \unit[0.307]{nm} enables the determination of the lattice constant of the embedded 3C-SiC precipitate.
+A lattice constant of \unit[4.34]{\AA} compared to \unit[4.36]{\AA} for bulk 3C-SiC is obtained.
+This is in accordance with the peak of Si-C pairs at a distance of \unit[0.188]{nm}.
Thus, the precipitate structure is slightly compressed compared to the bulk phase.
-This is a quite surprising result since due to the finite size of the c-Si surrounding a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
+This is a quite surprising result since due to the finite size of the c-Si surrounding, a non-negligible impact of the precipitate on the materializing c-Si lattice constant especially near the precipitate could be assumed.
However, it seems that the size of the c-Si host matrix is chosen large enough to even find the precipitate in a compressed state.
The absence of a compression of the c-Si surrounding is due to the possibility of the system to change its volume.
-Otherwise the increase of the lattice constant of the precipitate of roughly 4.31 \AA{} in the beginning up to 4.34 \AA{} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
-If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate the expected increase can be calculated by
+Otherwise, the increase of the lattice constant of the precipitate of roughly \unit[4.31]{\AA} in the beginning up to \unit[4.34]{\AA} in the relaxed precipitate configuration could not take place without an accompanying reduction of the lattice constant of the c-Si surrounding.
+If the total volume is assumed to be the sum of the volumes that are composed of Si atoms forming the c-Si surrounding and Si atoms involved forming the precipitate, the expected increase can be calculated by
\begin{equation}
\frac{V}{V_0}=
- \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si of precipitate configuration}}}+
- \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC of precipitate configuration}}}}
+ \frac{\frac{N^{\text{c-Si}}_{\text{Si}}}{8/a_{\text{c-Si prec}}}+
+ \frac{N^{\text{3C-SiC}}_{\text{Si}}}{4/a_{\text{3C-SiC prec}}}}
{\frac{N^{\text{total}}_{\text{Si}}}{8/a_{\text{plain c-Si}}}}
\end{equation}
-with the notation used in table \ref{table:md:sic_prec}.
-The lattice constant of plain c-Si at $20\,^{\circ}\mathrm{C}$ can be determined more accurately by the side lengthes of the simulation box of an equlibrated structure instead of using the radial distribution data.
-By this a value of $a_{\text{plain c-Si}}=5.439\text{ \AA}$ is obtained.
-The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si of precipitate configuration}}$ since peaks in the radial distribution match the ones of plain c-Si.
-Using $a_{\text{3C-SiC of precipitate configuration}}=4.34\text{ \AA}$ as observed from the radial distribution finally results in an increase of the initial volume by 0.12 \%.
-However, each side length and the total volume of the simulation box is increased by 0.20 \% and 0.61 \% respectively compared to plain c-Si at $20\,^{\circ}\mathrm{C}$.
-Since the c-Si surrounding resides in an uncompressed state the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
+with the notation used in Table~\ref{table:simulation:sic_prec}.
+Here, $a_{\text{c-Si prec}}$ denotes the lattice constant of the surrounding crystalline Si and $a_{\text{3C-SiC prec}}$ is the lattice constant of the precipitate.
+The lattice constant of plain c-Si at \unit[20]{$^{\circ}$C} can be determined more accurately by the side lengths of the simulation box of an equilibrated structure instead of using the radial distribution data.
+By this, a value of $a_{\text{plain c-Si}}=5.439\,\text{\AA}$ is obtained.
+The same lattice constant is assumed for the c-Si surrounding in the precipitate configuration $a_{\text{c-Si prec}}$ since peaks in the radial distribution match the ones of plain c-Si.
+Using $a_{\text{3C-SiC prec}}=4.34\,\text{\AA}$ as observed from the radial distribution finally results in an increase of the initial volume by \unit[0.12]{\%}.
+However, each side length and the total volume of the simulation box is increased by \unit[0.20]{\%} and \unit[0.61]{\%} respectively compared to plain c-Si at \unit[20]{$^{\circ}$C}.
+Since the c-Si surrounding resides in an uncompressed state, the excess increase must be attributed to relaxation of strain with the strain resulting from either the compressed precipitate or the 3C-SiC/c-Si interface region.
This also explains the possibly identified slight increase of the c-Si lattice constant in the surrounding as mentioned earlier.
-As the pressure is set to zero the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
-Apparently the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
+As the pressure is set to zero, the free energy is minimized with respect to the volume enabled by the Berendsen barostat algorithm.
+Apparently, the minimized structure with respect to the volume is a configuration of a small compressively stressed precipitate and a large amount of slightly stretched c-Si in the surrounding.
-In the following the 3C-SiC/c-Si interface is described in further detail.
+To finally draw some conclusions concerning the capabilities of the potential, the 3C-SiC/c-Si interface is now addressed.
One important size analyzing the interface is the interfacial energy.
-It is determined exactly in the same way than the formation energy as described in equation \eqref{eq:defects:ef2}.
-Using the notation of table \ref{table:md:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
+A good estimate of the interfacial energy should be obtained by utilizing the formula for determining the defect formation energy as described in equation \eqref{eq:basics:ef2}.
+Using the notation of Table~\ref{table:simulation:sic_prec} and assuming that the system is composed out of $N^{\text{3C-SiC}}_{\text{C}}$ C atoms forming the SiC compound plus the remaining Si atoms, the energy is given by
\begin{equation}
E_{\text{f}}=E-
- N^{\text{3C-SiC}}_{\text{C}} \mu_{\text{SiC}}-
+ N^{\text{3C-SiC}}_{\text{C}} E_{\text{coh}}^{\text{SiC}}-
\left(N^{\text{total}}_{\text{Si}}-N^{\text{3C-SiC}}_{\text{C}}\right)
- \mu_{\text{Si}} \text{ ,}
-\label{eq:md:ife}
+ \mu_{\text{coh}}^{\text{Si}} \text{ ,}
+\label{eq:simulation:ife}
\end{equation}
-with $E$ being the free energy of the precipitate configuration at zero temperature.
-An interfacial energy of 2267.28 eV is obtained.
-The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
-Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
-This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
-
-Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
-Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
-The precipitate structure is rapidly heated up to $2050\,^{\circ}\mathrm{C}$ with a heating rate of approximately $75\,^{\circ}\mathrm{C}/\text{ps}$.
-From that point on the heating rate is reduced to $1\,^{\circ}\mathrm{C}/\text{ps}$ and heating is continued to 120 \% of the Si melting temperature, that is 2940 K.
-\begin{figure}[!ht]
+where $E$ is the total energy of the precipitate configuration at zero temperature.
+An interfacial energy of \unit[2267.28]{eV} is obtained.
+The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of \unit[29.93]{\AA}.
+Thus, the interface tension, given by the energy of the interface divided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{-4}$]{J/cm$^2$}.
+This value perfectly fits within the experimentally estimated range of \unit[2--8$\times10^{-4}$]{J/cm$^2$}~\cite{taylor93}.
+Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
+Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
+
+\subsubsection{Stability of the precipitate}
+
+To investigate the stability of the precipitate, which is assumed to be stable even at temperatures above the Si melting temperature, the configuration is heated up beyond the critical point, at which the Si melting transition occurs.
+For this, the transition point of c-Si needs to be evaluated first.
+According to the authors of the potential, the Si melting point is \degk{2450}.
+However, melting is not predicted to occur instantly after exceeding the melting point due to the additionally required transition enthalpy and hysteresis behavior.
+To determine the transition point, c-Si is heated up using a heating rate of \unit[1]{$^{\circ}$C/ps}.
+\begin{figure}[tp]
\begin{center}
-\includegraphics[width=12cm]{fe_and_t_sic.ps}
+\includegraphics[width=0.7\textwidth]{fe_and_t.ps}
\end{center}
-\caption{Free energy and temperature evolution of a constructed 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
-\label{fig:md:fe_and_t_sic}
+\caption{Total energy and temperature evolution of c-Si at temperatures in the region around the melting transition.}
+\label{fig:simulation:fe_and_t}
\end{figure}
-Figure \ref{fig:md:fe_and_t_sic} shows the free energy and temperature evolution.
-The sudden increase of the free energy indicates possible melting occuring around 2840 K.
-\begin{figure}[!ht]
+Fig.~\ref{fig:simulation:fe_and_t} shows the total energy and temperature evolution in the region around the transition temperature.
+Indeed, a transition and the accompanied critical behavior of the total energy is first observed at approximately \degk{3125}, which corresponds to \unit[128]{\%} of the Si melting temperature.
+The difference in total energy is \unit[0.58]{eV} per atom corresponding to \unit[55.7]{kJ/mole}, which compares quite well to the Si enthalpy of melting of \unit[50.2]{kJ/mole}.
+
+The precipitate structure is heated up using the same heating rate.
+As can be seen in Fig.~\ref{fig:simulation:sic_melt}, which shows a cross-sectional image of the configuration at different temperatures, the precipitate is stable whereas melting of the surrounding Si host matrix starting at the interface region is observed.
+\begin{figure}[tp]
\begin{center}
-\includegraphics[width=12cm]{pc_500-fin.ps}
+\subfigure[]{\label{fig:simulation:sic_melt1}\includegraphics[width=7cm]{sic_prec/melt_01.eps}}
+\subfigure[]{\label{fig:simulation:sic_melt2}\includegraphics[width=7cm]{sic_prec/melt_02.eps}}
+\subfigure[]{\label{fig:simulation:sic_melt3}\includegraphics[width=7cm]{sic_prec/melt_03.eps}}
\end{center}
-\caption{Radial distribution of the constructed 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
-\label{fig:md:pc_500-fin}
+\caption{Cross section image of the 3C-SiC precipitate in c-Si at temperatures before (a), at the onset of (b) and after (c) the Si melting transition.}
+\label{fig:simulation:sic_melt}
\end{figure}
-Investigating the radial distribution function shown in figure \ref{fig:md:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the free energy plot.
-However the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
-Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
-This is surprising since the melting transition of plain c-Si is expected at temperatures around 3125 K, as discussed in section \ref{subsection:md:tval}.
-Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
-This is indeed verified by visualizing the atomic data.
-% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1
-\begin{figure}[!ht]
+This is verified by the radial distribution function shown in Fig.~\ref{fig:simulation:pc_500-fin}, which displays configurations before and after the Si transition occurs.
+\begin{figure}[tp]
\begin{center}
-\begin{minipage}{7cm}
-\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
-\end{minipage}
-\begin{minipage}{7cm}
-\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps}
-\end{minipage}
-\begin{minipage}{7cm}
-\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
-\end{minipage}
+\includegraphics[width=0.7\textwidth]{pc_500-fin.ps}
\end{center}
-\caption{Cross section image of atomic data gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at 200 ps (top left), 520 ps (top right) and 720 ps (bottom).}
-\label{fig:md:sic_melt}
+\caption{Radial distribution of a 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
+\label{fig:simulation:pc_500-fin}
\end{figure}
-Figure \ref{fig:md:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
-As can be seen from the image at 520 ps melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at 720 ps.
-As predicted from the radial distribution data the precipitate itself remains stable.
-
-For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
-To play safe the precipitate configuration at 100 \% of the Si melting temperature is chosen and cooled down to $20\,^{\circ}\mathrm{C}$ with a cooling rate of $1\,^{\circ}\mathrm{C}/\text{ps}$.
-However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
-Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.
+The precipitate itself is not involved in the transition, as indicated by the Si-C and C-C distribution, which essentially stays the same.
+It is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the respective Si-Si distributions.
+The temperature of transition is determined to be around \degk{2840}.
+This is surprising since the melting transition of c-Si for the same heating conditions is expected at temperatures around \degk{3125} as discussed above.
+Obviously, the precipitate lowers the transition point of the surrounding c-Si matrix.
+This is indeed verified by the cross-sectional images of the configurations shown in Fig.~\ref{fig:simulation:sic_melt}.
+Melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected.
+
+\subsubsection{Concluding remarks}
+
+To conclude, the obtained results, particularly the accurate value of the interface energy, the Si enthalpy of melting as well as the stability of the 3C-SiC structure give quite a good feeling concerning the applicability of the potential.
+
+%Since the precipitate configuration is artificially constructed, the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
+%Thus, annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
+%The precipitate structure is rapidly heated up to \unit[2050]{$^{\circ}$C} with a heating rate of approximately \unit[75]{$^{\circ}$C/ps}.
+%From that point on the heating rate is reduced to \unit[1]{$^{\circ}$C/ps} and heating is continued upto \unit[120]{\%} of the Si melting temperature of the potential, i.e.\ \unit[2940]{K}.
+%\begin{figure}[t]
+%\begin{center}
+%\includegraphics[width=0.7\textwidth]{fe_and_t_sic.ps}
+%\end{center}
+%\caption{Total energy and temperature evolution of a 3C-SiC precipitate embedded in c-Si at temperatures above the Si melting point.}
+%\label{fig:simulation:fe_and_t_sic}
+%\end{figure}
+%Figure~\ref{fig:simulation:fe_and_t_sic} shows the total energy and temperature evolution.
+%The sudden increase of the total energy indicates possible melting occuring around \unit[2840]{K}.
+%\begin{figure}[ht]
+%\begin{center}
+%\includegraphics[width=0.7\textwidth]{pc_500-fin.ps}
+%\end{center}
+%\caption{Radial distribution of a 3C-SiC precipitate embedded in c-Si at temperatures below and above the Si melting transition point.}
+%\label{fig:simulation:pc_500-fin}
+%\end{figure}
+%Investigating the radial distribution function shown in figure~\ref{fig:simulation:pc_500-fin}, which shows configurations below and above the temperature of the estimated transition, indeed supports the assumption of melting gained by the total energy plot in Fig.~\ref{fig:simulation:fe_and_t_sic}.
+%However, the precipitate itself is not involved, as can be seen from the Si-C and C-C distribution, which essentially stays the same for both temperatures.
+%Thus, it is only the c-Si surrounding undergoing a structural phase transition, which is very well reflected by the difference observed for the two Si-Si distributions.
+%This is surprising since the melting transition of plain c-Si for the same heating conditions is expected at temperatures around \unit[3125]{K}, as will be discussed later in section~\ref{subsection:md:tval}.
+%Obviously the precipitate lowers the transition point of the surrounding c-Si matrix.
+%This is indeed verified by visualizing the atomic data.
+%% ./visualize -w 640 -h 480 -d saves/sic_prec_120Tm_cnt1 -nll -11.56 -0.56 -11.56 -fur 11.56 0.56 11.56 -c -0.2 -24.0 0.6 -L 0 0 0.2 -r 0.6 -B 0.1
+%\begin{figure}[tp]
+%\begin{center}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_01.eps}
+%\end{minipage}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_02.eps}
+%\end{minipage}
+%\begin{minipage}{7cm}
+%\includegraphics[width=7cm,draft=false]{sic_prec/melt_03.eps}
+%\end{minipage}
+%\end{center}
+%\caption{Cross section image of the precipitate configuration gained by annealing simulations of the constructed 3C-SiC precipitate in c-Si at \unit[200]{ps} (top left), \unit[520]{ps} (top right) and \unit[720]{ps} (bottom).}
+%\label{fig:simulation:sic_melt}
+%\end{figure}
+%Fig.~\ref{fig:simulation:sic_melt} shows cross section images of the atomic structures at different times and temperatures.
+%As can be seen from the image at \unit[520]{ps} melting of the Si surrounding in fact starts in the defective interface region of the 3C-SiC precipitate and the c-Si surrounding propagating outwards until the whole Si matrix is affected at \unit[720]{ps}.
+%As predicted from the radial distribution data the precipitate itself indeed remains stable.
+%
+%For the rearrangement simulations temperatures well below the transition point should be used since it is very unlikely to recrystallize the molten Si surrounding properly when cooling down.
+%To play safe the precipitate configuration at \unit[100]{\%} of the Si melting temperature is chosen and cooled down to \unit[20]{$^{\circ}$C} with a cooling rate of \unit[1]{$^{\circ}$C/ps}.
+%However, an energetically more favorable interface is not obtained by quenching this structure to zero Kelvin.
+%Obviously the increased temperature run enables structural changes that are energetically less favorable but can not be exploited to form more favorable configurations by an apparently yet too fast cooling down process.
projects / lectures / latex.git / blobdiff