As discussed above, the HK and KS formulations are exact and so far no approximations except the adiabatic approximation entered the theory.
However, to make concrete use of the theory, effective approximations for the exchange and correlation energy functional $E_{\text{xc}}[n(\vec{r})]$ are required.
-Most simple and at the same time remarkably useful is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65}
+Most simple and at the same time remarkably useful, is the approximation of $E_{\text{xc}}[n(\vec{r})]$ by a function of the local density~\cite{kohn65},
\begin{equation}
E^{\text{LDA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}))n(\vec{r}) d\vec{r}
\text{ ,}
E^{\text{GGA}}_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(n(\vec{r}),|\nabla n(\vec{r})|)n(\vec{r}) d\vec{r}
\text{ .}
\end{equation}
-These functionals constitute the simplest extensions of LDA for inhomogeneous systems.
-At modest computational costs gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.
+This functional constitutes the simplest extension of LDA for inhomogeneous systems.
+At modest computational costs, gradient-corrected functionals very often yield much better results than the LDA with respect to cohesive and binding energies.
\subsection{Plane-wave basis set}
Finally, a set of basis functions is required to represent the one-electron KS wave functions.
-With respect to the numerical treatment it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions.
+With respect to the numerical treatment, it is favorable to approximate the wave functions by linear combinations of a finite number of such basis functions.
Convergence of the basis set, i.e.\ convergence of the wave functions with respect to the amount of basis functions, is most crucial for the accuracy of the numerical calculations.
Two classes of basis sets, the plane-wave and local basis sets, exist.
In fact, the employed \textsc{vasp} software is solving the KS equations within a plane-wave (PW) basis set.
The idea is based on the Bloch theorem~\cite{bloch29}, which states that in a periodic crystal each electronic wave function $\Phi_i(\vec{r})$ can be written as the product of a wave-like envelope function $\exp(i\vec{kr})$ and a function that has the same periodicity as the lattice.
The latter one can be expressed by a Fourier series, i.e.\ a discrete set of plane waves whose wave vectors just correspond to reciprocal lattice vectors $\vec{G}$ of the crystal.
-Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set
+Thus, the one-electron wave function $\Phi_i(\vec{r})$ associated with the wave vector $\vec{k}$ can be expanded in terms of a discrete PW basis set:
\begin{equation}
\Phi_i(\vec{r})=\sum_{\vec{G}
%, |\vec{G}+\vec{k}|<G_{\text{cut}}}
By construction, PW calculations require a periodic system.
This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
-As a consequence the number of atoms involved in the calculations are increased.
+As a consequence, the number of atoms involved in the calculations are increased.
To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
\label{section:basics:defects}
Point defects are defects that affect a single lattice site.
-At this site the crystalline periodicity is interrupted.
+At this site, the crystalline periodicity is interrupted.
An empty lattice site, which would be occupied in the perfect crystal structure, is called a vacancy defect.
If an additional atom is incorporated into the perfect crystal, this is called interstitial defect.
A substitutional defect exists, if an atom belonging to the perfect crystal is replaced with an atom of another species.
\subsection[C \hkl<1 0 0> dumbbell interstitial configuration]{\boldmath C \hkl<1 0 0> dumbbell interstitial configuration}
\label{subsection:100db}
-As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
+As the \ci{} \hkl<1 0 0> DB constitutes the ground-state configuration of a C atom incorporated into otherwise perfect c-Si, it is the most probable and, hence, one of the most important interstitial configurations of C in Si.
The structure was initially suspected by IR local vibrational mode absorption~\cite{bean70} and finally verified by electron paramagnetic resonance (EPR)~\cite{watkins76} studies on irradiated Si substrates at low temperatures.
Fig.~\ref{fig:defects:100db_cmp} schematically shows the \ci{} \hkl<1 0 0> DB structure and Table~\ref{tab:defects:100db_cmp} lists the details of the atomic displacements, distances and bond angles obtained by classical potential and quantum-mechanical calculations.
Present results suggest this configuration to correspond to a real local minimum.
In fact, an additional barrier has to be passed to reach this configuration starting from the \ci{} \hkl<1 0 0> interstitial configuration, which is investigated in section~\ref{subsection:100mig}.
After slightly displacing the C atom along the \hkl[1 0 0] (equivalent to a displacement along \hkl[0 1 0]), \hkl[0 0 1], \hkl[0 0 -1] and \hkl[1 -1 0] direction the distorted structures relax back into the BC configuration.
-As will be shown in subsequent migration simulations the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
+As will be shown in subsequent migration simulations, the same would happen to structures where the C atom is displaced along the migration direction, which approximately is the \hkl[1 1 0] direction.
These relaxations indicate that the BC configuration is a real local minimum instead of an assumed saddle point configuration.
Fig.~\ref{img:defects:bc_conf} shows the structure, charge density isosurface and Kohn-Sham levels of the BC configuration.
In fact, the net magnetization of two electrons is already suggested by simple molecular orbital theory considerations with respect to the bonding of the C atom.
Fig.~\ref{fig:defects:cp_bc_00-1_mig} shows the evolution of structure and energy along the \ci{} BC to \hkl[0 0 -1] DB transition.
Since the \ci{} BC configuration is unstable relaxing into the \hkl[1 1 0] DB configuration within this potential, the low kinetic energy state is used as a starting configuration.
Two different pathways are obtained for different time constants of the Berendsen thermostat.
-With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane
+With a time constant of \unit[1]{fs}, the C atom resides in the \hkl(1 1 0) plane
resulting in a migration barrier of \unit[2.4]{eV}.
However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
\label{fig:defects:110_mig}
\end{figure}
Fig.~\ref{fig:defects:110_mig} shows migration barriers of the \ci{} \hkl[1 1 0] DB to \hkl[0 0 -1], \hkl[0 -1 0] (in place) and BC configuration.
-As expected there is no maximum for the transition into the BC configuration.
+As expected, there is no maximum for the transition into the BC configuration.
As mentioned earlier, the BC configuration itself constitutes a saddle point configuration relaxing into the energetically more favorable \hkl[1 1 0] DB configuration.
An activation energy of \unit[2.2]{eV} is necessary to reorientate the \hkl[0 0 -1] into the \hkl[1 1 0] DB configuration, which is \unit[1.3]{eV} higher in energy.
Residing in this state another \unit[0.90]{eV} is enough to make the C atom form a \hkl[0 0 -1] DB configuration with the Si atom of the neighbored lattice site.
\end{table}
Obviously the EA potential properly describes the relative energies of formation.
Combined structures of C$_{\text{s}}$ and Si$_{\text{i}}$ T are energetically less favorable than the ground state C$_{\text{i}}$ \hkl<1 0 0> DB configuration.
-With increasing separation distance the energies of formation decrease.
+With increasing separation distance, the energies of formation decrease.
However, even for non-interacting defects, the energy of formation, which is then given by the sum of the formation energies of the separated defects (\unit[4.15]{eV}) is still higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB.
Unexpectedly, the structure of a Si$_{\text{i}}$ \hkl<1 1 0> DB and a neighbored C$_{\text{s}}$, which is the most favored configuration of a C$_{\text{s}}$ and Si$_{\text{i}}$ DB according to quantum-mechanical calculations, likewise constitutes an energetically favorable configuration within the EA description, which is even preferred over the two least separated configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ T.
This is attributed to an effective reduction in strain enabled by the respective combination.
However, it turns out that the BC configuration is not a saddle point configuration as proposed by Capaz et~al.~\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the $sp$ hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
-With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e.\ IBS.
+With an activation energy of \unit[0.9]{eV}, the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e.\ IBS.
Since the \ci{} \hkl<1 0 0> DB is the ground-state configuration and highly mobile, possible migration of these DBs to form defect agglomerates, as demanded by the model introduced in section~\ref{section:assumed_prec}, is considered possible.
Unfortunately the description of the same processes fails if classical potential methods are used.
\section{Long time scale simulations at maximum temperature}
-As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct} a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
+As discussed in section~\ref{section:md:limit} and~\ref{section:md:inct}, a further increase of the system temperature might help to overcome limitations of the short range potential and accelerate the dynamics involved in structural evolution.
Furthermore, these results indicate that increased temperatures are necessary to drive the system out of equilibrium enabling conditions needed for the formation of a metastable cubic polytype of SiC.
A maximum temperature to avoid melting is determined in section~\ref{section:md:tval} to be 120 \% of the Si melting point but due to defects lowering the transition point a maximum temperature of 95 \% of the Si melting temperature is considered useful.
\label{fig:sic:db_agglom}
\end{figure}
The incorporated C atoms form C-Si dumbbells on regular Si lattice sites.
-With increasing dose and proceeding time the highly mobile dumbbells agglomerate into large clusters.
+With increasing dose and proceeding time, the highly mobile dumbbells agglomerate into large clusters.
Finally, when the cluster size reaches a critical radius, the high interfacial energy due to the 3C-SiC/c-Si lattice misfit is overcome and precipitation occurs.
Due to the slightly lower silicon density of 3C-SiC, excessive silicon atoms exist, which will most probably end up as self-interstitials in the c-Si matrix since there is more space than in 3C-SiC.
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.
+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.
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.
+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.
To finally draw some conclusions concerning the capabilities of the potential, the 3C-SiC/c-Si interface is now addressed.
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