The Tersoff potential explicitly incorporates the dependence of bond order on local environments, permitting an improved description of covalent materials.
Due to the covalent character Tersoff restricted the interaction to nearest neighbor atoms accompanied by an increase in computational efficiency for the evaluation of forces and energy based on the short-range potential.
Tersoff applied the potential to silicon \cite{tersoff_si1,tersoff_si2,tersoff_si3}, carbon \cite{tersoff_c} and also to multicomponent systems like silicon carbide \cite{tersoff_m}.
-The basic idea is that, in real systems, the bond order, i.e. the strength of the bond, depends upon the local environment \cite{abell85}.
+The basic idea is that, in real systems, the bond order, i.e.\ the strength of the bond, depends upon the local environment \cite{abell85}.
Atoms with many neighbors form weaker bonds than atoms with only a few neighbors.
-Although the bond strength intricately depends on geometry, the focus on coordination, i.e. the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
+Although the bond strength intricately depends on geometry, the focus on coordination, i.e.\ the number of neighbors forming bonds, is well motivated qualitatively from basic chemistry since for every additional formed bond the amount of electron pairs per bond and, thus, the strength of the bonds is decreased.
If the energy per bond decreases rapidly enough with increasing coordination the most stable structure will be the dimer.
In the other extreme, if the dependence is weak, the material system will end up in a close-packed structure in order to maximize the number of bonds and likewise minimize the cohesive energy.
This suggests the bond order to be a monotonously decreasing function with respect to coordination and the equilibrium coordination being determined by the balance of bond strength and number of bonds.
\label{subsection:statistical_ensembles}
Using the above mentioned algorithms the most basic type of MD is realized by simply integrating the equations of motion of a fixed number of particles ($N$) in a closed volume $V$ realized by periodic boundary conditions (PBC).
-Providing a stable integration algorithm the total energy $E$, i.e. the kinetic and configurational energy of the particles, is conserved.
+Providing a stable integration algorithm the total energy $E$, i.e.\ the kinetic and configurational energy of the particles, is conserved.
This is known as the $NVE$, or microcanonical ensemble, describing an isolated system composed of microstates, among which the number of particles, volume and energy are held constant.
However, the successful formation of SiC dictates precise control of temperature by external heating.
Using this method the system does not behave like a true $NpT$ ensemble.
On average $T$ and $p$ correspond to the expected values.
-For large enough time constants, i.e. $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
+For large enough time constants, i.e.\ $\tau > 100 \delta t$, the method shows realistic fluctuations in $T$ and $p$.
The advantage of the approach is that the coupling can be decreased to minimize the disturbance of the system and likewise be adjusted to suit the needs of a given application.
It provides a stable algorithm that allows smooth changes of the system to new values of temperature or pressure, which is ideal for the investigated problem.
\label{section:dft}
Dirac declared that chemistry has come to an end, its content being entirely contained in the powerful equation published by Schr\"odinger in 1926 \cite{schroedinger26} marking the beginning of wave mechanics.
-Following the path of Schr\"odinger, the problem in quantum-mechanical modeling of describing the many-body problem, i.e. a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
+Following the path of Schr\"odinger, the problem in quantum-mechanical modeling of describing the many-body problem, i.e.\ a system of a large amount of interacting particles, is manifested in the high-dimensional Schr\"odinger equation for the wave function $\Psi({\vec{R}},{\vec{r}})$ that depends on the coordinates of all nuclei and electrons.
The Schr\"odinger equation contains the kinetic energy of the ions and electrons as well as the electron-ion, ion-ion and electron-electron interaction.
This cannot be solved exactly and finding approximate solutions requires several layers of simplification in order to reduce the number of free parameters.
Approximations that consider a truncated Hilbert space of single-particle orbitals yield promising results, however, with increasing complexity and demand for high accuracy the amount of Slater determinants to be evaluated massively increases.
\subsection{Kohn-Sham system}
-Inspired by the Hartree equations, i.e. a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}.
+Inspired by the Hartree equations, i.e.\ a set of self-consistent single-particle equations for the approximate solution of the many-electron problem \cite{hartree28}, which describe atomic ground states much better than the TF theory, Kohn and Sham presented a Hartree-like formulation of the Hohenberg and Kohn minimal principle \eqref{eq:basics:hkm} \cite{kohn65}.
However, due to a more general approach, the new formulation is formally exact by introducing the energy functional $E_{\text{xc}}[n(\vec{r})]$, which accounts for the exchange and correlation energy of the electron interaction $U$ and possible corrections due to electron interaction to the kinetic energy $T$.
The respective Kohn-Sham equations for the effective single-particle wave functions $\Phi_i(\vec{r})$ take the form
\begin{equation}
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.
-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.
+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.
-Local basis set functions usually are atomic orbitals, i.e. mathematical functions that describe the wave-like behavior of electrons, which are localized, i.e. centered on atoms or bonds.
+Local basis set functions usually are atomic orbitals, i.e.\ mathematical functions that describe the wave-like behavior of electrons, which are localized, i.e.\ centered on atoms or bonds.
Molecular orbitals can be represented by linear combinations of atomic orbitals (LCAO).
By construction, only a small number of basis functions is required to represent all of the electrons of each atom within reasonable accuracy.
Thus, local basis sets enable the implementation of methods that scale linearly with the number of atoms.
Another approach is to represent the KS wave functions by plane waves.
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.
+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
\begin{equation}
\Phi_i(\vec{r})=\sum_{\vec{G}
This way, the pseudo wave functions become smooth near the nuclei.
Most PPs satisfy four general conditions.
-The pseudo wave functions generated by the PP should not contain nodes, i.e. the pseudo wave functions should be smooth and free of wiggles in the core region.
+The pseudo wave functions generated by the PP should not contain nodes, i.e.\ the pseudo wave functions should be smooth and free of wiggles in the core region.
Outside the core region, the pseudo and real valence wave functions as well as the generated charge densities need to be identical.
The charge enclosed within the core region must be equal for both wave functions.
Last, almost redundantly, the valence all-electron and pseudopotential eigenvalues must be equal.
V_{\text{nl}}(\vec{r}) = \sum_{lm} | lm \rangle V_l(\vec{r}) \langle lm |
\text{ .}
\end{equation}
-Applying of the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $\mid lm \rangle$, i.e. the orbitals with azimuthal angular momentum $l$ and magnetic number $m$, which are then multiplied by the respective pseudopotential $V_l(\vec{r})$ for angular momentum $l$.
+Applying of the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $\mid lm \rangle$, i.e.\ the orbitals with azimuthal angular momentum $l$ and magnetic number $m$, which are then multiplied by the respective pseudopotential $V_l(\vec{r})$ for angular momentum $l$.
The standard generation procedure of pseudopotentials proceeds by varying its parameters until the pseudo eigenvalues are equal to the all-electron valence eigenvalues and the pseudo wave functions match the all-electron valence wave functions beyond a certain cut-off radius determining the core region.
Modified methods to generate ultra-soft pseudopotentials were proposed, which address the rapid convergence with respect to the size of the plane wave basis set \cite{vanderbilt90,troullier91}.
Since the values of the wave function within a small interval around $\vec{k}$ are almost identical, it is possible to approximate the infinite sum by a sum over an affordable number of $k$ points, each representing the respective region of the wave function in $\vec{k}$ space.
Methods have been derived for obtaining very accurate approximations by a summation over special sets of $\vec{k}$ points with distinct, associated weights \cite{baldereschi73,chadi73,monkhorst76}.
If present, symmetries in reciprocal space may further reduce the number of calculations.
-For supercells, i.e. repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurate results.
+For supercells, i.e.\ repeating unit cells that contain several primitive cells, restricting the sampling of the Brillouin zone (BZ) to the $\Gamma$ point can yield quite accurate results.
In fact, with respect to BZ sampling, calculating wave functions of a supercell containing $n$ primitive cells for only one $\vec{k}$ point is equivalent to the scenario of a single primitive cell and the summation over $n$ points in $\vec{k}$ space.
In general, finer $\vec{k}$ point meshes better account for the periodicity of a system, which in some cases, however, might be fictitious anyway.
To investigate equilibrium structures, however, the ionic subsystem must also be allowed to relax into a minimum energy configuration.
Local minimum configurations can be easily obtained in a MD-like way by moving the nuclei over small distances along the directions of the forces, as discussed in the MD chapter above.
Clearly, the conjugate gradient method constitutes a more sophisticated scheme, which will locate the equilibrium positions of the ions more rapidly.
-To find the global minimum, i.e. the absolute ground state, methods like simulated annealing or the Monte Carlo technique, which allow the system to escape local minima, have to be used for the search.
+To find the global minimum, i.e.\ the absolute ground state, methods like simulated annealing or the Monte Carlo technique, which allow the system to escape local minima, have to be used for the search.
The force on an ion is given by the negative derivative of the total energy with respect to the position of the ion.
-However, moving an ion, i.e. altering its position, changes the wave functions to the KS eigenstates corresponding to the new ionic configuration.
+However, moving an ion, i.e.\ altering its position, changes the wave functions to the KS eigenstates corresponding to the new ionic configuration.
Writing down the derivative of the total energy $E$ with respect to the position $\vec{R}_i$ of ion $i$
\begin{equation}
\frac{dE}{d\vec{R_i}}=
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.
The disturbance caused by these defects may result in the distortion of the surrounding atomic structure and is accompanied by an increase in configurational energy.
-Thus, next to the structure of the defect, the energy needed to create such a defect, i.e. the defect formation energy, is an important value characterizing the defect and likewise determining its relative stability.
+Thus, next to the structure of the defect, the energy needed to create such a defect, i.e.\ the defect formation energy, is an important value characterizing the defect and likewise determining its relative stability.
The formation energy of a defect is defined by
\begin{equation}
\label{eq:basics:ef2}
\end{equation}
where $E$ is the total energy of the interstitial structure involving $N_i$ atoms of type $i$ with chemical potential $\mu_i$.
-Here, the chemical potentials are determined by the chemical potential of the respective equilibrium bulk structure, i.e. the cohesive energy per atom for the fully relaxed structure at zero temperature and pressure.
+Here, the chemical potentials are determined by the chemical potential of the respective equilibrium bulk structure, i.e.\ the cohesive energy per atom for the fully relaxed structure at zero temperature and pressure.
Considering C interstitial defects in Si, the chemical potential for C could also be determined by the cohesive energies of Si and SiC according to the relation $\mu_{\text{C}}=\mu_{\text{SiC}}-\mu_{\text{Si}}$ of the chemical potentials.
In this way, SiC is chosen as a reservoir for the C impurity.
For defect configurations consisting of a single atom species the formation energy reduces to
\end{equation}
where the formation energies $E_{\text{f}}^{\text{comb}}$, $E_{\text{f}}^{1^{\text{st}}}$ and $E_{\text{f}}^{2^{\text{nd}}}$ are determined as discussed above.
Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.
-The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
+The interaction strength, i.e.\ the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.
Thus, $E_{\text{b}}$ indeed can be best thought of a binding energy, which is required to bring the defects to infinite separation.
The methods presented in the last two chapters can be used to investigate defect structures and energetics.
For the vacancy the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
No other intrinsic defect configuration, within the ones that are mentioned, is affected by spin polarization.
-In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e. if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
+In the case of the classical potential simulations bonds between atoms are displayed if there is an interaction according to the potential model, i.e.\ if the distance of two atoms is within the cut-off radius $S_{ij}$ introduced in equation \eqref{eq:basics:fc}.
For the tetrahedral and the slightly displaced configurations four bonds to the atoms located in the center of the planes of the unit cell exist in addition to the four tetrahedral bonds.
The length of these bonds are, however, close to the cut-off range and thus are weak interactions not constituting actual chemical bonds.
The same applies to the bonds between the interstitial and the upper two atoms in the \si{} \hkl<1 1 0> DB configuration.
The next energetically favorable defect configuration is the \hkl<1 1 0> C-Si DB interstitial.
Fig. \ref{fig:defects:110_mig_vasp} shows the migration barrier of the \hkl<1 1 0> C-Si DB to the BC, \hkl<0 0 -1> and \hkl<0 -1 0> (in place) transition.
Indeed less than \unit[0.7]{eV} are necessary to turn a \hkl<0 -1 0>- to a \hkl<1 1 0>-type C-Si DB interstitial.
-This transition is carried out in place, i.e. the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
+This transition is carried out in place, i.e.\ the Si DB pair is not changed and both, the Si and C atom share the initial lattice site.
Thus, this transition does not contribute to long-range diffusion.
Once the C atom resides in the \hkl<1 1 0> DB interstitial configuration it can migrate into the BC configuration requiring approximately \unit[0.95]{eV} of activation energy, which is only slightly higher than the activation energy needed for the \hkl<0 0 -1> to \hkl<0 -1 0> pathway as shown in Fig. \ref{fig:defects:00-1_0-10_mig}.
As already known from the migration of the \hkl<0 0 -1> to the BC configuration discussed in Fig. \ref{fig:defects:00-1_001_mig}, another \unit[0.25]{eV} are needed to turn back from the BC to a \hkl<1 0 0>-type interstitial.
Activation energies of roughly \unit[2.8]{eV} and \unit[2.7]{eV} are needed for migration.
The \ci{} \hkl[1 1 0] configuration seems to play a decisive role in all migration pathways in the classical potential calculations.
-As mentioned above, the starting configuration of the first migration path, i.e. the BC configuration, is fixed to be a transition point but in fact is unstable.
+As mentioned above, the starting configuration of the first migration path, i.e.\ the BC configuration, is fixed to be a transition point but in fact is unstable.
Further relaxation of the BC configuration results in the \ci{} \hkl[1 1 0] configuration.
Even the last two pathways show configurations almost identical to the \ci{} \hkl[1 1 0] configuration, which constitute local minima within the pathways.
Thus, migration pathways involving the \ci{} \hkl[1 1 0] DB configuration as a starting or final configuration are further investigated.
There is still a low interaction remaining, which is due to the equal orientation of the defects.
By changing the orientation of the second DB interstitial to the \hkl<0 -1 0>-type, the interaction is even more reduced resulting in an energy of \unit[-0.05]{eV} for a distance, which is the maximum that can be realized due to periodic boundary conditions.
Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.
-Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
+Antiparallel orientations of the second defect, i.e.\ \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.
In contrast, the parallel and particularly the twisted orientations constitute energetically favorable configurations, in which a vast reduction of strain is enabled by combination of these defects.
\begin{figure}[tp]
The two \ci{} atoms form a strong C-C bond, which is responsible for the large gain in energy resulting in a binding energy of \unit[-2.39]{eV}.
This bond has a length of \unit[1.38]{\AA} close to the next neighbor distance in diamond or graphite, which is approximately \unit[1.54]{\AA}.
The minimum of the binding energy observed for this configuration suggests preferred C clustering as a competing mechanism to the \ci{} DB interstitial agglomeration inevitable for the SiC precipitation.
-However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e. two times more often than the ground-state configuration, within the systematically investigated configuration space.
+However, the second most favorable configuration ($E_{\text{f}}=-2.25\,\text{eV}$) is represented four times, i.e.\ two times more often than the ground-state configuration, within the systematically investigated configuration space.
Thus, particularly at high temperatures that cause an increase of the entropic contribution, this structure constitutes a serious opponent to the ground state.
In fact, following results on migration simulations will reinforce the assumption of a low probability for C clustering by thermally activated processes.
The energetically most unfavorable configuration ($E_{\text{b}}=0.26\,\text{eV}$) is obtained for the \ci{} \hkl[0 0 1] DB, which is oppositely orientated with respect to the initial one.
A DB taking the same orientation as the initial one is less unfavorable ($E_{\text{b}}=0.04\,\text{eV}$).
Both configurations are unfavorable compared to far-off, isolated DBs.
-Nonparallel orientations, i.e. the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
+Nonparallel orientations, i.e.\ the \hkl[0 1 0], \hkl[0 -1 0] and its equivalents, result in binding energies of \unit[-0.12]{eV} and \unit[-0.27]{eV}, thus, constituting energetically favorable configurations.
The reduction of strain energy is higher in the second case, where the C atom of the second DB is placed in the direction pointing away from the initial C atom.
\begin{figure}[tp]
\label{fig:defects:comb_db110}
\end{figure}
The binding energy of these configurations with respect to the C-C distance is plotted in Fig.~\ref{fig:defects:comb_db110}.
-The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e. the ground-state configuration.
+The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the \ci{} DBs and saturates for the smallest possible separation, i.e.\ the ground-state configuration.
The ground-state configuration was ignored in the fitting process.
Not considering the previously mentioned elevated barriers for migration, an attractive interaction between the \ci{} \hkl<1 0 0> DB defects indeed is detected with a capture radius that clearly exceeds \unit[1]{nm}.
The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.
\label{fig:093-095}
\end{figure}
Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.
-By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
+By a single bond switch, i.e.\ the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
This configuration has been identified and described by spectroscopic experimental techniques \cite{song90_2} as well as theoretical studies \cite{leary97,capaz98}.
Configuration B is found to constitute the energetically slightly more favorable configuration.
However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.
Configuration $\alpha$ is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.
Nevertheless, the C and Si DB atoms remain threefold coordinated.
Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198--0.209]{nm}/\unit[0.189]{nm}).
-Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
+Again a single bond switch, i.e.\ the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.
The two C atoms form a \hkl[1 0 0] DB sharing the initial C$_{\text{s}}$ lattice site while the initial Si DB atom occupies its previously regular lattice site.
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground-state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.
This finding is in good agreement with a combined {\em ab initio} and experimental study of Liu et~al.~\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.
Configurations $\alpha$, A and B are not affected by spin polarization and show zero magnetization.
Mattoni et~al.~\cite{mattoni2002}, in contrast, find configuration $\beta$ less favorable than configuration A by \unit[0.2]{eV}.
Next to differences in the XC functional and plane-wave energy cut-off, this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.
-Indeed, investigating the migration path from configurations $\alpha$ to $\beta$ and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration $\beta$, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
+Indeed, investigating the migration path from configurations $\alpha$ to $\beta$ and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e.\ configuration $\beta$, is obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.
Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.
-However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.
+However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e.\ global, minimum with respect to electronic minimization.
%
% a b transition
A low activation energy of \unit[0.1]{eV} is observed for the a$\rightarrow$b transition.
Thus, configuration a is very unlikely to occur in favor of configuration b.
% repulsive along 110
-A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e. positions 1 (configuration a) and 5.
+A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0], i.e.\ positions 1 (configuration a) and 5.
This is due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom residing within the \hkl[1 1 0] bond chain.
This finding agrees well with results by Mattoni et~al.~\cite{mattoni2002}.
% all other investigated results: attractive interaction. stress compensation.
This structure is followed by C$_{\text{s}}$ located at position 2, the lattice site of one of the neighbor atoms below the two Si atoms that are bound to the C$_{\text{i}}$ DB atom.
As mentioned earlier, these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.
The latter is partially compensated by the C$_{\text{s}}$ atom.
-Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
+Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e.\ each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].
\begin{figure}[tp]
\begin{center}
\subfigure[\underline{$E_{\text{b}}=-0.51\,\text{eV}$}]{\label{fig:defects:051}\includegraphics[width=0.25\textwidth]{00-1dc/0-51.eps}}
The energetically most favorable configuration (configuration $\beta$) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.
Again, conclusions concerning the probability of formation are drawn by investigating respective migration paths.
Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.
-Pathways starting from the next most favored configuration, i.e. \cs{} located at position 2, into configuration $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
+Pathways starting from the next most favored configuration, i.e.\ \cs{} located at position 2, into configuration $\alpha$ and $\beta$ are investigated, which show activation energies above \unit[2.2]{eV} and \unit[2.5]{eV}.
The respective barriers and structures are displayed in Fig.~\ref{fig:051-xxx}.
For the transition into configuration $\beta$, as before, the non-magnetic configuration is obtained.
If not forced by the CRT algorithm, the structures beyond \perc{50} and below \perc{90} displacement of the transition approaching configuration $\alpha$ would settle into configuration $\beta$.
Strain reduced by this huge displacement is partially absorbed by tensile strain on Si atom number 1 originating from attractive forces of the C atom and the vacancy.
A binding energy of \unit[-0.50]{eV} is observed.
-The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e. the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
+The migration pathways of configuration \ref{fig:defects:314} and \ref{fig:defects:059} into the ground-state configuration, i.e.\ the \cs{} configuration, are shown in Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{314-539.ps}
\end{figure}
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground-state configuration.
Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.
-However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
+However, only \unit[0.77]{eV} are needed for the reverse process, i.e.\ the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.
Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.
\begin{figure}[tp]
\end{figure}
Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.
The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.
-Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance.
+Unable to model possible positive values of the binding energy, i.e.\ unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought of as a guide for the eye describing the decrease of the interaction strength, i.e.\ the absolute value of the binding energy, with increasing separation distance.
The binding energy quickly drops to zero.
The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}.
indicating a low interaction capture radius of the defect pair.
-%As can be seen, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
+%As can be seen, the interaction strength, i.e.\ the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
%Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS, highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.
For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.
However, this configuration is unstable involving a structural transition into the C$_{\text{i}}$ \hkl<1 1 0> DB interstitial, thus, not maintaining the tetrahedral Si nor the \cs{} defect.
Thus, the underestimated energy of formation of C$_{\text{s}}$ within the EA calculation does not pose a serious limitation in the present context.
-Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e. the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
+Since C is introduced into a perfect Si crystal and the number of particles is conserved in simulation, the creation of C$_{\text{s}}$ is accompanied by the creation of Si$_{\text{i}}$, which is energetically less favorable than the ground state, i.e.\ the C$_{\text{i}}$ \hkl<1 0 0> DB configuration, for both, the EA and {\em ab initio} treatment.
In either case, no configuration more favorable than the C$_{\text{i}}$ \hkl<1 0 0> DB has been found.
Thus, a proper description with respect to the relative energies of formation is assumed for the EA potential.
\ifnum1=0
Obtained results for separated point defects in Si are in good agreement to previous theoretical work on this subject, both for intrinsic defects \cite{leung99,al-mushadani03} as well as for C point defects \cite{dal_pino93,capaz94}.
-The ground-state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
+The ground-state configurations of these defects, i.e.\ the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, are reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$ \cite{leung99,al-mushadani03} as well as theoretical \cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental \cite{watkins76,song90} studies on C$_{\text{i}}$.
A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.~\cite{capaz94} to experimental values \cite{song90,lindner06,tipping87} ranging from \unit[0.70--0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si
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.
These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.
% which is elaborated in more detail within the comprehensive description in chapter~\ref{chapter:summary}.
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.
-Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.
+Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e.\ a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.
In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence.
In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.
Thus, the simulation is continued without adding more C atoms until the system temperature is equal to the chosen temperature again.
This is realized by the thermostat, which decouples excessive energy.
Every inserted C atom must exhibit a distance greater or equal to \unit[1.5]{\AA} to neighbored atoms to prevent the occurrence of too high forces.
-Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e. \unit[20]{$^{\circ}$C}, is reached.
+Once the total amount of C is inserted, the simulation is continued for \unit[100]{ps} followed by a cooling-down process until room temperature, i.e.\ \unit[20]{$^{\circ}$C}, is reached.
Fig.~\ref{fig:md:prec_fc} displays a flow chart of the applied steps involved in the simulation sequence.
\begin{figure}[tp]
\begin{center}
In addition the abrupt increase of Si pairs at \unit[0.29]{nm} can be attributed to the Si-Si cut-off radius of \unit[0.296]{nm} as used in the present bond order potential.
The cut-off function causes artificial forces pushing the Si atoms out of the cut-off region.
Without the abrupt increase, a maximum around \unit[0.31]{nm} gets even more conceivable.
-Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbor distance for low C concentrations, i.e. for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
+Analyses of randomly chosen configurations, in which distances around \unit[0.3]{nm} appear, identify \ci{} \hkl<1 0 0> DBs to be responsible for stretching the Si-Si next neighbor distance for low C concentrations, i.e.\ for the $V_1$ and early stages of $V_2$ and $V_3$ simulation runs.
This excellently agrees with the calculated value $r(13)$ in Table~\ref{tab:defects:100db_cmp} for a resulting Si-Si distance in the \ci \hkl<1 0 0> DB configuration.
\begin{figure}[tp]
Not only the peak locations but also the peak widths and heights become comprehensible.
The distinct peak at \unit[0.26]{nm}, which exactly matches the cut-off radius of the Si-C interaction, is again a potential artifact.
-For high C concentrations, i.e. the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
+For high C concentrations, i.e.\ the $V_2$ and $V_3$ simulation corresponding to a C density of about 8 atoms per c-Si unit cell, the defect concentration is likewise increased and a considerable amount of damage is introduced in the insertion volume.
The consequential superposition of these defects and the high amounts of damage generate new displacement arrangements for the C-C as well as for the Si-C pair distances, which become hard to categorize and trace and obviously lead to a broader distribution.
-Short range order indeed is observed, i.e. the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
+Short range order indeed is observed, i.e.\ the large amount of strong neighbored C-C bonds at \unit[0.15]{nm} as expected in graphite or diamond and Si-C bonds at \unit[0.19]{nm} as expected in SiC, but only hardly visible is the long range order.
This indicates the formation of an amorphous SiC-like phase.
In fact the resulting Si-C and C-C radial distribution functions compare quite well with these obtained by cascade amorphized and melt-quenched amorphous SiC using a modified Tersoff potential \cite{gao02}.
-In both cases, i.e. low and high C concentrations, the formation of 3C-SiC fails to appear.
+In both cases, i.e.\ low and high C concentrations, the formation of 3C-SiC fails to appear.
With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
However, sufficient defect agglomeration is not observed.
For high C concentrations, a rearrangement of the amorphous SiC structure, which is not expected at prevailing temperatures, and a transition into 3C-SiC is not observed either.
Vibrations of the covalent bond take place on the order of \unit[10$^{-14}$]{s}, of which the thermodynamic and kinetic properties are well described by MD simulations.
To avoid discretization errors, the integration time step needs to be chosen smaller than the fastest vibrational frequency in the system.
On the other hand, infrequent processes, such as conformational changes, reorganization processes during film growth, defect diffusion and phase transitions are processes undergoing long-term evolution in the range of microseconds.
-This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e. the system temperature.
+This is due to the existence of several local minima in the free energy surface separated by large energy barriers compared to the kinetic energy of the particles, i.e.\ the system temperature.
Thus, the average time of a transition from one potential basin to another corresponds to a great deal of vibrational periods, which in turn determine the integration time step.
Hence, time scales covering the necessary amount of infrequent events to observe long-term evolution are not accessible by traditional MD simulations, which are limited to the order of nanoseconds.
New methods have been developed to bypass the time scale problem.
As for low temperatures, order in the short range exists decreasing with increasing separation.
The increase of the amount of Si-C pairs at \distn{0.186} could be positively interpreted since this type of bond also exists in 3C-SiC.
On the other hand, the amount of next neighbored C atoms with a distance of approximately \distn{0.15}, which is the distance of C in graphite or diamond, is likewise increased.
-Thus, higher temperatures seem to additionally enhance a conflictive process, i.e. the formation of C agglomerates, obviously inconsistent with the desired process of 3C-SiC formation.
+Thus, higher temperatures seem to additionally enhance a conflictive process, i.e.\ the formation of C agglomerates, obviously inconsistent with the desired process of 3C-SiC formation.
This is supported by the C-C peak at \distn{0.252}, which corresponds to the second next neighbor distance in the diamond structure of elemental C.
Investigating the atomic data indeed reveals two C atoms, which are bound to and interconnected by a third C atom, to be responsible for this distance.
The C-C peak at about \distn{0.31}, which is slightly shifted to higher distances (\distn{0.317}) with increasing temperature still corresponds quite well to the next neighbor distance of C in 3C-SiC as well as a-SiC and indeed results from C-Si-C bonds.
\label{fig:sic:unit_cell}
\end{figure}
Its unit cell is shown in Fig.~\ref{fig:sic:unit_cell}.
-3C-SiC grows in zincblende structure, i.e. it is composed of two face-centered cubic (fcc) lattices, which are displaced by one quarter of the volume diagonal as in Si.
+3C-SiC grows in zincblende structure, i.e.\ it is composed of two face-centered cubic (fcc) lattices, which are displaced by one quarter of the volume diagonal as in Si.
However, in 3C-SiC, one of the fcc lattices is occupied by Si atoms while the other one is occupied by C atoms.
-Its lattice constant of \unit[0.436]{nm} compared to \unit[0.543]{nm} from that of Si results in a lattice mismatch of almost \unit[20]{\%}, i.e. four lattice constants of Si approximately match five SiC lattice constants.
-Thus, the Si density of SiC is only slightly lower, i.e. \unit[97]{\%} of plain Si.
+Its lattice constant of \unit[0.436]{nm} compared to \unit[0.543]{nm} from that of Si results in a lattice mismatch of almost \unit[20]{\%}, i.e.\ four lattice constants of Si approximately match five SiC lattice constants.
+Thus, the Si density of SiC is only slightly lower, i.e.\ \unit[97]{\%} of plain Si.
\section{Fabrication of silicon carbide}
%In the so called step-controlled epitaxy, lateral growth proceeds from atomic steps without the necessity of preceding nucleation events.
Investigations indicate that in the so-called step-controlled epitaxy, crystal growth proceeds through the adsorption of Si species at atomic steps and their carbonization by hydrocarbon molecules.
This growth mechanism does not require two-dimensional nucleation.
-Instead, crystal growth is governed by mass transport, i.e. the diffusion of reactants in a stagnant layer.
+Instead, crystal growth is governed by mass transport, i.e.\ the diffusion of reactants in a stagnant layer.
In contrast, layers of the 3C polytype are formed on exactly oriented \hkl(0 0 0 1) 6H-SiC substrates by two-dimensional nucleation on terraces.
These films show a high density of double positioning boundary (DPB) defects, which is a special type of twin boundary arising at the interface of regions that occupy one of the two possible orientations of the hexagonal stacking sequence, which are rotated by \unit[60]{$^{\circ}$} relative to each other, respectively.
However, lateral 3C-SiC growth was also observed on low tilt angle off-axis substrates originating from intentionally induced dislocations \cite{powell91}.
Additionally, 6H-SiC was observed on clean substrates even for a tilt angle as low as \unit[0.1]{$^{\circ}$} due to low surface mobilities that facilitate arriving molecules to reach surface steps.
Thus, 3C nucleation is assumed as a result of migrating Si and C containing molecules interacting with surface disturbances by a yet unknown mechanism, in contrast to a model \cite{ueda90}, in which the competing 6H versus 3C growth depends on the density of surface steps.
-Combining the fact of a well defined 3C lateral growth direction, i.e. the tilt direction, and an intentionally induced dislocation enables the controlled growth of a 3C-SiC film mostly free of DPBs \cite{powell91}.
+Combining the fact of a well defined 3C lateral growth direction, i.e.\ the tilt direction, and an intentionally induced dislocation enables the controlled growth of a 3C-SiC film mostly free of DPBs \cite{powell91}.
Lower growth temperatures, a clean growth ambient, in situ control of the growth process, layer-by-layer deposition and the possibility to achieve dopant profiles within atomic dimensions due to the reduced diffusion at low growth temperatures reveal MBE as a promising technique to produce SiC epitaxial layers.
Using alternating supply of the gas beams Si$_2$H$_6$ and C$_2$H$_2$ in GSMBE, 3C-SiC epilayers were obtained on 6H-SiC substrates at temperatures between \unit[850]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C} \cite{yoshinobu92}.
Although tremendous progress has been achieved in the above-mentioned growth methods during the last decades, available wafer dimensions and crystal qualities are not yet satisfactory.
Thus, alternative approaches to fabricate SiC have been explored.
-The ion beam synthesis (IBS) technique, i.e. high-dose ion implantation followed by a high-temperature annealing step, turned out to constitute a promising method to directly form compound layers of high purity and accurately controllable depth and stoichiometry.
+The ion beam synthesis (IBS) technique, i.e.\ high-dose ion implantation followed by a high-temperature annealing step, turned out to constitute a promising method to directly form compound layers of high purity and accurately controllable depth and stoichiometry.
A short chronological summary of the IBS of SiC and its origins is presented in the following.
High-dose carbon implantation into crystalline silicon (c-Si) with subsequent or in situ annealing was found to result in SiC microcrystallites in Si \cite{borders71}.
Instead defect annihilation is observed and the C-rich surface layer of the room temperature implant turns into a layer consisting of SiC precipitates, which, however, are not aligned with the Si matrix indicating a mechanism different to the one of the direct formation for the high-temperature implantation.
Based on these findings, a recipe was developed to form buried layers of single-crystalline SiC featuring an improved interface and crystallinity \cite{lindner99,lindner01,lindner02}.
-Therefore, the dose must not exceed the stoichiometry dose, i.e. the dose corresponding to \unit[50]{at.\%} C concentration at the implantation peak.
+Therefore, the dose must not exceed the stoichiometry dose, i.e.\ the dose corresponding to \unit[50]{at.\%} C concentration at the implantation peak.
Otherwise clusters of C are formed, which cannot be dissolved during post-implantation annealing at moderate temperatures below the Si melting point \cite{lindner96,calcagno96}.
Annealing should be performed for \unit[5--10]{h} at \unit[1250]{$^{\circ}$C} to enable the redistribution from the as-implanted Gaussian into a box-like C depth profile \cite{lindner95}.
The implantation temperature constitutes the most critical parameter, which is responsible for the structure after implantation and, thus, the starting point for subsequent annealing steps.
\end{equation}
The barrier of diffusion of substitutional C has been determined to be around \unit[3]{eV} \cite{newman61}.
-However, as suspected due to the substitutional position, the diffusion of C requires intrinsic point defects, i.e. Si self-interstitials and vacancies.
+However, as suspected due to the substitutional position, the diffusion of C requires intrinsic point defects, i.e.\ Si self-interstitials and vacancies.
Similar to phosphorous and boron, which exclusively use self-interstitials as a diffusion vehicle, the diffusion of C atoms is expected to obey the same mechanism.
Indeed, enhanced C diffusion was observed in the presence of self-interstitial supersaturation \cite{kalejs84} indicating an appreciable diffusion component involving self-interstitials and only a negligible contribution by vacancies.
Substitutional C and interstitial Si react into a C-Si complex forming a dumbbell structure oriented along a crystallographic \hkl<1 0 0> direction on a regular Si lattice site.
% lattice location of implanted carbon
Radiation damage introduced during implantation and a high concentration of the implanted species, which results in the reduction of the topological constraint of the host lattice imposed on the implanted species, can affect the manner of impurity incorporation.
-The probability of finding C, which will be most stable at sites for which the number of neighbors equals the natural valence, i.e. substitutionally on a regular Si site of a perfect lattice, is, thus, reduced at substitutional lattice sites and likewise increased at interstitial sites.
+The probability of finding C, which will be most stable at sites for which the number of neighbors equals the natural valence, i.e.\ substitutionally on a regular Si site of a perfect lattice, is, thus, reduced at substitutional lattice sites and likewise increased at interstitial sites.
Indeed, x-ray rocking curves reveal a positive lattice strain, which is decreased but still remains with increasing annealing temperature, indicating the location of the majority of implanted C atoms at interstitial sites \cite{isomae93}.
Due to the absence of dislocations in the implanted region interstitial C is assumed to prevent clustering of implantation-induced Si self-interstitials by agglomeration of C-Si interstitials or the formation of SiC precipitates accompanied by a relaxation of the lattice strain.
Thus, the same path explains the experimentally determined activation energies for reorientation of the DB ranging from \unit[0.77]{eV} \cite{watkins76} up to \unit[0.88]{eV} \cite{song90}.
Investigations based on the EA bond order potential suggest a migration involving an intermediate \ci{} \hkl<1 1 0> DB configuration.
Although different, starting and final configuration as well as the change in orientation of the \hkl<1 0 0> DB are equal to the identified pathway by the {\em ab initio} calculations.
-However, barrier heights, which are overestimated by a factor of 2.4 to 3.5 depending on the character of migration, i.e. a single step or two step process, compared to the DFT results, are obtained.
+However, barrier heights, which are overestimated by a factor of 2.4 to 3.5 depending on the character of migration, i.e.\ a single step or two step process, compared to the DFT results, are obtained.
Obviously, the EA potential fails to describe \ci{} diffusion yielding a drastically overestimated activation energy, which has to be taken into account in subsequent investigations.
Subsequent investigations focus on defect combinations exclusively by the first-principles description.
Simulations at temperatures used in IBS result in structures dominated by the C$_{\text{i}}$ \hkl<1 0 0> DB and its combinations if C is inserted into the total volume.
Incorporation into volumes $V_2$ and $V_3$, which correspond to the volume of the expected precipitate and the volume containing the necessary amount of Si, lead to an amorphous SiC-like structure within the respective volume.
Both results are not expected with respect to the outcome of the IBS experiments.
-In the first case, i.e. the low C concentration simulations, \ci{} \hkl<1 0 0> DBs are indeed formed.
+In the first case, i.e.\ the low C concentration simulations, \ci{} \hkl<1 0 0> DBs are indeed formed.
However, sufficient defect agglomeration is not observed.
-In the second case, i.e. the high C concentration simulations, crystallization of the amorphous structure, which is not expected at prevailing temperatures, is likewise not observed.
+In the second case, i.e.\ the high C concentration simulations, crystallization of the amorphous structure, which is not expected at prevailing temperatures, is likewise not observed.
Limitations of the MD technique in addition to overestimated bond strengths due to the short range potential are identified to be responsible.
The approach of using increased temperatures during C insertion is followed to work around this problem termed {\em potential enhanced slow phase space propagation}.
Thus, agglomeration of \ci{} in the absence of C clustering is expected.
These initial results suggest a conversion mechanism based on the agglomeration of \ci{} defects followed by a sudden precipitation once a critical size is reached.
However, subsequent investigations of structures that are particularly conceivable under conditions prevalent in IBS and at elevated temperatures show \cs{} to occur in all probability.
-The transition from the ground state of a single C atom incorporated into otherwise perfect c-Si, i.e. the \ci{} \hkl<1 0 0> DB, into a configuration of \cs{} next to a \si{} atom exhibits an activation energy lower than the one for the diffusion of the highly mobile \ci{} defect.
+The transition from the ground state of a single C atom incorporated into otherwise perfect c-Si, i.e.\ the \ci{} \hkl<1 0 0> DB, into a configuration of \cs{} next to a \si{} atom exhibits an activation energy lower than the one for the diffusion of the highly mobile \ci{} defect.
Considering additionally the likewise lower diffusion barrier of \si{}, configurations of separated \cs{} and \si{} will occur in all probability.
This is reinforced by the {\em ab initio} MD run at non-zero temperature, which shows structures of separating instead of recombining \cs{} and \si{} defects.
This suggests increased participation of \cs{} already in the initial stages of the implantation process.
-The highly mobile \si{} is assumed to constitute a vehicle for the rearrangement of other \cs{} atoms onto proper lattice sites, i.e. lattice sites of one of the the two fcc lattices composing the diamond structure.
+The highly mobile \si{} is assumed to constitute a vehicle for the rearrangement of other \cs{} atoms onto proper lattice sites, i.e.\ lattice sites of one of the the two fcc lattices composing the diamond structure.
This way, stretched SiC structures, which are coherently aligned to the c-Si host, are formed by agglomeration of \cs.
Precipitation into an incoherent and partially strain-compensated SiC nucleus occurs once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
As already assumed by Nejim~et~al.~\cite{nejim95}, \si{} serves as supply for subsequently implanted C atoms to form further SiC in the resulting free space due to the accompanied volume reduction.
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