In the following the simulation methods used within the scope of this study are introduced.
Enabling the investigation of the evolution of structure on the atomic scale, molecular dynamics (MD) simulations are chosen for modeling the behavior and precipitation of C introduced into an initially crystalline Si environment.
To be able to model systems with a large amount of atoms computational efficient classical potentials to describe the interaction of the atoms are most often used in MD studies.
-For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential~\cite{albe_sic_pot} an appropriate MD code called \textsc{posic}\footnote{\textsc{posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
+For reasons of flexibility in executing this non-standard task and in order to be able to use a novel interaction potential~\cite{albe_sic_pot}, an appropriate MD code called \textsc{posic}\footnote{\textsc{posic} is an abbreviation for {\bf p}recipitation {\bf o}f {\bf SiC}} including a library collecting respective MD subroutines was developed from scratch\footnote{Source code: http://www.physik.uni-augsburg.de/\~{}zirkelfr/posic}.
The basic ideas of MD in general and the adopted techniques as implemented in \textsc{posic} in particular are outlined in section~\ref{section:md}, while the functional form and derivative of the employed classical potential is presented in appendix~\ref{app:d_tersoff}.
An overview of the most important tools within the MD package is given in appendix~\ref{app:code}.
Although classical potentials are often most successful and at the same time computationally efficient in calculating some physical properties of a particular system, not all of its properties might be described correctly due to the lack of quantum-mechanical effects.
\subsection{Statistical ensembles}
\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).
+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.
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.
\end{equation}
where $\beta$ is the isothermal compressibility and $p$ corresponds to the current pressure, which is determined by equation \eqref{eq:basics:ps}.
-Using this method the system does not behave like a true $NpT$ ensemble.
+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 $p$.
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.
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}.
-Using PPs the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
+Using PPs, the rapid oscillations of the wave functions near the core of the atoms are removed considerably reducing the number of plane waves necessary to appropriately expand the wave functions.
More importantly, less accuracy is required compared to all-electron calculations to determine energy differences among ionic configurations, which almost totally appear in the energy of the valence electrons that are typically a factor $10^3$ smaller than the energy of the core electrons.
\subsection{Brillouin zone sampling}
\section{Silicon self-interstitials}
-For investigating the \si{} structures a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+For investigating the \si{} structures, a Si atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
The formation energies of \si{} configurations are listed in Table~\ref{tab:defects:si_self} for both methods used in this work as well as results obtained by other {\em ab initio} studies~\cite{al-mushadani03,leung99}.
\bibpunct{}{}{,}{n}{}{}
\begin{table}[tp]
\subsection{Defect structures in a nutshell}
-For investigating the \ci{} structures a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
+For investigating the \ci{} structures, a C atom is inserted or removed according to Fig.~\ref{fig:basics:ins_pos} of section~\ref{section:basics:defects}.
Formation energies of the most common C point defects in crystalline Si are summarized in Table~\ref{tab:defects:c_ints}.
The relaxed configurations are visualized in Fig.~\ref{fig:defects:c_conf}.
Again, the displayed structures are the results obtained by the classical potential calculations.
Intermediate configurations within the investigated turbulent pathway identify barrier heights of more than \unit[4]{eV} resulting in a low probability for the transition.
The high activation energy is attributed to the stability of such a low energy configuration, in which the C atom of the second DB is located close to the initial DB.
Due to an effective stress compensation realized in the respective low energy configuration, which will necessarily be lost during migration, a high energy configuration needs to get passed, which is responsible for the high barrier.
-Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g. the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
+Low barriers are only identified for transitions starting from energetically less favorable configurations, e.g.\ the configuration of a \hkl[-1 0 0] DB located at position 2 (\unit[-0.36]{eV}).
Starting from this configuration, an activation energy of only \unit[1.2]{eV} is necessary for the transition into the ground state configuration.
The corresponding migration energies and atomic configurations are displayed in Fig.~\ref{fig:036-239}.
\begin{figure}[tp]
The migration path is best described by the reverse process.
Starting at \unit[100]{\%}, energy is needed to break the bonds of Si atom 1 to its neighbored Si atoms as well as the bond of the C atom to Si atom number 5.
At \unit[50]{\%} displacement, these bonds are broken.
-Due to this, and due to the formation of new bonds, e.g. the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
+Due to this, and due to the formation of new bonds, e.g.\ the bond of Si atom number 1 to Si atom number 5, a less steep increase of configurational energy is observed.
In a last step, the just recently formed bond of Si atom number 1 to Si atom number 5 is broken up again as well as the bond of the initial Si DB atom and its Si neighbor in \hkl[-1 -1 -1] direction, which explains the repeated boost in energy.
Finally, the system gains some configurational energy by relaxation into the configuration corresponding to \unit[0]{\%} displacement.
%
The high breakdown field of SiC compared to Si allows the blocking voltage region of a device to be designed roughly 10 times thinner and 10 times heavier doped, resulting in a decrease of the blocking region resistance by a factor of 100 and a much faster switching behavior.
Thus, rectifier diodes and switching transistors with higher switching frequencies and much greater efficiencies can be realized and exploited in highly efficient power converters.
Therefore, SiC constitutes a promising candidate to become the key technology towards an extensive development and use of regenerative energies and electromobility.
-Beside the mentioned electrical capabilities the mechanical stability, which is almost as hard as diamond, and chemical inertness almost suggest SiC to be used in MEMS designs.
+Beside the mentioned electrical capabilities, the mechanical stability, which is almost as hard as diamond, and chemical inertness almost suggest SiC to be used in MEMS designs.
Among the different polytypes of SiC, the cubic phase shows a high electron mobility and the highest break down field as well as saturation drift velocity~\cite{neudeck95,wesch96}.
-In contrast to its hexagonal counterparts 3C-SiC exhibits isotropic mechanical and electronic properties.
-Additionally the smaller band gap is expected to be favorable concerning the interface state density in MOSFET devices fabricated on 3C-SiC~\cite{pensl00}.
+In contrast to its hexagonal counterparts, 3C-SiC exhibits isotropic mechanical and electronic properties.
+Additionally, the smaller band gap is expected to be favorable concerning the interface state density in MOSFET devices fabricated on 3C-SiC~\cite{pensl00}.
Thus, the cubic phase is most effective for highly efficient high-performance electronic devices.
\begin{figure}[t]
\begin{center}
Although the constituents of SiC are abundant and the compound is chemically and thermally stable, large deposits of SiC have never been found.
Due to the rarity, SiC is typically man-made.
The development of several methods was necessary to synthetically produce SiC crystals matching the needs of a respective application.
-The fact that natural SiC is almost only observed as individual presolar SiC stardust grains near craters of primitive meteorite impacts, already indicates the complexity involved in the synthesis process.
+The fact that natural SiC is almost only observed as individual presolar SiC stardust grains near craters of primitive meteorite impacts already indicates the complexity involved in the synthesis process.
The attractive properties and wide range of applications, however, have triggered extensive efforts to grow this material as a bulk crystal and as an epitaxial surface thin film.
In the following, the principal difficulties involved in the formation of crystalline SiC and the most recent achievements will be summarized.
Heating the hollow carbon cylinder to \unit[2500]{$^{\circ}$C} leads to sublimation of the material at the hot outer wall and diffusion through the porous graphite tube followed by an uncontrolled crystallization on the slightly cooler parts of the inner graphite cavity resulting in the formation of randomly sized, hexagonally shaped platelets, which exhibit a layered structure of various alpha (non-cubic) polytypes with equal \hkl{0001} orientation.
Subsequent research~\cite{tairov78,tairov81} resulted in the implementation of a seeded growth sublimation process wherein only one large crystal of a single polytype is grown.
-In the so called modified Lely or modified sublimation process nucleation occurs on a SiC seed crystal located at the top or bottom of a cylindrical growth cavity.
+In the so called modified Lely or modified sublimation process, nucleation occurs on a SiC seed crystal located at the top or bottom of a cylindrical growth cavity.
As in the Lely process, SiC sublimes at a temperature of \unit[2400]{$^{\circ}$C} from a polycrystalline source diffusing through a porous graphite retainer along carefully adjusted thermal and pressure gradients.
Controlled nucleation occurs on the SiC seed, which is held at approximately \unit[2200]{$^{\circ}$C}.
The growth process is commonly done in a high-purity argon atmosphere.
Although significant advances have been achieved in the field of SiC bulk crystal growth, a variety of problems remain.
The high temperatures required in PVT growth processes limit the range of materials used in the hot zones of the reactors, for which mainly graphite is used.
-The porous material constitutes a severe source of contamination, e.g. with the dopants N, B and Al, which is particularly effective at low temperatures due to the low growth rate.
+The porous material constitutes a severe source of contamination, e.g.\ with the dopants N, B and Al, which is particularly effective at low temperatures due to the low growth rate.
Since the vapor pressure of Si is much higher than that of C, a careful manipulation of the Si vapor content above the seed crystal is required.
Additionally, to preserve epitaxial growth conditions, graphitization of the seed crystal has to be avoided.
Avoiding defects constitutes a major difficulty.
Crystalline SiC layers have been grown by a large number of techniques on the surfaces of different substrates.
Most of the crystal growth processes are based on CVD, solid-source MBE (SSMBE) and gas-source MBE (GSMBE) on Si as well as SiC substrates.
In CVD as well as GSMBE, C and Si atoms are supplied by C containing gases like CH$_4$, C$_3$H$_8$, C$_2$H$_2$ or C$_2$H$_4$ and Si containing gases like SiH$_4$, Si$_2$H$_6$, SiH$_2$Cl$_2$, SiHCl$_3$ or SiCl$_4$ respectively.
-In the case of SSMBE atoms are provided by electron beam evaporation of graphite and solid Si or thermal evaporation of fullerenes.
+In the case of SSMBE, atoms are provided by electron beam evaporation of graphite and solid Si or thermal evaporation of fullerenes.
The following review will exclusively focus on CVD and MBE techniques.
The availability and reproducibility of Si substrates of controlled purity made it the first choice for SiC epitaxy.
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, 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}.
This is demonstrated by a shift in the IR absorption band and the disappearance of the C profile peak in RBS.
Implantations at different temperatures revealed a strong influence of the implantation temperature on the compound structure~\cite{edelman76}.
Temperatures below \unit[500]{$^{\circ}$C} result in amorphous layers, which are transformed into polycrystalline 3C-SiC after annealing at \unit[850]{$^{\circ}$C}.
-Otherwise single crystalline 3C-SiC is observed for temperatures above \unit[600]{$^{\circ}$C}.
+Otherwise, single crystalline 3C-SiC is observed for temperatures above \unit[600]{$^{\circ}$C}.
Annealing temperatures necessary for the onset of the amorphous to crystalline transition have been confirmed by further studies~\cite{kimura81,kimura82}.
Overstoichiometric doses result in the formation of clusters of C, which do not contribute to SiC formation during annealing up to \unit[1200]{$^{\circ}$C}~\cite{kimura82}.
The amount of formed SiC, however, increases with increasing implantation temperature.
Further studies revealed the possibility to form buried layers of SiC by IBS at moderate substrate and anneal temperatures~\cite{lindner95,lindner96}.
Different doses of C ions with an energy of \unit[180]{keV} were implanted at \unit[330--440]{$^{\circ}$C} and annealed at \unit[1200]{$^{\circ}$C} or \unit[1250]{$^{\circ}$C} for \unit[5--10]{h}.
For a critical dose, which was found to depend on the Si substrate orientation, the formation of a stoichiometric buried layer of SiC exhibiting a well-defined interface to the Si host matrix was observed.
-In case of overstoichiometric C concentrations the excess C is not redistributed.
+In case of overstoichiometric C concentrations, the excess C is not redistributed.
These investigations demonstrate the presence of an upper dose limit, which corresponds to a \unit[53]{at.\%} C concentration at the implantation peak, for the thermally induced redistribution of the C atoms from a Gaussian to a box-shaped depth profile upon annealing.
This is explained by the formation of strong graphitic C-C bonds for higher C concentrations~\cite{calcagno96}.
Increased temperatures exceeding the Si melting point are expected to be necessary for the dissociation of these C clusters.
% and extensive TEM investigations
, a recipe was developed to form buried layers of single-crystalline SiC featuring an improved interface and crystallinity~\cite{lindner99,lindner01,lindner02}.
Therefor, 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}.
+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.
Implantations at \unit[400]{$^{\circ}$C} resulted in buried layers of SiC subdivided into a polycrystalline upper and an epitaxial lower part.
\section{Substoichiometric concentrations of carbon in crystalline silicon}
-In the following some basic properties of C in crystalline Si are reviewed.
+In the following, some basic properties of C in crystalline Si are reviewed.
A lot of work has been done contributing to the understanding of C in Si either as an isovalent impurity as well as at concentrations exceeding the solid solubility limit.
A comprehensive survey on C-mediated effects in Si has been published by Skorupa and Yankov~\cite{skorupa96}.
\subsection{Carbon as an impurity in silicon}
Below the solid solubility, C impurities mainly occupy substitutionally Si lattice sites in Si~\cite{newman65}.
-Due to the much smaller covalent radius of C compared to Si every incorporated C atom leads to a decrease in the lattice constant corresponding to a lattice contraction of about one atomic volume~\cite{baker68}.
+Due to the much smaller covalent radius of C compared to Si, every incorporated C atom leads to a decrease in the lattice constant corresponding to a lattice contraction of about one atomic volume~\cite{baker68}.
The induced strain is assumed to be responsible for the low solid solubility of C in Si, which was determined~\cite{bean71} to be
\begin{equation}
c_{\text{s}}=4\times10^{24}\,\text{cm$^{-3}$}
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.
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.
+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.
% link to strain engineering
However, there is great interest to incorporate C onto substitutional lattice sites, which results in a contraction of the Si lattice due to the smaller covalent radius of C compared to Si~\cite{baker68}, causing tensile strain, which is applied to the Si lattice.
The tensile strain induced by the C atoms is found to compensate the compressive strain present due to the Ge atoms.
Studies on the thermal stability of Si$_{1-y}$C$_y$/Si heterostructures formed in the same way and equal C concentrations showed a loss of substitutional C accompanied by strain relaxation for temperatures ranging from \unit[810--925]{$^{\circ}$C} and the formation of spherical 3C-SiC precipitates with diameters of \unit[2--4]{nm}, which are incoherent but aligned to the Si host~\cite{strane94}.
During the initial stages of precipitation C-rich clusters are assumed, which maintain coherency with the Si matrix and the associated biaxial strain.
-Using this technique a metastable solubility limit was achieved, which corresponds to a C concentration exceeding the solid solubility limit at the Si melting point by nearly three orders of magnitude and, furthermore, a reduction of the defect density near the metastable solubility limit is assumed if the regrowth temperature is increased by rapid thermal annealing~\cite{strane96}.
+Using this technique, a metastable solubility limit was achieved, which corresponds to a C concentration exceeding the solid solubility limit at the Si melting point by nearly three orders of magnitude and, furthermore, a reduction of the defect density near the metastable solubility limit is assumed if the regrowth temperature is increased by rapid thermal annealing~\cite{strane96}.
Since high temperatures used in the solid-phase epitaxial regrowth method promotes SiC precipitation, other groups realized substitutional C incorporation for strained Si$_{1-y}$C$_y$/Si heterostructures~\cite{iyer92,fischer95,powell93,osten96,osten99,laveant2002} or partially to fully strain-compensated (even inversely distorted~\cite{osten94_2}) Si$_{1-x-y}$Ge$_x$C${_y}$ layers on Si~\cite{eberl92,powell93_2,osten94,dietrich94} by MBE.
Investigations reveal a strong temperature-dependence of the amount of substitutionally incorporated C, which is increased for decreasing temperature accompanied by deterioration of the crystal quality~\cite{osten96,osten99}.
While not being compatible to very-large-scale integration technology, C concentrations of \unit[2]{\%} and more have been realized~\cite{laveant2002}.
\section{Assumed silicon carbide conversion mechanisms}
\label{section:assumed_prec}
-Although high-quality films of single-crystalline 3C-SiC can be produced by means of IBS the precipitation mechanism in bulk Si is not yet fully understood.
+Although high-quality films of single-crystalline 3C-SiC can be produced by means of IBS, the precipitation mechanism in bulk Si is not yet fully understood.
Indeed, closely investigating the large amount of literature pulled up in the last two sections and a cautious combination of some of the findings reveals controversial ideas of SiC formation, which are reviewed in more detail in the following.
High resolution transmission electron microscopy (HREM) investigations of C-implanted Si at room temperature followed by rapid thermal annealing (RTA) indicate the formation of C-Si dumbbell agglomerates, which are stable up to annealing temperatures of about \unit[700--800]{$^{\circ}$C}, and a transformation into 3C-SiC precipitates at higher temperatures~\cite{werner96,werner97}.
The precipitation is manifested by the disappearance of the dark contrasts in favor of Moir\'e patterns (Fig.~\ref{fig:sic:hrem:sic}) due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and the Si host.
The insignificantly lower Si density of SiC of approximately \unit[3]{\%} compared to c-Si results in the emission of only a few excess Si atoms.
The same mechanism was identified by high resolution x-ray diffraction~\cite{eichhorn99}.
-For implantation temperatures of \unit[500]{$^{\circ}$C} C-Si dumbbells agglomerate in an initial stage followed by the additional appearance of aligned SiC precipitates in a slightly expanded Si region with increasing dose.
+For implantation temperatures of \unit[500]{$^{\circ}$C}, C-Si dumbbells agglomerate in an initial stage followed by the additional appearance of aligned SiC precipitates in a slightly expanded Si region with increasing dose.
The precipitation mechanism based on a preceding dumbbell agglomeration as indicated by the above-mentioned experiments is schematically displayed in Fig.~\ref{fig:sic:db_agglom}.
\begin{figure}[t]
\begin{center}
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