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.
Thus, in order to obtain more accurate results quantum-mechanical calculations from first principles based on density functional theory (DFT) were performed.
-The Vienna {\em ab initio} simulation package (\textsc{vasp}) \cite{kresse96} is used for this purpose.
+The Vienna {\em ab initio} simulation package (\textsc{vasp})~\cite{kresse96} is used for this purpose.
The relevant basics of DFT are described in section \ref{section:dft} while an overview of utilities mainly used to create input or parse output data of \textsc{vasp} is given in appendix \ref{app:code}.
The gain in accuracy achieved by this method, however, is accompanied by an increase in computational effort constraining the simulated system to be much smaller in size.
Thus, investigations based on DFT are restricted to single defects or combinations of two defects in a rather small Si supercell, their structural relaxation as well as some selected diffusion processes.
\begin{quotation}
\dq We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.\dq{}
\begin{flushright}
-{\em Marquis Pierre Simon de Laplace, 1814.} \cite{laplace}
+{\em Marquis Pierre Simon de Laplace, 1814.}~\cite{laplace}
\end{flushright}
\end{quotation}
\subsection{Introduction to molecular dynamics simulations}
Molecular dynamics (MD) simulation is a technique to compute a system of particles, referred to as molecules, with their positions, velocities and forces among each other evolving in time.
-The MD method was first introduced by Alder and Wainwright in 1957 \cite{alder57,alder59} to study the interactions of hard spheres.
+The MD method was first introduced by Alder and Wainwright in 1957~\cite{alder57,alder59} to study the interactions of hard spheres.
The basis of the approach are Newton's equations of motion to describe classically the many-body system.
MD is the numerical way of solving the $N$-body problem which cannot be solved analytically for $N>3$.
A potential is necessary to describe the interaction of the particles.
\subsubsection{The Tersoff bond order potential}
-Tersoff proposed an empirical interatomic potential for covalent systems \cite{tersoff_si1,tersoff_si2}.
+Tersoff proposed an empirical interatomic potential for covalent systems~\cite{tersoff_si1,tersoff_si2}.
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}.
+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}.
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.
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.
-Based on pseudopotential theory, the bond order term $b_{ijk}$ limiting the attractive pair interaction is of the form $b_{ijk}\propto Z^{-\delta}$ where $Z$ is the coordination number and $\delta$ a constant \cite{abell85}, which is $\frac{1}{2}$ in the second-moment approximation within the tight binding scheme \cite{horsfield96}.
+Based on pseudopotential theory, the bond order term $b_{ijk}$ limiting the attractive pair interaction is of the form $b_{ijk}\propto Z^{-\delta}$ where $Z$ is the coordination number and $\delta$ a constant~\cite{abell85}, which is $\frac{1}{2}$ in the second-moment approximation within the tight binding scheme~\cite{horsfield96}.
Tersoff incorporated the concept of bond order in a three-body potential formalism.
The interatomic potential is taken to have the form
\end{figure}
The angular dependence does not give a fixed minimum angle between bonds since the expression is embedded inside the bond order term.
The relation to the above discussed bond order potential becomes obvious if $\chi=1, \beta=1, n=1, \omega=1$ and $c=0$.
-Parameters with a single subscript correspond to the parameters of the elemental system \cite{tersoff_si3,tersoff_c} while the mixed parameters are obtained by interpolation from the elemental parameters by the arithmetic or geometric mean.
+Parameters with a single subscript correspond to the parameters of the elemental system~\cite{tersoff_si3,tersoff_c} while the mixed parameters are obtained by interpolation from the elemental parameters by the arithmetic or geometric mean.
The elemental parameters were obtained by fit with respect to the cohesive energies of real and hypothetical bulk structures and the bulk modulus and bond length of the diamond structure.
New parameters for the mixed system are $\chi$, which is used to finetune the strength of heteropolar bonds, and $\omega$, which is set to one for the C-Si interaction but is available as a feature to permit the application of the potential to more drastically different types of atoms in the future.
Although the Tersoff potential is one of the most widely used potentials, there are some shortcomings.
Describing the Si-Si interaction Tersoff was unable to find a single parameter set to describe well both, bulk and surface properties.
-Due to this and since the first approach labeled T1 \cite{tersoff_si1} turned out to be unstable \cite{dodson87}, two further parametrizations exist, T2 \cite{tersoff_si2} and T3 \cite{tersoff_si3}.
+Due to this and since the first approach labeled T1~\cite{tersoff_si1} turned out to be unstable~\cite{dodson87}, two further parametrizations exist, T2~\cite{tersoff_si2} and T3~\cite{tersoff_si3}.
While T2 describes well surface properties, T3 yields improved elastic constants and should be used for describing bulk properties.
However, T3, which is used in the Si/C potential, suffers from an underestimation of the dimer binding energy.
Similar behavior is found for the C-C interaction.
-For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction \cite{albe_sic_pot}.
+For this reason, Erhart and Albe provide a reparametrization of the Tersoff potential based on three independently fitted parameter sets for the Si-Si, C-C and Si-C interaction~\cite{albe_sic_pot}.
The functional form is similar to the one proposed by Tersoff.
Differences in the energy functional and the force evaluation routine are pointed out in appendix \ref{app:d_tersoff}.
Concerning Si the elastic properties of the diamond phase as well as the structure and energetics of the dimer are reproduced very well.
The potential succeeds in the description of the low as well as high coordinated structures.
The description of elastic properties of SiC is improved with respect to the potentials available in literature.
Defect properties are only fairly reproduced but the description is comparable to previously published potentials.
-It is claimed that the potential enables modeling of widely different configurations and transitions among these and has recently been used to simulate the inert gas condensation of Si-C nanoparticles \cite{erhart04}.
+It is claimed that the potential enables modeling of widely different configurations and transitions among these and has recently been used to simulate the inert gas condensation of Si-C nanoparticles~\cite{erhart04}.
Therefore, the Erhart/Albe (EA) potential is considered the superior analytical bond order potential to study the SiC precipitation and associated processes in Si.
\subsection{Verlet integration}
\label{subsection:integrate_algo}
-A numerical method to integrate Newton's equations of motion was presented by Verlet in 1967 \cite{verlet67}.
+A numerical method to integrate Newton's equations of motion was presented by Verlet in 1967~\cite{verlet67}.
The idea of the so-called Verlet and a variant, the velocity Verlet algorithm, which additionally generates directly the velocities, is explained in the following.
Starting point is the Taylor series for the particle positions at time $t+\delta t$ and $t-\delta t$
\begin{equation}
\section{Density functional theory}
\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.
+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.
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.
In contrast, instead of using the description by the many-body wave function, the key point in density functional theory (DFT) is to recast the problem to a description utilizing the charge density $n(\vec{r})$, which constitutes a quantity in real space depending only on the three spatial coordinates.
In the following sections the basic idea of DFT will be outlined.
-As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory \cite{thomas27,fermi27} and the self-consistent Hartree equations \cite{hartree28}.
-A nice review is given in the Nobel lecture of Kohn \cite{kohn99}, one of the inventors of DFT.
+As will be shown, DFT can formally be regarded as an exactification of the Thomas Fermi theory~\cite{thomas27,fermi27} and the self-consistent Hartree equations~\cite{hartree28}.
+A nice review is given in the Nobel lecture of Kohn~\cite{kohn99}, one of the inventors of DFT.
\subsection{Born-Oppenheimer approximation}
-Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom \cite{born27}.
+Born and Oppenheimer proposed a simplification enabling the effective decoupling of the electronic and ionic degrees of freedom~\cite{born27}.
Within the Born-Oppenheimer (BO) approximation the light electrons are assumed to move much faster and, thus, follow adiabatically to the motion of the heavy nuclei, if the latter are only slightly deflected from their equilibrium positions.
Thus, on the timescale of electronic motion the ions appear at fixed positions.
On the other way round, on the timescale of nuclear motion the electrons appear blurred in space adding an extra term to the ion-ion potential.
The answer to this question, whether the charge density completely characterizes a system, became the starting point of modern DFT.
Considering a system with a nondegenerate ground state there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
-In 1964 Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
+In 1964 Hohenberg and Kohn showed the opposite and far less obvious result~\cite{hohenberg64}.
Employing no more than the Rayleigh-Ritz minimal principle it is concluded by {\em reductio ad absurdum} that for a nondegenerate ground state the same charge density cannot be generated by different potentials.
Thus, the charge density of the ground state $n_0(\vec{r})$ uniquely determines the potential $V(\vec{r})$ and, consequently, the full Hamiltonian and ground-state energy $E_0$.
In mathematical terms the full many-electron ground state is a unique functional of the charge density.
In particular, $E$ is a functional $E[n(\vec{r})]$ of $n(\vec{r})$.
-The ground-state charge density $n_0(\vec{r})$ minimizes the energy functional $E[n(\vec{r})]$, its value corresponding to the ground-state energy $E_0$, which enables the formulation of a minimal principle in terms of the charge density \cite{hohenberg64,levy82}
+The ground-state charge density $n_0(\vec{r})$ minimizes the energy functional $E[n(\vec{r})]$, its value corresponding to the ground-state energy $E_0$, which enables the formulation of a minimal principle in terms of the charge density~\cite{hohenberg64,levy82}
\begin{equation}
E_0=\min_{n(\vec{r})}
\left\{
\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}
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{ ,}
\end{equation}
which is, thus, called local density approximation (LDA).
Here, the exchange-correlation energy per particle of the uniform electron gas of constant density $n$ is used for $\epsilon_{\text{xc}}(n(\vec{r}))$.
-Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods \cite{ceperley80} and corresponding parametrizations exist \cite{perdew81}.
+Although, even in such a simple case, no exact form of the correlation part of $\epsilon_{\text{xc}}(n)$ is known, highly accurate numerical estimates using Monte Carlo methods~\cite{ceperley80} and corresponding parametrizations exist~\cite{perdew81}.
Obviously exact for the homogeneous electron gas, the LDA was {\em a priori} expected to be useful only for densities varying slowly on scales of the local Fermi or TF wavelength.
Nevertheless, LDA turned out to be extremely successful in describing some properties of highly inhomogeneous systems accurately within a few percent.
Although LDA is known to overestimate the cohesive energy in solids by \unit[10--20]{\%}, the lattice parameters are typically determined with an astonishing accuracy of about \unit[1]{\%}.
-More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density \cite{kohn65}.
+More accurate approximations of the exchange-correlation functional are realized by the introduction of gradient corrections with respect to the density~\cite{kohn65}.
Respective considerations are based on the concept of an exchange-correlation hole density describing the depletion of the electron density in the vicinity of an electron.
-The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(\vec{r})]$ and an equivalent expression \cite{kohn96,kohn98}, which makes use of the electron density distribution $n(\tilde{\vec{r}})$ at positions $\tilde{\vec{r}}$ near $\vec{r}$, yielding the form
+The averaged hole density can be used to give a formally exact expression for $E_{\text{xc}}[n(\vec{r})]$ and an equivalent expression~\cite{kohn96,kohn98}, which makes use of the electron density distribution $n(\tilde{\vec{r}})$ at positions $\tilde{\vec{r}}$ near $\vec{r}$, yielding the form
\begin{equation}
E_{\text{xc}}[n(\vec{r})]=\int\epsilon_{\text{xc}}(\vec{r};[n(\tilde{\vec{r}})])n(\vec{r}) d\vec{r}
\end{equation}
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 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
\begin{equation}
It is worth to stress out one more time, that this is mainly due to the orthogonalization wiggles of the wave functions of valence electrons near the nuclei.
Thus, existing core states practically prevent the use of a PW basis set.
However, the core electrons, which are tightly bound to the nuclei, do not contribute significantly to chemical bonding or other physical properties of the solid.
-This fact is exploited in the pseudopotential (PP) approach \cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions.
+This fact is exploited in the pseudopotential (PP) approach~\cite{cohen70} by removing the core electrons and replacing the atom and the associated strong ionic potential by a pseudoatom and a weaker PP that acts on a set of pseudo wave functions rather than the true valance wave functions.
This way, the pseudo wave functions become smooth near the nuclei.
Most PPs satisfy four general conditions.
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.
-Pseudopotentials that meet the conditions outlined above are referred to as norm-conserving pseudopotentials \cite{hamann79}.
+Pseudopotentials that meet the conditions outlined above are referred to as norm-conserving pseudopotentials~\cite{hamann79}.
%Certain properties need to be fulfilled by PPs and the resulting pseudo wave functions.
%The pseudo wave functions should yield the same energy eigenvalues than the true valence wave functions.
%The PP is called norm-conserving if the pseudo and real charge contained within the core region matches.
%To guarantee transferability of the PP the logarithmic derivatives of the real and pseudo wave functions and their first energy derivatives need to agree outside of the core region.
-%A simple procedure was proposed to extract norm-conserving PPs obyeing the above-mentioned conditions from {\em ab initio} atomic calculations \cite{hamann79}.
+%A simple procedure was proposed to extract norm-conserving PPs obyeing the above-mentioned conditions from {\em ab initio} atomic calculations~\cite{hamann79}.
In order to achieve these properties different PPs are required for the different shapes of the orbitals, which are determined by their angular momentum.
Mathematically, a non-local PP, which depends on the angular momentum, has the form
\end{equation}
Applying the operator $V_{\text{nl}}(\vec{r})$ decomposes the electronic wave functions into spherical harmonics $| lm \rangle$, 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}.
+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.
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.
Following Bloch's theorem only a finite number of electronic wave functions need to be calculated for a periodic system.
However, to calculate quantities like the total energy or charge density, these have to be evaluated in a sum over an infinite number of $\vec{k}$ points.
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}.
+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.
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.
\vec{F}_i=-\sum_j \langle \Phi_j | \Phi_j\frac{\partial V}{\partial \vec{R}_i} \rangle
\text{ .}
\end{equation}
-This is called the Hellmann-Feynman theorem \cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.
+This is called the Hellmann-Feynman theorem~\cite{feynman39}, which enables the calculation of forces, called the Hellmann-Feynman forces, acting on the nuclei for a given configuration, without the need for evaluating computationally costly energy maps.
\section{Modeling of point defects}
\label{section:basics:defects}
\caption{Schematic of the constrained relaxation technique (a) and of a modified version (b) used to obtain migration pathways and corresponding configurational energies.}
\label{fig:basics:crt}
\end{figure}
-One possibility to compute the migration path from one stable configuration into another one is provided by the constrained relaxation technique (CRT) \cite{kaukonen98}.
+One possibility to compute the migration path from one stable configuration into another one is provided by the constrained relaxation technique (CRT)~\cite{kaukonen98}.
The atoms involving great structural changes in the diffusion process are moved stepwise from the starting to the final position and relaxation after each step is only allowed in the plane perpendicular to the direction of the vector connecting its starting and final position.
This is illustrated in Fig.~\ref{fig:basics:crto}.
The number of steps required for smooth transitions depends on the shape of the potential energy surface.
An interfacial energy of 2267.28 eV is obtained.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of 29.93 \AA.
Thus, the interface tension, given by the energy of the interface devided by the surface area of the precipitate is $20.15\,\frac{\text{eV}}{\text{nm}^2}$ or $3.23\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$.
-This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$ \cite{taylor93}.
+This is located inside the eperimentally estimated range of $2-8\times 10^{-4}\,\frac{\text{J}}{\text{cm}^2}$~\cite{taylor93}.
Since the precipitate configuration is artificially constructed the resulting interface does not necessarily correspond to the energetically most favorable configuration or to the configuration that is expected for an actually grown precipitate.
Thus annealing steps are appended to the gained structure in order to allow for a rearrangement of the atoms of the interface.
\section{Form of the Tersoff potential and its derivative}
-The Tersoff potential \cite{tersoff_m} is of the form
+The Tersoff potential~\cite{tersoff_m} is of the form
\begin{eqnarray}
E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\
V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .}
\subsection{Code realization}
-The implementation of the force evaluation shown in the following is applied to the potential designed by Erhart and Albe \cite{albe_sic_pot}.
+The implementation of the force evaluation shown in the following is applied to the potential designed by Erhart and Albe~\cite{albe_sic_pot}.
There are slight differences compared to the original potential by Tersoff:
\begin{itemize}
\item Difference in sign of the attractive part.
\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}.
-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}.
+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]
\begin{center}
\textsc{vasp} & 3.39 & 3.42 & 3.77 & 4.41 & 3.63 \\
\textsc{posic} & 4.39 & 4.48$^*$ & 3.40 & 5.42 & 3.13 \\
\multicolumn{6}{c}{Other {\em ab initio} studies} \\
-Ref. \cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
-Ref. \cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
+Ref.~\cite{al-mushadani03} & 3.40 & 3.45 & - & - & 3.53 \\
+Ref.~\cite{leung99} & 3.31 & 3.31 & 3.43 & - & - \\
\hline
\hline
\end{tabular}
This is nicely reproduced by the DFT calculations performed in this work.
It has turned out to be very difficult to capture the results of quantum-mechanical calculations in analytical potential models.
-Among the established analytical potentials only the environment-dependent interatomic potential (EDIP) \cite{bazant97,justo98} and Stillinger-Weber \cite{stillinger85} potential reproduce the correct order in energy of the defects.
+Among the established analytical potentials only the environment-dependent interatomic potential (EDIP)~\cite{bazant97,justo98} and Stillinger-Weber~\cite{stillinger85} potential reproduce the correct order in energy of the defects.
However, these potentials show shortcomings concerning the description of other physical properties and are unable to describe the C-C and C-Si interaction.
In fact the EA potential calculations favor the tetrahedral defect configuration.
This limitation is assumed to arise due to the cut-off.
In the tetrahedral configuration the second neighbors are only slightly more distant than the first neighbors, which creates the particular problem.
-Indeed, an increase of the cut-off results in increased values of the formation energies \cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
-The same issue has already been discussed by Tersoff \cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
+Indeed, an increase of the cut-off results in increased values of the formation energies~\cite{albe_sic_pot}, which is most significant for the tetrahedral configuration.
+The same issue has already been discussed by Tersoff~\cite{tersoff90} with regard to the description of the tetrahedral C defect using his potential.
While not completely rendering impossible further, more challenging empirical potential studies on large systems, the artifact has to be taken into account in the investigations of defect combinations later on in this chapter.
-The hexagonal configuration is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
+The hexagonal configuration is not stable opposed to results of the authors of the potential~\cite{albe_sic_pot}.
In the first two picoseconds, while kinetic energy is decoupled from the system, the \si{} seems to condense at the hexagonal site.
The formation energy of \unit[4.48]{eV} is determined by this low kinetic energy configuration shortly before the relaxation process starts.
The \si{} atom then begins to slowly move towards an energetically more favorable position very close to the tetrahedral one but slightly displaced along the three coordinate axes.
-The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work \cite{albe_sic_pot}.
+The formation energy of \unit[3.96]{eV} for this type of interstitial is equal to the result for the hexagonal one in the original work~\cite{albe_sic_pot}.
Obviously, the authors did not carefully check the relaxed results assuming a hexagonal configuration.
In Fig.~\ref{fig:defects:kin_si_hex} the relaxation process is shown on the basis of the kinetic energy plot.
\begin{figure}[tp]
\caption{Kinetic energy plot of the relaxation process of the hexagonal silicon self-interstitial defect simulation using the EA potential.}
\label{fig:defects:kin_si_hex}
\end{figure}
-To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the \textsc{parcas} MD code \cite{parcas_md}.
+To exclude failures in the implementation of the potential or the MD code itself the hexagonal defect structure was double-checked with the \textsc{parcas} MD code~\cite{parcas_md}.
The respective relaxation energetics are likewise plotted and look similar to the energetics obtained by \textsc{posic}.
In fact, the same type of interstitial arises using random insertions.
In addition, variations exist, in which the displacement is only along two \hkl<1 0 0> axes ($E_\text{f}=3.8\,\text{eV}$) or along a single \hkl<1 0 0> axes ($E_\text{f}=3.6\,\text{eV}$) successively approximating the tetrahedral configuration and formation energy.
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.
The type of reservoir of the C impurity to determine the formation energy of the defect is chosen to be SiC.
-This is consistent with the methods used in the articles \cite{tersoff90,dal_pino93}, which the results are compared to in the following.
+This is consistent with the methods used in the articles~\cite{tersoff90,dal_pino93}, which the results are compared to in the following.
Hence, the chemical potential of Si and C is determined by the cohesive energy of Si and SiC as discussed in section \ref{section:basics:defects}.
\begin{table}[tp]
\begin{center}
\textsc{posic} & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
\textsc{vasp} & Unstable & Unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
Other studies & & & & & & \\
- Tersoff \cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
- {\em Ab initio} \cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
+ Tersoff~\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
+ {\em Ab initio}~\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 & x+2.1 \\
\hline
\hline
\end{tabular}
\end{figure}
\cs{} occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration in energy for all potential models.
-An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$ \cite{bean71}.
+An experimental value of the formation energy of \cs{} was determined by a fit to solubility data yielding a concentration of $3.5 \times 10^{24} \exp{(-2.3\,\text{eV}/k_{\text{B}}T)} \text{ cm}^{-3}$~\cite{bean71}.
However, there is no particular reason for treating the prefactor as a free parameter in the fit to the experimental data.
It is simply given by the atomic density of pure silicon, which is $5\times 10^{22}\text{ cm}^{-3}$.
-Tersoff \cite{tersoff90} and Dal Pino et al. \cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
+Tersoff~\cite{tersoff90} and Dal Pino et al.~\cite{dal_pino93} pointed out that by combining this prefactor with the calculated values for the energy of formation ranging from \unit[1.6--1.89]{eV} an excellent agreement with the experimental solubility data within the entire temperature range of the experiment is obtained.
This reinterpretation of the solubility data, first proposed by Tersoff and later on reinforced by Dal~Pino~et~al. is in good agreement with the results of the quantum-mechanical calculations performed in this work.
Unfortunately the EA potential undervalues the formation energy roughly by a factor of two, which is a definite drawback of the potential.
Except for Tersoff's results for the tetrahedral configuration, the \ci{} \hkl<1 0 0> DB is the energetically most favorable interstitial configuration.
-As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in \cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3--10]{eV}.
+As mentioned above, the low energy of formation for the tetrahedral interstitial in the case of the Tersoff potential is believed to be an artifact of the abrupt cut-off set to \unit[2.5]{\AA} (see Ref. 11 and 13 in~\cite{tersoff90}) and the real formation energy is, thus, supposed to be located between \unit[3--10]{eV}.
Keeping these considerations in mind, the \ci{} \hkl<1 0 0> DB is the most favorable interstitial configuration for all interaction models.
This finding is in agreement with several theoretical~\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental~\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.
However, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.
The highest energy is observed for the hexagonal interstitial configuration using classical potentials.
Quantum-mechanical calculations reveal this configuration to be unstable, which is also reproduced by the EA potential.
In both cases a relaxation towards the \ci{} \hkl<1 0 0> DB configuration is observed.
-Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable \cite{tersoff90}.
+Opposed to results of the first-principles calculations, Tersoff finds this configuration to be stable~\cite{tersoff90}.
In fact, the stability of the hexagonal interstitial could not be reproduced in simulations performed in this work using the unmodified Tersoff potential parameters.
Unfortunately, apart from the modified parameters, no more conditions specifying the relaxation process are given in Tersoff's study on C point defects in Si.
\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.
-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.
+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.
For comparison, the obtained structures for both methods are visualized in Fig.~\ref{fig:defects:100db_vis_cmp}.
\label{img:defects:bc_conf}
\end{figure}
In the BC interstitial configuration the interstitial atom is located in between two next neighbored Si atoms forming linear bonds.
-In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possible migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one \cite{capaz94}.
+In a previous study this configuration was found to constitute an intermediate saddle point configuration determining the migration barrier of one possible migration path of a \ci{} \hkl<1 0 0> DB configuration into an equivalent one~\cite{capaz94}.
This is in agreement with results of the EA potential simulations, which reveal this configuration to be unstable relaxing into the \ci{} \hkl<1 1 0> configuration.
However, this fact could not be reproduced by spin polarized \textsc{vasp} calculations performed in this work.
Present results suggest this configuration to correspond to a real local minimum.
\label{fig:defects:00-1_0-10_mig}
\end{figure}
Fig.~\ref{fig:defects:00-1_0-10_mig} shows the migration barrier and structures of the \ci{} \hkl<0 0 -1> to \hkl<0 -1 0> DB transition.
-The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV} \cite{lindner06}, \unit[0.73]{eV} \cite{song90} and \unit[0.87]{eV} \cite{tipping87}.
+The resulting migration barrier of approximately \unit[0.9]{eV} is very close to the experimentally obtained values of \unit[0.70]{eV}~\cite{lindner06}, \unit[0.73]{eV}~\cite{song90} and \unit[0.87]{eV}~\cite{tipping87}.
\begin{figure}[tp]
\begin{center}
\end{figure}
The third migration path, in which the DB is changing its orientation, is shown in Fig.~\ref{fig:defects:00-1_0-10_ip_mig}.
An energy barrier of roughly \unit[1.2]{eV} is observed.
-Experimentally measured activation energies for reorientation range from \unit[0.77]{eV} to \unit[0.88]{eV} \cite{watkins76,song90}.
+Experimentally measured activation energies for reorientation range from \unit[0.77]{eV} to \unit[0.88]{eV}~\cite{watkins76,song90}.
Thus, this pathway is more likely to be composed of two consecutive steps of the second path.
Since the activation energy of the first and last migration path is much greater than the experimental value, the second path is identified to be responsible as a migration path for the most likely C interstitial in Si explaining both, annealing and reorientation experiments.
The activation energy of roughly \unit[0.9]{eV} nicely compares to experimental values reinforcing the correct identification of the C-Si DB diffusion mechanism.
Slightly increased values compared to experiment might be due to the tightened constraints applied in the modified CRT approach.
-Nevertheless, the theoretical description performed in this work is improved compared to a former study \cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
+Nevertheless, the theoretical description performed in this work is improved compared to a former study~\cite{capaz94}, which underestimates the experimental value by \unit[35]{\%}.
In addition, it is finally shown that the BC configuration, for which spin polarized calculations are necessary, constitutes a real local minimum instead of a saddle point configuration due to the presence of restoring forces for displacements in migration direction.
\begin{figure}[tp]
On the other hand, the activation energy obtained by classical potential simulations is tremendously overestimated by a factor of 2.4 to 3.5.
The overestimated barrier is due to the short range character of the potential, which drops the interaction to zero within the first and next neighbor distance.
Since the total binding energy is accommodated within a short distance, which according to the universal energy relation would usually correspond to a much larger distance, unphysical high forces between two neighbored atoms arise.
-This is explained in more detail in a previous study \cite{mattoni2007}.
+This is explained in more detail in a previous study~\cite{mattoni2007}.
Thus, atomic diffusion is wrongly described in the classical potential approach.
The probability of already rare diffusion events is further decreased for this reason.
However, agglomeration of C and diffusion of Si self-interstitials are an important part of the proposed SiC precipitation mechanism.
\caption[Relaxed structures of defect combinations obtained by creating {\hkl[1 0 0]} and {\hkl[0 -1 0]} DBs at position 1.]{Relaxed structures of defect combinations obtained by creating \hkl[1 0 0] (a) and \hkl[0 -1 0] (b) DBs at position 1.}
\label{fig:defects:comb_db_01}
\end{figure}
-Mattoni~et~al. \cite{mattoni2002} predict the ground-state configuration of \ci{} \hkl<1 0 0>-type defect pairs for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
+Mattoni~et~al.~\cite{mattoni2002} predict the ground-state configuration of \ci{} \hkl<1 0 0>-type defect pairs for a \hkl[1 0 0] or equivalently a \hkl[0 1 0] defect created at position 1 with both defects basically maintaining the as-isolated DB structure, resulting in a binding energy of \unit[-2.1]{eV}.
In the present study, a further relaxation of this defect structure is observed.
The C atom of the second and the Si atom of the initial DB move towards each other forming a bond, which results in a somewhat lower binding energy of \unit[-2.25]{eV}.
The corresponding defect structure is displayed in Fig.~\ref{fig:defects:225}.
\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.
-This configuration has been identified and described by spectroscopic experimental techniques \cite{song90_2} as well as theoretical studies \cite{leary97,capaz98}.
+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.
-Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV} \cite{song90_2}, reinforcing qualitatively correct results of previous theoretical studies on these structures.
+Present results show a difference in energy of states A and B, which exactly matches the experimental value of \unit[0.02]{eV}~\cite{song90_2}, reinforcing qualitatively correct results of previous theoretical studies on these structures.
% mattoni: A favored by 0.4 eV - NO, it is indeed B (reinforce Song and Capaz)!
%
% AB transition
-The migration barrier is identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV} \cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
+The migration barrier is identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV}~\cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.
Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected.
Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.
% not satisfactory!
\caption[Migration barrier of the \si{} {\hkl[1 1 0]} DB into the hexagonal and tetrahedral configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.]{Migration barrier of the \si{} \hkl[1 1 0] DB into the hexagonal (H) and tetrahedral (T) configuration as well as the hexagonal \si{} to tetrahedral \si{} transition.}
\label{fig:defects:si_mig2}
\end{figure}
-The obtained activation energies are of the same order of magnitude than values derived from other {\em ab initio} studies \cite{bloechl93,sahli05}.
+The obtained activation energies are of the same order of magnitude than values derived from other {\em ab initio} studies~\cite{bloechl93,sahli05}.
The low barriers indeed enable configurations of further separated \cs{} and \si{} atoms by the highly mobile \si{} atom departing from the \cs{} defect as observed in the previously discussed MD simulation.
% kept for nostalgical reason!
\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$_{\text{i}}$.
-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
+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$_{\text{i}}$.
+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.
First of all, a different pathway is suggested as the lowest energy path, which again might be attributed to the absence of quantum-mechanical effects in the classical interaction model.
Secondly, the activation energy is overestimated by a factor of 2.4 to 3.5 compared to the more accurate quantum-mechanical methods and experimental findings.
This is attributed to the sharp cut-off of the short range potential.
-As already pointed out in a previous study \cite{mattoni2007}, the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
+As already pointed out in a previous study~\cite{mattoni2007}, the short cut-off is responsible for overestimated and unphysical high forces of next neighbor atoms.
The overestimated migration barrier, however, affects the diffusion behavior of the C interstitials.
By this artifact, the mobility of the C atoms is tremendously decreased resulting in an inaccurate description or even absence of the DB agglomeration as proposed by one of the precipitation models.
Quantum-mechanical investigations of two \ci{} of the \hkl<1 0 0>-type and varying separations and orientations state an attractive interaction between these interstitials.
-Obtained results for the most part compare well with results gained in previous studies \cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment \cite{song90}.
+Obtained results for the most part compare well with results gained in previous studies~\cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment~\cite{song90}.
%
Depending on orientation, energetically favorable configurations are found, in which these two interstitials are located close together instead of the occurrence of largely separated and isolated defects.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
To conclude, the available results suggest precipitation by successive agglomeration of C$_{\text{s}}$.
However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time.
Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.
-It is worth to mention that there is no contradiction to results of the HREM studies \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
+It is worth to mention that there is no contradiction to results of the HREM studies~\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.
Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.
However, there is no particular reason for the C species to reside in the interstitial lattice.
Contrasts are also assumed for Si$_{\text{i}}$.
\chapter{Introduction}
Silicon carbide (SiC) has a number of remarkable physical and chemical properties that make it a promising new material in various fields of applications.
-The high electron mobility and saturation drift velocity as well as the high band gap and breakdown field in conjunction with its unique thermal stability and conductivity unveil SiC as the ideal candidate for high-power, high-frequency and high-temperature electronic and optoelectronic devices exceeding conventional silicon based solutions \cite{wesch96,morkoc94,casady96,capano97,pensl93}.
-Due to the large Si--C bonding energy SiC is a hard and chemical inert material suitable for applications under extreme conditions and capable for microelectromechanical systems, both as structural material and as a coating layer \cite{sarro00,park98}.
-Its radiation hardness allows the operation as a first wall material in nuclear reactors \cite{giancarli98} and as electronic devices in space \cite{capano97}.
+The high electron mobility and saturation drift velocity as well as the high band gap and breakdown field in conjunction with its unique thermal stability and conductivity unveil SiC as the ideal candidate for high-power, high-frequency and high-temperature electronic and optoelectronic devices exceeding conventional silicon based solutions~\cite{wesch96,morkoc94,casady96,capano97,pensl93}.
+Due to the large Si--C bonding energy, SiC is a hard and chemical inert material suitable for applications under extreme conditions and capable for microelectromechanical systems, both as structural material and as a coating layer~\cite{sarro00,park98}.
+Its radiation hardness allows the operation as a first wall material in nuclear reactors~\cite{giancarli98} and as electronic devices in space~\cite{capano97}.
The realization of silicon carbide based applications demands for reasonable sized wafers of high crystalline quality.
-Despite the tremendous progress achieved in the fabrication of high purity SiC employing techniques like the modified Lely process for bulk crystal growth \cite{tairov78,tsvetkov98} or chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) for homo- and heteroepitaxial growth \cite{kimoto93,powell90,fissel95}, available wafer dimensions and crystal qualities are not yet sufficient.
+Despite the tremendous progress achieved in the fabrication of high purity SiC employing techniques like the modified Lely process for bulk crystal growth~\cite{tairov78,tsvetkov98} or chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) for homo- and heteroepitaxial growth~\cite{kimoto93,powell90,fissel95}, available wafer dimensions and crystal qualities are not yet sufficient.
Another promising alternative to fabricate SiC is ion beam synthesis (IBS).
-High-dose carbon implantation at elevated temperatures into silicon with subsequent annealing results in the formation of buried SiC layers \cite{borders71,reeson87}.
-A two-temperature implantation technique was proposed to achieve single crystalline, epitaxial SiC layers and a sharp SiC/Si interface \cite{lindner99,lindner99_2,lindner01,lindner02}.
+High-dose carbon implantation at elevated temperatures into silicon with subsequent annealing results in the formation of buried SiC layers~\cite{borders71,reeson87}.
+A two-temperature implantation technique was proposed to achieve single crystalline, epitaxial SiC layers and a sharp SiC/Si interface~\cite{lindner99,lindner99_2,lindner01,lindner02}.
-Although high-quality SiC can be achieved by means of IBS the precipitation mechanism is not yet fully understood.
+Although high-quality SiC can be achieved by means of IBS, the precipitation mechanism is not yet fully understood.
High resolution transmission electron microscopy studies indicate the formation of C-Si interstitial complexes sharing conventional silicon lattice sites (C-Si dumbbells) during the implantation of carbon in silicon.
-These C-Si dumbbells agglomerate and once a critical radius is reached, the topotactic transformation into a SiC precipitate occurs \cite{werner97,lindner01}.
+These C-Si dumbbells agglomerate and once a critical radius is reached, the topotactic transformation into a SiC precipitate occurs~\cite{werner97,lindner01}.
In contrast, investigations of strained Si$_{1-y}$C$_y$/Si heterostructures formed by MBE~\cite{strane94,guedj98}, which incidentally involve the formation of SiC nanocrystallites, suggest an initial coherent precipitation by agglomeration of substitutional instead of interstitial C.
Coherency is lost once the increasing strain energy of the stretched SiC structure surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the Si substrate.
These two different mechanisms of precipitation might be attributed to the respective method of fabrication.
While in CVD and MBE surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal.
-However, in another IBS study \cite{nejim95} a topotactic transformation is proposed that is likewise based on the formation of substitutional C, which is accompanied by the emission of Si self-interstitial atoms that previously occupied the lattice sites and a compensating reduction of volume due to the lower lattice constant of SiC compared to Si.
+However, in another IBS study~\cite{nejim95} a topotactic transformation is proposed that is likewise based on the formation of substitutional C, which is accompanied by the emission of Si self-interstitial atoms that previously occupied the lattice sites and a compensating reduction of volume due to the lower lattice constant of SiC compared to Si.
The atomic migration involved in such a transformation is not clear.
For several reasons, solving the controversial view of SiC precipitation in Si is of fundamental interest.
A better understanding of the supposed SiC conversion mechanism and related carbon-mediated effects in silicon will enable significant technological progress in SiC thin film formation on the one hand and likewise offer perspectives for processes which rely upon prevention of precipitation events for improved silicon based devices on the other hand.
-Implanted carbon is known to suppress transient enhanced diffusion of dopant species like boron or phosphorus in the annealing step \cite{cowern96} which can be exploited to create shallow p-n junctions in submicron technologies.
-Si self-interstitials (Si$_{\text{i}}$), known as the transport vehicles for dopants \cite{fahey89,stolk95}, get trapped by reacting with the carbon atoms \cite{stolk97}.
-Furthermore, carbon incorporated in silicon is being used to fabricate strained silicon \cite{strane94,strane96,osten99} utilized in semiconductor industry for increased charge carrier mobilities in silicon \cite{chang05,osten97} as well as to adjust its band gap \cite{soref91,kasper91}.
+Implanted carbon is known to suppress transient enhanced diffusion of dopant species like boron or phosphorus in the annealing step~\cite{cowern96}, which can be exploited to create shallow p-n junctions in submicron technologies.
+Si self-interstitials (Si$_{\text{i}}$), known as the transport vehicles for dopants~\cite{fahey89,stolk95}, get trapped by reacting with the carbon atoms~\cite{stolk97}.
+Furthermore, carbon incorporated in silicon is being used to fabricate strained silicon~\cite{strane94,strane96,osten99} utilized in semiconductor industry for increased charge carrier mobilities in silicon~\cite{chang05,osten97} as well as to adjust its band gap~\cite{soref91,kasper91}.
Thus the understanding of carbon in silicon either as an isovalent impurity as well as at concentrations exceeding the solid solubility limit up to the stoichiometric ratio to form silicon carbide is of fundamental interest.
Due to the impressive growth in computer power on the one hand and outstanding progress in the development of new theoretical concepts, algorithms and computational methods on the other hand, computer simulations enable the modeling of increasingly complex systems.
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.
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 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.
With respect to the precipitation model, the formation of C$_{\text{i}}$ \hkl<1 0 0> DBs indeed occurs for low C concentrations.
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.
-The most famous approaches are hyperdynamics (HMD) \cite{voter97,voter97_2}, parallel replica dynamics \cite{voter98}, temperature accelerated dynamics (TAD) \cite{sorensen2000} and self-guided dynamics (SGMD) \cite{wu99}, which accelerate phase space propagation while retaining proper thermodynamic sampling.
+The most famous approaches are hyperdynamics (HMD)~\cite{voter97,voter97_2}, parallel replica dynamics~\cite{voter98}, temperature accelerated dynamics (TAD)~\cite{sorensen2000} and self-guided dynamics (SGMD)~\cite{wu99}, which accelerate phase space propagation while retaining proper thermodynamic sampling.
In addition to the time scale limitation, problems attributed to the short range potential exist.
-The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms \cite{tang95,mattoni2007}.
+The sharp cut-off function, which limits the interacting ions to the next neighbored atoms by gradually pushing the interaction force and energy to zero between the first and second next neighbor distance, is responsible for overestimated and unphysical high forces of next neighbored atoms~\cite{tang95,mattoni2007}.
This is supported by the overestimated activation energies necessary for C diffusion as investigated in section \ref{subsection:defects:mig_classical}.
Indeed, it is not only the strong C-C bond, which is hard to break, inhibiting C diffusion and further rearrangements.
This is also true for the low concentration simulations dominated by the occurrence of C-Si DBs spread over the whole simulation volume.
These unphysical effects inherent to this type of model potentials are solely attributed to their short range character.
While cohesive and formational energies are often well described, these effects increase for non-equilibrium structures and dynamics.
However, since valuable insights into various physical properties can be gained using this potentials, modifications mainly affecting the cut-off were designed.
-One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature \cite{mattoni2007}.
-Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC \cite{tang95} but might be rather ineffective for the challenge inherent to this study.
+One possibility is to simply skip the force contributions containing the derivatives of the cut-off function, which was successfully applied to reproduce the brittle propagation of fracture in SiC at zero temperature~\cite{mattoni2007}.
+Another one is to use variable cut-off values scaled by the system volume, which properly describes thermomechanical properties of 3C-SiC~\cite{tang95} but might be rather ineffective for the challenge inherent to this study.
To conclude the obstacle needed to get passed is twofold.
The sharp cut-off of the employed bond order model potential introduces overestimated high forces between next neighbored atoms enhancing the problem of slow phase space propagation immanent to MD simulations.
Due to this, pushing the time scale to the limits of computational resources or applying one of the above mentioned accelerated dynamics methods exclusively will not be sufficient enough.
Instead, the approach followed in this study, is the use of higher temperatures as exploited in TAD to find transition pathways of one local energy minimum to another one more quickly.
-Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock \cite{sorensen2000}.
+Since merely increasing the temperature leads to different equilibrium kinetics than valid at low temperatures, TAD introduces basin-constrained MD allowing only those transitions that should occur at the original temperature and a properly advancing system clock~\cite{sorensen2000}.
The TAD corrections are not applied in coming up simulations.
This is justified by two reasons.
First of all, a compensation of the overestimated bond strengths due to the short range potential is expected.
-Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS \cite{nejim95,lindner01}.
+Secondly, there is no conflict applying higher temperatures without the TAD corrections, since crystalline 3C-SiC is also observed for higher temperatures than \unit[450]{$^{\circ}$C} in IBS~\cite{nejim95,lindner01}.
It is therefore expected that the kinetics affecting the 3C-SiC precipitation are not much different at higher temperatures aside from the fact that it is occurring much more faster.
Moreover, the interest of this study is focused on structural evolution of a system far from equilibrium instead of equilibrium properties which rely upon proper phase space sampling.
On the other hand, during implantation, the actual temperature inside the implantation volume is definitely higher than the experimentally determined temperature tapped from the surface of the sample.
Increased temperatures are expected to compensate the overestimated diffusion barriers.
These are overestimated by a factor of 2.4 to 3.5.
Scaling the absolute temperatures accordingly results in maximum temperatures of \unit[1460--2260]{$^{\circ}$C}.
-Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K}) \cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
+Since melting already occurs shortly below the melting point of the potential (\unit[2450]{K})~\cite{albe_sic_pot} due to the presence of defects, temperatures ranging from \unit[450--2050]{$^{\circ}$C} are used.
The simulation sequence and other parameters except for the system temperature remain unchanged as in section \ref{section:initial_sims}.
Since there is no significant difference among the $V_2$ and $V_3$ simulations only the $V_1$ and $V_2$ simulations are carried on and referred to as low C and high C concentration simulations.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
Results of the MD simulations at different temperatures and C concentrations can be correlated to experimental findings.
-IBS studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates \cite{kimura82,eichhorn02}.
-In particular, the restructuring of strong C-C bonds is affected \cite{deguchi92}.
+IBS studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates~\cite{kimura82,eichhorn02}.
+In particular, the restructuring of strong C-C bonds is affected~\cite{deguchi92}.
These bonds preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements.
This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations.
The insertion of high amounts of C into a small volume within a short period of time results in essentially no time for the system to rearrange.
% rt implantation + annealing
-Furthermore, C implanted at room temperature was found to be able to migrate towards the surface and form C-rich clusters in contrast to implantations at elevated temperatures, which form stable epitaxially aligned 3C-SiC precipitates \cite{serre95}.
+Furthermore, C implanted at room temperature was found to be able to migrate towards the surface and form C-rich clusters in contrast to implantations at elevated temperatures, which form stable epitaxially aligned 3C-SiC precipitates~\cite{serre95}.
In simulation, low temperatures result in configurations of highly mobile \ci{} \hkl<1 0 0> DBs whereas elevated temperatures show configurations of \cs{}, which constitute an extremely stable configuration that is unlikely to migrate.
%
% added
-This likewise agrees to results of IBS experiments utilizing implantation temperatures of \degc{550}, which require annealing temperatures as high as \degc{1405} for C segregation due to the stability of \cs{} \cite{reeson87}.
+This likewise agrees to results of IBS experiments utilizing implantation temperatures of \degc{550}, which require annealing temperatures as high as \degc{1405} for C segregation due to the stability of \cs{}~\cite{reeson87}.
%
-Indeed, in the optimized recipe to form 3C-SiC by IBS \cite{lindner99}, elevated temperatures are used to improve the epitaxial orientation together with a low temperature implant to destroy stable SiC nanocrystals at the interface, which are unable to migrate during thermal annealing resulting in a rough surface.
+Indeed, in the optimized recipe to form 3C-SiC by IBS~\cite{lindner99}, elevated temperatures are used to improve the epitaxial orientation together with a low temperature implant to destroy stable SiC nanocrystals at the interface, which are unable to migrate during thermal annealing resulting in a rough surface.
Furthermore, the improvement of the epitaxial orientation of the precipitate with increasing temperature in experiment perfectly conforms to the increasing occurrence of \cs{} in simulation.
%
% todo add sync here starting from strane93, werner96 ...
-Moreover, implantations of an understoichiometric dose into preamorphized Si followed by an annealing step at \degc{700} result in Si$_{1-x}$C$_x$ layers with C residing on substitutional Si lattice sites \cite{strane93}.
-For implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing below and above \degc{700}, the formation of small C$_{\text{i}}$ agglomerates and SiC precipitates was observed respectively \cite{werner96}.
-However, increased implantation temperatures were found to be more efficient than postannealing methods resulting in the formation of SiC precipitates for implantations at \unit[450]{$^{\circ}$C} \cite{lindner99,lindner01}.
+Moreover, implantations of an understoichiometric dose into preamorphized Si followed by an annealing step at \degc{700} result in Si$_{1-x}$C$_x$ layers with C residing on substitutional Si lattice sites~\cite{strane93}.
+For implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing below and above \degc{700}, the formation of small C$_{\text{i}}$ agglomerates and SiC precipitates was observed respectively~\cite{werner96}.
+However, increased implantation temperatures were found to be more efficient than postannealing methods resulting in the formation of SiC precipitates for implantations at \unit[450]{$^{\circ}$C}~\cite{lindner99,lindner01}.
%
Thus, at elevated temperatures, implanted C is expected to occupy substitutionally usual Si lattice sites right from the start.
These findings, which are outlined in more detail within the comprehensive description in chapter~\ref{chapter:summary}, are in perfect agreement with previous results of the quantum-mechanical investigations.
Indeed, \si{} is observed in the direct surrounding of the stretched SiC structures.
Next to the rearrangement, \si{} is required as a supply for additional C atoms to form further SiC and to compensate strain, either within the coherent and stretched SiC structure as well as at the interface of the incoherent SiC precipitate and the Si host.
%
-In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$ \cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
+In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$~\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.
\begin{center}
\includegraphics[width=12cm]{si-c_phase.eps}
\end{center}
-\caption[Phase diagram of the C/Si system.]{Phase diagram of the C/Si system \cite{scace59}.}
+\caption[Phase diagram of the C/Si system.]{Phase diagram of the C/Si system~\cite{scace59}.}
\label{fig:sic:si-c_phase}
\end{figure}
SiC was first discovered by Henri Moissan in 1893 when he observed brilliant sparkling crystals while examining rock samples from a meteor crater in Arizona.
He mistakenly identified these crystals as diamond.
-Although they might have been considered \glqq diamonds from space\grqq{} Moissan identified them as SiC in 1904 \cite{moissan04}.
+Although they might have been considered \glqq diamonds from space\grqq{} Moissan identified them as SiC in 1904~\cite{moissan04}.
In mineralogy SiC is still referred to as moissanite in honor of its discoverer.
It is extremely rare and almost impossible to find in nature.
\hline
\end{tabular}
\end{center}
-\caption[Properties of SiC polytypes and other semiconductor materials.]{Properties of SiC polytypes and other semiconductor materials. Doping concentrations are $10^{16}\text{ cm}^{-3}$ (A) and $10^{17}\text{ cm}^{-3}$ (B) respectively. References: \cite[]{wesch96,casady96,park98}.}
+\caption[Properties of SiC polytypes and other semiconductor materials.]{Properties of SiC polytypes and other semiconductor materials. Doping concentrations are $10^{16}\text{ cm}^{-3}$ (A) and $10^{17}\text{ cm}^{-3}$ (B) respectively. References:~\cite[]{wesch96,casady96,park98}.}
\label{table:sic:properties}
\end{table}
\bibpunct{[}{]}{,}{n}{}{}
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.
-Though possible, melt growth processes \cite{nelson69} are complicated due to the small C solubility in Si at temperatures below \unit[2000]{$^{\circ}$C} and its small change with temperature \cite{scace59}.
+Though possible, melt growth processes~\cite{nelson69} are complicated due to the small C solubility in Si at temperatures below \unit[2000]{$^{\circ}$C} and its small change with temperature~\cite{scace59}.
High process temperatures are necessary and the evaporation of Si must be suppressed by a high-pressure inert atmosphere.
Crystals grown by this method are not adequate for practical applications with respect to their size as well as quality and purity.
The presented methods, thus, focus on vapor transport growth processes such as chemical vapor deposition (CVD) or molecular beam epitaxy (MBE) and the sublimation technique.
-Excellent reviews of the different SiC growth methods have been published by Wesch \cite{wesch96} and Davis~et~al. \cite{davis91}.
+Excellent reviews of the different SiC growth methods have been published by Wesch~\cite{wesch96} and Davis~et~al.~\cite{davis91}.
\subsection{SiC bulk crystal growth}
-The industrial Acheson process \cite{knippenberg63} is utilized to produce SiC on a large scale by thermal reaction of silicon dioxide (silica sand) and carbon (coal).
+The industrial Acheson process~\cite{knippenberg63} is utilized to produce SiC on a large scale by thermal reaction of silicon dioxide (silica sand) and carbon (coal).
The heating is accomplished by a core of graphite centrally placed in a furnace, which is heated up to a maximum temperature of \unit[2700]{$^{\circ}$C}, after which the temperature is gradually lowered.
Due to the insufficient and uncontrollable purity, material produced by this method, originally termed carborundum by Acheson, can hardly be used for device applications.
However, it is often used as an abrasive material and as seed crystals for subsequent vapor phase growth and sublimation processes.
-In the van Arkel apparatus \cite{arkel25}, Si and C containing gases like methylchlorosilanes \cite{moers31} and silicon tetrachloride \cite{kendall53} are pyrolitically decomposed and SiC is deposited on heated carbon rods in a vapor growth process.
+In the van Arkel apparatus~\cite{arkel25}, Si and C containing gases like methylchlorosilanes~\cite{moers31} and silicon tetrachloride~\cite{kendall53} are pyrolitically decomposed and SiC is deposited on heated carbon rods in a vapor growth process.
Typical deposition temperatures are in the range between \unit[1400]{$^{\circ}$C} and \unit[1600]{$^{\circ}$C} while studies up to \unit[2500]{$^{\circ}$C} have been performed.
The obtained polycrystalline material consists of small crystal grains with a size of several hundred microns stated to be mainly of the cubic polytype.
-A significant breakthrough was made in 1955 by Lely, who proposed a sublimation process for growing higher purity bulk SiC single crystals \cite{lely55}.
+A significant breakthrough was made in 1955 by Lely, who proposed a sublimation process for growing higher purity bulk SiC single crystals~\cite{lely55}.
In the so called Lely process, a tube of porous graphite is surrounded by polycrystalline SiC as gained by previously described processes.
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.
+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.
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.
-The method was successfully applied to grow 6H and 4H boules with diameters up to \unit[60]{mm} \cite{tairov81,barrett91,barrett93,stein93}.
-This refined versions of the physical vapor transport (PVT) technique enabled the reproducible boule growth of device quality SiC crystals, which were for instance used to fabricate blue light emitting diodes with increased quantum efficiencies \cite{hoffmann82}.
+The method was successfully applied to grow 6H and 4H boules with diameters up to \unit[60]{mm}~\cite{tairov81,barrett91,barrett93,stein93}.
+This refined versions of the physical vapor transport (PVT) technique enabled the reproducible boule growth of device quality SiC crystals, which were for instance used to fabricate blue light emitting diodes with increased quantum efficiencies~\cite{hoffmann82}.
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.
Additionally, to preserve epitaxial growth conditions, graphitization of the seed crystal has to be avoided.
Avoiding defects constitutes a major difficulty.
These defects include growth spirals (stepped screw dislocations), subgrain boundaries and twins as well as micropipes (micron sized voids extending along the c axis of the crystal) and 3C inclusions at the seed crystal in hexagonal growth systems.
-Micropipe-free growth of 6H-SiC has been realized by a reduction of the temperature gradient in the sublimation furnace resulting in near-equilibrium growth conditions in order to avoid stresses, which is, however, accompanied by a reduction of the growth rate \cite{schulze98}.
+Micropipe-free growth of 6H-SiC has been realized by a reduction of the temperature gradient in the sublimation furnace resulting in near-equilibrium growth conditions in order to avoid stresses, which is, however, accompanied by a reduction of the growth rate~\cite{schulze98}.
Further efforts have to be expended to find relations between the growth parameters, the kind of polytype and the occurrence and concentration of defects, which are of fundamental interest and might help to improve the purity of the bulk materials.
\subsection{SiC epitaxial thin film growth}
The heteroepitaxial growth of SiC on Si substrates has been stimulated for a long time due to a lack of suitable large substrates that could be adopted for homoepitaxial growth.
Furthermore, heteroepitaxy on Si substrates enables the fabrication of the advantageous 3C polytype, which constitutes a metastable phase and, thus, can be grown as a bulk crystal only with small sizes of a few mm.
The main difficulties in SiC heteroepitaxy on Si arise due to the lattice mismatch of Si and SiC by \unit[20]{\%} and the difference in the thermal expansion coefficient of \unit[8]{\%}.
-Thus, in most of the applied CVD and MBE processes, the SiC layer formation process is split into two steps, the surface carbonization and the growth step, as proposed by Nishino~et~al. \cite{nishino83}.
+Thus, in most of the applied CVD and MBE processes, the SiC layer formation process is split into two steps, the surface carbonization and the growth step, as proposed by Nishino~et~al.~\cite{nishino83}.
Cleaning of the substrate surface with HCl is required prior to carbonization.
During carbonization the Si surface is chemically converted into a SiC film with a thickness of a few nm by exposing it to a flux of C atoms and concurrent heating up to temperatures of about \unit[1400]{$^{\circ}$C}.
In a next step, the epitaxial deposition of SiC is realized by an additional supply of Si atoms at similar temperatures.
-Low defect densities in the buffer layer are a prerequisite for obtaining good quality SiC layers during growth, although defect densities decrease with increasing distance to the SiC/Si interface \cite{shibahara86}.
-Next to surface morphology defects such as pits and islands, the main defects in 3C-SiC heteroepitaxial layers are twins, stacking faults (SF) and antiphase boundaries (APB) \cite{shibahara86,pirouz87}.
-APB defects, which constitute the primary residual defects in thick layers, are formed near surface terraces that differ in a single-atom-height step resulting in domains of SiC separated by a boundary, which consists of either Si-Si or C-C bonds due to missing or disturbed sublattice information \cite{desjardins96,kitabatake97}.
-However, the number of such defects can be reduced by off-axis growth on a Si \hkl(0 0 1) substrate miscut towards \hkl[1 1 0] by \unit[2]{$^{\circ}$}-\unit[4]{$^{\circ}$} \cite{shibahara86,powell87_2}.
-This results in the thermodynamically favored growth of a single phase due to the uni-directional contraction of Si-C-Si bond chains perpendicular to the terrace steps edges during carbonization and the fast growth parallel to the terrace edges during growth under Si rich conditions \cite{kitabatake97}.
-By MBE, lower process temperatures than these typically employed in CVD have been realized \cite{hatayama95,henke95,fuyuki97,takaoka98}, which is essential for limiting thermal stresses and to avoid resulting substrate bending, a key issue in obtaining large area 3C-SiC surfaces.
+Low defect densities in the buffer layer are a prerequisite for obtaining good quality SiC layers during growth, although defect densities decrease with increasing distance to the SiC/Si interface~\cite{shibahara86}.
+Next to surface morphology defects such as pits and islands, the main defects in 3C-SiC heteroepitaxial layers are twins, stacking faults (SF) and antiphase boundaries (APB)~\cite{shibahara86,pirouz87}.
+APB defects, which constitute the primary residual defects in thick layers, are formed near surface terraces that differ in a single-atom-height step resulting in domains of SiC separated by a boundary, which consists of either Si-Si or C-C bonds due to missing or disturbed sublattice information~\cite{desjardins96,kitabatake97}.
+However, the number of such defects can be reduced by off-axis growth on a Si \hkl(0 0 1) substrate miscut towards \hkl[1 1 0] by \unit[2]{$^{\circ}$}-\unit[4]{$^{\circ}$}~\cite{shibahara86,powell87_2}.
+This results in the thermodynamically favored growth of a single phase due to the uni-directional contraction of Si-C-Si bond chains perpendicular to the terrace steps edges during carbonization and the fast growth parallel to the terrace edges during growth under Si rich conditions~\cite{kitabatake97}.
+By MBE, lower process temperatures than these typically employed in CVD have been realized~\cite{hatayama95,henke95,fuyuki97,takaoka98}, which is essential for limiting thermal stresses and to avoid resulting substrate bending, a key issue in obtaining large area 3C-SiC surfaces.
In summary, the almost universal use of Si has allowed significant progress in the understanding of heteroepitaxial growth of SiC on Si.
However, mismatches in the thermal expansion coefficient and the lattice parameter cause a considerably high concentration of various defects, which is responsible for structural and electrical qualities that are not yet satisfactory.
The alternative attempt to grow SiC on SiC substrates has shown to drastically reduce the concentration of defects in deposited layers.
-By CVD, both, the 3C \cite{kong88,powell90} as well as the 6H \cite{kong88_2,powell90_2} polytype could be successfully grown.
-In order to obtain the homoepitaxially grown 6H polytype, off-axis 6H-SiC wafers are required as a substrate \cite{kimoto93}.
+By CVD, both, the 3C~\cite{kong88,powell90} as well as the 6H~\cite{kong88_2,powell90_2} polytype could be successfully grown.
+In order to obtain the homoepitaxially grown 6H polytype, off-axis 6H-SiC wafers are required as a substrate~\cite{kimoto93}.
%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.
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.
-However, lateral 3C-SiC growth was also observed on low tilt angle off-axis substrates originating from intentionally induced dislocations \cite{powell91}.
+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}.
+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}.
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}.
+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}.
On \hkl(000-1) substrates twinned \hkl(-1-1-1) oriented 3C-SiC domains are observed, which suggest a nucleation driven rather than step-flow growth mechanism.
On \hkl(0-11-4) substrates, however, single crystalline \hkl(001) oriented 3C-SiC grows with the c axes of substrate and film being equal.
The beneficial epitaxial relation of substrate and film limits the structural difference between the two polytypes in two out of six layers with respect to the stacking sequence along the c axis.
-Homoepitaxial growth of 3C-SiC by GSMBE was realized for the first time by atomic layer epitaxy (ALE) utilizing the periodical change in the surface superstructure by the alternating supply of the source gases, which determines the growth rate giving atomic level control in the growth process \cite{fuyuki89}.
+Homoepitaxial growth of 3C-SiC by GSMBE was realized for the first time by atomic layer epitaxy (ALE) utilizing the periodical change in the surface superstructure by the alternating supply of the source gases, which determines the growth rate giving atomic level control in the growth process~\cite{fuyuki89}.
The cleaned substrate surface shows a C terminated $(2\times 2)$ pattern at \unit[1000]{$^{\circ}$C}, which turns into a $(3\times 2)$ pattern when Si$_2$H$_6$ is introduced and it is maintained after the supply is stopped.
-A more detailed investigation showed the formation of a preceding $(2\times 1)$ and $(5\times 2)$ pattern within the exposure to the Si containing gas \cite{yoshinobu90,fuyuki93}.
+A more detailed investigation showed the formation of a preceding $(2\times 1)$ and $(5\times 2)$ pattern within the exposure to the Si containing gas~\cite{yoshinobu90,fuyuki93}.
The $(3\times 2)$ superstructure contains approximately 1.7 monolayers of Si atoms, crystallizing into 3C-SiC with a smooth and mirror-like surface after C$_2$H$_6$ is inserted accompanied by a reconstruction of the surface into the initial C terminated $(2\times 2)$ pattern.
A minimal growth rate of 2.3 monolayers per cycle exceeding the value of 1.7 is due to physically adsorbed Si atoms not contributing to the superstructure.
To realize single monolayer growth precise control of the gas supply to form the $(2\times 1)$ structure is required.
-However, accurate layer-by-layer growth is achieved under certain conditions, which facilitate the spontaneous desorption of an additional layer of one atom species by supply of the other species \cite{hara93}.
-Homoepitaxial growth of the 6H polytype has been realized on off-oriented substrates utilizing simultaneous supply of the source gases \cite{tanaka94}.
+However, accurate layer-by-layer growth is achieved under certain conditions, which facilitate the spontaneous desorption of an additional layer of one atom species by supply of the other species~\cite{hara93}.
+Homoepitaxial growth of the 6H polytype has been realized on off-oriented substrates utilizing simultaneous supply of the source gases~\cite{tanaka94}.
Depending on the gas flow ratio either 3C island formation or step flow growth of the 6H polytype occurs, which is explained by a model including aspects of enhanced surface mobilities of adatoms on a $(3\times 3)$ reconstructed surface.
-Due to the strong adsorption of atomic hydrogen \cite{allendorf91} decomposed of the gas phase reactants at low temperatures, however, there seems to be no benefit of GSMBE compared to CVD.
-Next to lattice imperfections, incorporated hydrogen effects the surface mobility of the adsorbed species \cite{eaglesham93} setting a minimum limit for the growth temperature, which would preferably be further decreased in order to obtain sharp doping profiles.
+Due to the strong adsorption of atomic hydrogen~\cite{allendorf91} decomposed of the gas phase reactants at low temperatures, however, there seems to be no benefit of GSMBE compared to CVD.
+Next to lattice imperfections, incorporated hydrogen effects the surface mobility of the adsorbed species~\cite{eaglesham93} setting a minimum limit for the growth temperature, which would preferably be further decreased in order to obtain sharp doping profiles.
Thus, growth rates must be adjusted to be lower than the desorption rate of hydrogen, which leads to very low deposition rates at low temperatures.
SSMBE, by supplying the atomic species to be deposited by evaporation of a solid, presumably constitutes the preferred method in order to avoid the problems mentioned above.
-Although, in the first experiments, temperatures still above \unit[1100]{$^{\circ}$C} were necessary to epitaxially grow 3C-SiC films on 6H-SiC substrates \cite{kaneda87}, subsequent attempts succeeded in growing mixtures of twinned 3C-SiC and 6H-SiC films on off-axis \hkl(0001) 6H-SiC wafers at temperatures between \unit[800]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C} \cite{fissel95,fissel95_apl}.
-In the latter approach, as in GSMBE, excess Si atoms, which are controlled by the Si/C flux ratio, result in the formation of a Si adlayer and the formation of a non-stoichiometric, reconstructed surface superstructure, which influences the mobility of adatoms and, thus, has a decisive influence on the growth mode, polytype and crystallinity \cite{fissel95,fissel96,righi03}.
+Although, in the first experiments, temperatures still above \unit[1100]{$^{\circ}$C} were necessary to epitaxially grow 3C-SiC films on 6H-SiC substrates~\cite{kaneda87}, subsequent attempts succeeded in growing mixtures of twinned 3C-SiC and 6H-SiC films on off-axis \hkl(0001) 6H-SiC wafers at temperatures between \unit[800]{$^{\circ}$C} and \unit[1000]{$^{\circ}$C}~\cite{fissel95,fissel95_apl}.
+In the latter approach, as in GSMBE, excess Si atoms, which are controlled by the Si/C flux ratio, result in the formation of a Si adlayer and the formation of a non-stoichiometric, reconstructed surface superstructure, which influences the mobility of adatoms and, thus, has a decisive influence on the growth mode, polytype and crystallinity~\cite{fissel95,fissel96,righi03}.
Therefore, carefully controlling the Si/C ratio could be exploited to obtain definite heterostructures of different SiC polytypes providing the possibility for band gap engineering in SiC materials.
To summarize, much progress has been achieved in SiC thin film growth during the last few years.
However, the frequent occurrence of defects such as dislocations, twins and double positioning boundaries limit the structural and electrical quality of large SiC films.
-Solving this issue remains a challenging problem necessary to drive SiC for potential applications in high-performance electronic device production \cite{wesch96}.
+Solving this issue remains a challenging problem necessary to drive SiC for potential applications in high-performance electronic device production~\cite{wesch96}.
\subsection{Ion beam synthesis of cubic silicon carbide}
\label{subsection:ibs}
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}.
+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}.
Rutherford backscattering spectrometry (RBS) and infrared (IR) spectroscopy investigations indicate a \unit[10]{at.\%} C concentration peak and the occurrence of disordered C-Si bonds after implantation at room temperature (RT) followed by crystallization into SiC precipitates upon annealing.
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}.
+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}.
-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}.
+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.
The authors, thus, concluded that implantations at elevated temperatures lead to a reduction in the annealing temperatures required for the synthesis of homogeneous layers of SiC.
-In a comparative study of O, N and C implantation into Si, the absence of the formation of a stoichiometric SiC compound layer involving the transition of a Gaussian into a box-like C depth profile with respect to the implantation depth for the superstoichiometric C implantation and an annealing temperature of \unit[1200]{$^{\circ}$C} in contrast to the O and N implantations, which successfully form homogeneous layers, has been observed \cite{reeson86}.
+In a comparative study of O, N and C implantation into Si, the absence of the formation of a stoichiometric SiC compound layer involving the transition of a Gaussian into a box-like C depth profile with respect to the implantation depth for the superstoichiometric C implantation and an annealing temperature of \unit[1200]{$^{\circ}$C} in contrast to the O and N implantations, which successfully form homogeneous layers, has been observed~\cite{reeson86}.
This was attributed to the difference in the enthalpy of formation of the respective compound and the different mobility of the respective impurity in bulk Si.
Thus, higher annealing temperatures and longer annealing times were considered necessary for the formation of homogeneous SiC layers.
-Indeed, for the first time, buried homogeneous and stoichiometric epitaxial 3C-SiC layers embedded in single crystalline Si were obtained by the same group consequently applying annealing temperatures of \unit[1405]{$^{\circ}$C} for \unit[90]{min} and implantation temperatures of approximately \unit[550]{$^{\circ}$C} \cite{reeson87}.
+Indeed, for the first time, buried homogeneous and stoichiometric epitaxial 3C-SiC layers embedded in single crystalline Si were obtained by the same group consequently applying annealing temperatures of \unit[1405]{$^{\circ}$C} for \unit[90]{min} and implantation temperatures of approximately \unit[550]{$^{\circ}$C}~\cite{reeson87}.
The necessity of the applied extreme temperature (a few degrees below the Si melting point) and time scale is attributed to the stability of substitutional C within the Si matrix being responsible for high activation energies necessary to dissolve such precipitates and, thus, allow for redistribution of the implanted C atoms.
-In order to avoid extreme annealing temperatures close to the melting temperature of Si, triple-energy implantations in the range from \unit[180--190]{keV} with stoichiometric doses at a constant target temperature of \unit[860]{$^{\circ}$C} achieved by external substrate heating were performed \cite{martin90}.
+In order to avoid extreme annealing temperatures close to the melting temperature of Si, triple-energy implantations in the range from \unit[180--190]{keV} with stoichiometric doses at a constant target temperature of \unit[860]{$^{\circ}$C} achieved by external substrate heating were performed~\cite{martin90}.
It was shown that a thick buried layer of SiC is directly formed during implantation, which consists of small, only slightly misorientated but severely twinned 3C-SiC crystallites.
The authors assumed that due to the auxiliary heating rather than ion beam heating as employed in all the preceding studies, the complexity of the remaining defects in the synthesized structure is fairly reduced.
-Even better qualities by direct synthesis were obtained for implantations at \unit[950]{$^{\circ}$C} \cite{nejim95}.
+Even better qualities by direct synthesis were obtained for implantations at \unit[950]{$^{\circ}$C}~\cite{nejim95}.
Since no amorphous or polycrystalline regions have been identified, twinning is considered to constitute the main limiting factor in the IBS of SiC.
-Layers obtained by direct synthesis are characterized by rough surfaces of the buried layer and the substrate originating from the dendritic growth of SiC crystals at these temperatures \cite{lindner06}.
+Layers obtained by direct synthesis are characterized by rough surfaces of the buried layer and the substrate originating from the dendritic growth of SiC crystals at these temperatures~\cite{lindner06}.
-Further studies revealed the possibility to form buried layers of SiC by IBS at moderate substrate and anneal temperatures \cite{lindner95,lindner96}.
+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.
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}.
+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.
Furthermore, higher implantation energies were found to result in layers of variable composition exhibiting randomly distributed SiC precipitates.
-In another study \cite{serre95} high dose C implantations were performed at room temperature and \unit[500]{$^{\circ}$C} respectively.
+In another study~\cite{serre95} high dose C implantations were performed at room temperature and \unit[500]{$^{\circ}$C} respectively.
Implantations at room temperature lead to the formation of a buried amorphous carbide layer in addition to a thin C-rich film at the surface, which is attributed to the migration of C atoms towards the surface.
In contrast, implantations at elevated temperatures result in the exclusive formation of a buried layer consisting of 3C-SiC precipitates epitaxially aligned to the Si host, which obviously is more favorable than the C migration towards the surface.
Annealing at temperatures up to \unit[1150]{$^{\circ}$C} does not alter the C profile.
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}.
+Based on these findings 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}.
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}.
+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.
This corresponds to the region of randomly oriented SiC crystallites and epitaxially aligned precipitates surrounded by thin amorphous layers without crystalline SiC inclusions in the as-implanted state.
As expected, single-crystalline layers were achieved for an increased temperature of \unit[600]{$^{\circ}$C}.
However, these layers show an extremely poor interface to the Si top layer governed by a high density of SiC precipitates, which are not affected in the C redistribution during annealing and, thus, responsible for the rough interface.
Hence, to obtain sharp interfaces and single-crystalline SiC layers temperatures between \unit[400]{$^{\circ}$C} and \unit[600]{$^{\circ}$C} have to be used.
-Indeed, reasonable results were obtained at \unit[500]{$^{\circ}$C} \cite{lindner98} and even better interfaces were observed for \unit[450]{$^{\circ}$C} \cite{lindner99_2}.
-To further improve the interface quality and crystallinity a two-temperature implantation technique was developed \cite{lindner99}.
+Indeed, reasonable results were obtained at \unit[500]{$^{\circ}$C}~\cite{lindner98} and even better interfaces were observed for \unit[450]{$^{\circ}$C}~\cite{lindner99_2}.
+To further improve the interface quality and crystallinity a two-temperature implantation technique was developed~\cite{lindner99}.
To form a narrow, box-like density profile of oriented SiC nanocrystals, \unit[93]{\%} of the total dose of \unit[$8.5\cdot 10^{17}$]{cm$^{-2}$} is implanted at \unit[500]{$^{\circ}$C}.
The remaining dose is implanted at \unit[250]{$^{\circ}$C}, which leads to the formation of amorphous zones above and below the SiC precipitate layer and the destruction of SiC nanocrystals within these zones.
After annealing for \unit[10]{h} at \unit[1250]{$^{\circ}$C} a homogeneous, stoichiometric SiC layer with sharp interfaces is formed.
\begin{center}
\includegraphics[width=0.6\columnwidth]{ibs_3c-sic.eps}
\end{center}
-\caption[Bright field and \hkl(1 1 1) SiC dark field cross-sectional TEM micrographs of the buried SiC layer in Si created by the two-temperature implantation technique and subsequent annealing.]{Bright field (a) and \hkl(1 1 1) SiC dark field (b) cross-sectional TEM micrographs of the buried SiC layer in Si created by the two-temperature implantation technique and subsequent annealing as explained in the text \cite{lindner99_2}. The inset shows a selected area diffraction pattern of the buried layer.}
+\caption[Bright field and \hkl(1 1 1) SiC dark field cross-sectional TEM micrographs of the buried SiC layer in Si created by the two-temperature implantation technique and subsequent annealing.]{Bright field (a) and \hkl(1 1 1) SiC dark field (b) cross-sectional TEM micrographs of the buried SiC layer in Si created by the two-temperature implantation technique and subsequent annealing as explained in the text~\cite{lindner99_2}. The inset shows a selected area diffraction pattern of the buried layer.}
\label{fig:sic:hrem_sharp}
\end{figure}
To summarize, by understanding some basic processes, IBS nowadays has become a promising method to form thin SiC layers of high quality exclusively of the 3C polytype embedded in and epitaxially aligned to the Si host featuring a sharp interface.
-Due to the high areal homogeneity achieved in IBS, the size of the layers is only limited by the width of the beam-scanning equipment used in the implantation system as opposed to deposition techniques, which have to deal with severe wafer bending.
+Due to the high areal homogeneity achieved in IBS, the size of the layers is only limited by the width of the beam-scanning equipment used in the implantation system.
+% as opposed to deposition techniques, which have to deal with severe wafer bending.
+Buried layers can be revealed at the surface by additional implantation and etching steps~\cite{attenberger03}.
+They do not exhibit surface bending effects in contrast to these formed by the MBE or CVD technique.
This enables the synthesis of large area SiC films.
\section{Substoichiometric concentrations of carbon in crystalline silicon}
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}.
+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}.
-The induced strain is assumed to be responsible for the low solid solubility of C in Si, which was determined \cite{bean71} to be
+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}.
+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$^{-3}$}
\cdot\exp(-2.3\,\text{eV/$k_{\text{B}}T$})
\text{ .}
\end{equation}
-The barrier of diffusion of substitutional C has been determined to be around \unit[3]{eV} \cite{newman61}.
+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.
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.
+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.
-This structure, the so called C-Si \hkl<1 0 0> dumbbell structure, was initially suspected by local vibrational mode absorption \cite{bean70} and finally verified by electron paramagnetic resonance \cite{watkins76} studies on irradiated Si substrates at low temperatures.
-Measuring the annealing rate of the defect as a function of temperature reveals barriers for migration ranging from \unit[0.73]{eV} \cite{song90} to \unit[0.87]{eV} \cite{tipping87}, which is highly mobile compared to substitutional C.
+This structure, the so called C-Si \hkl<1 0 0> dumbbell structure, was initially suspected by local vibrational mode absorption~\cite{bean70} and finally verified by electron paramagnetic resonance~\cite{watkins76} studies on irradiated Si substrates at low temperatures.
+Measuring the annealing rate of the defect as a function of temperature reveals barriers for migration of \unit[0.70]{eV}~\cite{lindner06}, \unit[0.73]{eV}~\cite{song90} and \unit[0.87]{eV}~\cite{tipping87}, which is highly mobile compared to substitutional C.
% diffusion pathway?
% expansion of the lattice (positive strain)
\subsection{Suppression of transient enhanced diffusion of dopant species}
-The predominant diffusion mechanism of most dopants in Si based on native self-interstitials \cite{fahey89} has a large impact on the diffusion behavior of dopants that have been implanted in Si.
+The predominant diffusion mechanism of most dopants in Si based on native self-interstitials~\cite{fahey89} has a large impact on the diffusion behavior of dopants that have been implanted in Si.
The excess population of Si self-interstitials created by low-energy implantations of dopants for shallow junction formation in submicron technologies may enhance the diffusion of the respective dopant during annealing by more than one order of magnitude compared to normal diffusion.
-This kind of diffusion, labeled transient enhanced diffusion (TED), which is driven by the presence of non-equilibrium concentrations of point defects, was first discovered for implantations of boron in Si \cite{hofker74} and is well understood today \cite{michel87,cowern90,stolk95,stolk97}.
-The TED of B was found to be inhibited in the presence of a sufficient amount of incorporated C \cite{cowern96}.
-This is due to the reduction of the excess Si self-interstitials with substitutional C atoms forming the C-Si interstitial complex \cite{stolk97,zhu98}.
+This kind of diffusion, labeled transient enhanced diffusion (TED), which is driven by the presence of non-equilibrium concentrations of point defects, was first discovered for implantations of boron in Si~\cite{hofker74} and is well understood today~\cite{michel87,cowern90,stolk95,stolk97}.
+The TED of B was found to be inhibited in the presence of a sufficient amount of incorporated C~\cite{cowern96}.
+This is due to the reduction of the excess Si self-interstitials with substitutional C atoms forming the C-Si interstitial complex~\cite{stolk97,zhu98}.
Therefore, incorporation of C provides a promising method for suppressing TED enabling an improved shallow junction formation in future Si devices.
% in general: high c diffusion in areas of high damage, low diffusion for substitutional or even sic prec
% 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.
-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}.
+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.
% 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.
-Thus, substitutional C enables strain engineering of Si and Si/Si$_{1-x}$Ge$_x$ heterostructures \cite{yagi02,chang05,kissinger94,osten97}, which is used to increase charge carrier mobilities in Si as well as to adjust its band structure \cite{soref91,kasper91}.
+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.
+Thus, substitutional C enables strain engineering of Si and Si/Si$_{1-x}$Ge$_x$ heterostructures~\cite{yagi02,chang05,kissinger94,osten97}, which is used to increase charge carrier mobilities in Si as well as to adjust its band structure~\cite{soref91,kasper91}.
% increase of C at substitutional sites
-Epitaxial layers with \unit[1.4]{at.\%} of substitutional C have been successfully synthesized in preamorphized Si$_{0.86}$Ge$_{0.14}$ layers, which were grown by CVD on Si substrates, using multiple-energy C implantation followed by solid-phase epitaxial regrowth at \unit[700]{$^{\circ}$C} \cite{strane93}.
+Epitaxial layers with \unit[1.4]{at.\%} of substitutional C have been successfully synthesized in preamorphized Si$_{0.86}$Ge$_{0.14}$ layers, which were grown by CVD on Si substrates, using multiple-energy C implantation followed by solid-phase epitaxial regrowth at \unit[700]{$^{\circ}$C}~\cite{strane93}.
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}.
+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}.
-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 dependence of the growth temperature on 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}.
+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.
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) show 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}.
+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 precipitates with diameters between \unit[2]{nm} and \unit[5]{nm} are incorporated in the Si matrix without any remarkable strain fields, which is explained by the nearly equal atomic density of C-Si agglomerates and the SiC unit cell.
-Implantations at \unit[500]{$^{\circ}$C} likewise suggest an initial formation of C-Si dumbbells on regular Si lattice sites, which agglomerate into large clusters \cite{lindner99_2}.
-The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential \cite{werner96,werner97}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}.
+Implantations at \unit[500]{$^{\circ}$C} likewise suggest an initial formation of C-Si dumbbells on regular Si lattice sites, which agglomerate into large clusters~\cite{lindner99_2}.
+The agglomerates of such dimers, which do not generate lattice strain but lead to a local increase of the lattice potential~\cite{werner96,werner97}, are indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM, as can be seen in Fig.~\ref{fig:sic:hrem:c-si}.
\begin{figure}[t]
\begin{center}
\subfigure[]{\label{fig:sic:hrem:c-si}\includegraphics[width=0.25\columnwidth]{tem_c-si-db.eps}}
\subfigure[]{\label{fig:sic:hrem:sic}\includegraphics[width=0.25\columnwidth]{tem_3c-sic.eps}}
\end{center}
-\caption[High resolution transmission electron microscopy (HREM) micrographs of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes and equally sized Moir\'e patterns indicating 3C-SiC precipitates.]{High resolution transmission electron microscopy (HREM) micrographs \cite{lindner99_2} of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes (a) and equally sized Moir\'e patterns indicating 3C-SiC precipitates (b).}
+\caption[High resolution transmission electron microscopy (HREM) micrographs of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes and equally sized Moir\'e patterns indicating 3C-SiC precipitates.]{High resolution transmission electron microscopy (HREM) micrographs~\cite{lindner99_2} of agglomerates of C-Si dimers showing dark contrasts and otherwise undisturbed Si lattice fringes (a) and equally sized Moir\'e patterns indicating 3C-SiC precipitates (b).}
\label{fig:sic:hrem}
\end{figure}
A topotactic transformation into a 3C-SiC precipitate occurs once a critical radius of \unit[2]{nm} to \unit[4]{nm} is reached.
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}.
+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.
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]
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.
-In contrast, IR spectroscopy and HREM investigations on the thermal stability of strained Si$_{1-y}$C$_y$/Si heterostructures formed by solid-phase epitaxy (SPE) \cite{strane94} and MBE \cite{guedj98}, which finally involve the incidental formation of SiC nanocrystallites, suggest a coherent initiation of precipitation by agglomeration of substitutional instead of interstitial C.
+In contrast, IR spectroscopy and HREM investigations on the thermal stability of strained Si$_{1-y}$C$_y$/Si heterostructures formed by solid-phase epitaxy (SPE)~\cite{strane94} and MBE~\cite{guedj98}, which finally involve the incidental formation of SiC nanocrystallites, suggest a coherent initiation of precipitation by agglomeration of substitutional instead of interstitial C.
These experiments show that the C atoms, which are initially incorporated substitutionally at regular lattice sites, form C-rich clusters maintaining coherency with the Si lattice during annealing above a critical temperature prior to the transition into incoherent 3C-SiC precipitates.
Increased temperatures in the annealing process enable the diffusion and agglomeration of C atoms.
Coherency is lost once the increasing strain energy of the stretched SiC structure surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the Si substrate.
-Estimates of the SiC/Si interfacial energy \cite{taylor93} and the consequent critical size correspond well with the experimentally observed precipitate radii within these studies.
+Estimates of the SiC/Si interfacial energy~\cite{taylor93} and the consequent critical size correspond well with the experimentally observed precipitate radii within these studies.
This different mechanism of precipitation might be attributed to the respective method of fabrication.
While in CVD and MBE, surface effects need to be taken into account, SiC formation during IBS takes place in the bulk of the Si crystal.
-However, in another IBS study Nejim et~al. \cite{nejim95} propose a topotactic transformation that is likewise based on substitutional C, which replaces four of the eight Si atoms in the Si unit cell accompanied by the generation of four Si interstitials.
-Since the emerging strain due to the expected volume reduction of \unit[48]{\%} would result in the formation of dislocations, which, however, are not observed, the interstitial Si is assumed to react with further implanted C atoms in the released volume.
+However, in another IBS study Nejim et~al.~\cite{nejim95} propose a
+% topotactic
+transformation that is likewise based on substitutional C, which replaces four of the eight Si atoms in the Si unit cell accompanied by the generation of four Si interstitials.
+The replacement of a Si unit cell by a 3C-SiC unit cell is accompanied by a volume reduction of \unit[48]{\%} due to the \unit[20]{\%} lower lattice constant.
+Since the emerging strain caused by the expected volume reduction would result in the formation of dislocations, which, however, are not observed, the interstitial Si is assumed to react with further implanted C atoms in the released volume.
The resulting strain due to the slightly lower Si density of SiC compared to Si of about \unit[3]{\%} is sufficiently small to legitimate the absence of dislocations.
+However, the exact atomic rearrangement involved within this topotactic transformation is not identified.
Furthermore, IBS studies of Reeson~et~al.~\cite{reeson87}, in which implantation temperatures of \unit[500]{$^{\circ}$C} were employed, revealed the necessity of extreme annealing temperatures for C redistribution, which is assumed to result from the stability of substitutional C and consequently high activation energies required for precipitate dissolution.
The results support a mechanism of an initial coherent precipitation based on substitutional C that is likewise valid for the IBS of 3C-SiC by C implantation into Si at elevated temperatures.
The fact that the metastable cubic phase instead of the thermodynamically more favorable hexagonal $\alpha$-SiC structure is formed and the alignment of these cubic precipitates within the Si matrix, which can be explained by considering a topotactic transformation by C atoms occupying substitutionally Si lattice sites of one of the two fcc lattices that make up the Si crystal, reinforce the proposed mechanism.
\section{DFT calculations}
\label{section:simulation:dft_calc}
-The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp}) \cite{kresse96}.
-The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91) \cite{perdew86,perdew92}.
-The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp} \cite{vanderbilt90}.
+The first-principles DFT calculations are performed with the plane-wave-based Vienna {\em ab initio} simulation package (\textsc{vasp})~\cite{kresse96}.
+The Kohn-Sham equations are solved using the GGA utilizing the exchange-correlation functional proposed by Perdew and Wang (GGA-PW91)~\cite{perdew86,perdew92}.
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials as implemented in \textsc{vasp}~\cite{vanderbilt90}.
An energy cut-off of \unit[300]{eV} is used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone is restricted to the $\Gamma$ point.
Spin polarization has been fully accounted for.
-The electronic ground state is calculated by an iterative Davidson scheme \cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
+The electronic ground state is calculated by an iterative Davidson scheme~\cite{davidson75} until the difference in total energy of two subsequent iterations is below \unit[$10^{-4}$]{eV}.
Defect structures and the migration paths have been modeled in cubic supercells of type 3 containing 216 Si atoms.
The conjugate gradient algorithm is used for ionic relaxation.
Migration paths are determined by the modified version of the CRT method as explained in section \ref{section:basics:migration}.
-The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman \cite{parrinello81} in order to realize constant pressure simulations.
+The cell volume and shape is allowed to change using the pressure control algorithm of Parrinello and Rahman~\cite{parrinello81} in order to realize constant pressure simulations.
Due to restrictions by the \textsc{vasp} code, {\em ab initio} MD could only be performed at constant volume.
In MD simulations the equations of motion are integrated by a fourth order predictor corrector algorithm for a time step of \unit[1]{fs}.
\label{fig:simulation:ef_ss}
\end{figure}
To estimate a critical size the formation energies of several intrinsic defects in Si with respect to the system size are calculated.
-An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh \cite{monkhorst76} is used.
+An energy cut-off of \unit[250]{eV} and a $4\times4\times4$ Monkhorst-Pack $k$-point mesh~\cite{monkhorst76} is used.
The results are displayed in Fig.~\ref{fig:simulation:ef_ss}.
The formation energies converge fast with respect to the system size.
Thus, investigating supercells containing more than 56 primitive cells or $112\pm1$ atoms should be reasonably accurate.
Throughout this work sampling of the BZ is restricted to the $\Gamma$ point.
The calculation is usually two times faster and half of the storage needed for the wave functions can be saved since $c_{i,q}=c_{i,-q}^*$, where the $c_{i,q}$ are the Fourier coefficients of the wave function.
As discussed in section \ref{subsection:basics:bzs} this does not pose a severe limitation if the supercell is large enough.
-Indeed, it was shown \cite{dal_pino93} that already for calculations involving only 32 atoms energy values obtained by sampling the $\Gamma$ point differ by less than \unit[0.02]{eV} from calculations using the Baldereschi point \cite{baldereschi73}, which constitutes a mean-value point in the BZ.
+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.
\subsection{Energy cut-off}
To find the most suitable combination of potential and XC functional for the C/Si system a $2\times2\times2$ supercell of type 3 of Si and C, both in the diamond structure, as well as 3C-SiC is equilibrated for different combinations of the available potentials and XC functionals.
To exclude a possibly corrupting influence of the other parameters highly accurate calculations are performed, i.e.\ an energy cut-off of \unit[650]{eV} and a $6\times6\times6$ Monkhorst-Pack $k$-point mesh is used.
-Next to the ultra-soft pseudopotentials \cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW) \cite{bloechl94} to describe the ion-electron interaction.
-The two XC functionals included in the test are of the LDA \cite{ceperley80,perdew81} and GGA \cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
+Next to the ultra-soft pseudopotentials~\cite{vanderbilt90} \textsc{vasp} offers the projector augmented-wave method (PAW)~\cite{bloechl94} to describe the ion-electron interaction.
+The two XC functionals included in the test are of the LDA~\cite{ceperley80,perdew81} and GGA~\cite{perdew86,perdew92} type as implemented in \textsc{vasp}.
\begin{table}[t]
\begin{center}
Thus, the method is capable of performing structural optimizations on large systems and MD calculations may be used to model a system over long time scales.
Defect structures are modeled in a cubic supercell (type 3) of nine Si lattice constants in each direction containing 5832 Si atoms.
Reproducing the SiC precipitation was attempted in cubic c-Si supercells, which have a size up to 31 Si unit cells in each direction consisting of 238328 Si atoms.
-A Tersoff-like bond order potential by Erhart and Albe (EA) \cite{albe_sic_pot} is used to describe the atomic interaction.
-Constant pressure simulations are realized by the Berendsen barostat \cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
-The temperature is controlled by the Berendsen thermostat \cite{berendsen84} with a time constant of \unit[100]{fs}.
-Integration of the equations of motion is realized by the velocity Verlet algorithm \cite{verlet67} using a fixed time step of \unit[1]{fs}.
+A Tersoff-like bond order potential by Erhart and Albe (EA)~\cite{albe_sic_pot} is used to describe the atomic interaction.
+Constant pressure simulations are realized by the Berendsen barostat~\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
+The temperature is controlled by the Berendsen thermostat~\cite{berendsen84} with a time constant of \unit[100]{fs}.
+Integration of the equations of motion is realized by the velocity Verlet algorithm~\cite{verlet67} using a fixed time step of \unit[1]{fs}.
For structural relaxation of defect structures the same algorithm is utilized with the temperature set to zero Kelvin.
This also applies for the relaxation of structures within the CRT calculations to find migration pathways.
In the latter case the time constant of the Berendsen thermostat is set to \unit[1]{fs} in order to achieve direct velocity scaling, which corresponds to a steepest descent minimization driving the system into a local minimum, if the temperature is set to zero Kelvin.
Defect structures as well as the simulations modeling the SiC precipitation are performed in the isothermal-isobaric $NpT$ ensemble.
In addition to the bond order formalism the EA potential provides a set of parameters to describe the interaction in the C/Si system, as discussed in section \ref{subsection:interact_pot}.
-There are basically no free parameters, which could be set by the user and the properties of the potential and its parameters are well known and have been extensively tested by the authors \cite{albe_sic_pot}.
+There are basically no free parameters, which could be set by the user and the properties of the potential and its parameters are well known and have been extensively tested by the authors~\cite{albe_sic_pot}.
Therefore, test calculations are restricted to the time step used in the Verlet algorithm to integrate the equations of motion.
Nevertheless, a further and rather uncommon test is carried out to roughly estimate the capabilities of the EA potential regarding the description of 3C-SiC precipitation in c-Si.
An interfacial energy of \unit[2267.28]{eV} is obtained.
The amount of C atoms together with the observed lattice constant of the precipitate leads to a precipitate radius of \unit[29.93]{\AA}.
Thus, the interface tension, given by the energy of the interface divided by the surface area of the precipitate is \unit[20.15]{eV/nm$^2$} or \unit[$3.23\times 10^{-4}$]{J/cm$^2$}.
-This value perfectly fits within the experimentally estimated range of \unit[2--8$\times10^{-4}$]{J/cm$^2$} \cite{taylor93}.
+This value perfectly fits within the experimentally estimated range of \unit[2--8$\times10^{-4}$]{J/cm$^2$}~\cite{taylor93}.
Thus, the EA potential is considered an appropriate choice for the current study concerning the accurate description of the energetics of interfaces.
Furthermore, since the calculated interfacial energy is located in the lower part of the experimental range, the obtained interface structure might resemble an authentic configuration of an energetically favorable interface structure of a 3C-SiC precipitate in c-Si.
Next to the shortcomings of the potential, quirks inherent to MD are discussed and a workaround is proposed.
Although direct formation of SiC fails to appear, the obtained results indicate a mechanism of precipitation, which is consistent with previous quantum-mechanical conclusions as well as experimental findings.
-Quantum-mechanical results of intrinsic point defects in Si are in good agreement to previous theoretical work on this subject \cite{leung99,al-mushadani03}.
+Quantum-mechanical results of intrinsic point defects in Si are in good agreement to previous theoretical work on this subject~\cite{leung99,al-mushadani03}.
The \si{} \hkl<1 1 0> DB defect is reproduced as the ground-state configuration followed by the hexagonal and tetrahedral defect.
Spin polarized calculations are required for the \si{} \hkl<1 0 0> DB and vacancy whereas no other of the investigated intrinsic defects is affected.
For the \si{} \hkl<1 0 0> DB, the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms.
Results obtained by calculations utilizing the classical EA potential yield formation energies, which are of the same order of magnitude.
However, EA predicts the tetrahedral configuration to be most stable.
The particular problem is due to the cut-off and the fact that the second neighbors are only slightly more distant than the first neighbors within the tetrahedral configuration.
-Furthermore, the hexagonal defect structure is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
+Furthermore, the hexagonal defect structure is not stable opposed to results of the authors of the potential~\cite{albe_sic_pot}.
The obtained structure after relaxation, which is similar to the tetrahedral configuration, exhibits a formation energy equal to the one given by the authors for the hexagonal one.
Obviously, the authors did not check the relaxed structure still assuming a hexagonal configuration.
The actual structure is equal to the tetrahedral configuration, which is slightly displaced along the three coordinate axes.
Variations exist with displacements along two or a single \hkl<1 0 0> direction indicating a potential artifact.
However, finite temperature simulations are not affected by this artifact due to a low activation energy necessary for a transition into the energetically more favorable tetrahedral configuration.
-Next to the known problem of the underestimated formation energy of the tetrahedral configuration \cite{tersoff90}, the energetic sequence of the defect structures is well reproduced by the EA calculations.
-Migration barriers of \si{} investigated by quantum-mechanical calculations are found to be of the same order of magnitude than values derived in other {\em ab initio} studies \cite{bloechl93,sahli05}.
+Next to the known problem of the underestimated formation energy of the tetrahedral configuration~\cite{tersoff90}, the energetic sequence of the defect structures is well reproduced by the EA calculations.
+Migration barriers of \si{} investigated by quantum-mechanical calculations are found to be of the same order of magnitude than values derived in other {\em ab initio} studies~\cite{bloechl93,sahli05}.
Defects of C in Si are well described by both methods.
-The \ci{} \hkl<1 0 0> DB is found to constitute the most favorable interstitial configuration in agreement with several theoretical \cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental \cite{watkins76,song90} investigations.
+The \ci{} \hkl<1 0 0> DB is found to constitute the most favorable interstitial configuration in agreement with several theoretical~\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental~\cite{watkins76,song90} investigations.
Almost equal formation energies are predicted by the EA and DFT calculations for this defect.
A small discrepancy in the resulting equilibrium structure with respect to the DFT method exists due to missing quantum-mechanical effects within the classical treatment.
The high formation energies of the tetrahedral and hexagonal configuration obtained by classical potentials act in concert with the fact that these configurations are found unstable by the first-principles description.
The BC configuration turns out to be unstable relaxing into the \ci{} \hkl<1 1 0> DB configuration within the EA approach.
-This is supported by another {\em ab initio} study \cite{capaz94}, which in turn finds the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
+This is supported by another {\em ab initio} study~\cite{capaz94}, which in turn finds the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
By quantum-mechanical calculations performed in this work, however, it turns out that the BC configuration constitutes a real local minimum if the electron spin is fully accounted for.
Indeed, spin polarization is absolutely necessary for the BC configuration resulting in a net magnetization of two electrons accompanied by a reduction of the total energy by \unit[0.3]{eV}.
The resulting spin up density is localized in a torus around the C perpendicular to the linear Si-C-Si bond.
Thus, the underestimated formation energy does not pose a serious limitation.
Based on the above findings, it is concluded that modeling of the SiC precipitation by the EA potential might lead to trustable results.
-Quantum-mechanical investigations of the mobility of the \ci{} \hkl<1 0 0> DB yield a migration barrier of \unit[0.9]{eV}, which excellently agrees to experimental values ranging from \unit[0.70]{eV} to \unit[0.87]{eV} \cite{lindner06,song90,tipping87}.
+Quantum-mechanical investigations of the mobility of the \ci{} \hkl<1 0 0> DB yield a migration barrier of \unit[0.9]{eV}, which excellently agrees to experimental values ranging from \unit[0.70]{eV} to \unit[0.87]{eV}~\cite{lindner06,song90,tipping87}.
The respective path corresponds to a \ci{} \hkl[0 0 -1] DB migrating towards the next neighbored Si atom located in \hkl[1 1 -1] direction forming a \ci{} \hkl[0 -1 0] DB.
The identified migration path involves a change in orientation of the DB.
-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}.
+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.
Results of combinations of \ci{} and \cs{} revealed two additional metastable structures different to these obtained by a naive relaxation.
Small displacements result in a structure of a \hkl<1 1 0> C-C DB and in a structure of a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
Both structures are lower in energy compared to the respective counterparts.
-These results, for the most part, compare well with results gained in previous studies \cite{leary97,capaz98,liu02} and show an astonishingly good agreement with experiment \cite{song90_2}.
+These results, for the most part, compare well with results gained in previous studies~\cite{leary97,capaz98,liu02} and show an astonishingly good agreement with experiment~\cite{song90_2}.
Again, spin polarized calculations are revealed necessary.
A net magnetization of two electrons is observed for the \hkl<1 1 0> C-C DB configuration, which constitutes the ground state.
A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0] due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom.
The approach of using increased temperatures during C insertion is followed to work around this problem termed {\em potential enhanced slow phase space propagation}.
Higher temperatures are justified for several reasons.
Elevated temperatures are expected to compensate the overestimated diffusion barriers and accelerate structural evolution.
-In addition, formation of SiC is also observed at higher implantation temperatures \cite{nejim95,lindner01} and temperatures in the implantation region is definitely higher than the temperature determined experimentally at the surface of the sample.
+In addition, formation of SiC is also observed at higher implantation temperatures~\cite{nejim95,lindner01} and temperatures in the implantation region is definitely higher than the temperature determined experimentally at the surface of the sample.
Furthermore, the present study focuses on structural transitions in a system far from equilibrium.
No significant change is observed for high C concentrations at increased temperatures.
These findings as well as the derived conclusion on the precipitation mechanism involving an increased participation of \cs{} agree well with experimental results.
% low t high mobility
% high t stable config, no redistr
-C implanted at room temperature was found to be able to migrate towards the surface in contrast to implantations at \degc{500}, which do not show redistribution of the C atoms \cite{serre95}.
+C implanted at room temperature was found to be able to migrate towards the surface in contrast to implantations at \degc{500}, which do not show redistribution of the C atoms~\cite{serre95}.
This excellently conforms to the results of the MD simulations at different temperatures, which show the formation of highly mobile \ci{} \hkl<1 0 0> DBs for low and much more stable \cs{} defects for high temperatures.
-This is likewise suggested by IBS experiments utilizing implantation temperatures of \degc{550} followed by incoherent lamp annealing at temperatures as high as \degc{1405} required for the C segregation due to the stability of \cs{} \cite{reeson87}.
+This is likewise suggested by IBS experiments utilizing implantation temperatures of \degc{550} followed by incoherent lamp annealing at temperatures as high as \degc{1405} required for the C segregation due to the stability of \cs{}~\cite{reeson87}.
% high imp temps more effective to achieve ?!? ...
-Furthermore, increased implantation temperatures were found to be more efficient than high temperatures in the postannealing step concerning the formation of topotactically aligned 3C-SiC precipitates \cite{kimura82,eichhorn02}.
+Furthermore, increased implantation temperatures were found to be more efficient than high temperatures in the postannealing step concerning the formation of topotactically aligned 3C-SiC precipitates~\cite{kimura82,eichhorn02}.
%
-Particularly strong C-C bonds, which are hard to break by thermal annealing, were found to effectively aggravate the restructuring process of such configurations \cite{deguchi92}.
+Particularly strong C-C bonds, which are hard to break by thermal annealing, were found to effectively aggravate the restructuring process of such configurations~\cite{deguchi92}.
These bonds preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements in regions exhibiting a large amount of C atoms.
This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations.
%
%Considering the efficiency of high implantation temperatures, experimental arguments exist, which point to the precipitation mechanism based on the agglomeration of \cs.
-More substantially, understoichiometric implantations at room temperature into preamorphized Si followed by a solid-phase epitaxial regrowth step at \degc{700} result in Si$_{1-x}$C$_x$ layers in the diamond cubic phase with C residing on substitutional Si lattice sites \cite{strane93}.
+More substantially, understoichiometric implantations at room temperature into preamorphized Si followed by a solid-phase epitaxial regrowth step at \degc{700} result in Si$_{1-x}$C$_x$ layers in the diamond cubic phase with C residing on substitutional Si lattice sites~\cite{strane93}.
The strained structure is found to be stable up to \degc{810}.
Coherent clustering followed by precipitation is suggested if these structures are annealed at higher temperatures.
%
-Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C} \cite{werner96} annealing temperature.
-Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples \cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
+Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C}~\cite{werner96} annealing temperature.
+Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples~\cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
According to this, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
%
%
% low t - randomly ...
% high t - epitaxial relation ...
-Moreover, implantations below the optimum temperature for the IBS of SiC show regions of randomly oriented SiC crystallites whereas epitaxial crystallites are found for increased temperatures \cite{lindner99}.
+Moreover, implantations below the optimum temperature for the IBS of SiC show regions of randomly oriented SiC crystallites whereas epitaxial crystallites are found for increased temperatures~\cite{lindner99}.
The results of the MD simulations can be interpreted in terms of these experimental findings.
The successive occupation of regular Si lattice sites by \cs{} atoms as observed in the high temperature MD simulations and assumed from results of the quantum-mechanical investigations perfectly satisfies the epitaxial relation of substrate and precipitate.
In contrast, there is no obvious reason for a topotactic transition of \ci{} \hkl<1 0 0> DB agglomerates, as observed in the low temperature MD simulations, into epitaxially aligned precipitates.
The latter transition would necessarily involve a much more profound change in structure.
% amorphous region for low temperatures
-Experimentally, randomly oriented precipitates might also be due to SiC nucleation within the arising amorphous matrix \cite{lindner99}.
+Experimentally, randomly oriented precipitates might also be due to SiC nucleation within the arising amorphous matrix~\cite{lindner99}.
In simulation, an amorphous SiC phase is formed for high C concentrations.
This is due to high amounts of introduced damage within a short period of time resulting in essentially no time for structural evolution, which is comparable to the low temperature experiments, which lack the kinetic energy necessary for recrystallization of the highly damaged region.
Indeed, the complex transformation of agglomerated \ci{} DBs as suggested by results of the low C concentration simulations could involve an intermediate amorphous phase probably accompanied by the loss of alignment with respect to the Si host matrix.
In any case, the precipitation mechanism by accumulation of \cs{} obviously satisfies the experimental finding of identical \hkl(h k l) planes of substrate and precipitate.
% no contradictions, something in interstitial lattice, projected potential ...
-Finally, it is worth to point out that the precipitation mechanism based on \cs{} does not necessarily contradict to results of the HREM studies \cite{werner96,werner97,lindner99_2}, which propose precipitation by agglomeration of \ci.
+Finally, it is worth to point out that the precipitation mechanism based on \cs{} does not necessarily contradict to results of the HREM studies~\cite{werner96,werner97,lindner99_2}, which propose precipitation by agglomeration of \ci.
In these studies, regions of dark contrasts are attributed to C atoms that reside in the interstitial lattice in an otherwise undisturbed Si lattice.
The \ci{} atoms lead to a local increase of the crystal potential, which is responsible for the dark contrast.
However, there is no particular reason for the C species to reside in the interstitial lattice.
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