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\title{Combined ab initio and classical potential simulation study on the silicon carbide precipitation in silcon}
\author{F. Zirkelbach}
\author{B. Stritzker}
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}
\author{K. Nordlund}
\affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}
\author{J. K. N. Lindner}
\author{W. G. Schmidt}
\author{E. Rauls}
\affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}

\begin{abstract}
Atomistic simulations on the silicon carbide precipitation in bulk silicon employing both, classical potential and first-principles methods are presented.
For the quantum-mechanical treatment basic processes assumed in the precipitation process are mapped to feasible systems of small size.
Results of the accurate first-principles calculations on the carbon diffusion in silicon are compared to results of calssical potential simulations revealing significant limitations of the latter method.
An approach to work around this problem is proposed.
Finally results of the classical potential molecular dynamics simulations of large systems are discussed.
\end{abstract}

\keywords{point defects, migration, interstitials, first-principles calculations, classical potentials, ... more ...}
\pacs{61.72.uf,66.30.-h,31.15.A-,34.20.Cf, ... more ...}
\maketitle

%  --------------------------------------------------------------------------------
\section{Introduction}

% TOOD: redo complete intro!

The wide band gap semiconductor silicon carbide (SiC) is well known for its outstanding physical and chemical properties.
The high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with the unique thermal and mechanical stability as well as radiation hardness makes SiC a suitable material for high-temperature, high-frequency and high-power devices operational in harsh and radiation-hard environments\cite{edgar92,morkoc94,wesch96,capano97,park98}.
Different modifications of SiC exist, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
Different polytypes exhibit different properties, in which the cubic phase of SiC (3C-SiC) shows increased values for the thermal conductivity and breakdown field compared to other polytypes\cite{wesch96}, which is, thus, most effective for high-performance electronic devices.
Much progress has been made in 3C-SiC thin film growth by chemical vapor deposition (CVD) and molecular beam epitaxy (MBE) on hexagonal SiC\cite{powell90,fissel95,fissel95_apl} and Si\cite{nishino83,nishino87,kitabatake93,fissel95_apl} substrates.
Howeve, the frequent occurrence of defects such as twins, dislocations and double position boundaries remains a challenging problem. 
Next to these methods, 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}.
Utilized and enhanced, ion beam synthesis (IBS) has become a promising method to form thin SiC layers of high quality and exclusively of the 3C polytype embedded in and epitactically aligned to the Si host featuring a sharp interface\cite{lindner99,lindner01,lindner02}.

However, only little is known about the SiC conversion in C implanted Si.
High resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} suggest the formation of C-Si dimers (dumbbells) on regular Si lattice sites, which agglomerate into large clusters indicated by dark contrasts and otherwise undisturbed Si lattice fringes in HREM.
A topotactic transformation into a 3C-SiC precipitate occurs once a critical radius of 2 nm to 4 nm is reached, which is manifested by the disappearance of the dark contrasts in favor of Moir\'e patterns due to the lattice mismatch of \unit[20]{\%} of the 3C-SiC precipitate and c-Si.
The insignificantly lower Si density of SiC ($\approx \unit[4]{\%}$) compared to c-Si results in the emission of only a few excess Si atoms.
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 Nejim et~al.\cite{nejim95} propose a topotactic transformation that is likewise based on the formation of substitutional C.
The formation of substitutional C, however, is accompanied by Si self-interstitial atoms that previously occupied the lattice sites and concurrently by a reduction of volume due to the lower lattice constant of SiC compared to Si.
Both processes are believed to compensate each other.

Solving this controversy and understanding the effective underlying processes will enable significant technological progress in 3C-SiC thin film formation driving the superior polytype for potential applications in high-performance electronic device production.
It will likewise offer perspectives for processes that rely upon prevention of precipitation events, e.g. the fabrication of strained pseudomorphic Si$_{1-y}$C$_y$ heterostructures\cite{strane96,laveant2002}.

Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.
In particular, molecular dynamics (MD) constitutes a suitable technique to investigate the dynamical and structural properties of some material.
Modelling the processes mentioned above requires the simulation of a large amount of atoms ($\approx 10^5-10^6$), which inevitably dictates the atomic interaction to be described by computationally efficient classical potentials.
These are, however, less accurate compared to quantum-mechnical methods and theire applicability for the description of the physical problem has to be verified first.
The most common empirical potentials for covalent systems are the Stillinger-Weber\cite{stillinger85} (SW), Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic potential (EDIP)\cite{bazant96,bazant97,justo98}.
These potentials are assumed to be reliable for large-scale simulations\cite{balamane92,huang95,godet03} on specific problems under investigation providing insight into phenomena that are otherwise not accessible by experimental or first-principles methods.
Until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system within the mentioned interaction models did only exist for the Tersoff\cite{tersoff_m} and related potentials, e.g. the one by Gao and Weber\cite{gao02} as well as the one by Erhart and Albe\cite{albe_sic_pot}.
All these potentials are short range potentials employing a cut-off function,  which drops the atomic interaction to zero inbetween the first and second next neighbor distance.
In a combined ab initio and empirical potential study it was shown that the Tersoff potential properly describes binding energies of combinations of C defects in Si\cite{mattoni2002}.
However, investigations of brittleness in covalent materials\cite{mattoni2007} identified the short range character of these potentials to be responsible for overestimated forces necessary to snap the bond of two next neighbored atoms.
In a previous study\cite{zirkelbach10a} we approved explicitly the influence on the migration barrier for C diffusion in Si.
Using the Erhart/Albe (EA) potential\cite{albe_sic_pot} an overestimated barrier height compared to ab initio calculations and experiment is obtained.
A proper edscription of C diffusion, however, is crucial for the problem under study.

In this work, a combined ab initio and empirical potential simulation study on the initially mentioned SiC precipitation mechanism has been performed.
High accurate quantum-mechanical results have been used to identify shortcomings of the classical potentials, which are then taken into account in these type of simulations.

%  --------------------------------------------------------------------------------
\section{Methodology}
% ----- DFT ------
The first-principles DFT calculations have been performed with the plane-wave based Vienna ab initio Simulation package (VASP)\cite{kresse96}.
The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.
The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.
Throughout this work, an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.
Sampling of the Brillouin zone was restricted to the $\Gamma$-point.
The defect structures and the migration paths have been modeled in cubic supercells containing 216 Si atoms.
The ions and cell shape were allowed to change in order to realize a constant pressure simulation.
Spin polarization has been fully accounted for.

% ------ Albe potential ---------
For the classical potential calculations, defect structures were modeled in a supercell of nine Si lattice constants in each direction consisting of 5832 Si atoms.
Reproducing the SiC precipitation was attempted by the successive insertion of 6000 C atoms (the number necessary to form a 3C-SiC precipitate with a radius of $\approx 3.1$ nm) into the Si host, which has a size of 31 Si unit cells in each direction consisting of 238328 Si atoms.
At constant temperature 10 atoms are inserted at a time.
Three different regions within the total simulation volume are considered for a statistically distributed insertion of the C atoms: $V_1$ corresponding to the total simulation volume, $V_2$ corresponding to the size of the precipitate and $V_3$, whih holds the necessary amount of Si atoms of the precipitate.
A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor distance.
The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
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 kept constant by the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs}.
Integration of equations of motion is realized by the velocity Verlet algorithm\cite{verlet67} and a fixed time step of \unit[1]{fs}.
For structural relaxation of defect structures the same algorith is used with the temperature set to 0 K.

The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ of a defect configuration is defined by chosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.
Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.
Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.

\section{Results}

\subsection{Carbon and silicon defect configurations}

Table~\ref{tab:defects} summarizes the formation energies of relevant defect structures for the EA and DFT calculations.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
 & C$_{\text{i}}$ \hkl<1 0 0> DB & C$_{\text{s}}$ & C$_{\text{i}}$ \hkl<1 1 0> DB & C$_{\text{i}}$ BC & Si$_{\text{i}}$ \hkl<1 1 0> DB & Si$_{\text{i}}$ T\\
\hline
 VASP & 3.72 & 1.95 &  4.16 & 4.66 & 3.39 & 3.77 \\
 Erhart/Albe & 3.88 & 0.75 & 5.18 & 5.59$^*$ & 4.39 & 3.40
\end{tabular}
\end{ruledtabular}
\caption{Formation energies of C and Si point defects in c-Si determined by classical potential and ab initio methods. The formation energies are given in electron volt. T denotes the tetrahedral and BC the bond-centered configuration. Subscript i and s indicates the interstitial and substitutional configuration. Dumbbell configurations are abbreviated by DB. Formation energies for unstable configurations obtained by classical potential MD are marked by an asterisk and determined by using the low kinetic energy configuration shortly before the relaxation into the more favorable configuration starts.}
\label{tab:defects}
\end{table*}
Although discrepancies exist, both methods depict the correct order of the formation energies with regard to C defects in Si.
Substitutional C (C$_{\text{s}}$) constitutes the energetically most favorable defect configuration.
Since the C atom occupies an already vacant Si lattice site, C$_{\text{s}}$ is not an interstitial defect.
The quantum-mechanical result agrees well with the result of another ab initio study\cite{dal_pino93}.
Clearly, the empirical potential underestimates the C$_{\text{s}}$ formation energy.
The C interstitial defect with the lowest energy of formation has been found to be the C-Si \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB), which, thus, constitutes the ground state of an additional C impurity in otherwise perfect c-Si.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Astonishingly EA and DFT predict almost equal formation energies.
There are, however, geometric differences with regard to the DB position within the tetrahedron spanned by the four next neighbored Si atoms, as already reported in a previous study\cite{zirkelbach10a}.
Since the energetic description is considered more important than the structural description minor discrepancies of the latter are assumed non-problematic.
The second most favorable configuration is the C$_{\text{i}}$ \hkl<1 1 0> DB followed by the C$_{\text{i}}$ bond-centered (BC) configuration.
For both configurations EA overestimates the energy of formation by approximately \unit[1]{eV} compared to DFT.
Thus, nearly the same difference in energy has been observed for these configurations in both methods.
However, we have found the BC configuration to constitute a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
Due to the high formation energy of the BC defect resulting in a low probability of occurence of this defect, the wrong description is not posing a serious limitation of the EA potential.
A more detailed discussion of C defects in Si modeled by EA and DFT including further defect configurations are presented in a previous study\cite{zirkelbach10a}.

Regarding intrinsic defects in Si, both methods predict energies of formation that are within the same order of magnitude.
However discrepancies exist.
Quantum-mechanical results reveal the Si$_{\text{i}}$ \hkl<1 1 0> DB to compose the energetically most favorabe configuration, which is the consensus view for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.
The EA potential does not reproduce the correct ground state.
Instead the tetrahedral defect configuration is favored.
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 terahedral 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 following investigations of defect combinations.
%This artifact does not necessarily render impossible further challenging empirical potential studies on large systems.
%However, it has to be taken into account in the following investigations of defect combinations.

\subsection{Defect combinations}

\subsection{Mobility}

A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable position to another.
The stable defect geometries have been discussed in the previous subsection.
In the following the migration of the most stable configuration, i.e. C$_{\text{i}}$, from one site of the Si host lattice to a neighboring site has been investigated by both, EA and DFT calculations utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
Three migration pathways are investigated.
The starting configuration for all pathways was the \hkl[0 0 -1] dumbbell interstitial configuration.
In path~1 and 2 the final configuration is a \hkl[0 0 1] and \hkl[0 -1 0] dumbbell interstitial respectively, located at the next neighbored Si lattice site displaced by $\frac{a_{\text{Si}}}{4}\hkl[1 1 -1]$, where $a_{\text{Si}}$ is the Si lattice constant.
In path~1 the C atom resides in the \hkl(1 1 0) plane crossing the BC configuration whereas in path~2 the C atom moves out of the \hkl(1 1 0) plane.
Path 3 ends in a \hkl[0 -1 0] configuration at the initial lattice site and, for this reason, corresponds to a reorientation of the dumbbell, a process not contributing to long range diffusion.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{path2_vasp_s.ps}
\end{center}
\caption{Migration barrier and structures of the \hkl[0 0 -1] dumbbell (left) to the \hkl[0 -1 0] dumbbell (right) transition as obtained by first-principles methods. The activation energy of \unit[0.9]{eV} agrees well with experimental findings of \unit[0.70]{eV}\cite{lindner06}, \unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}.}
\label{fig:vasp_mig}
\end{figure} 
The lowest energy path (path~2) as detected by the first-principles approach is illustrated in Fig.~\ref{fig:vasp_mig}, in which the \hkl[0 0 -1] dumbbell migrates towards the next neighbored Si atom escaping the $(1 1 0)$ plane forming a \hkl[0 -1 0] dumbbell.
The activation energy of \unit[0.9]{eV} excellently agrees with experimental findings ranging from \unit[0.70]{eV} to \unit[0.87]{eV}\cite{lindner06,song90,tipping87}.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{path1_albe_s.ps}
\end{center}
\caption{Migration barrier and structures of the bond-centered (left) to \hkl[0 0 -1] dumbbell (right) transition utilizing the classical potential method. Two different pathways are obtained for different time constants of the Berendsen thermostat. The lowest activation energy is \unit[2.2]{eV}.}
\label{fig:albe_mig}
\end{figure}
Calculations based on the EA potential yield a different picture.
Fig.~\ref{fig:albe_mig} shows the evolution of structure and energy along the lowest energy migration path (path~1) based on the EA potential.
Due to symmetry it is sufficient to merely consider the migration from the BC to the C$_{\text{i}}$ configuration.
Two different pathways are obtained for different time constants of the Berendsen thermostat.
With a time constant of \unit[1]{fs} the C atom resides in the \hkl(1 1 0) plane resulting in a migration barrier of \unit[2.4]{eV}.
However, weaker coupling to the heat bath realized by an increase of the time constant to \unit[100]{fs} enables the C atom to move out of the \hkl(1 1 0) plane already at the beginning, which is accompanied by a reduction in energy, approaching the final configuration on a curved path.
The energy barrier of this path is \unit[0.2]{eV} lower in energy than the direct migration within the \hkl(1 1 0) plane.
It should be noted that the BC configuration is actually not a local minimum configuration in EA based calculations since a relaxation into the \hkl<1 1 0> dumbbell configuration occurs.
However, investigating further migration pathways involving the \hkl<1 1 0> interstitial did not yield lower migration barriers.
Thus, the activation energy should at least amount to \unit[2.2]{eV}.

\section{Discussion}

The first-principles results are in good agreement to previous work on this subject\cite{burnard93,leary97,dal_pino93,capaz94}.
The C-Si \hkl<1 0 0> dumbbell interstitial is found to be the ground state configuration of a C defect in Si.
The lowest migration path already proposed by Capaz et~al.\cite{capaz94} is reinforced by an additional improvement of the quantitative conformance of the barrier height calculated in this work (\unit[0.9]{eV}) with experimentally observed values (\unit[0.70]{eV} -- \unit[0.87]{eV})\cite{lindner06,song90,tipping87}.
However, it turns out that the bond-centered configuration is not a saddle point configuration as proposed by Capaz et~al.\cite{capaz94} but constitutes a real local minimum if the electron spin is properly accounted for.
A net magnetization of two electrons, which is already clear by simple molecular orbital theory considerations on the bonding of the sp hybridized C atom, is settled.
By investigating the charge density isosurface it turns out that the two resulting spin up electrons are localized in a torus around the C atom.
With an activation energy of \unit[0.9]{eV} the C$_{\text{i}}$ carbon interstitial can be expected to be highly mobile at prevailing temperatures in the process under investigation, i.e. IBS.

We found that the description of the same processes fails if classical potential methods are used.
Already the geometry of the most stable dumbbell configuration differs considerably from that obtained by first-principles calculations.
The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
%ref mod: language - energy / order 
%Nevertheless, both methods predict the same type of interstitial as the ground state configuration, and also the order in energy of the remaining defects is reproduced fairly well.
Nevertheless, both methods predict the same type of interstitial as the ground state configuration.
Furthermore, the relative energies of the other defects are reproduced fairly well.
From this, a description of defect structures by classical potentials looks promising.
% ref mod: language - changed
%However, focussing on the description of diffusion processes the situation is changing completely.
However, focussing on the description of diffusion processes the situation has changed completely.
Qualitative and quantitative differences exist.
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 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.
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 dumbbell agglomeration as proposed by the precipitation model.

\section{Summary}

To conclude, we have shown that ab initio calculations on interstitial carbon in silicon are very close to the results expected from experimental data.
The calculations presented in this work agree well with other theoretical results.
So far, the best quantitative agreement with experimental findings has been achieved concerning the interstitial carbon mobility.
For the first time, we have shown that the bond-centered configuration indeed constitutes a real local minimum configuration resulting in a net magnetization if spin polarized calculations are performed.
Classical potentials, however, fail to describe the selected processes.
This has been shown to have two reasons, i.e. the overestimated barrier of migration due to the artificial interaction cut-off on the one hand, and on the other hand the lack of quantum-mechanical effects which are crucial in the problem under study. 
% ref mod: language - being investigated
%In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently being performed.

% ----------------------------------------------------
\section*{Acknowledgment}
We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05) and the Deutsche Forschungsgemeinschaft (DFG SCHM 1361/11).

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