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% ref mod: no capital letters (by editors)
\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}

The wide band gap semiconductor silicon carbide (SiC) has a number of remarkable physical and chemical properties.
Its high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with its unique thermal and mechanical stability as well as radiation hardness makes it 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 (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.

Thin films of 3C-SiC can be fabricated 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.
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 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 anocrystallites, 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 cocurrently 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\cite{wesch96}.
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}.
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 ... influence on migration ... crucial for problem under study.
However ... considered good, especially non-zero temperatures(?) ... 

HIER WEITER ...

An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.
Despite their shortcomings 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.
Remaining shortcomings have frequently been resolved by modifying the interaction\cite{tang95,gao02a,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.

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 then are 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 XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\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 modelled 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, a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms has been used.
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.
% ref mod: extension for short distances
The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
% ref mod: time constants
%Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84}.
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.
% ref mod: time constants + language (Verlet)
%Structural relaxation in the MD run is achieved by the velocity Verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[1]{fs} resulting in direct velocity scaling and the temperature set to zero Kelvin.
Structural relaxation in the MD run is achieved by the Velocity Verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs} and the temperature set to zero Kelvin.
Additionally, a time constant of \unit[1]{fs} resulting in direct velocity scaling was used for relaxation within the mobility calculations.
% ref mod: time step
A fixed time step of \unit[1]{fs} for integrating the equations of motion was used.

\section{Results}

\subsection{Carbon interstitials in various geometries}

Table~\ref{tab:defects} summarizes the formation energies of defect structures for the EA and DFT calculations performed in this work as well as further results from literature.
The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined in the same way as in the articles used for comparison\cite{tersoff90,dal_pino93} chosing SiC as a reservoir for the carbon impurity in order to determine $\mu_{\text{C}}$.
Relaxed geometries are displayed in Fig.~\ref{fig:defects}.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
 & T & H & \hkl<1 0 0> dumbbell & \hkl<1 1 0> dumbbell & S & BC \\
\hline
 Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\
 VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\
 Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\
 ab initio\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\
\end{tabular}
\end{ruledtabular}
\caption{Formation energies of carbon point defects in crystalline silicon determined by classical potential and ab initio methods. The formation energies are given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. S corresponds to substitutional C. 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*}
\begin{figure}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Tetrahedral}\\
\includegraphics[width=\columnwidth]{tet.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Hexagonal}\\
\includegraphics[width=\columnwidth]{hex.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{\hkl<1 0 0> dumbbell}\\
\includegraphics[width=\columnwidth]{100.eps}
\end{minipage}\\
\begin{minipage}[t]{0.32\columnwidth}
\underline{\hkl<1 1 0> dumbbell}\\
\includegraphics[width=\columnwidth]{110.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Substitutional}\\[0.05cm]
\includegraphics[width=\columnwidth]{sub.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\underline{Bond-centered}\\
\includegraphics[width=\columnwidth]{bc.eps}
\end{minipage}
\caption{Configurations of carbon point defects in silicon. The silicon/carbon atoms and the bonds (only for the interstitial atom) are illustrated by yellow/grey spheres and blue lines. Bonds are drawn for atoms located within a certain distance and do not necessarily correspond to chemical bonds.}
\label{fig:defects}
\end{figure} 

Substitutional carbon (C$_{\text{sub}}$) occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration with regard to energy for all potential models.
DFT calculations performed in this work are in good agreement with results obtained by classical potential simulations by Tersoff\cite{tersoff90} and ab initio calculations done by Dal Pino et~al\cite{dal_pino93}.
% ref mod: typo - underestim_a_tes
%However, the EA potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.
However, the EA potential dramatically underestimates the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.

Except for the Tersoff potential the \hkl<1 0 0> dumbbell (C$_{\text{i}}$) is the energetically most favorable interstital configuration, in which the C and Si dumbbell atoms share a Si lattice site.
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.
Tersoff as well, considers C$_{\text{i}}$ to be the ground state configuration and believes an artifact due to the abrupt C-Si cut-off used in the potential to be responsible for the small value of the tetrahedral formation energy\cite{tersoff90}.
It should be noted that EA and DFT predict almost equal formation energies.
However, there is a qualitative difference: while the C-Si distance of the dumbbell atoms is almost equal for both methods, the position along \hkl[0 0 1] of the dumbbell inside the tetrahedron spanned by the four next neighbored Si atoms differs significantly.
The dumbbell based on the EA potential is almost centered around the regular Si lattice site as can be seen in Fig.~\ref{fig:defects} whereas for DFT calculations it is translated upwards with the C atom forming an almost collinear bond to the two Si atoms of the top face of the tetrahedron and the bond angle of the Si dumbbell atom to the two bottom face Si atoms approaching \unit[120]{$^\circ$}.
This indicates predominant sp and sp$^2$ hybridization for the C and Si dumbbell atom respectively.
Obviously the classical potential is not able to reproduce the clearly quantum-mechanically dominated character of bonding.

Both, EA and DFT reveal the hexagonal configuration unstable relaxing into the C$_{\text{i}}$ ground state structure.
Tersoff finds this configuration stable, though it is the most unfavorable.
Thus, the highest formation energy observed by the EA potential is the tetrahedral configuration, which turns out to be unstable in DFT calculations.
The high formation energy of this defect involving a low probability to find such a defect in classical potential MD acts in concert with finding it unstable by the more accurate quantum-mechnical description.

The \hkl<1 1 0> dumbbell constitutes the second most favorable configuration, reproduced by both methods.
It is followed by the bond-centered (BC) configuration.
However, even though EA yields the same difference in energy with respect to the \hkl<1 1 0> defect as DFT does, the BC configuration is found to be a saddle point within the EA description relaxing into the \hkl<1 1 0> configuration.
Tersoff indeed predicts a metastable BC configuration.
However,  it is not in the correct order and lower in energy than the \hkl<1 1 0> dumbbell.
Please note, that Capaz et~al.\cite{capaz94} in turn found this configuration to be a saddle point, which is about \unit[2.1]{eV} higher in energy than the C$_{\text{i}}$ configuration.
This is assumed to be due to the neglection of the electron spin in these calculations.
Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate states for the BC configuration.
This problem is resolved by spin polarized calculations resulting in a net spin of one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.
All other configurations are not affected.

To conclude, we observed discrepancies between the results from classical potential calculations and those obtained from first principles.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further comparative studies.
Both methods (EA and DFT) predict the \hkl<1 0 0> dumbbell interstitial configuration to be most stable.
%ref mod: language - energetical order
%Also the remaining defects and their energetical order are described fairly well.
Also the remaining defects and their relative energies are described fairly well.
It is thus concluded that -- so far -- modelling of the SiC precipitation by the EA potential might lead to trustable results.

\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).

% --------------------------------- references -------------------

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