\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.
+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.
+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 ...}
+\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.
+% TOOD: redo complete intro!
-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.
+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 exclusively of the 3C polytype embedded in and epitactically aligned to the Si host featuring a sharp interface\cite{lindner99,lindner01,lindner02}.
+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 anocrystallites, suggest an initial coherent precipitation by agglomeration of substitutional instead of interstitial C.
+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 cocurrently by a reduction of volume due to the lower lattice constant of SiC compared to Si.
+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\cite{wesch96}.
+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.
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.
+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}.
% --------------------------------------------------------------------------------
\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 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.
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.
-The formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing SiC as a reservoir for the carbon impurity in order to determine $\mu_{\text{C}}$.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
- & C-Si \hkl<1 0 0> dumbbell & C$_{\text{s}}$ & C-Si \hkl<1 1 0> dumbbell & C$_{\text{i}}$ bond-centered & Si$_{\text{i}}$ \hkl<1 1 0> dumbbell & Si$_{\text{i}}$ T\\
+ & 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
- 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 \\
+ 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, 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.}
+\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*}
-\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}
+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.
+This 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}.
+
-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 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.
+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
\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}.}
+\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 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}
\section{Discussion}
-The first principles results are in good agreement to previous work on this subject\cite{burnard93,leary97,dal_pino93,capaz94}.
+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.
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.
+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.
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