\usepackage{amssymb}
% additional stuff
-% \usepackage{miller}
+\usepackage{miller}
\begin{document}
\title{Defects in Carbon implanted Silicon calculated by classical potentials and first principles methods}
-\author{F. Zirkelbach} \author{B. Stritzker}
+\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}
-% \affiliation{Experimentelle Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
-\author{W. G. Schmidt} \author{E. Rauls}
-% \affiliation{Theoretische Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
+\author{W. G. Schmidt}
+\author{E. Rauls}
\affiliation{Department Physik, Universit\"at Paderborn, 33095 Paderborn, Germany}
\begin{abstract}
Silicon carbide (SiC) has a number of remarkable physical and chemical properties.
The wide band gap semiconductor (2.3 eV - 3.3 eV) exhibiting a high breakdown field, saturated electron drift velocity and thermal conductivity in conjunction with its unique thermal and mechanical stability as well as radiation hardness is a suitable material for high-temperature, high-frequency and high-power devices\cite{wesch96,morkoc94}, which are moreover deployable in harsh and radiation-hard environments\cite{capano97}.
-%SiC, which forms fourfold coordinated covalent bonds, tends to crystallize into many different modifications, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
SiC, which forms fourfold coordinated mostly covalent bonds, tends to crystallize into many different modifications, which solely differ in the one-dimensional stacking sequence of identical, close-packed SiC bilayers\cite{fischer90}.
-% J mod end
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.
-%Next to the fabrication of 3C-SiC layers 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, 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}.
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}.
-% KN mod end
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,lindner99_2} 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.
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 (EDIP)\cite{bazant96,bazant97,justo98} potential.
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}.
Whether such potentials are appropriate for the description of the physical problem has, however, to be verified first by applying classical and quantum-mechanical methods to relevant processes that can be treated by both methods.
-%For instance, by combination of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations, SW turned out to be best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} important in ion implantation, while the Tersoff potential yielded a qualitative agreement for the interaction of Si self-interstitials with substitutional C\cite{mattoni2002}.
For instance, a comparison of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations showed that SW is best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} in Si important in ion implantation, while the Tersoff potential yielded a qualitative agreement for the interaction of Si self-interstitials with substitutional C\cite{mattoni2002}.
-% KN mod end
-%Antisite pairs and defects in SiC have been investigated, both classically\cite{posselt06,gao04} employing the Gao/Weber potential\cite{gao02} and quantum-mechanically\cite{rauls03a,gao07,gao2001,bockstedte03}, which, both, agree very well with experimental results\cite{gali03,chen98,weber01}.
Antisite pairs and defects in SiC have been investigated, both classically\cite{posselt06,gao04} employing the Gao/Weber potential\cite{gao02} and quantum-mechanically\cite{rauls03a,gao07,gao2001,bockstedte03}, which, both, agree very well with experimental results\cite{gali03,chen98,weber01}.
-% KN mod end
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}.
\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 energies are 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.
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 $\mu_{\text{C}}$.
-% J mod end
Relaxed geometries are displayed in Fig.~\ref{fig:defects}.
-%Astonishingly there is only little literature present to compare with.
\begin{table*}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c}
-%\hline
-%\hline
- & T & H & $\langle 1 0 0\rangle$ dumbbell & $\langle1 1 0\rangle$ dumbbell & S & BC \\
+ & 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} \\
- % there is no more ab initio data!
-%\hline
-%\hline
\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.}
\includegraphics[width=\columnwidth]{hex.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
-\underline{$\langle1 0 0\rangle$ dumbbell}\\
+\underline{\hkl<1 0 0> dumbbell}\\
\includegraphics[width=\columnwidth]{100.eps}
\end{minipage}\\
\begin{minipage}[t]{0.32\columnwidth}
-\underline{$\langle1 1 0\rangle$ dumbbell}\\
+\underline{\hkl<1 1 0> dumbbell}\\
\includegraphics[width=\columnwidth]{110.eps}
\end{minipage}
\begin{minipage}[t]{0.32\columnwidth}
\label{fig:defects}
\end{figure}
-%Substitutional carbon (C$_{\text{sub}}$) in silicon, which is in fact not an interstitial defect, is found to be the lowest configuration with regard to energy for all potential models.
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.
-% J mod end
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}.
However, the EA potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.
-Except for the Tersoff potential the $\langle1 0 0\rangle$ dumbbell (C$_{\text{i}}$) is the energetically most favorable interstital configuration, in which the C and Si dumbbell atoms share a Si lattice site.
+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 $\langle0 0 1\rangle$ of the dumbbell inside the tetrahedron spanned by the four next neighbored Si atoms differs significantly.
+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]{$^\circ$}.
-% maybe transfer to discussion chapter later
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.
-% substitute 'dominated'
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.
-% maybe transfer to discussion chapter later
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 $\langle1 1 0\rangle$ dumbbell constitutes the second most favorable configuration, reproduced by both methods.
+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 $\langle1 1 0\rangle$ defect as DFT does, the BC configuration is found to be a saddle point within the EA description relaxing into the $\langle1 1 0\rangle$ 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 $\langle1 1 0\rangle$ dumbbell.
+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.
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.
-%Nevertheless, both methods (EA and DFT) predict the $\langle1 0 0\rangle$ dumbbell interstitial configuration to be most stable.
-Both methods (EA and DFT) predict the $\langle1 0 0\rangle$ dumbbell interstitial configuration to be most stable.
-% KN mod end
+Both methods (EA and DFT) predict the \hkl<1 0 0> dumbbell interstitial configuration to be most stable.
Also the remaining defects and their energetical order 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.
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 $[0 0 -1]$ dumbbell interstitial configuration.
-In path~1 and 2 the final configuration is a $[0 0 1]$ and $[0 -1 0]$ dumbbell interstitial respectively, located at the next neighbored Si lattice site displaced by $\frac{a_{\text{Si}}}{4}[1 1 -1]$, where $a_{\text{Si}}$ is the Si lattice constant.
-In path~1 the C atom resides in the $(1 1 0)$ plane crossing the BC configuration whereas in path~2 the C atom moves out of the $(1 1 0)$ plane.
-Path 3 ends in a $[0 -1 0]$ 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.
+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.ps}
\includegraphics[width=\columnwidth]{path2_vasp_s.ps}
\end{center}
-\caption{Migration barrier and structures of the $\langle0 0 -1\rangle$ dumbbell (left) to the $\langle0 -1 0\rangle$ 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 $\langle0 0 -1\rangle$ dumbbell migrates towards the next neighbored Si atom escaping the $(1 1 0)$ plane forming a $\langle0 -1 0\rangle$ 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}
\begin{center}
-%\includegraphics[width=\columnwidth]{path1_albe.ps}
\includegraphics[width=\columnwidth]{path1_albe_s.ps}
\end{center}
-\caption{Migration barrier and structures of the bond-centered (left) to $\langle0 0 -1\rangle$ 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}.}
+\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 C$_{\text{i}}$ configuration.
Due to symmetry it is sufficient to merely consider the migration from the BC to the C$_{\text{i}}$ configuration.
-% KN mod end
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 $(1 1 0)$ 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 $(1 1 0)$ 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 $(1 1 0)$ 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 $\langle1 1 0\rangle$ dumbbell configuration occurs.
-However, investigating further migration pathways involving the $\langle1 1 0\rangle$ interstitial did not yield lower migration barriers.
+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 $\langle1 0 0\rangle$ dumbbell interstitial is found to be the ground state configuration of a C defect in Si.
+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, adjusts.
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.
-% J mod end
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.
-%The description of the same processes obviously fails if classical potential methods are used.
We found that the description of the same processes fails if classical potential methods are used.
-% KN mod end
Already the geometry of the most stable dumbbell configuration differs considerably from that obtained by first principles calculations.
-%Obviously the classical approach is unable to reproduce the correct character of bonding due to a too short treatment of quantum-mechanical effects in the potential.
-%The classical approach is unable to reproduce the correct character of bonding due to no treatment of quantum-mechanical effects in the potential.
The classical approach is unable to reproduce the correct character of bonding due to the deficiency of quantum-mechanical effects in the potential.
-% J mod end
-% KN mod end
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.
From this, a description of defect structures by classical potentials looks promising.
However, focussing on the description of diffusion processes the situation is changing completely.
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 neighbored 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.
-% KN mod end
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.
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.
-%Further investigations, i.e. the structure and energetics of defect combinations, still small enough to be treated by DFT have been accomplished in order to draw conclusions regarding the precipitation mechanism, will be published elsewhere.
In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
% ----------------------------------------------------
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