[lectures/latex.git] / posic / publications / c_defects_in_si.tex

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\begin{document}

\title{Description of Defects in Carbon implanted Silicon -- a comparison of classical potentials and first principles methods}
\author{F. Zirkelbach} \author{B. Stritzker}
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, D-86135 Augsburg, Germany}
\author{K. Nordlund}
\affiliation{Accelerator Laboratory, 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}

\begin{abstract}
A comparative theoretical investigation of carbon interstitials in silicon is presented.
Calculations using classical potentials are put aside first principles density functional theory calculations of the geometries, formation and activation energies of the carbon dumbbell interstitial, showing the importance of a quantum-mechanical description of this system.
By first principles the migration path of interstitial carbon is identified exhibiting an activation energy that -- for the first time -- excellently matches experimental findings.
The bond-centered interstitial configuration is found to show a net magnetization of two electrons illustrating the imperative of spin polarized calculations.

\end{abstract}

\keywords{point defects, migration, interstitials, first principles calculations, classical potentials}
\pacs{find out later...}
\maketitle

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

%Frank:  Idea: description of 3C-SiC-precipitation in C-implanted silicon.\\ 
%        cite and describe briefly experimental work - why is this material important/better than other SiC).  \\ 
%        Describe the precipitation process in brief.\\ 
%        Sum up literature where classical potentials have been used (a) successful, and (b) failed. Also add citations of Nordlunds papers. Not only on silicon or SiC!\\

% there should be a short motivation for the material system!
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}.
% there are different polytpes with different properties and 3c-sic in special
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}.
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 produced by ibs
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}.
% maybe split CVD and MBE from IBS and explain remaining problems:
% - on 6H-SiC: twin boundaries
% - on Si: structural defects due to thermal conductivity and lattice mismatch
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}.
% precipitation model
However, only little is known on 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.
Once a critical radius of 2 nm to 4 nm is reached a topotactic transformation into a 3C-SiC precipitate occurs.
The transformation is manifested by the disappearance of 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.
% motivation to understand the precipitation and link to atomistic simulations
A detailed understanding of the underlying processes will enable significant technological progress in 3C-SiC thin film formation and likewise offer perspectives for processes which 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.
Relevant structures consisting of $\approx 10^4$ atoms for the nanocrystal and even more atoms for a reasonably sized Si host matrix are too large to be completely described by high accuracy quantum mechanical methods.
Directly modelling the dynamics of the processes mentioned above almost inevitably requires the atomic interaction to be described by less accurate though computationally more efficient classical potentials.
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.
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}.
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,mattoni2007} or extending it\cite{devanathan98_2} with data gained from ab initio calculations\cite{nordlund97}.

In this work, the applicability of a Tersoff-like bond order potential\cite{albe_sic_pot}  to some basic processes involved in the initially mentioned SiC precipitation mechanism is investigated by comparing results gained by classical and ab inito calculations.
In the following a comparative investigation of density functional theory studies and classical potential calculations of the structure, energetics and mobility of carbon defects in silicon is presented.

%  --------------------------------------------------------------------------------
\section{Methodology}
% ----- DFT ------
The first-principles DFT calculations are performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.
The Kohn-Sham equations are 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 is restricted to the $\Gamma$-point.
The defect structures and the migration paths are modelled in cubic supercells containing 216 Si atoms.
The ions and cell shape are allowed to change in order to realize a constant pressure simulation.
Spin polarization is fully accounted for.
Only neutral defects are considered.

% ------ Albe potential ---------
%% Frank:  Setup/short description of the potential ?
For the classical potential calculations a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms is used.
A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} is utilized, which accounts for nearest neighbour interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbour distance.
Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84}.
Structural relaxation in the MD run is achieved by the verlocity 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.

\section{Results}

% ... wer sagt das ...
%After ion implantation, carbon interstitials are the most common defects in the silicon sample.
According to the assumed SiC precipitation model described in the introductary part, carbon interstitial defects form and agglomerate into large clusters.
Thus, it is of crucial importance to investigate the various possible structures of carbon defects and the mobility of the lowest energy, hence most probable, defect configuration in crystalline silicon.
%Their mobility is the crucial quantity to be investigated.
%We thus started our comparative investigations by comparing the stability and the mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take.

\subsection{Carbon interstitials in various geometries}

Table~\ref{tab:defects} summarizes the formation energies of defect structures for the EA and VASP 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.
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 & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\
\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, B the bond-centered and S the substitutional interstitial configuration. The 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} 

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.
VASP 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 \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 VASP predict almost equal formation energies.
% pick up again later, that this is why erhart/albe is more promising for the specific problem under investigation
However, a qualitative difference is observed investigating the dumbbell structures.
While the C-Si distance of the dumbbell atoms is almost equal for both methods, the vertical position of the dumbbell inside the tetrahedra spanned by the four next neighboured 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 VASP 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 tetrahedra 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 VASP 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 VASP 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 \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 repsect to the \hkl<1 1 0> defect as VASP 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.
% due to missing accounting for electron spin ...
This is assumed to be due to the neglection of the electron spin in these calculations.
Another VASP calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerated 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, discrepancies are observed between the results from classical potential calculations and those obtained from first principles.
Within the classical potentials EA outperforms Tersoff, which is, thus, used for further comparative studies.
Nevertheless, both methods (EA and VASP) 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.
% sth like that ... defects might still be ok but when it comes to diffusion ...
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 neighbored site is investigated by both, EA and VASP calculations utilizing the constrained conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.
Three migration pathways are investigated.
The starting configuration for all pathways is 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 neighboured Si lattice site displaced by $\frac{a_{\text{Si}}}{4}$\hkl[1 1 -1], whereat $a_{\text{Si}}$ 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 0 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_plot.ps}\\
\includegraphics[width=\columnwidth]{path2_conf.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[0.73]{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 neighboured Si atom escaping the \hkl(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[0.73]{eV}\cite{song90} to \unit[0.87]{eV}\cite{tipping87}.
% not the path you expected!
%This path is not surprising -- a similar behavior was e.g. found earlier for the carbon split interstitial \cite{rauls03a} and the phosphorus interstitial \cite{rauls03b,gerstmann03} in SiC.

\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth]{path1_plot.ps}\\
\includegraphics[width=\columnwidth]{path1_conf.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 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}.
% lower / weaker / less strong ?
However, lower 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 lowest activation energy must be assumed to be higher than or equal to \unit[2.2]{eV}.
% experimental findings much lower, overestimated by a factor of 2.4

\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[0.73]{eV} -- \unit[0.87]{eV})\cite{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.
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.
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.
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.
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 neighboured 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, it is shown that ab initio calculations are very close to the results expected from experimental data.
Furthermore, they 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 it is 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. 

% ----------------------------------------------------
\section*{Acknowledgment}
%The calculations were done using grants of computer time from the 
%Paderborn Center for Parallel Computing (PC$^2$) and the 
%H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche 
%Forschungsgemeinschaft is acknowledged for financial support.
One of us (F.Z.) wants to acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05).

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

%\bibliography{../../bibdb/bibdb}{}
%\bibliographystyle{h-physrev3}

% didnt know how to include these ...

%\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).
%\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). 
%\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).


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\end{document}