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+\usepackage{miller}\r
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\begin{document}\r
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% there should be a short motivation for the material system!\r
Silicon carbide (SiC) has a number of remarkable physical and chemical properties.\r
-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}, which are moreover deployable in harsh environments\cite{capano97}.\r
+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}.\r
% there are different polytpes with different properties and 3c-sic in special\r
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}.\r
-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.\r
+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.\r
\r
% (thin films of) 3c-sic can be produced by ibs\r
-Next to 3C-SiC epitaxial layer growth by chemical vapor deposition (CVD) on 6H-SiC substrates\cite{powell90} or molecular beam epitaxy on silicon\cite{} as well as 6H-SiC\cite{}, ion beam synthesis (IBS) constitutes a promising method to produce 3C-SiC epitaxial layers of high quality in crystalline silicon\cite{lindner02} (c-Si).\r
-Highly energetic carbon ions are ...\r
-\r
-The relevant structures are with $\approx$ 20000 atoms/nanocrystal way too large to be completely be described with high accuracy \r
-quantum mechanical methods. Modelling the processes described above require the use of less accurate methods, like e.g. classical \r
-potentials (Erhard/Albe\cite{albe},Stillinger-Weber\cite{stillinger},...). Whether such potentials are appropriate for the description of the \r
-physical problem has, however, to be verified first by applying both methods to relevant processes that can be treated by both methods. \r
-In this work, we have implemented and compared the applicability of several (?) classical potentials to ab initio results for one \r
-of the most important processes of our original question. \r
-\r
-In the following we will present a comparative investigation of density functional theory (DFT) studies and \r
-classical potential calculations of the structure, energetics and mobility of carbon defects in silicon. \r
+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}.\r
+% maybe split CVD and MBE from IBS and explain remaining problems:\r
+% - on 6H-SiC: twin boundaries\r
+% - on Si: structural defects due to thermal conductivity and lattice mismatch\r
+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}.\r
+% precipitation model\r
+However, only little is known on the SiC conversion in C implanted Si.\r
+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.\r
+Once a critical radius of 2 nm to 4 nm is reached a topotactic transformation into a 3C-SiC precipitate occurs.\r
+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.\r
+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.\r
+% motivation to understand the precipitation and link to atomistic simulations\r
+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}.\r
+\r
+Atomistic simulations offer a powerful tool to study materials on a microscopic level providing detailed insight not accessible by experiment.\r
+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.\r
+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.\r
+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.\r
+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.\r
+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.\r
+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}.\r
+An extensive comparison\cite{balamane92} concludes that each potential has its strengths and limitations and none of them is clearly superior to others.\r
+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.\r
+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}.\r
+\r
+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.\r
+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.\r
\r
% --------------------------------------------------------------------------------\r
\section{Methodology}\r
% ----- DFT ------\r
-The first-principles DFT calculations were performed with the plane-wave based \r
-Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}. The Kohn-Sham equations were solved \r
-using the generalized-gradient XC-functional approximation proposed by Perdew and \r
-Wang (GGA-PW91)\cite{perdew92}. \r
-The electron-ion interaction was described by the projector-augmented wave (PAW) method\cite{bloechel94,kresse99}. \r
-In the PAW data scalar relativistic corrections are contained. Throughout this work an \r
-energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis. \r
-For the sampling of the Brillouin zone, only the $\Gamma$-point was used. \r
-The defect structures and the migration paths were modelled in cubic supercells containing 216 Si-atoms. Spin polarization has been \r
-fully accounted for. \r
+The first-principles DFT calculations are performed with the plane-wave based Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}.\r
+The Kohn-Sham equations are solved using the generalized-gradient XC-functional approximation proposed by Perdew and Wang (GGA-PW91)\cite{perdew86,perdew92}.\r
+The electron-ion interaction is described by norm-conserving ultra-soft pseudopotentials\cite{hamann79} as implemented in VASP\cite{vanderbilt90}.\r
+Throughout this work an energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis.\r
+Sampling of the Brillouin zone is restricted to the $\Gamma$-point.\r
+The defect structures and the migration paths are modelled in cubic supercells containing 216 Si atoms.\r
+The ions and cell shape are allowed to change in order to realize a constant pressure simulation.\r
+Spin polarization is fully accounted for.\r
+Only neutral defects are considered.\r
\r
% ------ Albe potential ---------\r
%% Frank: Setup/short description of the potential ?\r
-For the calculations with the classical potentials... \r
-\r
+For the classical potential calculations a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms is used.\r
+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.\r
+Constant pressure simulations are realized by the Berendsen barostat\cite{brendsen84}.\r
+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.\r
\r
\section{Results}\r
\r
-After ion implantation, carbon interstitials are the most common defects in the silicon sample. Their mobility is the \r
-crucial quantity to be investigated. We thus started our comparative investigations by comparing the stability and the \r
-mobility of an isolated carbon interstitial in silicon bulk in the various possible geometries it can take. \r
+% ... wer sagt das ...\r
+%After ion implantation, carbon interstitials are the most common defects in the silicon sample.\r
+According to the assumed SiC precipitation model described in the introductary part, carbon interstitial defects form and agglomerate into large clusters.\r
+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.\r
+%Their mobility is the crucial quantity to be investigated.\r
+%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.\r
\r
\subsection{Carbon interstitials in various geometries}\r
\r
-Several geometries have been calculated to be stable for the carbon interstitial. Fig.\ref{fig:interstitials} shows all these \r
-structures. However, there are some discrepancies between the results from classical potential calculations and those obtained \r
-from first principles. \r
-Table \ref{table:formation} summarizes the formation energies of the interstitial geometries for both methods used in this work \r
-and compares the results to literature values. (...check references for more data, ..) \r
-\r
-% Tables: like in the talk, but add further literature data and give the references/citations (also to bibliography \r
-% at the end!) \r
-%\begin{figure}\r
-%\includegraphics[width=1.0\columnwidth]{models.eps}\r
-%\caption{\label{fig:interstitials} Molecular model of the possible carbon interstitials. }\r
-%\end{figure} \r
-\r
-While the Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... \r
- Nevertheless, both methods predict the (110) dumb bell configuration to be the most stable... (?) \r
-\r
+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.\r
+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.\r
+Relaxed geometries are displayed in Fig.~\ref{fig:defects}.\r
+Astonishingly there is only little literature present to compare with.\r
+\begin{table*}\r
+\begin{ruledtabular}\r
+\begin{tabular}{l c c c c c c}\r
+%\hline\r
+%\hline\r
+ & T & H & \hkl<1 0 0> DB & \hkl<1 1 0> DB & S & B \\\r
+\hline\r
+ Erhart/Albe & 6.09 & 9.05$^*$ & 3.88 & 5.18 & 0.75 & 5.59$^*$ \\\r
+ VASP & unstable & unstable & 3.72 & 4.16 & 1.95 & 4.66 \\\r
+ Tersoff\cite{tersoff90} & 3.8 & 6.7 & 4.6 & 5.9 & 1.6 & 5.3 \\\r
+ ab initio\cite{dal_pino93,capaz94} & - & - & x & - & 1.89 \cite{dal_pino93} & x+2.1 \cite{capaz94} \\\r
+ % there is no more ab initio data!\r
+%\hline\r
+%\hline\r
+\end{tabular}\r
+\end{ruledtabular}\r
+\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.}\r
+\label{tab:defects}\r
+\end{table*}\r
+\begin{figure}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Tetrahedral}\\\r
+\includegraphics[width=\columnwidth]{01.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Hexagonal}\\\r
+\includegraphics[width=\columnwidth]{02.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{\hkl<1 0 0> dumbbell}\\\r
+\includegraphics[width=\columnwidth]{03.eps}\r
+\end{minipage}\\\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{\hkl<1 1 0> dumbbell}\\\r
+\includegraphics[width=\columnwidth]{04.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Substitutional}\\[0.05cm]\r
+\includegraphics[width=\columnwidth]{05.eps}\r
+\end{minipage}\r
+\begin{minipage}[t]{0.32\columnwidth}\r
+\underline{Bond-centered}\\\r
+\includegraphics[width=\columnwidth]{06.eps}\r
+\end{minipage}\r
+\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.}\r
+\label{fig:defects}\r
+\end{figure} \r
+\r
+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.\r
+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}.\r
+However, the EA potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.\r
+\r
+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.\r
+This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94} and experimental\cite{watkins76,song90} investigations.\r
+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}.\r
+It should be noted that EA and VASP predict almost equal formation energies.\r
+% pick up again later, that this is why erhart/albe is more promising for the specific problem under investigation\r
+However, a qualitative difference is observed investigating the dumbbell structures.\r
+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.\r
+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]{$^\circ$}.\r
+% maybe transfer to discussion chapter later\r
+This indicates predominant sp and sp$^2$ hybridization for the C and Si dumbbell atom respectively.\r
+Obviously the classical potential is not able to reproduce the clearly quantum-mechanically dominated character of bonding.\r
+% substitute 'dominated'\r
+\r
+Both, EA and VASP reveal the hexagonal configuration unstable relaxing into the C$_{\text{I}}$ ground state structure.\r
+Tersoff finds this configuration stable, though it is the most unfavorable.\r
+Thus, the highest formation energy observed by the EA potential is the tetrahedral configuration, which turns out to be unstable in VASP calculations.\r
+% maybe transfer to discussion chapter later\r
+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.\r
+\r
+The \hkl<1 1 0> dumbbell constitutes the second most favorable configuration, reproduced by both methods.\r
+It is followed by the bond-centered (BC) configuration.\r
+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.\r
+Tersoff indeed predicts a metastable BC configuration.\r
+However it is not in the correct order and lower in energy than C$_{\text{I}}$.\r
+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.\r
+% due to missing accounting for electron spin ...\r
+This is assumed to be due to the neglection of the electron spin in these calculations.\r
+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.\r
+This problem is resolved by spin polarized calculations resulting in a net spin one accompanied by a reduction of the total energy by \unit[0.3]{eV} and the transformation into a metastable local minimum configuration.\r
+All other configurations are not affected.\r
+\r
+To conclude, discrepancies are observed between the results from classical potential calculations and those obtained from first principles.\r
+Within the classical potentials EA outperforms Tersoff, which is, thus, used for further comparative studies.\r
+Nevertheless, both methods (EA and VASP) predict the \hkl<1 0 0> dumbbell interstitial configuration to be most stable.\r
+Also the remaining defects and their energetical order are described fairly well.\r
+% sth like that ... defects might still be ok but when it comes to diffusion ...\r
+It is thus concluded that -- so far -- modelling of the SiC precipitation by the EA potential might lead to trustable results.\r
\r
\subsection{Mobility}\r
-A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable \r
-position to another. The stable defect geometries have been discussed in the previous subsection. We now investigate \r
-the migration from the most stable structure (...should be named somehow...) on one site of the silicon host lattice to \r
-a neighbored site. \r
-On the lowest energy path (first principles), the carbon atom starts to move along (110)..(check that!)... The center of the line connecting \r
-initial and final structure has been found to be a local minimum and not a saddle point as could be expected. The two \r
-saddle points shortly before and behind this local minimum are slightly displaced out of the (110) plane by ... {\AA}. ..(check that!)..\r
-This path is not surprising -- a similar behavior was e.g. found earlier for the carbon split interstitial \cite{rauls03a} and the phosphorus \r
-interstitial \cite{rauls03b,gerstmann03} in SiC. However, an interesting effect is the change of the spin state from zero at the (110) dumb bell \r
-configuration to one at the local minimum. By this, the energy of the local minimum is lowered by 0.3 eV (... check it!!..). \r
-%\begin{figure}\r
-%\includegraphics[width=1.0\columnwidth]{path-DFT.eps}\r
-%\caption{\label{fig:path-DFT} Energy of the carbon interstitial during migration from ... to ... calculated from first principles. The \r
-% activation energy of 0.9 eV (?) agrees well with experimental findings (0.7-0.9 eV?). }\r
-%\end{figure} \r
-Fig.\ref{fig:path-DFT} shows the energy along this lowest energy migration path. The activation energy of 0.9 eV (?) agrees well \r
-with experimental findings (0.7-0.9 eV?).\r
+\r
+A measure for the mobility of the interstitial carbon is the activation energy for the migration path from one stable position to another.\r
+The stable defect geometries have been discussed in the previous subsection.\r
+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}.\r
+Three migration pathways are investigated.\r
+The starting configuration for all pathways is the \hkl<0 0 -1> dumbbell interstitial configuration.\r
+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.\r
+Path 3 ends in a \hkl<0 -1 0> configuration at the initial lattice site and, for this reason, corrsponds to a reorientation of the dumbbell, a process not contributing to long range diffusion.\r
+\r
+\begin{figure}\r
+\begin{center}\r
+\includegraphics[width=\columnwidth]{00-1_0-10_nosym_sp_fullct.ps}\\[1.0cm]\r
+\begin{picture}(0,0)(90,0)\r
+\includegraphics[width=0.2\columnwidth]{00-1_a.eps}\r
+\end{picture}\r
+\begin{picture}(0,0)(10,0)\r
+\includegraphics[width=0.2\columnwidth]{00-1_0-10_sp.eps}\r
+\end{picture}\r
+\begin{picture}(0,0)(-70,0)\r
+\includegraphics[width=0.2\columnwidth]{0-10.eps}\r
+\end{picture}\r
+\begin{picture}(0,0)(15,15)\r
+\includegraphics[width=0.2\columnwidth]{100_arrow.eps}\r
+\end{picture}\r
+\begin{picture}(0,0)(130,0)\r
+\includegraphics[height=0.2\columnwidth]{001_arrow.eps}\r
+\end{picture}\r
+\end{center}\r
+\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 (\unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}).}\r
+\label{fig:vasp_mig}\r
+\end{figure} \r
+The lowest energy path (path 2) as detected by the first principles approach is illustrated in Fig. \ref{fig:vasp_mig}.\r
+The activation energy of \unit[0.9]{eV} agrees well with experimental findings (\unit[0.73]{eV}\cite{song90} and \unit[0.87]{eV}\cite{tipping87}).\r
+% not the path you expected!\r
+%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.\r
\r
Calculations with the Albe potential yield a different picture. \r
%\begin{figure}\r
\r
% ----------------------------------------------------\r
\section*{Acknowledgment}\r
-The calculations were done using grants of computer time from the \r
-Paderborn Center for Parallel Computing (PC$^2$) and the \r
-H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche \r
-Forschungsgemeinschaft is acknowledged for financial support.\r
+%The calculations were done using grants of computer time from the \r
+%Paderborn Center for Parallel Computing (PC$^2$) and the \r
+%H\"ochstleistungs-Rechenzentrum Stuttgart. The Deutsche \r
+%Forschungsgemeinschaft is acknowledged for financial support.\r
+One of us (F.Z.) wants to acknowledge financial support by the Bayerische Forschungsstiftung (DPA-61/05).\r
\r
% --------------------------------- references -------------------\r
-\begin{thebibliography}{99}\r
-\bibitem{kresse96} G. Kresse and J. Furthm\"uller, \r
- Comput. Mater. Sci. {\bf 6}, 15 (1996). \r
-\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, \r
- Phys. Rev. B {\bf 46}, 6671 (1992). \r
-\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, \r
- Phys. Rev. Lett. {\bf 45}, 556 (1980). \r
-\bibitem{perdew81} J. P. Perdew and A. Zunger, \r
- Phys. Rev. B {\bf 23}, 5048 (1981). \r
-\bibitem{bloechel94} P. E. Bl\"ochl, \r
- Phys. Rev. B {\bf 50}, 17953 (1994).\r
-\bibitem{kresse99} G. Kresse and D. Joubert, \r
- Phys. Rev. B {\bf 59}, 1758 (1999).\r
-\bibitem{monk76} H. J. Monkhorst and J. D. Pack, \r
- Phys. Rev. B {\bf 13}, 5188 (1976). \r
-\bibitem{albe} Albe potential\r
-\bibitem{stillinger} Stillinger-Weber potential \r
-\bibitem{joannopoulos} Joannopoulos\r
-\bibitem{xyz} who else? \r
-\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).\r
-\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). \r
-\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).\r
-\r
-\end{thebibliography}\r
+\bibliography{../../bibdb/bibdb}{}\r
+%\begin{thebibliography}{99}\r
+%\bibitem{kresse96} G. Kresse and J. Furthm\"uller, \r
+% Comput. Mater. Sci. {\bf 6}, 15 (1996). \r
+%\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, \r
+% Phys. Rev. B {\bf 46}, 6671 (1992). \r
+%\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, \r
+% Phys. Rev. Lett. {\bf 45}, 556 (1980). \r
+%\bibitem{perdew81} J. P. Perdew and A. Zunger, \r
+% Phys. Rev. B {\bf 23}, 5048 (1981). \r
+%\bibitem{bloechel94} P. E. Bl\"ochl, \r
+% Phys. Rev. B {\bf 50}, 17953 (1994).\r
+%\bibitem{kresse99} G. Kresse and D. Joubert, \r
+% Phys. Rev. B {\bf 59}, 1758 (1999).\r
+%\bibitem{monk76} H. J. Monkhorst and J. D. Pack, \r
+% Phys. Rev. B {\bf 13}, 5188 (1976). \r
+%\bibitem{albe} Albe potential\r
+%\bibitem{stillinger} Stillinger-Weber potential \r
+%\bibitem{joannopoulos} Joannopoulos\r
+%\bibitem{xyz} who else? \r
+%\bibitem{rauls03a} E. Rauls, A. Gali, P. De´ak, and Th. Frauenheim, Phys. Rev. B, 68, 155208 (2003).\r
+%\bibitem{rauls03b} E. Rauls, U. Gerstmann, H. Overhof, and Th. Frauenheim, Physica B, Vols. 340-342, p. 184-189 (2003). \r
+%\bibitem{gerstmann03} U. Gerstmann, E. Rauls, Th. Frauenheim, and H. Overhof, Phys. Rev. B, 67, 205202, (2003).\r
+%\end{thebibliography}\r
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\end{document}\r
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