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\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-86153 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}
We present a comparative theoretical investigation of carbon interstitials in silicon.
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.
\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}, which are moreover deployable in harsh 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 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).
Highly energetic carbon ions are ...

The relevant structures are with $\approx$ 20000 atoms/nanocrystal way too large to be completely be described with high accuracy 
quantum mechanical methods. Modelling the processes described above require the use of less accurate methods, like e.g. classical 
potentials (Erhard/Albe\cite{albe},Stillinger-Weber\cite{stillinger},...).  Whether such potentials are appropriate for the description of the 
physical problem has, however, to be verified first by applying both methods to relevant processes that can be treated by both methods.     
In this work, we have implemented and compared the applicability of several (?) classical potentials to ab initio results for one 
of the most important processes of our original question.   

In the following we will present a comparative investigation of density functional theory (DFT) studies and 
classical potential calculations of the structure, energetics and mobility of carbon defects in silicon. 

%  --------------------------------------------------------------------------------
\section{Methodology}
% ----- DFT ------
The first-principles DFT calculations were performed with the plane-wave based 
Vienna Ab-initio Simulation Package (VASP)\cite{kresse96}. The Kohn-Sham equations were solved 
using the generalized-gradient XC-functional approximation proposed by Perdew and 
Wang (GGA-PW91)\cite{perdew92}. 
The electron-ion interaction was described by the projector-augmented wave (PAW) method\cite{bloechel94,kresse99}. 
In the PAW data scalar relativistic corrections are contained. Throughout this work an 
energy cut-off of \unit[300]{eV} was used to expand the wave functions into the plane-wave basis. 
For the sampling of the Brillouin zone, only the $\Gamma$-point was used. 
The defect structures and the migration paths were modelled in cubic supercells containing 216 Si-atoms. Spin polarization has been 
fully accounted for.    

% ------ Albe potential ---------
%% Frank:  Setup/short description of the potential ?
For the calculations with the classical potentials... 


\section{Results}

After ion implantation, carbon interstitials are the most common defects in the silicon sample. 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}

Several geometries have been calculated to be stable for the carbon interstitial. Fig.\ref{fig:interstitials} shows all these 
structures. However, there are some discrepancies between the results from classical potential calculations and those obtained 
from first principles.    
Table \ref{table:formation} summarizes the formation energies of the interstitial geometries for both methods used in this work 
and compares the results to literature values. (...check references for more data, ..) 

% Tables:  like in the talk, but add further literature data and give the references/citations (also to bibliography 
% at the end!) 
%\begin{figure}
%\includegraphics[width=1.0\columnwidth]{models.eps}
%\caption{\label{fig:interstitials} Molecular model of the possible carbon interstitials. }
%\end{figure} 

While the Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)...    
 Nevertheless, both methods predict the (110) dumb bell configuration to  be the most stable...  (?) 


\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.   We now investigate 
the migration from the most stable structure (...should be named somehow...) on one site of the silicon host lattice to 
a neighbored site.  
On the lowest energy path (first principles), the carbon atom starts to move along (110)..(check that!)...   The center of the line connecting 
initial and final structure has been found to be a local minimum and not a saddle point as could be expected. The two 
saddle points shortly before and behind this local minimum are slightly displaced out of the (110) plane by ... {\AA}.  ..(check that!)..
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.  However, an interesting effect is the change of the spin state from zero at the (110) dumb bell 
configuration to one at the local minimum.  By this, the energy of the local minimum is lowered by 0.3 eV  (... check it!!..). 
%\begin{figure}
%\includegraphics[width=1.0\columnwidth]{path-DFT.eps}
%\caption{\label{fig:path-DFT} Energy of the carbon interstitial during migration from ... to ... calculated from first principles. The 
%         activation energy of 0.9 eV (?) agrees well with experimental findings (0.7-0.9 eV?). }
%\end{figure} 
Fig.\ref{fig:path-DFT} shows the energy along this lowest energy migration path.  The activation energy of 0.9 eV (?) agrees well 
with experimental findings (0.7-0.9 eV?).

Calculations with the Albe potential yield a different picture.  
%\begin{figure}
%\includegraphics[width=1.0\columnwidth]{path-Albe.eps}
%\caption{\label{fig:path-Albe} Energy of the carbon interstitial during migration from ... to ... calculated using the classical potential 
%          method. Here, the activation energy is 2.2 eV (?). }
%\end{figure} 
Fig.\ref{fig:path-Albe} shows the energy along the lowest energy migration path found by this method. The activation energy of 2.2 eV (?) 
is way too high to explain the experimental findings (0.7-0.9 eV?).  (...further discussion...) 

\section{Discussion}
The first principles results are in good agreement to previous work on this subject \cite{joannopoulos,xyz}  (...add some references!...).   With an 
activation energy of 0.9 eV, the carbon interstitial can be expected to be mobile at temperatures in the range of... (?).    
The description of the same processes obviously fails if we use the classical potential method.  
Already the geometry of the most stable dumb bell configuration differs considerably from that of the first principles calculated 
structure.  (..... add description, the two main angles and bond lengths and an explanation...)
Formation energies are throughout too high... (...reason?...)

A reason for this failure of the classical description is most likely...  (cut-off, neglect of quantum mechanical effects,...)


\section{Summary}
In summary, we have shown that ab initio calculations are very close to the results expected from experimental data.  
Furthermore, they agree well with other theoretical results. (...some results - later...)  The classical potentials, however, fail to describe the 
selected processes. This has been shown to have two reasons, i.e. the artificial cut-off of the next nearest neighbor 
interaction on the one hand, on the other hand the 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.

% --------------------------------- references -------------------
\begin{thebibliography}{99}
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  Comput. Mater. Sci. {\bf 6}, 15 (1996). 
\bibitem{perdew92} J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh and C. Fiolhais, 
  Phys. Rev. B {\bf 46}, 6671 (1992). 
\bibitem{ceperley80} D. M. Ceperley and B. J. Alder, 
  Phys. Rev. Lett. {\bf 45}, 556 (1980). 
\bibitem{perdew81} J. P. Perdew and A. Zunger, 
  Phys. Rev. B {\bf 23}, 5048 (1981). 
\bibitem{bloechel94} P. E. Bl\"ochl, 
  Phys. Rev. B {\bf 50}, 17953 (1994).
\bibitem{kresse99} G. Kresse and D. Joubert, 
  Phys. Rev. B {\bf 59}, 1758 (1999).
\bibitem{monk76} H. J. Monkhorst and J. D. Pack, 
  Phys. Rev. B {\bf 13}, 5188 (1976). 
\bibitem{albe}  Albe potential
\bibitem{stillinger} Stillinger-Weber potential 
\bibitem{joannopoulos} Joannopoulos
\bibitem{xyz} who else?  
\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).

\end{thebibliography}


\end{document}

