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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,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 20 \% of the 3C-SiC precipitate and c-Si.
The insignificantly lower Si density of SiC ($\approx 4$ \%) 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}, Brenner\cite{brenner90}, Tersoff\cite{tersoff_si3} and environment-dependent interatomic (EDIP)\cite{bazant96,bazant97,justo98} potential.
From the mentioned potentials, until recently\cite{lucas10}, a parametrization to describe the C-Si multicomponent system did only exist for the Tersoff potential\cite{tersoff_m}.
Description successful ... and failed ...
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.
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 (DFT) 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 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}


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