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{mattoni02}.\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
% ------ Albe potential ---------\r
%% Frank: Setup/short description of the potential ?\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\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
+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
\r
\subsection{Carbon interstitials in various geometries}\r
\r
-Table~\ref{tab:defects} summarizes the formation energies of defect structures for the Erhart/Albe and VASP calculations performed in this work as well as further results from literature.\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
\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 Erhart/Albe potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.\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 is the energetically most favorable interstital configuration, in which the C and Si dumbbell atoms share a Si lattice site.\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 the \hkl<1 0 0> 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
-A qualitative difference is observed investigating the dumbbell structures.\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 Erhart/Albe 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
+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-mechanical character of bonding.\r
-% empirical potential adjusts according to minimum of angular function, no QM!\r
-\r
-\r
-% pick up again later, that this is why erhart/albe is more promising for the specific problem under investigation\r
-\r
-\r
-\r
-\r
-While the Erhart/Albe potential predicts ... as stable, DFT does not. ...(further comparisons, trend "too high/low" E-formation,...)... \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
-Nevertheless, both methods predict the \hkl<1 0 0> dumbbell configuration to be most stable.\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
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