\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.
+% ref mod: language - put aside
+%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.
+Calculations using classical potentials are compared to 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.
In contrast to previous studies, the present first principles calculations of the interstitial carbon migration path yield an activation energy that
excellently matches the experiment.
-The bond-centered interstitial configuration shows a net magnetization of two electrons, illustrating the imperative of spin polarized calculations.
+%ref mod: language - imperative
+%The bond-centered interstitial configuration shows a net magnetization of two electrons, illustrating the imperative of spin polarized calculations.
+The bond-centered interstitial configuration shows a net magnetization of two electrons, illustrating the need for spin polarized calculations.
\end{abstract}
Thin films of 3C-SiC can be fabricated 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.
Next to these methods, 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}.
-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}.
+% ref mod: language - omit first part (pretty much liked it!)
+%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}.
+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}.
However, only little is known about 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.
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.
+% ref mod: language EDIP
+%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.
+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 potential (EDIP)\cite{bazant96,bazant97,justo98}.
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, e.g. the one by Gao and Weber\cite{gao02}.
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, a comparison of empirical potential molecular dynamics (MD) and density functional theory (DFT) calculations showed that SW is best suited for simulations of dislocation nucleation processes\cite{godet03} and threshold displacement energy calculations\cite{holmstroem08} in Si important in ion implantation, while the Tersoff potential yielded a qualitative agreement for the interaction of Si self-interstitials with substitutional C\cite{mattoni2002}.
% ------ Albe potential ---------
For the classical potential calculations, a supercell of 9 Si lattice constants in each direction consisting of 5832 Si atoms has been used.
A Tersoff-like bond order potential by Erhart and Albe (EA)\cite{albe_sic_pot} has been utilized, which accounts for nearest neighbor interactions only realized by a cut-off function dropping the interaction to zero in between the first and second next neighbor 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.
+% ref mod: extension for short distances
+The potential was used as is, i.e. without any repulsive potential extension at short interatomic distances.
+% ref mod: time constants
+%Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84}.
+Constant pressure simulations are realized by the Berendsen barostat\cite{berendsen84} using a time constant of \unit[100]{fs} and a bulk modulus of \unit[100]{GPa} for Si.
+% ref mod: time constants + language (Verlet)
+%Structural relaxation in the MD run is achieved by the velocity 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.
+Structural relaxation in the MD run is achieved by the Velocity Verlet algorithm\cite{verlet67} and the Berendsen thermostat\cite{berendsen84} with a time constant of \unit[100]{fs} and the temperature set to zero Kelvin.
+Additionally, a time constant of \unit[1]{fs} resulting in direct velocity scaling was used for relaxation within the mobility calculations.
+% ref mod: time step
+A fixed time step of \unit[1]{fs} for integrating the equations of motion was used.
\section{Results}
Substitutional carbon (C$_{\text{sub}}$) occupying an already vacant Si lattice site, which is in fact not an interstitial defect, is found to be the lowest configuration with regard to energy for all potential models.
DFT 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.
+% ref mod: typo - underestim_a_tes
+%However, the EA potential dramatically underestimtes the C$_{\text{sub}}$ formation energy, which is a definite drawback of the potential.
+However, the EA potential dramatically underestimates 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.
To conclude, we observed discrepancies between the results from classical potential calculations and those obtained from first principles.
Within the classical potentials EA outperforms Tersoff and is, therefore, used for further comparative studies.
Both methods (EA and DFT) 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.
+%ref mod: language - energetical order
+%Also the remaining defects and their energetical order are described fairly well.
+Also the remaining defects and their relative energies are described fairly well.
It is thus concluded that -- so far -- modelling of the SiC precipitation by the EA potential might lead to trustable results.
\subsection{Mobility}
We found that the description of the same processes 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.
The classical approach is unable to reproduce the correct character of bonding due to the deficiency 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.
+%ref mod: language - energy / order
+%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.
+Nevertheless, both methods predict the same type of interstitial as the ground state configuration, and also the relative energies of the remaining defects are 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.
+% ref mod: language - changed
+%However, focussing on the description of diffusion processes the situation is changing completely.
+However, focussing on the description of diffusion processes the situation is changed 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.
For the first time, we have 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.
-In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
+% ref mod: language - being investigated
+%In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently investigated.
+In order to get more insight on the SiC precipitation mechanism, further ab initio calculations are currently being investigated.
% ----------------------------------------------------
\section*{Acknowledgment}
projects / lectures / latex.git / commitdiff