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\begin{document}\r
\r
-%\title{Mobility of Carbon in Silicon -- a first-principles study}\r
\title{First-principles study of defects in carbon implanted silicon}\r
\author{F. Zirkelbach}\r
\author{B. Stritzker}\r
The Kohn-Sham equations were solved using the generalized-gradient exchange-correlation (XC) functional approximation proposed by Perdew and Wang\cite{perdew86,perdew92}.\r
The electron-ion interaction was 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 was restricted to the $\Gamma$-point.\r
+To reduce the computational effort sampling of the Brillouin zone was restricted to the $\Gamma$-point, which has been shown to yield reliable results\cite{dal_pino93}.\r
The defect structures and the migration paths were modelled in cubic supercells with a side length of \unit[1.6]{nm} containing $216$ Si atoms.\r
+Formation energies and structures are reasonably converged with respect to the system size.\r
The ions and cell shape were allowed to change in order to realize a constant pressure simulation.\r
+The observed changes in volume were less than \unit[0.2]{\%} of the volume indicating a rather low dependence of the results on the ensemble choice.\r
Ionic relaxation was realized by the conjugate gradient algorithm.\r
Spin polarization has been fully accounted for.\r
\r
Migration and recombination pathways have been investigated utilizing the constraint conjugate gradient relaxation technique (CRT)\cite{kaukonen98}.\r
+While not guaranteed to find the true minimum energy path, the method turns out to identify reasonable pathways for the investigated structures.\r
The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by choosing SiC as a particle reservoir for the C impurity, i.e. the chemical potentials are determined by the cohesive energies of a perfect Si and SiC supercell after ionic relaxation.\r
+%In the same way defect formation energies are determined in the article\cite{dal_pino93} used for comparison.\r
+This corresponds to the definition utilized in another study on C defects in Si\cite{dal_pino93} that we compare our results to.\r
The binding energy of a defect pair is given by the difference of the formation energy of the complex and the sum of the two separated defect configurations.\r
-Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.\r
+%Accordingly, energetically favorable configurations show binding energies below zero while non-interacting isolated defects result in a binding energy of zero.\r
+Accordingly, energetically favorable configurations result in binding energies below zero while unfavorable configurations show positive values for the binding energy.\r
+The interaction strength, i.e. the absolute value of the binding energy, approaches zero for increasingly non-interacting isolated defects.\r
\r
\section{Results}\r
\r
Due to the low activation energy this process must be considered to be activated without much effort either thermally or by introduced energy of the implantation process.\r
\r
\begin{figure}\r
-\includegraphics[width=\columnwidth]{c_sub_si110.ps}\r
-\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The binding energies of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}\r
+%\includegraphics[width=\columnwidth]{c_sub_si110.ps}\r
+\includegraphics[width=\columnwidth]{c_sub_si110_data.ps}\r
+\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance.}\r
+%\caption{Binding energies of combinations of a C$_{\text{s}}$ and a Si$_{\text{i}}$ DB with respect to the separation distance. The interaction strength of the defect pairs are well approximated by a Lennard-Jones 6-12 potential, which is used for curve fitting.}\r
\label{fig:dc_si-s}\r
\end{figure}\r
Fig.~\ref{fig:dc_si-s} shows the binding energies of pairs of C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB with respect to the separation distance.\r
-The interaction of the defects is well approximated by a Lennard-Jones 6-12 potential, which was used for curve fitting.\r
-The binding energy quickly drops to zero.\r
-The Lennard-Jones fit estimates almost zero interaction already at \unit[0.6]{nm}, indicating a low interaction capture radius of the defect pair.\r
+%The interaction of the defects is well approximated by a Lennard-Jones (LJ) 6-12 potential, which is used for curve fitting.\r
+%Unable to model possible positive values of the binding energy, i.e. unfavorable configurations, located to the right of the minimum, the LJ fit should rather be thought as a guide for the eye describing the decrease of the interaction strength, i.e. the absolute value of the binding energy, with increasing separation distance.\r
+%The binding energy quickly drops to zero.\r
+%The LJ fit estimates almost zero interaction already at \unit[0.6]{nm}, indicating a low interaction capture radius of the defect pair.\r
+As can be seen, the interaction strength, i.e. the absolute value of the binding energy, quickly drops to zero with increasing separation distance.
+Almost zero interaction may be assumed already at distances about \unit[0.5-0.6]{nm}, indicating a low interaction capture radius of the defect pair.
In IBS highly energetic collisions are assumed to easily produce configurations of defects exhibiting separation distances exceeding the capture radius.\r
For this reason C$_{\text{s}}$ without a Si$_{\text{i}}$ DB located within the immediate proximity, which is, thus, unable to form the thermodynamically stable C$_{\text{i}}$ \hkl<1 0 0> DB, constitutes a most likely configuration to be found in IBS.\r
\r
\r
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\end{document}\r
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