\r
\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
\affiliation{Experimentalphysik IV, Universit\"at Augsburg, 86135 Augsburg, Germany}\r
-\author{K. Nordlund}\r
-\affiliation{Department of Physics, University of Helsinki, 00014 Helsinki, Finland}\r
\author{J. K. N. Lindner}\r
\author{W. G. Schmidt}\r
\author{E. Rauls}\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
Ref.\cite{dal_pino93,capaz94} & - & - & - & - & - & 1.89\cite{dal_pino93} & x & - & x+2.1\cite{capaz94}\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Formation energies of silicon and carbon point defects in crystalline silicon given in electron Volt. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB.}\r
+\caption{Formation energies of silicon and carbon point defects in crystalline silicon given in eV. T denotes the tetrahedral, H the hexagonal and BC the bond-centered interstitial configuration. V corresponds to the vacancy configuration. Dumbbell configurations are abbreviated by DB.}\r
\label{table:sep_eof}\r
\end{table*}\r
Results obtained by the present study compare well with results from literature\cite{leung99,al-mushadani03,dal_pino93,capaz94}.\r
-Regarding intrinsic defects in Si, the \hkl<1 1 0> self-interstitial dumbbell (Si$_{\text{i}}$ \hkl<1 1 0> DB) is found to be the ground state configuration tersely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
+Regarding intrinsic defects in Si, the \hkl<1 1 0> self-interstitial dumbbell (Si$_{\text{i}}$ \hkl<1 1 0> DB) is found to be the ground state configuration closely followed by the hexagonal and tetrahedral configuration, which is consensus for Si$_{\text{i}}$\cite{leung99,al-mushadani03}.\r
In the case of a C impurity, next to the C$_{\text{s}}$ configuration, in which a C atom occupies an already vacant Si lattice site, the C \hkl<1 0 0> interstitial dumbbell (C$_{\text{i}}$ \hkl<1 0 0> DB) constitutes the energetically most favorable configuration, in which the C and Si dumbbell atoms share a regular Si lattice site.\r
This finding is in agreement with several theoretical\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental\cite{watkins76,song90} investigations, which all predict this configuration to be the ground state.\r
%However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations has yet been explicitly stated in literature.\r
\hkl[1 0 0] & -2.25 & -2.16 & -0.10 & -0.27 & -1.38 & -0.06\\\r
\end{tabular}\r
\end{ruledtabular}\r
-\caption{Binding energies in electron Volt of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}\r
+\caption{Binding energies in eV of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}\r
\label{table:dc_c-c}\r
\end{table}\r
-Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of these type of defects.\r
+Most of the obtained configurations result in binding energies well below zero indicating a preferable agglomeration of this type of defects.\r
For increasing distances of the defect pair the binding energy approaches to zero (R in Table~\ref{table:dc_c-c}) as it is expected for non-interacting isolated defects.\r
Energetically favorable and unfavorable configurations can be explained by stress compensation and increase respectively based on the resulting net strain of the respective configuration of the defect combination.\r
Antiparallel orientations of the second defect, i.e. \hkl[0 0 1] for positions located below the \hkl(0 0 1) plane with respect to the initial one (positions 1, 2 and 4) form the energetically most unfavorable configurations.\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
The investigation of defect pairs indicated a general trend of defect agglomeration mainly driven by the potential of strain reduction.\r
Obtained results for the most part compare well with results gained in previous studies\cite{leary97,capaz98,mattoni2002,liu02} and show an astonishingly good agreement with experiment\cite{song90}.\r
-For configurations involving two C impurities the ground state configurations have been found to to consist of C-C bonds, which are responsible for the vast gain in energy.\r
+For configurations involving two C impurities the ground state configurations have been found to consist of C-C bonds, which are responsible for the vast gain in energy.\r
However, based on investigations of possible migration pathways, these structures are less likely to arise than structures, in which both C atoms are interconnected by another Si atom, which is due to high activation energies of the respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations.\r
Thus, agglomeration of C$_{\text{i}}$ is expected while the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.\r
\r
Agglomeration of C$_{\text{i}}$ is energetically favored and enabled by a low activation energy for migration.\r
Although ion implantation is a process far from thermodynamic equilibrium, which might result in phases not described by the Si/C phase diagram, i.e. a C phase in Si, high activation energies are believed to be responsible for a low probability of the formation of C-C clusters.\r
\r
-Unrolling these findings on the initially stated controversy present in the precipitation model, an increased participation of C$_{\text{s}}$ already in the initial stage must be assumed due to its high probability of incidence.\r
+In the context of the initially stated controversy present in the precipitation model, these findings suggest an increased participation of C$_{\text{s}}$ already in the initial stage due to its high probability of incidence.\r
In addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.\r
The associated emission of Si$_{\text{i}}$ serves two needs: as a vehicle for other C$_{\text{s}}$ atoms and as a supply of Si atoms needed elsewhere to form the SiC structure.\r
As for the vehicle, Si$_{\text{i}}$ is believed to react with C$_{\text{s}}$ turning it into highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.\r
% ----------------------------------------------------\r
\section*{Acknowledgment}\r
We gratefully acknowledge financial support by the Bayerische Forschungsstiftung (Grant No. DPA-61/05) and the Deutsche Forschungsgemeinschaft (Grant No. DFG SCHM 1361/11).\r
+Prof. Kai Nordlund is greatly acknowledged for useful comments on the present manuscript.\r
\r
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\end{document}\r
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