\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 influence of a nearby vacancy, another carbon interstitial and a substitutional defect as well as a silicon self-interstitial has been investigated systematically.\r
Interactions of various combinations of defects have been characterized including a couple of selected migration pathways within these configurations.\r
Almost all of the investigated pairs of defects tend to agglomerate allowing for a reduction in strain.\r
-The formation of structures involving strong carbon-carbon bonds was found to occur very unlikely.\r
-In contrast, substitutional carbon was found to occur in all probability.\r
+The formation of structures involving strong carbon-carbon bonds turns out to be very unlikely.\r
+%In contrast, substitutional carbon occurs in all probability.\r
+In contrast, substitutional carbon occurs.\r
A long range capture radius has been observed for pairs of interstitial carbon as well as interstitial carbon and vacancies.\r
-A rather small capture radius has been identified for substitutional carbon and silicon self-interstitials.\r
+A rather small capture radius is predicted for substitutional carbon and silicon self-interstitials.\r
We derive conclusions on the precipitation mechanism of silicon carbide in bulk silicon and discuss conformability to experimental findings.\r
\end{abstract}\r
\r
% Frank: Intro: hier kuerzer als in dem anderen Paper, dieselben (und mehr) Zitate bzgl. der Defekte (s. letzte Mail). SiC-precipitation würde ich schon erwähnen, aber nicht so detailliert.\r
\r
Silicon carbide (SiC) is a promising material for high-temperature, high-power and high-frequency electronic and optoelectronic devices employable under extreme conditions\cite{edgar92,morkoc94,wesch96,capano97,park98}.\r
-Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing constitutes a promising technique to fabricate nano-sized precipitates and thin films of cubic SiC (3C-SiC) topotactically aligned to and embedded in the silicon host\cite{borders71,lindner99,lindner01,lindner02}.\r
+Ion beam synthesis (IBS) consisting of high-dose carbon implantation into crystalline silicon (c-Si) and subsequent or in situ annealing is a promising technique to fabricate nano-sized precipitates and thin films of cubic SiC (3C-SiC) topotactically aligned to and embedded in the silicon host\cite{borders71,lindner99,lindner01,lindner02}.\r
However, the process of the formation of SiC precipitates in Si during C implantation is not yet fully understood.\r
Based on experimental high resolution transmission electron microscopy (HREM) studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03} it is assumed that incorporated C atoms form C-Si dimers (dumbbells) on regular Si lattice sites.\r
The highly mobile C interstitials agglomerate into large clusters followed by the formation of incoherent 3C-SiC nanocrystallites once a critical size of the cluster is reached.\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
-The defect formation energy $E-N_{\text{Si}}\mu_{\text{Si}}-N_{\text{C}}\mu_{\text{C}}$ is defined by chosing 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
+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
The implantation of highly energetic C atoms results in a multiplicity of possible defect configurations.\r
Next to individual Si$_{\text{i}}$, C$_{\text{i}}$, V and C$_{\text{s}}$ defects, combinations of these defects and their interaction are considered important for the problem under study.\r
First of all, structure and energetics of separated defects are presented.\r
-The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC transition.\r
+The investigations proceed with pairs of the ground state and, thus, most probable defect configurations that are believed to be fundamental in the Si to SiC conversion.\r
\r
\subsection{Separated defects in silicon}\r
\label{subsection:sep_def}\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 the consensus view 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 as 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
+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
+However, to our best knowledge, no energy of formation for this type of defect based on first-principles calculations is available.\r
\r
Instead, Capaz et al.\cite{capaz94}, investigating migration pathways of the C$_{\text{i}}$ \hkl<1 0 0> DB, find this defect to be \unit[2.1]{eV} lower in energy than the bond-centered (BC) configuration.\r
The BC configuration is claimed to constitute the saddle point within the C$_{\text{i}}$ \hkl[0 0 -1] DB migration path residing in the \hkl(1 1 0) plane and, thus, interpreted as the barrier of migration for the respective path.\r
Another DFT calculation without fully accounting for the electron spin results in the smearing of a single electron over two non-degenerate Kohn-Sham states and an increase of the total energy by \unit[0.3]{eV} for the BC configuration.\r
Regardless of the rather small correction of \unit[0.3]{eV} due to the spin, the difference we found is much smaller (\unit[0.94]{eV}), which would nicely compare to experimentally observed migration barriers of \unit[0.70-0.87]{eV}\cite{lindner06,tipping87,song90}.\r
However, since the BC configuration constitutes a real local minimum another barrier exists which is about \unit[1.2]{eV} in height.\r
-Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Indeed Capaz et al. propose another path and find it to be the lowest in energy\cite{capaz94}, in which a C$_{\text{i}}$ \hkl[0 0 -1] DB migrates to a C$_{\text{i}}$ \hkl[0 -1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.\r
Calculations in this work reinforce this path by an additional improvement of the quantitative conformance of the barrier height (\unit[0.90]{eV}) to experimental values.\r
A more detailed description can be found in a previous study\cite{zirkelbach10a}.\r
\r
In the Si$_{\text{i}}$ \hkl<1 0 0> DB configuration the net spin up density is localized in two caps at each of the two DB atoms perpendicularly aligned to the bonds to the other two Si atoms respectively.\r
No other configuration, within the ones that are mentioned, is affected.\r
\r
-Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the next neighbored Si lattice site in \hkl[1 1 -1] direction.\r
+Concerning the mobility of the ground state Si$_{\text{i}}$, we found an activation energy of \unit[0.67]{eV} for the transition of the Si$_{\text{i}}$ \hkl[0 1 -1] to \hkl[1 1 0] DB located at the neighbored Si lattice site in \hkl[1 1 -1] direction.\r
Further investigations revealed a barrier of \unit[0.94]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ H, \unit[0.53]{eV} for the Si$_{\text{i}}$ \hkl[1 1 0] DB to Si$_{\text{i}}$ T and \unit[0.35]{eV} for the Si$_{\text{i}}$ H to Si$_{\text{i}}$ T transition.\r
-Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.\r
+%Obtained values are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.\r
+These are of the same order of magnitude than values derived from other ab initio studies\cite{bloechl93,sahli05}.\r
\r
\subsection{Pairs of C$_{\text{i}}$}\r
\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 eelctron 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
% => low probability of C-C clustering ?!?\r
%\r
% should possibly be transfered to discussion section\r
-Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, mass C clustering is not expected.\r
+Since thermally activated C clustering is, thus, only possible by traversing energetically unfavored configurations, extensive C clustering is not expected.\r
Furthermore, the migration barrier of \unit[1.2]{eV} is still higher than the activation energy of \unit[0.9]{eV} observed for a single C$_{\text{i}}$ \hkl<1 0 0> DB in c-Si.\r
The migration barrier of a C$_{\text{i}}$ DB in a complex system is assumed to approximate the barrier of a DB in a separated system with increasing defect separation.\r
Accordingly, lower migration barriers are expected for pathways resulting in larger separations of the C$_{\text{i}}$ DBs.\r
In both cases the configuration yielding a binding energy of \unit[-2.25]{eV} is promising.\r
First of all, it constitutes the second most energetically favorable structure.\r
Secondly, a migration path with a barrier as low as \unit[0.47]{eV} exists starting from a configuration of largely separated defects exhibiting a low binding energy (\unit[-1.88]{eV}).\r
-The migration barrier and correpsonding structures are shown in Fig.~\ref{fig:188-225}.\r
+The migration barrier and corresponding structures are shown in Fig.~\ref{fig:188-225}.\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{188-225.ps}\r
\caption{Migration barrier and structures of the transition of a C$_{\text{i}}$ \hkl[0 -1 0] DB at position 5 (left) into a C$_{\text{i}}$ \hkl[1 0 0] DB at position 1 (right). An activation energy of \unit[0.47]{eV} is observed.}\r
% alternatively: ... considered period of time (of the IBS process).\r
%\r
% ?!?\r
-% look for precapture mechnism (local minimum in energy curve)\r
+% look for precapture mechanism (local minimum in energy curve)\r
% also: plot energy all confs with respect to C-C distance\r
% maybe a pathway exists traversing low energy confs ?!?\r
\r
% point out that configurations along 110 were extended up to the 6th NN in that direction\r
-The binding energies of the energetically most favorable configurations with the seocnd DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
+The binding energies of the energetically most favorable configurations with the second DB located along the \hkl[1 1 0] direction and resulting C-C distances of the relaxed structures are summarized in Table~\ref{table:dc_110}.\r
\begin{table}\r
\begin{ruledtabular}\r
\begin{tabular}{l c c c c c c }\r
\label{fig:dc_110}\r
\end{figure}\r
The interaction is found to be proportional to the reciprocal cube of the C-C distance for extended separations of the C$_{\text{i}}$ and saturates for the smallest possible separation, i.e. the ground state configuration.\r
-Not considering the previously mentioned elevated barriers for migration an attractive interaction between the C$_{\text{i}}$ defects indeed is detected with a capture radius that clearly exceeds the \unit[1]{nm} mark.\r
-The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, inbetween the two lowest separation distances of the defects.\r
-This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clsutering.\r
+Not considering the previously mentioned elevated barriers for migration an attractive interaction between the C$_{\text{i}}$ defects indeed is detected with a capture radius that clearly exceeds \unit[1]{nm}.\r
+The interpolated graph suggests the disappearance of attractive interaction forces, which are proportional to the slope of the graph, in between the two lowest separation distances of the defects.\r
+This finding, in turn, supports the previously established assumption of C agglomeration and absence of C clustering.\r
\r
\begin{table}\r
\begin{ruledtabular}\r
%./visualize_contcar -w 640 -h 480 -d results/c_00-1_c3_csub_B -nll -0.20 -0.4 -0.1 -fur 0.9 0.6 0.9 -c 0.5 -1.5 0.375 -L 0.5 0 0.3 -r 0.6 -A -1 2.465\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{093-095.ps}\r
-\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located inbetween two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}\r
+\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 3 (left) into a configuration of a twofold coordinated Si$_{\text{i}}$ located in between two C$_{\text{s}}$ atoms occupying the lattice sites of the initial DB and position 3 (right). An activation energy of \unit[0.44]{eV} is observed.}\r
\label{fig:093-095}\r
\end{figure}\r
-Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a next neighbor.\r
-By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located inbetween two substitutional C atoms residing on regular Si lattice sites.\r
+Configuration A consists of a C$_{\text{i}}$ \hkl[0 0 -1] DB with threefold coordinated Si and C DB atoms slightly disturbed by the C$_{\text{s}}$ at position 3, facing the Si DB atom as a neighbor.\r
+By a single bond switch, i.e. the breaking of a Si-Si in favor of a Si-C bond, configuration B is obtained, which shows a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.\r
This configuration has been identified and described by spectroscopic experimental techniques\cite{song90_2} as well as theoretical studies\cite{leary97,capaz98}.\r
Configuration B is found to constitute the energetically slightly more favorable configuration.\r
However, the gain in energy due to the significantly lower energy of a Si-C compared to a Si-Si bond turns out to be smaller than expected due to a large compensation by introduced strain as a result of the Si interstitial structure.\r
%\r
% AB transition\r
The migration barrier was identified to be \unit[0.44]{eV}, almost three times higher than the experimental value of \unit[0.16]{eV}\cite{song90_2} estimated for the neutral charge state transition in p- and n-type Si.\r
-Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected and disappointing.\r
+Keeping in mind the formidable agreement of the energy difference with experiment, the overestimated activation energy is quite unexpected.\r
Obviously, either the CRT algorithm fails to seize the actual saddle point structure or the influence of dopants has exceptional effect in the experimentally covered diffusion process being responsible for the low migration barrier.\r
% not satisfactory!\r
\r
\caption{Migration barrier and structures of the transition of the initial C$_{\text{i}}$ \hkl[0 0 -1] DB and C$_{\text{s}}$ at position 1 (left) into a C-C \hkl[1 0 0] DB occupying the lattice site at position 1 (right). An activation energy of \unit[0.1]{eV} is observed.}\r
\label{fig:026-128}\r
\end{figure}\r
-Configuration a is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a next neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.\r
+Configuration a is similar to configuration A, except that the C$_{\text{s}}$ atom at position 1 is facing the C DB atom as a neighbor resulting in the formation of a strong C-C bond and a much more noticeable perturbation of the DB structure.\r
Nevertheless, the C and Si DB atoms remain threefold coordinated.\r
Although the C-C bond exhibiting a distance of \unit[0.15]{nm} close to the distance expected in diamond or graphite should lead to a huge gain in energy, a repulsive interaction with a binding energy of \unit[0.26]{eV} is observed due to compressive strain of the Si DB atom and its top neighbors (\unit[0.230]{nm}/\unit[0.236]{nm}) along with additional tensile strain of the C$_{\text{s}}$ and its three neighboring Si atoms (\unit[0.198-0.209]{nm}/\unit[0.189]{nm}).\r
Again a single bond switch, i.e. the breaking of the bond of the Si atom bound to the fourfold coordinated C$_{\text{s}}$ atom and the formation of a double bond between the two C atoms, results in configuration b.\r
The transition is accompanied by a large gain in energy as can be seen in Fig.~\ref{fig:026-128}, making it the ground state configuration of a C$_{\text{s}}$ and C$_{\text{i}}$ DB in Si yet \unit[0.33]{eV} lower in energy than configuration B.\r
This finding is in good agreement with a combined ab initio and experimental study of Liu et~al.\cite{liu02}, who first proposed this structure as the ground state identifying an energy difference compared to configuration B of \unit[0.2]{eV}.\r
% mattoni: A favored by 0.2 eV - NO! (again, missing spin polarization?)\r
-A net magnetization of two spin up electrons, which are euqally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.\r
-In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two next neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles inbetween its bonds.\r
+A net magnetization of two spin up electrons, which are equally localized as in the Si$_{\text{i}}$ \hkl<1 0 0> DB structure is observed.\r
+In fact, these two configurations are very similar and are qualitatively different from the C$_{\text{i}}$ \hkl<1 0 0> DB that does not show magnetization but a nearly collinear bond of the C DB atom to its two neighbored Si atoms while the Si DB atom approximates \unit[120]{$^{\circ}$} angles in between its bonds.\r
Configurations a, A and B are not affected by spin polarization and show zero magnetization.\r
Mattoni et~al.\cite{mattoni2002}, in contrast, find configuration b less favorable than configuration A by \unit[0.2]{eV}.\r
-Next to differences in the XC functional and plane-wave energy cut-off this discrepancy might be attributed to the missing accounting for spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.\r
+Next to differences in the XC functional and plane-wave energy cut-off this discrepancy might be attributed to the neglect of spin polarization in their calculations, which -- as has been shown for the C$_{\text{i}}$ BC configuration -- results in an increase of configurational energy.\r
Indeed, investigating the migration path from configurations a to b and, in doing so, reusing the wave functions of the previous migration step the final structure, i.e. configuration b, was obtained with zero magnetization and an increase in configurational energy by \unit[0.2]{eV}.\r
-Obviously a different energy minimum of the electronic system is obatined indicating hysteresis behavior.\r
+Obviously a different energy minimum of the electronic system is obtained indicating hysteresis behavior.\r
However, since the total energy is lower for the magnetic result it is believed to constitute the real, i.e. global, minimum with respect to electronic minimization.\r
%\r
% a b transition\r
This finding agrees well with results by Mattoni et~al.\cite{mattoni2002}.\r
% all other investigated results: attractive interaction. stress compensation.\r
In contrast, all other investigated configurations show attractive interactions.\r
-The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper next neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.\r
+The most favorable configuration is found for C$_{\text{s}}$ at position 3, which corresponds to the lattice site of one of the upper neighbored Si atoms of the DB structure that is compressively strained along \hkl[1 -1 0] and \hkl[0 0 1] by the C-Si DB.\r
The substitution with C allows for most effective compensation of strain.\r
-This structure is followed by C$_{\text{s}}$ located at position 2, the next neighbor atom below the two Si atoms bound to the C$_{\text{i}}$ DB atom.\r
+This structure is followed by C$_{\text{s}}$ located at position 2, the lattice site of one of the neighbor atoms below the two Si atoms that are bound to the C$_{\text{i}}$ DB atom.\r
As mentioned earlier these two lower Si atoms indeed experience tensile strain along the \hkl[1 1 0] bond chain, however, additional compressive strain along \hkl[0 0 1] exists.\r
The latter is partially compensated by the C$_{\text{s}}$ atom.\r
Yet less of compensation is realized if C$_{\text{s}}$ is located at position 4 due to a larger separation although both bottom Si atoms of the DB structure are indirectly affected, i.e. each of them is connected by another Si atom to the C atom enabling the reduction of strain along \hkl[0 0 1].\r
\r
% c agglomeration vs c clustering ... migs to b conf\r
% 2 more migs: 051 -> 128 and 026! forgot why ... probably it's about probability of C clustering\r
-Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for seprations along one of the \hkl<1 1 0> directions.\r
-The eneregtically most favorable configuration (configuration b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.\r
+Obviously agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ is energetically favorable except for separations along one of the \hkl<1 1 0> directions.\r
+The energetically most favorable configuration (configuration b) forms a strong but compressively strained C-C bond with a separation distance of \unit[0.142]{nm} sharing a Si lattice site.\r
Again, conclusions concerning the probability of formation are drawn by investigating migration paths.\r
Since C$_{\text{s}}$ is unlikely to exhibit a low activation energy for migration the focus is on C$_{\text{i}}$.\r
Pathways starting from the two next most favored configurations were investigated, which show activation energies above \unit[2.2]{eV} and \unit[3.5]{eV} respectively.\r
In the last subsection configurations of a C$_{\text{i}}$ DB with C$_{\text{s}}$ occupying a vacant site have been investigated.\r
Additionally, configurations might arise in IBS, in which the impinging C atom creates a vacant site near a C$_{\text{i}}$ DB, but does not occupy it.\r
Resulting binding energies of a C$_{\text{i}}$ DB and a nearby vacancy are listed in the second row of Table~\ref{table:dc_c-sv}.\r
-All investigated structures are prefered compared to isolated largely separated defects.\r
+All investigated structures are preferred compared to isolated largely separated defects.\r
In contrast to C$_{\text{s}}$ this is also valid for positions along \hkl[1 1 0] resulting in an entirely attractive interaction between defects of these types.\r
Even for the largest possible distance (R) achieved in the calculations of the periodic supercell a binding energy as low as \unit[-0.31]{eV} is observed.\r
The ground state configuration is obtained for a V at position 1.\r
The C atom of the DB moves towards the vacant site forming a stable C$_{\text{s}}$ configuration resulting in the release of a huge amount of energy.\r
The second most favorable configuration is accomplished for a V located at position 3 due to the reduction of compressive strain of the Si DB atom and its two upper Si neighbors present in the C$_{\text{i}}$ DB configuration.\r
-This configuration is follwed by the structure, in which a vacant site is created at position 2.\r
+This configuration is followed by the structure, in which a vacant site is created at position 2.\r
Similar to the observations for C$_{\text{s}}$ in the last subsection a reduction of strain along \hkl[0 0 1] is enabled by this configuration.\r
Relaxed structures of the latter two defect combinations are shown in the bottom left of Fig.~\ref{fig:314-539} and \ref{fig:059-539} respectively together with their energetics during transition into the ground state.\r
\begin{figure}\r
Activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} are observed.\r
In the first case the Si and C atom of the DB move towards the vacant and initial DB lattice site respectively.\r
In total three Si-Si and one more Si-C bond is formed during transition.\r
-In the second case the lowest barrier is found for the migration of Si number 1 , which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.\r
+In the second case the lowest barrier is found for the migration of Si number 1, which is substituted by the C$_{\text{i}}$ atom, towards the vacant site.\r
A net amount of five Si-Si and one Si-C bond are additionally formed during transition.\r
The direct migration of the C$_{\text{i}}$ atom onto the vacant lattice site results in a somewhat higher barrier of \unit[1.0]{eV}.\r
In both cases, the formation of additional bonds is responsible for the vast gain in energy rendering almost impossible the reverse processes.\r
\r
-In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations indpendent of orientation and separation direction of the defects.\r
+In summary, pairs of C$_{\text{i}}$ DBs and Vs, like no other before, show highly attractive interactions for all investigated combinations independent of orientation and separation direction of the defects.\r
Furthermore, small activation energies, even for transitions into the ground state exist.\r
Based on these results, a high probability for the formation of C$_{\text{s}}$ must be concluded.\r
\r
There are good reasons for the existence of regions exhibiting such configurations with regard to the IBS process.\r
Highly energetic C atoms are able to kick out a Si atom from its lattice site, resulting in a Si self-interstitial accompanied by a vacant site, which might get occupied by another C atom that lost almost all of its kinetic energy.\r
%Thus, configurations of C$_{\text{s}}$ and Si self-interstitials are investigated in the following.\r
-Provided that the first C atom, which created the V and Si$_{\text{i}}$ pair had enough kinetic energy to escape the affected region, we are left ... WEITER\r
+Provided that the first C atom, which created the V and Si$_{\text{i}}$ pair has enough kinetic energy to escape the affected region, the C$_{\text{s}}$-Si$_{\text{i}}$ pair can be described as a separated defect complex.\r
The Si$_{\text{i}}$ \hkl<1 1 0> DB, which was found to exhibit the lowest energy of formation within the investigated self-interstitial configurations, is assumed to provide the energetically most favorable configuration in combination with C$_{\text{s}}$.\r
\r
\begin{table}\r
4 & \RM{4} & \RM{7} & \RM{9} & \RM{10} & \RM{10} & \RM{9} \\\r
5 & \RM{5} & \RM{8} & \RM{6} & \RM{2} & \RM{6} & \RM{2} \\\r
\end{tabular}\r
-\caption{Equivalent configurations labeld \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}\r
+\caption{Equivalent configurations labeled \RM{1}-\RM{10} of \hkl<1 1 0>-type Si$_{\text{i}}$ DBs created at position I and C$_{\text{s}}$ created at positions 1 to 5 according to Fig.~\ref{fig:combos_si}. The respective orientation of the Si$_{\text{i}}$ DB is given in the first row.}\r
\label{table:dc_si-s}\r
\end{ruledtabular}\r
\end{table}\r
$E_{\text{b}}$ [eV] & -0.97 & -0.08 & 0.22 & -0.02 & -0.23 & -0.25 & -0.02 & -0.06 & 0.05 & -0.03 \\\r
$r$ [nm] & 0.292 & 0.394 & 0.241 & 0.453 & 0.407 & 0.408 & 0.452 & 0.392 & 0.456 & 0.453\\\r
\end{tabular}\r
-\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of the combinational C$_{\text{s}}$ and Si$_{\text{i}}$ configurations as defined in Table~\ref{table:dc_si-s}. Energies are given in eV while the separation is given in nm.}\r
+\caption{Formation energies $E_{\text{f}}$, binding energies $E_{\text{b}}$ and C$_{\text{s}}$-Si$_{\text{i}}$ separation distances of configurations combining C$_{\text{s}}$ and Si$_{\text{i}}$ as defined in Table~\ref{table:dc_si-s}.}\r
\label{table:dc_si-s_e}\r
\end{ruledtabular}\r
\end{table*}\r
Corresponding formation as well as binding energies and the separation distances of the C$_{\text{s}}$ atom and the Si$_{\text{i}}$ DB lattice site are listed in Table~\ref{table:dc_si-s_e}.\r
In total ten different configurations exist within the investigated range.\r
Configuration \RM{1} constitutes the energetically most favorable structure exhibiting a formation energy of \unit[4.37]{eV}.\r
-Obviously the configuration of a \hkl[1 1 0] Si$_{\text{i}}$ DB and a next neighbored C$_{\text{s}}$ in the same direction as the alignment of the DB, as displayed in the bottom right of Fig.~\ref{fig:162-097}, enables the largest possible reduction of strain.\r
-The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si DB atom closest to the C atom does no longer form bonds to its top Si neighbors, but to the second next neighbored Si atom along \hkl[1 1 0].\r
-However, this configuration is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.\r
+Obviously the configuration of a Si$_{\text{i}}$ \hkl[1 1 0] DB and a neighbored C$_{\text{s}}$ atom along the bond chain, which has the same direction as the alignment of the DB, enables the largest possible reduction of strain.\r
+The relaxed structure is displayed in the bottom right of Fig.~\ref{fig:162-097}.\r
+Compressive strain originating from the Si$_{\text{i}}$ is compensated by tensile strain inherent to the C$_{\text{s}}$ configuration.\r
+The Si$_{\text{i}}$ DB atoms are displaced towards the lattice site occupied by the C$_{\text{s}}$ atom in such a way that the Si$_{\text{i}}$ DB atom closest to the C atom does no longer form bonds to its top Si neighbors, but to the next neighbored Si atom along \hkl[1 1 0].\r
+\r
+However, the configuration is energetically less favorable than the \hkl<1 0 0> C$_{\text{i}}$ DB, which, thus, remains the ground state of a C atom introduced into otherwise perfect c-Si.\r
The transition involving the latter two configurations is shown in Fig.~\ref{fig:162-097}.\r
\begin{figure}\r
\includegraphics[width=\columnwidth]{162-097.ps}\r
-\caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} is observed.}\r
+\caption{Migration barrier and structures of the transition of a \hkl[1 1 0] Si$_{\text{i}}$ DB next to C$_{\text{s}}$ (right) into the C$_{\text{i}}$ \hkl[0 0 -1] DB configuration (left). An activation energy of \unit[0.12]{eV} and \unit[0.77]{eV} for the reverse process is observed.}\r
\label{fig:162-097}\r
\end{figure}\r
An activation energy as low as \unit[0.12]{eV} is necessary for the migration into the ground state configuration.\r
-Thus, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.\r
+Accordingly, the C$_{\text{i}}$ \hkl<1 0 0> DB configuration is assumed to occur more likely.\r
However, only \unit[0.77]{eV} are needed for the reverse process, i.e. the formation of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB out of the ground state.\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 with the fit estimating almost zero interaction at \unit[0.6]{nm}.\r
-This indicates a low interaction capture radius of the defect pair.\r
-In IBS highly energetic collisions are assumed to easily produce configurations of these defects with separation distances exceeding the capture radius.\r
-For this reason C$_{\text{s}}$ without a nearby interacting Si$_{\text{i}}$ DB, which are, 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
-As mentioned above, configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ DBs might be especially important at higher temperatures due to the low activation energy necessary for its formation.\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
+Similar to what was previously mentioned, configurations of C$_{\text{s}}$ and a Si$_{\text{i}}$ DB might be particularly important at higher temperatures due to the low activation energy necessary for its formation.\r
At higher temperatures the contribution of entropy to structural formation increases, which might result in a spatial separation even for defects located within the capture radius.\r
-Indeed an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 next neighbor distances realized in a repeated migration mechnism of annihilating and arising Si DBs.\r
+Indeed, an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from configuration \RM{1}, which -- based on the above findings -- is assumed to recombine into the ground state configuration, results in a separation of the C$_{\text{s}}$ and Si$_{\text{i}}$ DB by more than 4 neighbor distances realized in a repeated migration mechanism of annihilating and arising Si$_{\text{i}}$ DBs.\r
The atomic configurations for two different points in time are shown in Fig.~\ref{fig:md}.\r
-Si atoms 1 and 2, which form the initial DB, occupy usual Si lattice sites in the final configuration while atom 3 occupies an interstitial site.\r
+Si atoms 1 and 2, which form the initial DB, occupy Si lattice sites in the final configuration while Si atom 3 is transferred from a regular lattice site into the interstitial lattice.\r
\begin{figure}\r
\begin{minipage}{0.49\columnwidth}\r
\includegraphics[width=\columnwidth]{md01.eps}\r
$t=\unit[2900]{fs}$\r
\end{center}\r
\end{minipage}\r
-\caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms.}\r
+\caption{Atomic configurations of an ab initio molecular dynamics run at \unit[900]{$^{\circ}$C} starting from a configuration of C$_{\text{s}}$ located next to a Si$_{\text{i}}$ \hkl[1 1 0] DB (atoms 1 and 2). Equal atoms are marked by equal numbers. Bonds are drawn for substantial atoms only.}\r
\label{fig:md}\r
\end{figure}\r
\r
The ground state configurations of these defects, i.e. the Si$_{\text{i}}$ \hkl<1 1 0> and C$_{\text{i}}$ \hkl<1 0 0> DB, have been reproduced and compare well to previous findings of theoretical investigations on Si$_{\text{i}}$\cite{leung99,al-mushadani03} as well as theoretical\cite{dal_pino93,capaz94,burnard93,leary97,jones04} and experimental\cite{watkins76,song90} studies on C$_{\text{i}}$.\r
A quantitatively improved activation energy of \unit[0.9]{eV} for a qualitatively equal migration path based on studies by Capaz et.~al.\cite{capaz94} to experimental values\cite{song90,lindner06,tipping87} ranging from \unit[0.70-0.87]{eV} reinforce their derived mechanism of diffusion for C$_{\text{i}}$ in Si.\r
\r
-The investigation of defect pairs indicates a general trend of defect agglomeration mainly driven by the potential of strain reduction.\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
-Configurations involving two C impurities indeed exhibit the ground state for structures consisting 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 with less high activation energies, which, however, involve intermediate unfavorable configurations.\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
-In contrast, C$_{\text{i}}$ and V were found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in C$_{\text{s}}$ configurations.\r
-In addition, a highly attractive interaction exhibiting a large capture radius was observed, effective independent of the orientation and the direction of separation of the defects.\r
-Thus, the formation of C$_{\text{s}}$ is very likely to occur.\r
+In contrast, C$_{\text{i}}$ and Vs were found to efficiently react with each other exhibiting activation energies as low as \unit[0.1]{eV} and \unit[0.6]{eV} resulting in stable C$_{\text{s}}$ configurations.\r
+In addition, we observed a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects.\r
+Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.\r
Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.\r
\r
-Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$ formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.\r
+Investigating configurations of C$_{\text{s}}$ and Si$_{\text{i}}$, formation energies higher than that of the C$_{\text{i}}$ \hkl<1 0 0> DB were obtained keeping up previously derived assumptions concerning the ground state of C$_{\text{i}}$ in otherwise perfect Si.\r
However, a small capture radius was identified for the respective interaction that might prevent the recombination of defects exceeding a separation of \unit[0.6]{nm} into the ground state configuration.\r
In addition, a rather small activation energy of \unit[0.77]{eV} allows for the formation of a C$_{\text{s}}$-Si$_{\text{i}}$ pair originating from the C$_{\text{i}}$ \hkl<1 0 0> DB structure by thermally activated processes.\r
Thus, elevated temperatures might lead to configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is supported by the result of the molecular dynamics run.\r
\r
% add somewhere: nearly same energies of C_i -> Si_i + C_s, Si_i mig and C_i mig\r
\r
-% add somewhere: controversy c_i vs c_s agglomeration, we suggest both!\r
-\r
These findings allow to draw conclusions on the mechanisms involved in the process of SiC conversion in Si.\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 equlibrium, 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
+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$_{\text{s}}$ 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 a highly mobile C$_{\text{i}}$ again, allowing for the rearrangement of the C atom.\r
-The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the fcc lattices that build up the diamond lattice as expected in 3C-SiC.\r
-On the other hand the conversion of some region of Si into SiC by substitutional C is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si.\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
+The rearrangement is crucial to end up in a configuration of C atoms only occupying substitutionally the lattice sites of one of the two fcc lattices that build up the diamond lattice.\r
+% TODO: add SiC structure info to intro\r
+On the other hand, the conversion of some region of Si into SiC by substitutional C is accompanied by a reduction of the volume since SiC exhibits a \unit[20]{\%} smaller lattice constant than Si.\r
The reduction in volume is compensated by excess Si$_{\text{i}}$ serving as building blocks for the surrounding Si host or a further formation of SiC.\r
\r
-It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$.\r
+We conclude that precipitation occurs by successive agglomeration of C$_{\text{s}}$.\r
However, the agglomeration and rearrangement of C$_{\text{s}}$ is only possible by mobile C$_{\text{i}}$, which has to be present at the same time.\r
-Thus, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.\r
+Accordingly, the process is governed by both, C$_{\text{s}}$ accompanied by Si$_{\text{i}}$ as well as C$_{\text{i}}$.\r
It is worth to mention that there is no contradiction to results of the HREM studies\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}.\r
Regions showing dark contrasts in an otherwise undisturbed Si lattice are attributed to C atoms in the interstitial lattice.\r
However, there is no particular reason for the C species to reside in the interstitial lattice.\r
Contrasts are also assumed for Si$_{\text{i}}$.\r
-Once precipitation occurs regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.\r
-Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials too small to be detected in HREM.\r
-In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain either in the stretched SiC or at the interface of the contracted SiC and the Si host.\r
+Once precipitation occurs, regions of dark contrasts disappear in favor of Moir\'e patterns indicating 3C-SiC in c-Si due to the mismatch in the lattice constant.\r
+Until then, however, these regions are either composed of stretched coherent SiC and interstitials or of already contracted incoherent SiC surrounded by Si and interstitials, where the latter is too small to be detected in HREM.\r
+In both cases Si$_{\text{i}}$ might be attributed a third role, which is the partial compensation of tensile strain that is present either in the stretched SiC or at the interface of the contracted SiC and the Si host.\r
\r
-In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is statisfied by the mechanism of successive positioning of C$_{\text{s}}$.\r
+In addition, the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate is satisfied by the mechanism of successive positioning of C$_{\text{s}}$.\r
In contrast, there is no obvious reason for the topotactic orientation of an agglomerate consisting exclusively of C-Si dimers, which would necessarily involve a much more profound change in structure for the transition into SiC.\r
\r
\section{Summary}\r
\r
In summary, C and Si point defects in Si, combinations of these defects and diffusion processes within such configurations have been investigated.\r
-It is shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.\r
-Investigations of migration pathways, however, allow to conclude that C clustering fails to appear by thermally activated processes due to high activation energies of the respective diffusion processes.\r
+We have shown that C interstitials in Si tend to agglomerate, which is mainly driven by a reduction of strain.\r
+Investigations of migration pathways, however, allow to conclude that C clustering is hindered due to high activation energies of the respective diffusion processes.\r
A highly attractive interaction and a large capture radius has been identified for the C$_{\text{i}}$ \hkl<1 0 0> DB and the vacancy indicating a high probability for the formation of C$_{\text{s}}$.\r
In contrast, a rapidly decreasing interaction with respect to the separation distance has been identified for C$_{\text{s}}$ and a Si$_{\text{i}}$ \hkl<1 1 0> DB resulting in a low probability of defects exhibiting respective separations to transform into the C$_{\text{i}}$ \hkl<1 0 0> DB, which constitutes the ground state configuration for a C atom introduced into otherwise perfect Si. \r
-Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.\r
-It is concluded that the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.\r
+%Based on these findings conclusions on basic processes involved in the SiC precipitation in bulk Si are drawn.\r
+Obviously, the precipitation process is governed by the formation of C$_{\text{s}}$ already in the initial stages.\r
Agglomeration and rearrangement of C$_{\text{s}}$, however, is only possible by mobile C$_{\text{i}}$, which, thus, needs to be present at the same time.\r
Si$_{\text{i}}$ constitutes the vehicle for the rearrangement of C$_{\text{s}}$.\r
\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
% --------------------------------- references -------------------\r
\r
-\bibliography{../../bibdb/bibdb}{}\r
-\bibliographystyle{h-physrev3}\r
-\r
+%\bibliography{../../bibdb/bibdb}{}\r
+%\bibliographystyle{h-physrev3}\r
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\end{document}\r
\r
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