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
+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
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
\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 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
\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
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
\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
+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
+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
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 next 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
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
\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
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
\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$_{\text{i}}$ 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 next 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 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
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 indicatet 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
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
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
\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 addition, thermally activated, C$_{\text{i}}$ might turn into C$_{\text{s}}$.\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
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