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 next 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 second 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
\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
+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
+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
-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
+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 next neighbor distances realized in a repeated migration mechnism 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 indicatet 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 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
\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
-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
+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 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
\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
+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 fails to appear by thermally activated processes 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
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