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 has been found to occur very unlikely.\r
-In contrast, substitutional carbon occurs 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 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 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
\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 tersely 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
\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
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 electron volt of C$_{\text{i}}$ \hkl<1 0 0>-type defect pairs. Equivalent configurations exhibit equal energies. Column 1 lists the orientation of the second defect, which is combined with the initial C$_{\text{i}}$ \hkl[0 0 -1] DB. The position index of the second defect is given in the first row according to Fig.~\ref{fig:combos}. R corresponds to the position located at $\frac{a_{\text{Si}}}{2}\hkl[3 2 3]$ relative to the initial defect position, which is the maximum realizable defect separation distance ($\approx \unit[1.3]{nm}$) due to periodic boundary conditions.}\r
+\caption{Binding energies in 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
% => 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
\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
+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
%\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
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
+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 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
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
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
+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
We conclude that precipitation occurs by successive agglomeration of C$_{\text{s}}$.\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
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
+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
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