\chapter{Summary and conclusions}
\label{chapter:summary}
-{\setlength{\parindent}{0pt}
-%\paragraph{To summarize,}
{\bf To summarize},
in a short review of the C/Si compound and the fabrication of the technologically promising semiconductor SiC by IBS, two controversial assumptions of the precipitation mechanism of 3C-SiC in c-Si are elaborated.
-}
-These propose the precipitation of SiC by agglomeration of \ci{} DBs followed by a sudden formation of SiC and otherwise a formation by successive accumulation of \cs{} via intermediate stretched SiC structures, which are coherent to the Si lattice.
+These propose precipitation of SiC by agglomeration of \ci{} DBs followed by a sudden formation of SiC and otherwise a formation by successive accumulation of \cs{} via intermediate stretched SiC structures, which are coherent to the Si lattice.
To solve this controversy and contribute to the understanding of SiC precipitation in c-Si, a series of atomistic simulations is carried out.
In the first part, intrinsic and C related point defects in c-Si as well as some selected diffusion processes of the C defect are investigated by means of first-principles quantum-mechanical calculations based on DFT and classical potential calculations employing a Tersoff-like analytical bond order potential.
Shortcomings of the computationally efficient though less accurate classical potential approach compared to the quantum-mechanical treatment are revealed.
The study proceeds investigating combinations of defect structures and related diffusion processes exclusively by the first-principles method.
The applicability of the utilized bond order potential for subsequent MD simulations is discussed.
-Conclusions on the precipitation based on the DFT results are drawn.
-In the second part, classical potential MD simulations are performed, which try to directly reproduce the precipitation.
+Conclusions on the precipitation mechanism based on the DFT results are drawn.
+In the second part, classical potential MD simulations are performed, which try to directly reproduce the precipitation process.
Next to the shortcomings of the potential, quirks inherent to MD are discussed and a workaround is proposed.
Although direct formation of SiC fails to appear, the obtained results indicate a mechanism of precipitation, which is consistent with previous quantum-mechanical conclusions as well as experimental findings.
-Quantum-mechanical results of intrinsic point defects in Si are in good agreement to previous theoretical work on this subject \cite{leung99,al-mushadani03}.
+Quantum-mechanical results of intrinsic point defects in Si are in good agreement to previous theoretical work on this subject~\cite{leung99,al-mushadani03}.
The \si{} \hkl<1 1 0> DB defect is reproduced as the ground-state configuration followed by the hexagonal and tetrahedral defect.
Spin polarized calculations are required for the \si{} \hkl<1 0 0> DB and vacancy whereas no other of the investigated intrinsic defects is affected.
For the \si{} \hkl<1 0 0> DB, 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.
Results obtained by calculations utilizing the classical EA potential yield formation energies, which are of the same order of magnitude.
However, EA predicts the tetrahedral configuration to be most stable.
The particular problem is due to the cut-off and the fact that the second neighbors are only slightly more distant than the first neighbors within the tetrahedral configuration.
-Furthermore, the hexagonal defect structure is not stable opposed to results of the authors of the potential \cite{albe_sic_pot}.
+Furthermore, the hexagonal defect structure is not stable opposed to results of the authors of the potential~\cite{albe_sic_pot}.
The obtained structure after relaxation, which is similar to the tetrahedral configuration, exhibits a formation energy equal to the one given by the authors for the hexagonal one.
Obviously, the authors did not check the relaxed structure still assuming a hexagonal configuration.
The actual structure is equal to the tetrahedral configuration, which is slightly displaced along the three coordinate axes.
Variations exist with displacements along two or a single \hkl<1 0 0> direction indicating a potential artifact.
However, finite temperature simulations are not affected by this artifact due to a low activation energy necessary for a transition into the energetically more favorable tetrahedral configuration.
-Next to the known problem of the underestimated formation energy of the tetrahedral configuration \cite{tersoff90}, the energetic sequence of the defect structures is well reproduced by the EA calculations.
-Migration barriers of \si{} investigated by quantum-mechanical calculations are found to be of the same order of magnitude than values derived in other {\em ab initio} studies \cite{bloechl93,sahli05}.
+Next to the known problem of the underestimated formation energy of the tetrahedral configuration~\cite{tersoff90}, the energetic sequence of the defect structures is well reproduced by the EA calculations.
+Migration barriers of \si{} investigated by quantum-mechanical calculations are found to be of the same order of magnitude than values derived in other {\em ab initio} studies~\cite{bloechl93,sahli05}.
Defects of C in Si are well described by both methods.
-The \ci{} \hkl<1 0 0> DB is found to constitute the most favorable interstitial configuration in agreement with several theoretical \cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental \cite{watkins76,song90} investigations.
+The \ci{} \hkl<1 0 0> DB is found to constitute the most favorable interstitial configuration in agreement with several theoretical~\cite{burnard93,leary97,dal_pino93,capaz94,jones04} and experimental~\cite{watkins76,song90} investigations.
Almost equal formation energies are predicted by the EA and DFT calculations for this defect.
A small discrepancy in the resulting equilibrium structure with respect to the DFT method exists due to missing quantum-mechanical effects within the classical treatment.
The high formation energies of the tetrahedral and hexagonal configuration obtained by classical potentials act in concert with the fact that these configurations are found unstable by the first-principles description.
The BC configuration turns out to be unstable relaxing into the \ci{} \hkl<1 1 0> DB configuration within the EA approach.
-This is supported by another {\em ab initio} study \cite{capaz94}, which in turn finds the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
+This is supported by another {\em ab initio} study~\cite{capaz94}, which in turn finds the BC configuration to be an intermediate saddle point structure of a possible migration path, which is \unit[2.1]{eV} higher than the \ci{} \hkl<1 0 0> DB structure.
By quantum-mechanical calculations performed in this work, however, it turns out that the BC configuration constitutes a real local minimum if the electron spin is fully accounted for.
Indeed, spin polarization is absolutely necessary for the BC configuration resulting in a net magnetization of two electrons accompanied by a reduction of the total energy by \unit[0.3]{eV}.
The resulting spin up density is localized in a torus around the C perpendicular to the linear Si-C-Si bond.
Thus, the underestimated formation energy does not pose a serious limitation.
Based on the above findings, it is concluded that modeling of the SiC precipitation by the EA potential might lead to trustable results.
-Quantum-mechanical investigations of the mobility of the \ci{} \hkl<1 0 0> DB yield a migration barrier of \unit[0.9]{eV}, which excellently agrees to experimental values ranging from \unit[0.70]{eV} to \unit[0.87]{eV} \cite{lindner06,song90,tipping87}.
+Quantum-mechanical investigations of the mobility of the \ci{} \hkl<1 0 0> DB yield a migration barrier of \unit[0.9]{eV}, which excellently agrees to experimental values ranging from \unit[0.70]{eV} to \unit[0.87]{eV}~\cite{lindner06,song90,tipping87}.
The respective path corresponds to a \ci{} \hkl[0 0 -1] DB migrating towards the next neighbored Si atom located in \hkl[1 1 -1] direction forming a \ci{} \hkl[0 -1 0] DB.
The identified migration path involves a change in orientation of the DB.
-Thus, the same path explains the experimentally determined activation energies for reorientation of the DB ranging from \unit[0.77]{eV} \cite{watkins76} up to \unit[0.88]{eV} \cite{song90}.
+Thus, the same path explains the experimentally determined activation energies for reorientation of the DB ranging from \unit[0.77]{eV}~\cite{watkins76} up to \unit[0.88]{eV}~\cite{song90}.
Investigations based on the EA bond order potential suggest a migration involving an intermediate \ci{} \hkl<1 1 0> DB configuration.
Although different, starting and final configuration as well as the change in orientation of the \hkl<1 0 0> DB are equal to the identified pathway by the {\em ab initio} calculations.
-However, barrier heights, which are overestimated by a factor of 2.4 to 3.5 depending on the character of migration, i.e. a single step or two step process, compared to the DFT results, are obtained.
+However, barrier heights, which are overestimated by a factor of 2.4 to 3.5 depending on the character of migration, i.e.\ a single step or two step process, compared to the DFT results, are obtained.
Obviously, the EA potential fails to describe \ci{} diffusion yielding a drastically overestimated activation energy, which has to be taken into account in subsequent investigations.
-Subsequent investigations focus on defect combinations exclusively by the first-principles description.
+A series of investigations focuses on defect combinations exclusively by the first-principles description.
These configurations are constructed in such a way as to allow for a quantum-mechanical treatment.
-Investigations of two \ci{} defects of the \hkl<1 0 0>-type for varying separations and orientations state a rather attractive interaction between these interstitials.
+Investigations of two \ci{} defects of the \hkl<1 0 0>-type for varying separations and orientations state a fairly attractive interaction between these interstitials.
The capture radius is found to be rather large compared to other defect combinations.
-Mostly energetically favorable configurations of two interstitials are found.
+For the most part, energetically favorable configurations of two interstitials are found.
This is due to strain compensation enabled by the combination of such defects in certain orientations.
An interaction energy proportional to the reciprocal cube of the distance in the far field regime is found supporting the assumption of \ci{} DB agglomeration.
The energetically most favorable configuration consists of a C-C bond.
Results of combinations of \ci{} and \cs{} revealed two additional metastable structures different to these obtained by a naive relaxation.
Small displacements result in a structure of a \hkl<1 1 0> C-C DB and in a structure of a twofold coordinated Si atom located in between two substitutional C atoms residing on regular Si lattice sites.
Both structures are lower in energy compared to the respective counterparts.
-These results, for the most part, compare well with results gained in previous studies \cite{leary97,capaz98,liu02} and show an astonishingly good agreement with experiment \cite{song90_2}.
+These results, for the most part, compare well with results gained in previous studies~\cite{leary97,capaz98,liu02} and show an astonishingly good agreement with experiment~\cite{song90_2}.
Again, spin polarized calculations are revealed necessary.
A net magnetization of two electrons is observed for the \hkl<1 1 0> C-C DB configuration, which constitutes the ground state.
A repulsive interaction is observed for C$_{\text{s}}$ at lattice sites along \hkl[1 1 0] due to tensile strain originating from both, the C$_{\text{i}}$ DB and the C$_{\text{s}}$ atom.
% maybe preliminary conclusions here ...
Classical potential MD calculations targeting the direct simulation of SiC precipitation in Si are adopted.
-Therefore, the necessary amount of C is gradually incorporated into a large c-Si host.
+Therefor, the necessary amount of C is gradually incorporated into a large c-Si host.
Simulations at temperatures used in IBS result in structures dominated by the C$_{\text{i}}$ \hkl<1 0 0> DB and its combinations if C is inserted into the total volume.
Incorporation into volumes $V_2$ and $V_3$, which correspond to the volume of the expected precipitate and the volume containing the necessary amount of Si, lead to an amorphous SiC-like structure within the respective volume.
Both results are not expected with respect to the outcome of the IBS experiments.
-In the first case, i.e. the low C concentration simulations, \ci{} \hkl<1 0 0> DBs are indeed formed.
+In the first case, i.e.\ the low C concentration simulations, \ci{} \hkl<1 0 0> DBs are indeed formed.
However, sufficient defect agglomeration is not observed.
-In the second case, i.e. the high C concentration simulations, crystallization of the amorphous structure, which is not expected at prevailing temperatures, is likewise not observed.
+In the second case, i.e.\ the high C concentration simulations, crystallization of the amorphous structure, which is not expected at prevailing temperatures, is likewise not observed.
Limitations of the MD technique in addition to overestimated bond strengths due to the short range potential are identified to be responsible.
The approach of using increased temperatures during C insertion is followed to work around this problem termed {\em potential enhanced slow phase space propagation}.
Higher temperatures are justified for several reasons.
Elevated temperatures are expected to compensate the overestimated diffusion barriers and accelerate structural evolution.
-In addition, formation of SiC is also observed at higher implantation temperatures \cite{nejim95,lindner01} and temperatures in the implantation region is definitely higher than the temperature determined experimentally at the surface of the sample.
+In addition, formation of SiC is also observed at higher implantation temperatures~\cite{nejim95,lindner01} and temperatures in the implantation region is definitely higher than the temperature determined experimentally at the surface of the sample.
Furthermore, the present study focuses on structural transitions in a system far from equilibrium.
No significant change is observed for high C concentrations at increased temperatures.
Indeed, utilizing increased temperatures is assumed to constitute a necessary condition to simulate IBS of 3C-SiC in c-Si.
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-% todo - sync with respective conclusion chapter
-%
% conclusions 2nd part
-%\paragraph{Conclusions}
{\bf Conclusions}
concerning the SiC conversion mechanism are derived from results of both, first-principles and classical potential calculations.
-Although classical potential MD calculations fail to directly simulate the precipitation of SiC, obtained results, on the one hand, reinforce previous findings of the first-principles investigations and, on the other hand, allow further conclusions on the SiC precipitation in Si.
+Although classical potential MD calculations fail to directly simulate precipitation of SiC, obtained results, on the one hand, reinforce previous findings of the first-principles investigations and, on the other hand, allow further conclusions on the SiC precipitation in Si.
Initially, quantum-mechanical investigations suggest agglomeration of \ci{} defects that form energetically favorable configurations by an effective stress compensation.
Low barriers of migration are found except for transitions into the ground-state configuration, which is composed of a strong C-C bond.
Thus, agglomeration of \ci{} in the absence of C clustering is expected.
These initial results suggest a conversion mechanism based on the agglomeration of \ci{} defects followed by a sudden precipitation once a critical size is reached.
However, subsequent investigations of structures that are particularly conceivable under conditions prevalent in IBS and at elevated temperatures show \cs{} to occur in all probability.
-The transition from the ground state of a single C atom incorporated into otherwise perfect c-Si, i.e. the \ci{} \hkl<1 0 0> DB, into a configuration of \cs{} next to a \si{} atom exhibits an activation energy lower than the one for the diffusion of the highly mobile \ci{} defect.
+The transition from the ground state of a single C atom incorporated into otherwise perfect c-Si, i.e.\ the \ci{} \hkl<1 0 0> DB, into a configuration of \cs{} next to a \si{} atom exhibits an activation energy lower than the one for the diffusion of the highly mobile \ci{} defect.
Considering additionally the likewise lower diffusion barrier of \si{}, configurations of separated \cs{} and \si{} will occur in all probability.
This is reinforced by the {\em ab initio} MD run at non-zero temperature, which shows structures of separating instead of recombining \cs{} and \si{} defects.
This suggests increased participation of \cs{} already in the initial stages of the implantation process.
-The highly mobile \si{} is assumed to constitute a vehicle for the rearrangement of other \cs{} atoms onto proper lattice sites, i.e. lattice sites of one of the the two fcc lattices composing the diamond structure.
+The highly mobile \si{} is assumed to constitute a vehicle for the rearrangement of other \cs{} atoms onto proper lattice sites, i.e.\ lattice sites of one of the the two fcc lattices composing the diamond structure.
This way, stretched SiC structures, which are coherently aligned to the c-Si host, are formed by agglomeration of \cs.
Precipitation into an incoherent and partially strain-compensated SiC nucleus occurs once the increasing strain energy surpasses the interfacial energy of the incoherent 3C-SiC precipitate and the c-Si substrate.
As already assumed by Nejim~et~al.~\cite{nejim95}, \si{} serves as supply for subsequently implanted C atoms to form further SiC in the resulting free space due to the accompanied volume reduction.
These findings as well as the derived conclusion on the precipitation mechanism involving an increased participation of \cs{} agree well with experimental results.
% low t high mobility
% high t stable config, no redistr
-C implanted at room temperature was found to be able to migrate towards the surface in contrast to implantations at \degc{500}, which do not show redistribution of the C atoms \cite{serre95}.
+C implanted at room temperature was found to be able to migrate towards the surface in contrast to implantations at \degc{500}, which do not show redistribution of the C atoms~\cite{serre95}.
This excellently conforms to the results of the MD simulations at different temperatures, which show the formation of highly mobile \ci{} \hkl<1 0 0> DBs for low and much more stable \cs{} defects for high temperatures.
-This is likewise suggested by IBS experiments utilizing implantation temperatures of \degc{550} followed by incoherent lamp annealing at temperatures as high as \degc{1405} required for the C segregation due to the stability of \cs{} \cite{reeson87}.
+This is likewise suggested by IBS experiments utilizing implantation temperatures of \degc{550} followed by incoherent lamp annealing at temperatures as high as \degc{1405} required for the C segregation due to the stability of \cs{}~\cite{reeson87}.
% high imp temps more effective to achieve ?!? ...
-Furthermore, increased implantation temperatures were found to be more efficient than high temperatures in the postannealing step concerning the formation of topotactically aligned 3C-SiC precipitates \cite{kimura82,eichhorn02}.
+Furthermore, increased implantation temperatures were found to be more efficient than high temperatures in the postannealing step concerning the formation of topotactically aligned 3C-SiC precipitates~\cite{kimura82,eichhorn02}.
%
-Particularly strong C-C bonds, which are hard to break by thermal annealing, were found to effectively aggravate the restructuring process of such configurations \cite{deguchi92}.
+Particularly strong C-C bonds, which are hard to break by thermal annealing, were found to effectively aggravate the restructuring process of such configurations~\cite{deguchi92}.
These bonds preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements in regions exhibiting a large amount of C atoms.
This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations.
%
%Considering the efficiency of high implantation temperatures, experimental arguments exist, which point to the precipitation mechanism based on the agglomeration of \cs.
-More substantially, understoichiometric implantations at room temperature into preamorphized Si followed by a solid-phase epitaxial regrowth step at \degc{700} result in Si$_{1-x}$C$_x$ layers in the diamond cubic phase with C residing on substitutional Si lattice sites \cite{strane93}.
+More substantially, understoichiometric implantations at room temperature into preamorphized Si followed by a solid-phase epitaxial regrowth step at \degc{700} result in Si$_{1-x}$C$_x$ layers in the diamond cubic phase with C residing on substitutional Si lattice sites~\cite{strane93}.
The strained structure is found to be stable up to \degc{810}.
Coherent clustering followed by precipitation is suggested if these structures are annealed at higher temperatures.
%
-Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C} \cite{werner96} annealing temperature.
-Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples \cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
+Similar, implantations of an understoichiometric dose into c-Si at room temperature followed by thermal annealing result in small spherical sized C$_{\text{i}}$ agglomerates below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size above \unit[700]{$^{\circ}$C}~\cite{werner96} annealing temperature.
+Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected and indeed observed for as-implanted samples~\cite{lindner99,lindner01} in implantations performed at \unit[450]{$^{\circ}$C}.
According to this, implanted C is likewise expected to occupy substitutionally regular Si lattice sites right from the start for implantations into c-Si at elevated temperatures.
%
%
% low t - randomly ...
% high t - epitaxial relation ...
-Moreover, implantations below the optimum temperature for the IBS of SiC show regions of randomly oriented SiC crystallites whereas epitaxial crystallites are found for increased temperatures \cite{lindner99}.
+Moreover, implantations below the optimum temperature for the IBS of SiC show regions of randomly oriented SiC crystallites whereas epitaxial crystallites are found for increased temperatures~\cite{lindner99}.
The results of the MD simulations can be interpreted in terms of these experimental findings.
The successive occupation of regular Si lattice sites by \cs{} atoms as observed in the high temperature MD simulations and assumed from results of the quantum-mechanical investigations perfectly satisfies the epitaxial relation of substrate and precipitate.
In contrast, there is no obvious reason for a topotactic transition of \ci{} \hkl<1 0 0> DB agglomerates, as observed in the low temperature MD simulations, into epitaxially aligned precipitates.
The latter transition would necessarily involve a much more profound change in structure.
% amorphous region for low temperatures
-Experimentally, randomly oriented precipitates might also be due to SiC nucleation within the arising amorphous matrix \cite{lindner99}.
+Experimentally, randomly oriented precipitates might also be due to SiC nucleation within the arising amorphous matrix~\cite{lindner99}.
In simulation, an amorphous SiC phase is formed for high C concentrations.
This is due to high amounts of introduced damage within a short period of time resulting in essentially no time for structural evolution, which is comparable to the low temperature experiments, which lack the kinetic energy necessary for recrystallization of the highly damaged region.
Indeed, the complex transformation of agglomerated \ci{} DBs as suggested by results of the low C concentration simulations could involve an intermediate amorphous phase probably accompanied by the loss of alignment with respect to the Si host matrix.
In any case, the precipitation mechanism by accumulation of \cs{} obviously satisfies the experimental finding of identical \hkl(h k l) planes of substrate and precipitate.
% no contradictions, something in interstitial lattice, projected potential ...
-Finally, it is worth to point out that the precipitation mechanism based on \cs{} does not necessarily contradict to results of the HREM studies \cite{werner96,werner97,lindner99_2}, which propose precipitation by agglomeration of \ci.
+Finally, it is worth to point out that the precipitation mechanism based on \cs{} does not necessarily contradict to results of the HREM studies~\cite{werner96,werner97,lindner99_2}, which propose precipitation by agglomeration of \ci.
In these studies, regions of dark contrasts are attributed to C atoms that reside in the interstitial lattice in an otherwise undisturbed Si lattice.
The \ci{} atoms lead to a local increase of the crystal potential, which is responsible for the dark contrast.
However, there is no particular reason for the C species to reside in the interstitial lattice.
This way, C can be easily rearranged in order to end up in a configuration of C atoms that occupy substitutionally the lattice sites of one of the fcc lattices of the diamond structure.
Stretched SiC structures arise, which are coherently aligned to the Si matrix.
\si{} is believed to likewise compensate the tensile strain within these structures.
-This is followed by the precipitation into incoherent 3C-SiC once the strain energy of the coherent structure surpasses the interfacial energy of the incoherent precipitate and the c-Si substrate.
+This is followed by precipitation into incoherent 3C-SiC once the strain energy of the coherent structure surpasses the interfacial energy of the incoherent precipitate and the c-Si substrate.
The associated volume reduction is compensated by \si{} that may serve as a supply for further SiC or as a building block for the surrounding Si host and likewise reduce existing strain in the interface region.
%
Results of the atomistic simulation study that indicate the respective precipitation mechanism conform well with other experimental findings.
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