\chapter{Summary and conclusions}
\label{chapter:summary}

\paragraph{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.
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 quatum-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.
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}.
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.
For the vacancy, the net spin up electron density is localized in caps at the four surrounding Si atoms directed towards the vacant site.
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}.
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 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.
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 calssical 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 inito} 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.
No other configuration is affected by spin polarization.
The underestimated formation energy of \cs{} is a definite drawback of the classical potential.
However, the creation of \cs{} is necessarily accompanied by a \si{} in a perfect Si crystal, in which a C atom is incorporated.
Fortunately, the energetics of combinations of \cs{} and \si{} are quite well described by the EA potential.
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}.
The respective path correpsonds 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} upto \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.
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.
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.
The capture radius is found to be rather large compared to other defect combinations.
Mostly 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.
However, due to high activation energies of respective pathways or alternative pathways featuring less high activation energies, which, however, involve intermediate unfavorable configurations, this structure is less likely to arise than structures of C atoms that are interconnected by another Si atom.
Thus, agglomeration of C$_{\text{i}}$ is expected whereas the formation of C-C bonds is assumed to fail to appear by thermally activated diffusion processes.

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 respetive 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}.
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.
All other investigated configurations show attractive interactions, which suggest an energetically favorable agglomeration of C$_{\text{i}}$ and C$_{\text{s}}$ except for separations along one of the \hkl<1 1 0> directions.
Although the most favorable configuration exhibits a C-C bond, migration paths show large barriers exceeding \unit[2.2]{eV} for transitions into the ground state.
As before, structures other than the ground-state configuration are assumed to arise more likely.
Thus, agglomeration of C defects in contrast to C clustering is again reinforced by these findings.

C$_{\text{i}}$ and vacancies are 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.
In addition, a highly attractive interaction exhibiting a large capture radius, effective independent of the orientation and the direction of separation of the defects, is observed.
Accordingly, the formation of C$_{\text{s}}$ is very likely to occur.
Comparatively high energies necessary for the reverse process reveal this configuration to be extremely stable.
Thus, C interstitials and vacancies located close together are assumed to end up in a configuration of \cs{}.

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.
However, a small capture radius is 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.
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.
Low diffusion barriers of \si{} enable further separation of the defect pair.
Thus, elevated temperatures might lead to configurations of C$_{\text{s}}$ and a remaining Si atom in the near interstitial lattice, which is likewise supported by the result of the MD run.

% 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.
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.
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.

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 severeal reasons.
Elevated temperatures are expected to compensate the overestimated diffusion barriers and accelerate strcutural evolution.
In addition, formation of SiC is also observed at higher implantation temperatures \cite{nejim95,lindner01} and temperatures in the implantation region is definetly 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.
The amorphous phase is maintained.
The huge amount of damage hampers identification of investigated structures, which in many cases lost the alignment to the c-Si host.
Obviously, inccorporation of a high quantity of C into a small volume within a short period of time creates damage, which decelerates structural evolution.
For the low C concentrations, time scales are still too low to observe C agglomeration.
However, a phase transition of the C$_{\text{i}}$-dominated into a clearly C$_{\text{s}}$-dominated structure is observed.
The amount of \cs{} increases with increasing temperature.
Diamond and graphite like bonds as well as the artificial bonds due to the cut-off are reduced.
Loose structures of stretched SiC, which are adjusted to the Si lattice with respect to the lattice constant and alignment, are identified.
\si{} is often found in the direct surrounding.
Entropic contributions are assumed to be responsible for these structures at elevated temperatures that deviate from the ground state at 0 K.
Indeed, utilizing increased temperatures is assumed to constitute a necessary condition to simulate IBS of 3C-SiC in c-Si.

% conclusions 2nd part
\paragraph{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.

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 clsutering is expected.
These initial results suggest a conversion mechansim 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 prevalant 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.
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{} defetcs.
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.
This way, stretched SiC strcutures, 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.

Several conclusions based on results obtained by classical potential MD simulations are drawn.
First of all, increased temperatures are considered a necessary condition to simulate the IBS of epitaxially aligned 3C-SiC in Si, which constitutes a process far from thermodynamic equilibrium.
The strong deviation from equilibrium by elevated temperatures enables the formation of \cs{}-\si{} structures as observed in the quantum-mechanical calculations.
In contrast, structures of \ci{} \hkl<1 0 0> DBs, which constitute the thermodynamic ground state, appear at low temperatures.
%
Secondly, in configurations of stretched SiC composed by \cs, the accompanied \si{} defect may be assigned further functionality.
Next to that as a vehicle that is able to rearrange \cs{} and a building block for the surrounding Si host or further SiC, the analyzed configurations suggest \si{} to be required for stress compensation.
As evidently observed in these structures, \si{} reduces tensile strain by capturing a position near one of the C atoms within a configuration of two C atoms that basically reside on Si lattice sites.
Furthermore, \si{} might compensate strain in the interface region of an incoherent, nucleated SiC precipitate and the c-Si matrix.
This could be achieved by \ci{} \hkl<1 0 0> DBs in the Si region slightly contracting the Si atoms next to the C atom to better match the spacing of Si atoms present in 3C-SiC.
Indeed, combinations of \cs{} and \ci{} \hkl<1 0 0> DBs are observed.
%
Further conclusions are derived from results of the high C concentration simulations, in which a large amount of C atoms is incorporated into a small volume within a short period of time, which results in essentially no time for the system to rearrange.
Due to this, the formation of strong C-C bonds and the production of a vast amount of damage is observed, which finally results in the formation of an amorphous phase.
The strong bonds and damage obviously decelerate structural evolution.
The short time, which is not sufficient for structural evolution, can be mapped to a system of low temperature, which lacks the kinetic energy required for the restructuring process.


HIER WEITER ...


To conclude, results of the present study indicate a precipitation of SiC in Si by successive agglomeration of \cs.
\si{}, which is likewise existent, serves several needs:
... Incoherent but epitaxially aligned SiC precipitates are ...


Experimental studies revealed increased implantation temperatures to be more efficient than postannealing methods for the formation of topotactically aligned precipitates \cite{kimura82,eichhorn02}.
In particular, restructuring of strong C-C bonds is affected \cite{deguchi92}, which preferentially arise if additional kinetic energy provided by an increase of the implantation temperature is missing to accelerate or even enable atomic rearrangements.
This is assumed to be related to the problem of slow structural evolution encountered in the high C concentration simulations due to the insertion of high amounts of C into a small volume within a short period of time resulting in essentially no time for the system to rearrange.
% rt implantation + annealing
Implantations of an understoichiometric dose at room temperature followed by thermal annealing results in small spherical sized C$_{\text{i}}$ agglomerates at temperatures below \unit[700]{$^{\circ}$C} and SiC precipitates of the same size at temperatures above \unit[700]{$^{\circ}$C} \cite{werner96}.
Since, however, the implantation temperature is considered more efficient than the postannealing temperature, SiC precipitates are expected -- and indeed are observed for as-implanted samples \cite{lindner99,lindner01} -- in implantations performed at \unit[450]{$^{\circ}$C}.
Implanted C is therefore expected to occupy substitutionally usual Si lattice sites right from the start.

Thus, we propose an increased participation of C$_{\text{s}}$ already in the initial stages of the implantation process at temperatures above \unit[450]{$^{\circ}$C}, the temperature most applicable for the formation of SiC layers of high crystalline quality and topotactical alignment\cite{lindner99}.
Thermally activated, C$_{\text{i}}$ is enabled to turn into C$_{\text{s}}$ accompanied by Si$_{\text{i}}$.
The associated emission of Si$_{\text{i}}$ is needed for several reasons.
For the agglomeration and rearrangement of C, Si$_{\text{i}}$ is needed to turn C$_{\text{s}}$ into highly mobile C$_{\text{i}}$ again.
Since the conversion of a coherent SiC structure, i.e. C$_{\text{s}}$ occupying the Si lattice sites of one of the two fcc lattices that build up the c-Si diamond lattice, into incoherent SiC is accompanied by a reduction in volume, large amounts of strain are assumed to reside in the coherent as well as at the surface of the incoherent structure.
Si$_{\text{i}}$ serves either as a supply of Si atoms needed in the surrounding of the contracted precipitates or as an interstitial defect minimizing the emerging strain energy of a coherent precipitate.
The latter has been directly identified in the present simulation study, i.e. structures of two C$_{\text{s}}$ atoms and Si$_{\text{i}}$ located in the vicinity.

It is, thus, concluded that precipitation occurs by successive agglomeration of C$_{\text{s}}$ as already proposed by Nejim et~al.~\cite{nejim95}.
This agrees well with a previous ab initio study on defects in C implanted Si\cite{zirkelbach11a}, which showed C$_{\text{s}}$ to occur in all probability.
However, agglomeration and rearrangement is enabled by mobile C$_{\text{i}}$, which has to be present at the same time and is formed by recombination of C$_{\text{s}}$ and Si$_{\text{i}}$.
In contrast to assumptions of an abrupt precipitation of an agglomerate of C$_{\text{i}}$\cite{werner96,werner97,eichhorn99,lindner99_2,koegler03}, however, structural evolution is believed to occur by a successive occupation of usual Si lattice sites with substitutional C.
This mechanism satisfies the experimentally observed alignment of the \hkl(h k l) planes of the precipitate and the substrate, whereas 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.

