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+slide 1
+
+thank you very much and welcome everybody.
+as the title suggests / as already mentioned ...
+... i am going to present theoretical results of investigations
+of defect structures and mobilities in silicon.
+
+slide 2
+
+of course there is an experimental / practical motivation,
+which is the ion beam synthesis (IBS) of thin films of epitaxial 3C-SiC in Si.
+IBS consists of high-dose C implantation in Si followed by an annealing step,
+which, if properly done, results in buried homogeneous thin films of SiC
+as can bee seen in the XTEM image.
+however, the precipitation in the first step is not yet fully understood.
+
+this will be adressed in this study.
+after sketching controversial ideas of the mechanism of precipitation,
+the utilized simulation techniques are explained
+followed by a summary of the most important results of these calculations.
+
+slide 3
+
+one assumed mechanism is schematically displayed here.
+incorporated carbon atoms form C-Si dumbbells on regular Si lattice sites.
+with increasing dose and time these dumbbells agglomerate into large clusters,
+indicated by dark contrasts in the otherwise undisturbed lattice in hrtem.
+once a critical radius of 2-4 nm is reached,
+the interfacial energy due to the lattice mismatch is overcome
+and precipitation occurs.
+this is manifested by the disappearance of the dark contrasts in favor of
+moire patterns, again due to the lattice mismatch of SiC and silicon.
+the excess silicon atoms are released in the silicon host,
+since there is more space.
+#it is worth to note that the hkl planes of substrate and SiC match.
+
+slide 4
+
+however, controversial findings exist in the literature.
+instead of a carbon interstitial (Ci) based mechanism,
+nejim et al propose a transformation based on substitutionally incorporated
+carbon (Cs) and the generation of interstitial silicon,
+which reacts with further impanted carbon in the cleared volume.
+investigations of the annealing behavior of implantations at low and high
+temperatures show high and almost zero carbon diffusion respectively.
+this suggests the formation of mobile Ci at low temperatures
+opposed to much more stable Cs configurations at elevated temperatures.
+furthermore, investigations of strained SiC/Si heterostructures,
+find initial coherent SiC structures, which, in this case,
+incidentally transform into incoherent SiC nanocrystals
+accompanied by strain relaxation.
+
+these findings suggest a mechanism based on the agglomeration of substitutional
+instead of interstitial carbon.
+
+slide 6
+
+to understand the precipitation mechanism
+in the context of these controversial results
+atomistic simulations are performed.
+
+HIER WEITER
+
+in md, a system of n particles is described
+by numerically integrating newtons equations of motion.
+the particle interaction is given by an analytical interaction potential.
+observables are obtained by taking time or ensemble averages.
+
+roughly 6000 atoms were used to investigate defect structures
+and nearly a quater of a million for the precipitation simulations.
+the equations of motion are integrated by the velocity verlet algorithm
+with a time step of 1 fs.
+the interaction is decribed by a Tersoff-like short-range bond order potential,
+developed by erhart and albe.
+the short range character is achieved by a cutoff function,
+which drops the interaction to zero inbetween the first and next neighbor atom.
+simulations are performed in the isothermal-isobaric ensemble
+realized by the berendsen thermostat and barostat.
+
+the basic concept of dft is the hohenberg kohn (hk) theorem, which states that
+the ground-state wavefunction is a unique functional of the ground-state
+electron density, which minimizes the energy,
+i.e. it has the variational property.
+now, the kohn sham (ks) approach constitutes a hartree-like formulation
+of the hk minimal principle, which maps the system of interacting electrons to
+an auxillary system of non-interacting electrons in an effective potential.
+however formally exact by introducing an energy functional,
+which accounts for exchange and correlation.
+the kohn sham equations need to be solved in a self consistency loop.
+
+the vasp code is used for this purpose.
+it utilizes plane waves to expand the ks wavefunctions.
+an energy cut-off of 300 eV is employed.
+the electron-ion interaction is described by ultrasoft pseudopotentials.
+the generalized gradient approximation is used to solve the ks equations.
+sampling in k space is restricted to the gamma point.
+the supercell consists of 216 atoms.
+
+slide 8
+
+defect structures are obtained by creating a supercell of crystalline silicon.
+the interstitial carbon or silicon atom is inserted,
+for example at the tetrahedral or heexagonal site,
+followed by structural relaxation into a local minimum configuration.
+
+next to the structure, defects can be characterized by the formation energy,
+which is defined by this formula.
+
+combinations of defects can be characterized by the binding energy,
+the difference of the formation energy of the defect combination and
+the isolated defects.
+this way, binding energies below zero correspond to energetically favorable
+configurations whereas the binding energy for non-interacting isolated defects
+approaches zero.
+
+migration barriers from one stable configuration into another
+are obtained by the constrained relaxation technique.
+the diffusing atom is displaced stepwise from the starting
+to the final position and relaxation is only allowed
+perpendicular to the displacement direction.
+each step the configurational energy is recorded.
+
+slide 9
+
+this has been used to investigate, amongst others,
+carbon interstitial defects in silicon.
+both methods provide the correct order of the formation energies
+and find the 100 db to be the ground state.
+the hexagonal defect is unstable relaxing into the ground state.
+the tetrahedral configuration is found to be unstable
+in contrast to the prediction of the classical potential, which, however,
+shows a high energy of formation making this defect very unlikely to occur.
+the opposite is found for the bond-centered configuration, which constitutes
+a stable configuration but is found unstable in the classical description,
+relaxing into the 110 db configuration.
+however, again, the formation energy is quite high and, thus,
+the wrong description is not posing a serious limitation.
+the substitutional defect, which is not an interstitial defect,
+has the lowest formation energy for both methods, although,
+it is drastically underestimated in the empirical approach.
+regarding the problem addressed in this study, this might constitute a problem.
+however, it turns out, that combination of Cs and Si_i are very well described
+by the ea potential, with formation energies higher than the ground state.
+
+slide 10
+
+as a next step, the Ci mobility is determined by the quantum mechanical method.
+two of the intuitively guessed migration pathways of a carbon 00-1 db are shown.
+
+in number one, the carbon atom resides in the 110 plane
+crossing the bc configuration.
+due to symmetry it is sufficient to consider only the first half
+of the transition path.
+an activation energy of 1.2 eV is obtained.
+actually another barrier exists to reach a ground-state configuration.
+
+in path two, the carbon atom moves towards the same silicon atom, however,
+it escapes the 110 plane and forms a 0-10 oriented db.
+the obtained actiavtion energy of 0.9 eV excellently matches experiment.
+thus, there is no doubt, the migration mechanism is identified.
+
+slide 11
+
+the situation changes completely for the classical description.
+path number one, shows the lowermost migration barrier of 2.2 eV.
+next to the fact, that this is a different pathway,
+the barrier is overestimated by a factor of 2.4.
+
+moreover, the ea description predicts the bc configuration to be unstable
+relaxing into the 110 db configuration.
+additionally, the observed minimum in the classical 00-1 to 0-10 transition,
+likewise relaxes into the 110 db structure without constraints.
+
+this suggests to investigate the transition involving the 110 configuration.
+this migration is displayed here,
+the 00-1 db turns into a 110 type followed by a final rotation into the 0-10 db
+configuration.
+barriers of 2.2 eV and 0.9 eV are obtained.
+these activation energies are 2.4 to 3.4 times higher than the ab initio ones.
+however, due to the above reasons, this is considered the most probable
+migration path in the ea description.
+and after all, the expected change of the db orientation is fullfilled.
+
+nevertheless, diffusion barriers are drastically overestimated
+by the classical potentials, a problem, which needs to be addressed later on.
+
+slide 12
+
+implantation of highly energetic carbon atoms results in a multiplicity
+of possible point defects and respective combinations.
+thus, in the following, defect combinations of an initial carbon interstitial
+and further types of defects,
+created at certain neighbor positions, numbered 1-5, are investigated.
+the investigations are restricted to dft calculations.
+energetically favorable and unfavorable configurations,
+determined by the binding energies,
+can be explained by stress compensation and increase respetively.
+
+as can be seen, the agglomeration of interstitial carbon is energetically
+favorable.
+the most favorable configuration shows a strong C-C bond.
+however, a high migration barrier is necessary to obtain this configuration
+in contrast to the second most favorable configuration,
+which additionally is represented 2 times more often in the systematically
+investigated configuration space.
+
+this suggests that agglomeration of Ci indeed is expected, but no C clustering.
+
+slide 13
+
+this is reinforced by the plot of the binding energy of dumbbells
+separated along the 110 direction.
+a capture radius clearly exceeding 1 nm is observed.
+however, the interpolated graph suggests the disappearance of attractive forces
+between the two lowest separation distances.
+
+this supports the assumption of C agglomeration and the absence of C clustering.
+
+slide 14
+
+if a vacancy is created next to the Ci defect,
+a situation absolutely conceivable in ibs,
+structures are obtained, which exhibit low migration barriers
+for the transition into the Cs configuration.
+in contrast, high barriers are necessary for the reverse process.
+
+based on this, a high probability of stable Cs configurations must be concluded.
+
+slide 15
+
+in addition, it is instructive to look at combinations of Cs and Si_i.
+the most favorable configuration is obtained for
+Cs located right next to the 110 Si db within the 110 chain.
+this configuration is still less favorable than the Ci 100 ground state.
+however, the interaction of C_s and Si_i drops quickly to zero
+indicating a low capture radius.
+in ibs, configurations exceedinig this separation distance are easily produced.
+
+moreover, a low transition barrier is found from the ground state
+into the configuration of separated defects.
+the barrier is even smaller than migration barrier for carbon.
+in addition, the low migration barrier of interstitial silicon,
+enables configurations of further separated Cs and Si_i defects.
+
+in total, these findings demonstrate that configurations of Cs and Si_i,
+instead of the thermodynamic ground state, play an important role in ibs,
+which indeed constitutes a process far from equilibrium.
+
+slide 16
+
+this is supported by results of an ab inito md simulation at 900 dc.
+the initial configuration of Cs and Si_i does not recombine into the gs,
+instead, the defects are separated by more than 4 neighbor distances
+realized in a repeated migration mechanism of annihilating and arising Si_i dbs.
+
+clearly, at higher temperatures, the contribution of entropy
+to structural formation increases, which results in a spatial separation,
+even for defects located within the capture radius.
+
+!!!
+to conclude, the results of the investigations of defect combinations
+suggest an increased participation of Cs already in the initial stage
+of precipitation due to its high probability of incidence.
+
+slide 17
+
+as a last task, reproducing the SiC precipitation is attempted
+by successive insertion of 6000 C atoms,
+the number necessary to form a minimal precipitate,
+into a supercell consisting of 31 Si unit cells in each direction.
+insertion is realized at constant temperature.
+due to the high amount of particles,
+the classical potential must be used.
+since low carbon diffusion due to the overestimated barriers is expected,
+insertion volumes v2 and v3 next to the total volume v1 are considered.
+v2 corresponds to the minimal precipiatte size.
+v3 contains the amount of silicon atoms to form such a minimal precipitate.
+after insertion, the simulation is continued for 100 ps
+follwed by a cooling sequence downto 20 degrees celsius.
+
+slide 18
+
+the radial distribution function of simulations at 450 dc,
+an operative and efficient temperature in ibs, are shown.
+
+for the low C concentration simulation,
+a clearly 100 C-Si db dominated structure is obtained,
+which is obvious by comparing it to the
+reference distribution generated by a single Ci defect.
+the second peak is an artifact due to the cut-off.
+the C-C peak at 0.31 nm, as expected in cubic SiC,
+is generated by concatenated, differently oriented Ci dbs.
+the same distance is also expected for the Si atoms, and, indeed,
+the db structure stretches the Si-Si next neighbor distance,
+which is represented by nonzero values in the correlation function.
+
+so, the formation of Ci dumbbells indeed occurs.
+even the C atoms are already found in a separation as expected in cubic SiC.
+
+turning to the high C concentration simulations,
+a high amount of strongly bound C-C bonds
+as in graphite or diamond is observed.
+due to increased defect and damage densities
+defect arrangemnets are hard to categorize and trace.
+only short range order is observed.
+and, indeed, by comparing to other distribution data,
+an amorphous SiC-like phase is identified.
+
+slide 19
+
+to summarize, the formation of cubic SiC fails to appear.
+neither agglomeration of C interstitials
+nor a transition into SiC can be identified.
+
+slide 20
+
+having a closer look, there are two obvious reasons for this obstacle.
+
+first of all, there is the time scale problem inherent to md in general,
+which results in a slow phase space propagation due to
+a large amount of local minima separated by large energy barriers.
+accelerated methods, like temperature accelerated MD and so on, exist
+to bypass the time scale problem while retaining proper thermodynamic sampling.
+
+however, in addition, the overestimated diffusion barriers,
+due to the short range character of the potential,
+intensify this problem, which I termed:
+potential enhanced slow phase space propagation.
+
+the approach used in this study is to simply increase the temperature, however,
+without possible corrections.
+accelerated methods or higher time scales applied exclusively
+are assumed to be not sufficient.
+anyways, in this case,
+structural evolution instead of equilibrium properties are matter of interest.
+
+slide 21
+
+and indeed, promising changes are observed by comparing,
+again the radial distribution data for temperatures up to 2050 dc.
+first of all, the cut-off artifact disappears.
+more important, a transition into a clearly Cs dominated structure takes place,
+as can be seen by direct comparison with the respective reference peaks of Cs.
+
+the rising Si-Si peak is due to stretched Si-C-Si structures
+along a 110 direction.
+
+the C-C next neighbor pairs are reduced,
+which is mandatory for SiC formation.
+the peak at roughly 0.3 nm gets slightly shifted to higher distances,
+due to a decrease of interstitial carbon combinations accompanied by an
+increase in interstitial and substitutional as well as pure substitutional
+combinations.
+increasing values in this range
+correspond to bonds of Cs and another Cs with a nearby Si_i atom.
+
+slide 22
+
+to conclude, stretched coherent structures are directly observed.
+therefore, it is expected that Cs is extensively involved
+in the precipitation process for implantations at elevated temperatures.
+
+the emission of Si_i serves several needs:
+as a vehicle to rearrange stable Cs,
+as a building block for the surrounding Si host or further SiC formation.
+and for strain compensation either at the Si/SiC interface
+or in the stretched SiC structure, which, again,
+was diretly observed in simulation.
+
+this perfectly explains the results of the annealing experiments
+stated in the beginning of this talk.
+at low temperatures highly mobile Ci
+whereas at high temperatures stable Cs configurations are formed.
+
+thus, it is further concluded that high temperatures are necessary to model
+ibs conditions, which are far from equilibrium.
+the high temperatures deviate the system from thermodynamic equilibrium
+enabling Ci to turn into Cs.
+
+slide 23
+
+to summarize and conclude ...
+point defects were investigated by both methods.
+the interstitial carbon mmigration path was identified.
+it turned out that the diffusion barrier is drastically overestimated
+within the ea description.
+
+combinations of defects were investigated by first principles methods.
+the agglomeration of point defects is energetically favorable.
+however, substitutional carbon arises in all probability.
+even transitions from the ground state are very likely to occur.
+
+concerning the precipitation simulations, the problem of
+potential enhanced slow phase space propagation was discussed.
+high temperatures are assumed necessary to simulate ibs conditions.
+at low temperatures a dumbbell dominated structure is obatined
+whereas
+it is expected that
+Stretched structures of SiC were observed at elevated temperatures.
+it is thus concluded that
+substitutional carbon is heavily involved in the precipitation process.
+the role of the Si_i was outlined.
+
+in total, these results suggest,
+that cubic SiC precipitation occurs by successive agglomeration of Cs.
+
+slide 24
+
+finally, I would like to thank all of the people listed on this slide,
+categorized by location.
+
+thank you for your attention!
+
+
+
+
+
+slide X polytypes
+
+although the local order of the silicon and carbon atoms
+characterized by the tetrahedral bond is always the same,
+more than 250 different polytypes exist,
+which differ in the one-dimensional stacking sequence of
+identical, close-packed SiC bilayers,
+the stacking sequence of the most important polytypes is displayed here.
+the 3c polytype is the only cubic polytype.
+
+different polytypes exhibit different properties,
+which are listed in the table
+and compared to other technologically relevant semiconductor materials.
+SiC clearly outperforms silicon.
+among the different polytypes, the cubic phase shows the highest
+break down field and saturation drift velocity.
+additionally, these properties are isotropic.
+thus, the cubic polytype is considered most effective for highly efficient
+high-performance electronic devices.
+
+slide X silicon self interstitials
+
+in the following, structures and formation energies
+of silicon self-interstitial defects are shown.
+the classical potential and ab initio method predicts formation energies,
+which are within the same order of magnitude.
+however, discrepancies exist.
+quantum-mechanical results reveal the silicon 110 interstitial dumbbell (db)
+as the ground state closely followed by the hexagonal and tetrahedral
+configuration, which is the consensus view for silicon interstitials.
+in contrast, the ea potential favors the tetrahedral configuration,
+a known problem, which arises due to the cut-off
+underestimating the closely located second next neighbors.
+the hexagonal defect is not stable
+opposed to results of the authors of the potential.
+first, it seems to condense at the hexagonal site but suddenly
+begins to move towards a more favoarble position,
+close to the tetrahedral one but slightly displaced along all 3 coordinate axes.
+this energy is equal to the formation energy given in the original work.
+this artificial configuration, however, turns out to have negligible influence
+in finite temperature simulations due to a low migration barrier into the
+tetrahedral configuration.
+nevertheless, all these discrepancies have to be taken into account
+in the following investigations of defect combinations.
+
+slide X quantum mechanical details of 100 and bc
+
+it is worth to note that there are differences in the 100 defect geometries
+obtained by both methods.
+while the carbon-silicon distance of the db is equal,
+the db position inside the tetrahedron differs significantly.
+of course, the classical potential is not able to reproduce
+the clearly quantum mechanically dominated character of bonding.
+
+more important, the bc configuration is found to constitute
+a local minimum configuration and not a saddle point as found in another study.
+this is due to the neglection of spin in these calculations, which,
+however, is necessary as can already be seen from simple molecular orbital
+considerations, assuming a sp hybridized carbon atom due to the linear bond.
+this assumption turns to be right as indicated by the charge density isosurface
+which shows a net spin up density located in a torus around the C atom.
+
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