dear examiners, dear colleagues.
welcome everybody to the the defense of my doctor's thesis entitled ...
as usual, i would like to start with a small motivation,
-which in this case is a motivation with respect to the materials system, SiC.
+which in this case focuses on the materials system, SiC.
+and, thereby, approach the problem to be investigated within this study, i.e.
+a controversy concerning the precipitation mechanism present in the literature.
slide 2
-the semiconductor material SiC ...
+the semiconductor material SiC has remarkable physical and chemical properties,
+which make it a promising new material in various fields of applications.
+the wide band gap and high breakdown field
+as well as the high electron mobility and saturation drift velocity
+in conjunction with its unique thermal stability and conductivity
+unveil SiC as the ideal candidate for
+high-temperature, high-power and high-frequency electronic
+and opto-electronic devices.
+
+in fact light emission from SiC crystal rectifiers was observed
+already in the very beginning of the 20th century
+constituting the brirth of solid state optoelectronics.
+and indeed, the first blue light emitting diodes in 1990 were based on SiC.
+(nowadays superceded by direct band gap materials like GaN).
+
+the focus of SiC based applications, however,
+is in the area of solid state electronic devices
+experiencing revolutionary performance improvements enabled by its capabilities.
+devices can be designed much thinner with increased dopant concentrations
+resulting in highly efficient rectifier diodes and switching transistors.
+one example is displayed: a SiC based inverter with an efficiency of 98.5%
+designed by the frauenhofer institute for solar energy systems.
+therefore, SiC constitutes a promising candidate to become the key technology
+towards an extensive development and use of regenerative energies and emobility.
+
+moreover, due to the large bonding energy,
+SiC is a hard and chemical inert material
+suitable for applications under extreme conditions.
+its radiation hardness allows the operation as a first wall reactor material
+and as electronic devices in space.
slide 3
+
+the stoichiometric composition of silicon and carbon
+is the only stable compound in the C/Si system.
+SiC is a mainly covalent material in which both,
+the Si and C atom are sp3 hybridized.
+the local order of the silicon and carbon atoms
+characterized by the tetrahedral bond is always the same.
+however, more than 250 different polytypes exist,
+which differ in the one-dimensional stacking sequence of
+identical, close-packed SiC bilayers,
+which can be situated on one of three possible positions (abbreviated a,b,c).
+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.
+despite the lower charge carrier mobilities for low electric fields,
+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 4
+
+SiC is rarely found in nature and, thus, must be synthesized.
+it was first observed by moissan from a meteor crater in arizona.
+the fact that natural SiC is almost only observed
+as individual presolar SiC stardust grains near craters of meteorite impacts
+already indicates the complexity involved in the synthesis process.
+
+however, nowadays, much progress has been achieved in thin film growth
+by molecular beam epitaxy and chemical vapor deposition.
+indeed, commerically available semiconductor devices based on alpha SiC exist,
+although these are still extremely expensive.
+however, production of the advantageous cubic type is less advanced,
+mainly due to the
+mismatches in the thermal expansion coefficient and the lattice parameter
+(with respect to the substrate)
+which cause a considerable amount of defects,
+that is responsible for structural and electrical qualities
+that are not yet satisfactory.
+
+next to CVD and MBE, the ion beam synthesis technique, which consists of
+high dose ion implantation followed by a high-temperature annealing step
+turned out to constitute a promising method to form buried layers of SiC in Si
+as indicated in this sketch.
+due to the high areal homogenity achieved in ibs
+the size is only limited by the beam scanning equipment
+and sythesized films do not exhibit surface bending effects
+in contrast these formed by cvd and mbe.
+this enables the synthesis of large are SiC films.
+
slide 5
+
+the ibs synthesis of SiC was extensively investigated and optimized
+here in augsburg in the group of joerg lindner.
+a two-step implantation process was suggested.
+the trick is to destroy stable precipitates at the layer interface
+by implanting a remaining low amount of the dose at lower temperatures
+to enable redistribution of the C profile during annealing,
+which results in a homogeneous SiC layers with a sharp interface
+as you can see in this cross section tem image.
+
+however, the precipitation itself is not yet fully understood.
+understanding the effective underlying processes of precipitation
+will enable significant progress in thin film formation of cubic SiC
+and likewise offer perspectives for processes that rely upon prevention
+of SiC precipitation, for example the fabrication of strained silicon.
+
slide 6
+
+there is an assumed mechanism of precipitation based on the formation and
+agglomeration of interstitial carbon.
+first note, however, that silicon as well as SiC consists of two fcc lattices
+displaced by one quater of the volume diagonal.
+in the case of SiC one of the fcc lattice atoms is replaced by carbon atoms.
+4 lattice constants of silicon correspond to 5 lattice constants of SiC.
+thus, in total, the silicon density is only slightly lower in SiC.
+
+the mechanism is schematically displayed here.
+a pair of black dots represent two atoms of the two fcc lattices.
+the incorporated carbon atoms form C-Si dumbbells
+situated on regular silicon lattice sites.
+with increasing doese these dumbbells agglomerate into large clusters,
+indicated by dark contrasts and an 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.
+due to the slightly lower silicon density of SiC,
+precipitation is accompanied by the emission of a few excess silicon atoms
+into the silicon host, since there is more space.
+it is worth to note that the hkl planes of substrate and SiC match.
+
slide 7
+
+however, controversial findings and conclusions 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 different temperatures showed high and zero carbon diffusion
+for the room temperature and elevated temperature implantations 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 atoms.
+the task of the present study is to understand the precipitation mechanism
+in the context of these controversial results.
+
slide 8
+
+therefore, atomistic simulations are utilized,
+to gain insight on a microscopic level not accessible by experiment.
+namely, molecular dynamics (md) simulations and density functional theory (dft)
+calculations, which are explained in the following, are used
+to investigate carbon and silicon defect configurations as well as to
+directly model SiC precipitation.
+finally, after these results are presented,
+i would like to give a short summary and conclusion.
+
slide 9
+
+in md, a system of n particles is described on the microscopic level
+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.
+
+in this case roughly 6000 atoms were used to investigate defect structures
+and nearly a quater of a million atoms 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 inbetween the first and next neighbor atom.
+the potential consists of a repulsive and an attractive part associated with
+the bonding, which is limited by the bond order term, which takes
+into consideration all atoms k influencing the bond of atoms i and j.
+simulations are performed in the isothermal-isobaric ensemble
+realized by the berendsen thermostat and barostat.
+
+furthermore, highly accurate quantum mechanical calculations
+based on dft are used.
+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.
+in that way, the many body problem can be described by the electron density,
+which depends only on the 3 spatial coordinates.
+now, the kohn sham (ks) approach constitutes a hartree-like formulation
+of the hk minimal principle, which maps the system of interacting particles to
+an auxillary system of non-interacting electrons in an effective potential.
+however formally exact by introducing an energy functional,
+which accounts for the exchange and correlation energy.
+the effective potential yields a ground-state density
+for non-interacting electrons, which is equal to that for interacting electrons
+in the external potential.
+the kohn sham equations need to be solved in a self consistency loop.
+
+the vasp code was 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.
+brillouin zone sampling is restricted to the gamma point.
+the supercell consists of 216 atoms, 3 silicon unit cells in each direction,
+of course much less atoms compared to the highly efficient md technique.
+
slide 10
+
+defect structures are obtained by creating a supercell of crystalline silicon
+with periodic boundary conditions and temperature and pressure set to zero.
+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 formation energies,
+which is defined by this formula, where the chemical potential
+is taken to be the cohesive energy per atom for the fully relaxed structure.
+
+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 while 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.
+atoms involving great structural changes are displaced stepwise
+from the starting to the final position and relaxation is only allowed
+perpendicular to the displacement direction.
+each step the configurational energy of the relaxed structure is recorded.
+
slide 11
+
+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, these artificats have to be taken into account
+in the following investigations of defect combinations.
+
slide 12
+
+the situation is much better for carbon defects.
+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.
+this might be a problem concerning the clarification of the controversial views
+of participation of Cs in the precipitation mechanism.
+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 13
+
+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.
+
slide 14
+
+here, two of the intuitively obvious migration pathways of a carbon 00-1 db,
+and the corresponding activation energies
+for the highly accurate quantum mechnaical calculations are shown.
+
+in number one, the carbon atom resides in the 110 plane
+crossing the bc configuration.
+due to symmetry it is sufficient to merely consider the migration into the bc
+configuration.
+an activation energy of 1.2 eV is obtained.
+
+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.
+
+a simple reorientation process was also calculated.
+however, an energy of 1.2 eV was obtained.
+thus, reorientation is most probably composed of two consecutive processes of
+the above type.
+
slide 15
+
+the situation changes completely for the classical description.
+path number one, from the 00-1 to bc configuration
+shows the lowermost migration barrier of 2.2 eV.
+next to the fact, that this is a different pathway,
+the barrier is 2.4 times higher than the experimental and ab inito results.
+
+moreover, the ea description predicts the bc configuration to be unstable
+relaxing into the 110 db configuration.
+indeed, the observed minima in the 00-1 to 0-10 transition,
+is close to the 110 db structure.
+
+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.
+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 16
+
+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.
+indeed, the most favorable configuration shows a strong C-C bond.
+however, due to high migration barriers or energetically unfavorable
+intermediate configurations to obtain this configuration,
+only a low probability is assumed for C-C clustering.
+
+in contrast, for the second most favorable configuration,
+a migration path with a low barrier exists.
+moreover, within the systematically investigated configuration space,
+this type of defect pair is represented two times more often
+than the ground state.
+
+the results suggest that agglomeration of Ci indeed is expected.
+
+slide 17
+
+this is reinforced by the plot of the binding energy of Ci dbs
+separated along the 110 direction with respect to the C-C distance.
+the interaction is found to be proportional to the reciprocal cube
+of the distance for extended separations and saturates for the smallest
+possible distance, i.e. the ground state.
+a capture radius clearly extending 1 nm is observed.
+the interpolated graph suggests the disappearance of attractive forces
+between the two lowest separation distances of the defects.
+
+this supports the assumption of C agglomeration and the absence of C clustering.
+
+slide 18
+
+if a vacancy is created next to the Ci defect,
+a situation absolutely conceivable in ibs,
+the obtained structure will most likely turn into the Cs configuration.
+if the vacancy is created at position 1, the Cs configuration is directly
+obtained in the relaxation process.
+if it is created at other positions, e.g. 2 and 3,
+only low barriers into the Cs configuration exist
+and high barriers are necessary for the reverse process.
+
+based on this, a high probability for the formation of Cs,
+which is found to be extremely stable, must be concluded.
+
+slide 19
+
+in addition, it is instructive to look at combinations of Cs and Si_i,
+again, a situation which is very likely to arise due to implantation.
+Cs located right next to the 110 Si db within the 110 chain
+constitutes the energetically most favirable configuration,
+which, however, is still less favorable than the Ci 100 db,
+in which the silicon and carbon atom share a single lattice site.
+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.
+thus, Cs and Si_i, which do not react into the ground state,
+constitute most likely configurations to be found in ibs.
+
+this is supported by a low migration barrier necessary for the transition
+from the ground state Ci 100 db into the configuration of Cs and Si_i.
+in addition, a low migration barrier of the interstitial silicon,
+enables configurations of further separated Cs and Si_i defects.
+
+in total, these findings demonstrate that configurations of Cs and a Si_i db,
+instead of the thermodynamic ground state, play an important role in ibs,
+which indeed constitutes a process far from equilibrium.
+
+slide 20
+
+once more, 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 might result 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 in the precipitation process.
+
+slide 21
+
+as a last task, reproducing the SiC precipitation is attempted
+by successive insertion of 6000 C atoms,
+the number necessary to form a precipitate with a radius of approximately 3 nm,
+into a supercell consisting of 31 Si unit cells in each direction.
+insertion is realized at constant temperature.
+after insertion, the simulation is continued for 100 ps
+follwed by a cooling sequence downto 20 degrees celsius.
+due to the high amount of particles,
+the classical potential is exclusively 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.
+
+slide 22
+
+the radial distribution Si-C, C-C and Si-Si bonds of simulations,
+in which C was inserted at 450 dc,
+an operative and efficient temperature in ibs, are shown.
+
+for the low C concentration simulation, i.e. the v1 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 a cut-off artifact,
+correpsonding to the Si-C cut-off distance of 0.26 nm.
+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, i.e. the v1 and v2 simulation,
+a high amount of strongly bound C-C bonds
+as in graphite or diamond are observed.
+an increased defect and damage density is obtained,
+which makes it hard to categorize and trace defect arrangements.
+only short range orde is observed.
+and, indeed, comparing to other distribution data, an amorphous SiC-like
+phase is obtained.
+
+slide 23
+
+to summarize, the formation of cubic SiC fails to appear.
+in the v1 simulation, formation of Ci indeed occurs, however,
+agglomeration is missing.
+in the high concentration simulation, an amorphous SiC-like structure,
+which is not expected at 450 dc, is obtained.
+no rearrangemnt into crystalline cubic SiC is indicated.
+
+slide 24
+
+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.
+to minimize the integration error, the time step must be chosen smaller
+than the reciprocal of the fastes vibrational mode.
+several local minima exist, which are separated by large energy barriers.
+due to the low probability for escaping such a local minimum,
+a transition event correpsonds to a multiple of vibrational periods.
+a phase transition, in turn, consists of many such infrequent transition events.
+new accelerated methods, like temperature accelerated MD and so on,
+have been developed to bypass the time scale problem while retaining proper
+thermodynamic sampling.
+
+in addition, the overestimated diffusion barriers,
+due to the short range character of the potential,
+intensify this problem, which I called:
+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 oto be not sufficient.
+moreover, to legitimate the usage of increased temperatures:
+cubic SiC is also observed for higher temperatures,
+there is definitely a higher temperature inside the sample, and, anyways,
+structural evolution instead of equilibrium properties are matter of interest.
+
+slide 25
+
+and indeed, promising changes are observed by comparing,
+again the radial distribution data of Si-C, Si-Si and C-C bonds
+for temperatures up to 2050 dc.
+first of all, the cut-off artifact disappears.
+more important, a transition a 100 db into a Cs dominated structure takes place,
+as can be seen by direct comparison with the respective reference peaks.
+
+the Si-Si rising peak at 0.325 nm is due to two Si atoms next to a Cs atom.
+
+the C-C next neighbor pairs are reduced,
+which is mandatory for cubic SiC formation.
+the peak at roughly 0.3 nm gets slightly shifter to higher distances.
+the amount of bonds due to Ci 100 combinations, represented by dashed arrows,
+decreases accompanied by an increase of bonds due to combinations of
+Ci 100 and Cs and pure Cs combinations, represented by the dashed line and
+solid arrows respectively.
+increasing values in the range between the dashed line and first solid arrow
+correpsond to bonds of a Cs and another Cs with a nearby Si_i atom.
+
+slide 26
+
+to conclude, stretched coherent structures of SiC embedded in the Si host
+are directly observed.
+therefore, an increased participation of Cs is suggested
+for implantations at elevated temperatures,
+which simulate the conditions prevalent in ibs that deviate the system
+from thermodynamic equilibrium enabling Ci to turn into Cs.
+
+the emission of Si_i serves several needs:
+as a vehicle to rearrange the Cs,
+realized by recombination into the highly mobile Ci configuration.
+furthermore, it serves as a building block for the surrounding Si host
+or further SiC formation.
+finally, it may compensate stress 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.
+
+to summarize, the results suggest that Cs plays an important role
+in the precipitation process.
+moreover, high temperatures are necessary to model ibs conditions,
+which are far from equilibrium.
+
+slide 27
+
+to summarize and conclude
+
+slide 28
+
+in the end, I would like to say thank you.
+
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