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.
+... i am going to present results of theoretical investigations
+of defects and defect mobilities in silicon.
slide 2
-of course there is an experimental / practical motivation,
+there is of course 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
these findings suggest a mechanism based on the agglomeration of substitutional
instead of interstitial carbon.
-slide 6
+slide 5
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.
+namely, molecular dynamics simulations,
+employing an empirical Tersoff-like short range bond order potential
+developed by Erhart and Albe.
+a large amount of atoms can be simulated.
-slide 8
+moreover, the investigations are extended by first-principles calculations
+based on dft using the plane wave pseudopotgential vasp code.
+of course limited to smaller systems.
-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.
+slide 6
-next to the structure, defects can be characterized by the formation energy,
-which is defined by this formula.
+using these methods we can now investigate single defect structures,
+which can be characterized by the formation energy.
-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.
+Defect combinations can be described by the binding energy,
+the difference of the formation energy of the defect combination
+and the isoltaed defects.
-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.
+to acquire the mobilities migration barriers
+are obtained by a constrained relaxation technique.
-slide 9
+slide 7
+
+now let's turn to the results ...
+... of carbon interstitial defects in silicon.
-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.
+
+it is worth to note that the bond centered configuration
+is unstable only within the empirical description, relaxing into the 110 DB.
+however, the formation energy is quite high
+so this does not pose 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.
+has the lowest formation energy and is drastically underestimated within EA.
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 8
-slide 11
+concerning the defect mobility, by first-principles methods,
+a migration path is found, the 00-1 to 0-10 transition,
+with a barrier that excellently matches experimental values.
+the migration path is identified, it involves a change in orientation of the DB.
-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.
+related to the just mentioned instability of the BC configuration,
+the most probable transition for the empirical potential
+involves an intermediate 110 DB configuration.
+this results in a barrier, which is up to 3.4 times higher than the ab initio
+or experimental value.
+At least, there is the same change in orientation, a qualitative agreement.
-slide 12
+slide 9
-implantation of highly energetic carbon atoms results in a multiplicity
+implantation of 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.
+and further types of defects created in the vicinity are inestigated by dft.
-slide 13
+concerning combinations of 100-type interstitials,
+there are lots of negative values for the binding energy,
+so the agglomeration of C_i is indeed energetically favorable,
+mainly due to a reduction of strain.
-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.
+a capture radius clearly exceeding 1 nm is observed
+for the interaction of DBs along the 110 direction.
however, the interpolated graph suggests the disappearance of attractive forces
between the two lowest separation distances.
+so this suggests agglomeration of C but the absence of C clustering.
-this supports the assumption of C agglomeration and the absence of C clustering.
-
-slide 14
+slide 10
if a vacancy is created next to the Ci defect,
a situation absolutely conceivable in ibs,
based on this, a high probability of stable Cs configurations must be concluded.
-slide 15
+slide 11
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 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,
+these findings suggest 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
+slide 12
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,
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.
+to structural formation increases, resulting in configurations of C_s and Si_i.
+
+slide 13
+
+these findings are supported by results of empirical potential MD simulations
+employed to directly simulate precipitation.
+
+6000 C atoms are inserted at constant temperature into a Si volume
+consisting of 31 Si unit cells in each direction.
+
+smaller insertion volumes were also considered due to an expected low diffusion.
+however - here - we only consider the total volume.
+
after insertion, the simulation is continued for 100 ps
follwed by a cooling sequence downto 20 degrees celsius.
-slide 18
+slide 14
-the radial distribution function of simulations at 450 dc,
+the radial distribution function of Si-C bonds 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.
+
+so, the formation of Ci dumbbells indeed occurs
+but no agglomeration is observed.
+
+one reason is the drastically overestimated dissufion barrier
+within the empirical potential description as outlined earlier.
+due to this, simulations are performed at increased temperatures.
+agglomeration and precipitation is still not observed, however,
+a phase transition into a clearly Cs dominated structure
+can be observed with increasing temperature
+by comparing with the reference peak.
+stretched coherent structures of SiC are directly observed
+and the Si_i could be attributed the role of strain reduction.
+
+slide 15
+
+i would like to conclude.
+based on both, ...
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