[lectures/latex.git] / posic / talks / defense.txt

slide 1

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 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 has remarkable physical and chemical properties,
which make it a promising new material in various fields of applications.
most important, 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 highly efficient
high-temperature, high-power and high-frequency electronic
and opto-electronic devices, which can operate in harsh environments.

#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#(nowadays superceded by direct band gap materials like GaN).

as an example a SiC based inverter with an efficiency of 98.5%
designed by the frauenhofer institute for solar energy systems is displayed.
therefore, SiC constitutes a promising candidate to become the key technology
towards an extensive development and use of regenerative energies and emobility.

slide 3

one method to fabricate the 3C-SiC, the cubic phase of SiC, is ibs,
i.e. high dose ion implantation followed by a high-temperature annealing step,
as extensively investigated and optimzed here in augsburg
in the group of joerg lindner.
an optimized two-step implantation process was suggested.
the trick is to destroy stable precipitates
formed at the layer interface during the first implantation step
by implanting the low remaining amount of the regular dose at lower temperatures
to enable redistribution of the C atoms during annealing,
which results in a homogeneous SiC layer with a sharp interface
as you can see in this cross section tem image.

however, the precipitation, at elevated temperatures,
is not yet fully understood.
detailed understanding of the effective underlying processes of precipitation
might 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 4

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 sites is occupied by carbon atoms.
4 lattice constants of silicon correspond to 5 lattice constants of SiC.
in total, this results in a only slightly lower silicon density for SiC.

the mechanism is schematically displayed.
a pair of black dots represent two atoms of the two fcc lattices.
the incorporated carbon atoms form C-Si dumbbells
sharing regular silicon lattice sites.
with increasing dose and time these dumbbells agglomerate into large clusters,
indicated by dark contrasts in 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 only 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 5

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 show 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 6

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 on the following slides, are used
to investigate carbon and silicon defect configurations as well as to
directly model SiC precipitation.
finally, after these results are presented,
a short summary and conclusion is given.

slide 7

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.

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 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 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.
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 8

defect structures are obtained by creating a supercell of crystalline silicon
with 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 of 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 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.
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 9

the method 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.
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 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,
and the corresponding activation energies 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.
actually another barrier exists to reach a 100 type 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, 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.

however, 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,
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 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, 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 exceeding 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 are necessary for a transition into the Cs configuration
whereas 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 already in the initial stage
of precipitation due to its high probability of incidence.

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

finally, I would like to say thank you.





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.