added text for emrs2012

[lectures/latex.git] / posic / talks / emrs2012.txt

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.