From: hackbard Date: Tue, 15 May 2012 20:36:05 +0000 (+0200) Subject: added text for emrs2012 X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=baf3f5e756290f1ffa841971cbcef9771a408892 added text for emrs2012 --- diff --git a/posic/talks/emrs2012.txt b/posic/talks/emrs2012.txt new file mode 100644 index 0000000..676db77 --- /dev/null +++ b/posic/talks/emrs2012.txt @@ -0,0 +1,479 @@ +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. +