Kinetic Monte Carlo Simulation of Epitaxial Growth Giovanni Russo, DMI, Catania In collaboration with: Peter Smereka, Len Sander, J. De Vita, University of Michigan A. La Magna, IMM, Catania A. Terrasi, G. Foti, ..., Dep. Physics, Catania

Basic model in Molecular Beam Epitaxy Technique for growing crystals one layer at a time. Useful in production of microelectronic and opto- electronic devices, due to the high degree of control. Atoms are deposited at random on a substrate ...

Simple Model and basic KMC Simplified model simple cubic lattice Solid-on-Solid Model (SOS). Atoms then move according to a rate Eb : bond energy, Nb: # neighbors, : attempt frequency. Basic rejection KMC Pick an atom at random Compute its number of bonds Pick a random number, r U[0,]: If r<exp(- NbEb/kBT ) then move the atom

Properties of basic KMC Simple Flexible Speed can be improved by reducing rejection Easily parallelizable (domain decomposition) R_1 R1 R6 R2 R4 R5 R3 Computational domain divided into regions associated with processors, communicating through the boundaries

Heteroepitaxial growth Layer by layer growth: energetically favoured for homoepitaxy. In heteroepitaxy the contribution of elastic energy may favour island and valleys. Prototype: Germanium on Silicon misfit = (aGe-aSi)/aGe= 4%

Experimental results (InAs/GaAs on GaAs) From the Mirecki-Millunchick group, Dept. of Mater. Sci. Eng., Univ. of Michigan. transition to 3D islands substrate 3D islands cooperative growth TEM image of dislocations in very thick layers. zig-zag pattern

Atomic Force Microscopy of Ge on Si (001) 6.4 Monolayers “pyramids” “domes” (A. Terrasi, Department of Physics, Catania)

Different morphologies 600 °C “huts” pyramid “pyramids” dome “domes” hut

Model for heteroepitaxial growth SOS type model with a cubic lattice. Nearest and next nearest neighbor bonds. Elastic effects are modeled using a linear ball and spring model with springs connecting nearest and next nearest neighbor atoms Hopping Rate is  exp(-(Eb Nb+ Es)/kB T) Nb = number of bonds, Eb = bond energy,  Es = change in elastic energy that occurs if the atom is completely removed

Simulation method: KMC + elastic Elastic computation: multigrid-Fourier efficient method for the computation of equilibrium configuration of Ge and first row of Si [Smereka & R., JCP 2006] Further simplifications (e.g. elastic computation only if hop is accepted) Constants of the spring artificially increased by one order of magnitude

Simulation results – misfit 2% 0.5 monolayers 1.5 monolayers

Simulation results – misfit 4% 0.5 monolayers 1.5 monolayers

Simulation results – misfit 4% 2.5 monolayers 3.5 monolayers

Simulation results – misfit 4% 3.5 monolayers 4.5 monolayers

Simulation results – misfit 6% 0.5 monolayers 1.5 monolayers

Computational considerations State of the art algorithm: multigrid-Fourier method for the computation of the elastic energy 2D representation of atoms: system is embedded in rectangular 2x2 grid.

Computational cost of elastic solve 128x128x20 grid 512x512x20 grid

Conclusions and perspectives In spite of the optimal algorithm, several approximations are necessary (e.g. energy evaluation) The use of realistic spring strengths (and island size) requires at least two orders of magnitude increase in speed More sophisticated domain decomposition techniques are required to parallelize the elastic computation More physical effects: FCC lattice, intermixing, alloys, defects, chemistry, ...