1. 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

2. 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 ...

3. 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

4. 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

5. 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%

6. 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

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

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

9. 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

10. 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

11. Simulation results – misfit 2%
0.5 monolayers
1.5 monolayers

12. Simulation results – misfit 4%
0.5 monolayers
1.5 monolayers

13. Simulation results – misfit 4%
2.5 monolayers
3.5 monolayers

14. Simulation results – misfit 4%
3.5 monolayers
4.5 monolayers

15. Simulation results – misfit 6%
0.5 monolayers
1.5 monolayers

16. 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.

17. Computational cost of elastic solve
128x128x20 grid
512x512x20 grid

18. 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, ...