The Level-Set Method for Modeling Epitaxial Growth

Christian Ratsch

C. RATSCH, R.E. Caflisch, S. Chen, M. Kang, B. Merriman, S.J. Osher, M. Petersen, UCLA, Los Angeles, CA; M.F. Gyure, HRL Laboratories, Malibu, CA; and D.D. Vvedensky, Imperial College, London, UK

We develop an island dynamics model that employs the level-set technique [1] to describe epitaxial growth. This method is essentially continuous in the plane, yet it retains atomic disrcreteness in the height [2]. It is thus ideally suited to describe for example growth of next generation electronic devices which might be nanometer to micrometer in lateral size, but have a thickness of only a few atomic layers.

In this method, the surface morphology is described by defining the island boundaries as the set psi = 0 of the so-called level-set function. Islands are nucleated on the surface and their boundaries are moved at rates that are determined by the adatom density, which is obtained from solving the diffusion equation [3].

Scaled island size distributions in the submonolayer aggregation regime are compared to those obtained from a kinetic Monte Carlo (KMC) simulation for irreversible as well as reversible aggregation. Excellent agreement is obtained. We identify spatial fluctuations in the seeding of islands as the only essential source of noise, while all other stochastic elements can be averaged [4]. Effects of edge diffusion can be incorporated by adding a curvature dependent term to the velocity of the island boundaries.

We also show that the level-set method can naturally be extended to multilayer growth; here, the set psi = n-1 corresponds to the nth layer. Roughening and Coarsening of the surface will be discussed. In particular, we will study the evolution of the step edge density, which is related to the RHEED signal in experiment. A qualtitative and quantitative comparison to KMC simulations will be given.


[1] S. Osher and J.A. Sethian, J. Comput. Phys. 79, 12 (1988).

[2] R.E. Caflisch, M.F. Gyure, B. Merriman, S.J. Osher, C. Ratsch, D.D. Vvedensky, J.J. Zinck, Appl. Math. Lett. 12, 13 (1999).

[3] S.Chen, M. Kang, B. Merriman, R.E. Caflisch, C. Anderson, C. Ratsch, Fedkiw, M.F. Gyure, S. Osher, J. Comput. Phys., in press.

[4] C. Ratsch, M.F. Gyure, S. Chen, M. Kang, and D.D. Vvedensky, Phys. Rev. B 61, R10598 (2000).

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