An application of multigrid methods for a discrete elastic model for epitaxial systems

  • Authors:

  • R. E. Caflisch;Y. -J. Lee;S. Shu;Y. -X. Xiao;J. Xu

  • Affiliations:

  • Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA;Department of Mathematics, UCLA, 520 Portola Plaza, Los Angeles, CA 90095, USA;Institute for Computational and Applied Mathematics, Xiangtan University, Xiangtan 411105, PR China;Institute of Fundamental Mechanic and Material Engineering, Xiangtan University, Xiangtan 411105, PR China;Department of Mathematics and Center for Computational Mathematics and Application of Pennsylvania State University, PA, USA

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2006

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Abstract

We apply an efficient and fast algorithm to simulate the atomistic strain model for epitaxial systems, recently introduced by Schindler et al. [Phys. Rev. B 67, 075316 (2003)]. The discrete effects in this lattice statics model are crucial for proper simulation of the influence of strain for thin film epitaxial growth, but the size of the atomistic systems of interest is in general quite large and hence the solution of the discrete elastic equations is a considerable numerical challenge. In this paper, we construct an algebraic multigrid method suitable for efficient solution of the large scale discrete strain model. Using this method, simulations are performed for several representative physical problems, including an infinite periodic step train, a layered nanocrystal, and a system of quantum dots. The results demonstrate the effectiveness and robustness of the method and show that the method attains optimal convergence properties, regardless of the problem size, the geometry and the physical parameters. The effects of substrate depth and of invariance due to traction-free boundary conditions are assessed. For a system of quantum dots, the simulated strain energy density supports the observations that trench formation near the dots provides strain relief.