Phase Field Modeling of Surface Instabilities Induced by Stresses
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U2.10.1
Phase Field Modeling of Surface Instabilities Induced by Stresses D. J. Seol1, S. Y. Hu1, Z. K. Liu1, S. G. Kim2, W. T. Kim3, K. H. Oh4 and L. Q. Chen1 1
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA 2 Department of Materials Science and Engineering, Kunsan National University, Kunsan 573701, Korea 3 Division of Applied Science, Chongju University, Chongju 360-764, Korea 4 School of Materials Science and Engineering, Seoul National University, Seoul 151-742, Korea ABSTRACT In this work, we developed a phase field model describing surface evolution dynamics of a strained solid in contact with gas phase. Elastic solutions are solved with an iteration method for the system in which a solid film, strained by the mismatch strain between the film and substrate, has anisotropic elastic constants and the vapor phase has zero elastic constants. Elastic solutions are incorporated into the phase field evolution equation. The model predicts stable morphology at a given mismatch strain and perturbation wavelength.
INTRODUCTION Stress-induced surface instability is associated with the dislocation-free Stranski-Krastanov (SK) growth mode [1-4]. During the SK growth, the growth mode transition from a layer-bylayer growth to a three-dimensional island growth occurs mostly by the competition between the strain energy and surface energy of a film. It is essential to understand the effect of each factor on the stability and evolution dynamics of island structures in producing controllable quantum dot arrays for many novel optical or electronic applications. Several modeling techniques have been developed coupling kinetic equation with the finite element method (FEM) [5-7]. In this work, a phase-field model is developed to describe the surface evolution dynamics of a strained solid film in contact with a gas phase. One of the main advantages of the phase-field approach is its ability to predict the island formation and coarsening without any a priori assumption with regard to the transient, metastable, and stable morphologies [8]. The effect of different interface energies at triple junctions on the island formation and shape evolution is studied.
THEORY Phase field model We introduce 3 phase field variables to define a system consisting of a substrate, a film and vapor. A total free energy of the system is defined as follows [9]:
U2.10.2
3
3
F = ∫ ∑φi f i ( ρ i ) − ∑ V i
α ij2
i, j i≠ j
3 1 3 ∇φ i ∇φ j + ∑ ω ij φ i φ j + e el + λ L (∑ φ i − 1)dV i, j 2 i 4
(1)
i≠ j
3
where ρ = ∑ φ i ρ i is the density of a film material, α ij is the gradient energy coefficient, ω ij i
is the height of double-well potential, and λ L is the Lagrange multiplier accounting for the constraint, ∑ φ i = 1.0 . Following the interface field concept of Steinbach and Pezzola [10], we get the governing equation for the phase fields, 1 n
3
δF
j ≠i
δφ i
φ&i = − ∑ si s j M ij
−
δF δφ j
(2)
where n is the number of phases present at a given point, and
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