Analytical validation of a $2+1$ dimensional continuum model for epitaxial growth with elastic substrate

Abstract

An analiyical validation is obtained for the evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h], introduced in [18] by W.T. Tekalign and B.J. Spencer to describe the heteroepitaxial growth of a two-dimensional thin film on an elastic substrate. In the expression above, h denotes the surface height of the film, F is the Fourier transform, and a, E, r are positive material constants. Existence, uniqueness, and Lipschitz regularity in time for weak solutions are proved, under suitable assumptions on the initial datum. Introduction The epitaxial deposition of a thin film on a relatively thick substrate has gained much interest in the recent years due to its applications to semiconductor electronics and quantum dots. Roughly speaking, the morphology of the film is known to be the result of a competition between the elastic energy associated to the mismatch between film and substrate, and the surface mass transport due to the film deposition. An extensive mathematical analysis of the mechanism associated to epitaxial film growth has been carried out in [2, 3, 10, 11, 15, 16] in the context of plane linear elasticity, and regularity results have been established for volume-constrained minimizers. Short time existence for a surface diffusion type geometric evolution equation keeping into account elasticity has first been analyzed in [12] in a two-dimensional setting (see also [17]). The previous result has been recently extended in [13] to the three-dimensional case. The central aim of this work is to study existence and Lipschitz regularity in time of weak solutions to the 2+1 dimensional evolution equation ht = ∆[F−1(−aEF(h))− r/h2 −∆h] (0.1) derived by W.T. Tekalign and B.J. Spencer in [18], where h denotes the surface height of the film and F is the Fourier transform. The quantities a, E, r are positive material constants. To be precise, a (resp. r) is the wavenumber (resp. wetting coefficient) associated to the equation, and E is defined as E := 2μ F (1 + ν )(1− ν) (1− νF )μS , where μ and ν (resp. μ and ν) are the elastic shear modulus and the Poisson’s 2010 Mathematics Subject Classification. 35K55, 35K67, 44A15, 74K35.