Equilibrium Configurations of Epitaxially Strained Crystalline Films: Existence and Regularity Results

Abstract

Strained epitaxial films grown on a relatively thick substrate are considered in the context of plane linear elasticity. The total free energy of the system is assumed to be the sum of the energy of the free surface of the film and the strain energy. Because of the lattice mismatch between film and substrate, flat configurations are in general energetically unfavourable and a corrugated or islanded morphology is the preferred growth mode of the strained film. After specifying the functional setup where the existence problem can be properly framed, a study of the qualitative properties of the solutions is undertaken. New regularity results for volume constrained local minimizers of the total free energy are established, leading, as a byproduct, to a rigorous proof of the zero contact-angle condition between islands and wetting layers.

Scaling Instability in Buckling of Axially Compressed Cylindrical Shells

“…We remark that the scaling of the critical strain λ * (h) ∼ h implies that the elastic energy stored in the critically strained cylinder is of order h 3 , since the stress remains proportional to the strain at the onset of buckling. Thus, the Γ-limit theorem from [7] applies. However, that theorem misses the structure of low energy sequences, since the set of W 2,2 isometries of the cylindrical surface consists of rigid motions, and the limiting energy is zero.…”

Scaling Law and Reduced Models for Epitaxially Strained Crystalline Films

“…We study a free energy functional introduced in [41] (see also [8,19,25]) to model the epitaxial deposition of a film on a rigid substrate when there is a crystallographic misfit between the two solids. The energy consists of the stored elastic energy in the film and the interfacial energy of its free surface.…”

“…This condition forces the film to be strained. The free energy density W is nonnegative and In recent years, the mathematical analysis of this model has been devoted to the geometrically linear small strain approximation (see [19,25,8]), corresponding to small deformations, in which W depends only on the symmetrized gradient E(u) := ∇u + ∇ T u. Since we are mainly interested in the regime of large mismatch e 0 , the small strain hypothesis is not applicable and instead we focus on the geometrically nonlinear case.…”

“…In [19] the authors prove that minimizing configurations are smooth outside of a finite number of cusps and cuts. Subsequently, it was proven in [25] that for small values of the mismatch, the flat configuration is always minimizing no matter how thick is the film.…”

Evolution of elastic thin films with curvature regularization via minimizing movements

“…In order to release the resulting elastic energy, the atoms in the film move and reorganize themselves in more convenient configurations. In analogy with [10,18,21] and with the surface diffusion case (see [19]), we work in the context of the elasticity theory for small deformations. Hence, fixing a time in [0, T ], the linearized strain is represented by E(u) = 1 2 (∇u + ∇ T u), where u defined on Ω h denotes the planar displacement of the bulk material that is assumed to be in (quasistatic) equilibrium, and the bulk elastic energy is for a positive definite fourth-order tensor C. Furthermore, we model the displacement of the film atoms at the interface with the substrate using the Dirichlet boundary condition u(x, 0) = (e 0 x, 0), where the constant e 0 > 0 measures the mismatch between the crystalline lattices.…”