Continuum modeling of surface roughening in heteroepitaxial structures based on phase field theory

Abstract

Surface roughening is essential for the design of new nanostructures. In this paper, a continuum model based on phase field theory is developed to study the surface roughening in heteroepitaxial structures. A long-range order parameter is introduced to represent a diffusional free surface profile. By minimizing the system total free energy under constraints, the coupled field equations of quasi-static mechanical equilibrium and diffusion-controlled surface evolution are obtained. In contrast to the existing phase field models of free surface which are mostly based on the analytical Green’s function, the Galerkin finite element formulation is derived in this paper to solve the weak form of the coupled field equations. The advantage is that it is very convenient to deal with the elastic inhomogeneities and complex boundary conditions without additional complications. The proposed model is applied to study the nonlinear surface morphology evolution of heteroepitaxial thin films. The simulation shows that the heteroepitaxial film with an initial random fluctuation develops to form islands and coarsens during annealing. The kinetics of coarsening and the calculated average wavelength are consistent with the existing experimental results. It also shows that the initial island array formed during annealing consists of islands with similar size, shape and spacing. The influences of film thickness and misfit strain on the formation of initial island array are studied.