Nonlinear evolution of a morphological instability in a strained epitaxial film

Abstract

A strained epitaxial film deposited on a deformable substrate undergoes a morphological instability relaxing the elastic energy by surface diffusion. The nonlinear and nonlocal dynamical equations of such films with wetting interactions are derived and solved numerically in two and three dimensions. Above some critical thickness, the surface evolves towards an array of islands separated by a wetting layer. The island chemical potential decreases with its volume, so that the system experiences a non-interrupted coarsening described by power laws with a marked dimension dependence.PACS numbers: 81.15.Aa, 68.35.Ct The dynamics of semiconductor thin films is under active scrutiny due to its importance for both fundamental science and technological applications [1,2]. Indeed, thin film elastic instabilities lead to the self-organization of nanostructures [3] potentially useful e.g. for quantum dots, wires and electronic devices with specific confinement properties [4]. A notorious experimental example is Si/Ge films on a Si substrate which exhibit a variety of structures such as pre-pyramids, pyramids, domes and huts [5,6]. Such epitaxial films experience an elastic stress due to the misfit with the substrate which is relaxed by a morphological instability similar to the Asaro-Tiller-Grinfeld thermodynamical instability in solid-liquid interfaces [7]. This instability was first observed in experiments in helium at low temperature [8] and more generally in various solid interfaces [9,10,11].Although the evolution of epitaxial films involves many complex phenomena regarding surface energy, intermixing and kinetic processes, we focus here on the main effects ruling the dynamics of the morphological instability in strained films. The dynamics is ruled here by surface diffusion driven by the interplay between isotropic surface energy and elastic energy [12,13]. When the film is infinitely thick or when the substrate is infinitely rigid, different theoretical [14,15] and numerical [16,17,18,19] approaches revealed finite-time singularities enforced by elastic stress concentration which account for experiments in thick films [8,9] where dislocations can finally develop. However, these models can not describe experiments of thin films in the Stranski-Krastanov type of growth [5,6] where the surface organizes smoothly into islands separated by a wetting layer and evolving with a coarsening dynamics under annealing [6]. A crucial issue for these systems is the wetting of the substrate by the film [20,21] which is a good candidate for regularizing the dynamics of the instability. Indeed, crack singularities were circumvented near the instability threshold by considering slope dependent wetting effects [22]. However, the interplay between elastic relaxation, surface energy and wetting interactions is still under active study [23,24] and the description of the long term dynamics of the morphological instability in a thin strained film is an open issue. In this Letter, we present a model based on continuum elasticity which we ...

Atomic-Scale Wear in UHV: Connection Between AFM Induced-Abrasion and Debris Recrystallization

“…We have not yet explicated a form for the surface energy; nevertheless, at the cluster-surface boundary the surface energy generally has a drastic variation that implies a severe change in wetting potential, this can lead to the self-organization of dots with almost uniform sizes if the wetting potential is strongly dependent on the surface slope. Many authors have investigated the stability of regular dot patterns with respect to patterns having other symmetries [35][36][37]. It is shown that, with the increase of the parameter a, a regular dot pattern can become unstable leading to stripe patterns.…”

“…where E, a g , v g are the Young modulus, lattice parameter and Poisson ratio, respectively, of the growth structure and a s is the lattice parameter of the substrate, and (a g -a s )/a s is the possible misfit at the interface) is the elastic energy density, k is the free surface mean curvature, c(h) is the isotropic surface energy and U(h,|rh| 2 ) = (dc/dh) 1 þ rh j j 2 À1=2 is the surface chemical potential related to wetting interactions with the substrate (wetting chemical potential), and it is a function of the height and slope [36,37]. The morphology of a surface, where a scratch test takes place, needs to be described with an abrasion process and a localised accumulations of atoms due to surface diffusion, therefore combining Eq.…”

“…Number density of growing clusters dependence on the normal load was demonstrated elsewhere [21] One main problem in Eq. 11 is the specification of the chemical potential, and one possible choice is given manipulating such equation by using the thermodynamics of stressed solids with a reference state for the film of zero strain, and under the additional assumptions of isotropic surface energy, the gradient of the chemical potential can be written as [34][35][36] …”

Continuum modelling of semiconductor heteroepitaxy: an applied perspective

“…in Refs. [53,[114][115][116][117][118][119]). More accurate calculations involving the numerical solution of the mechanical equilibrium equations by Finite Element Method (FEM) have also been exploited [120,121].…”

“…in Refs. [117][118][119]135,136], profiting of highly optimized algorithms, such as the Fast Fourier Transform. However, height function methods are not applicable to complex structures with overhangs, e.g.…”