Shape and coarsening dynamics of strained islands

Abstract

We investigate the formation and the coarsening dynamics of islands in a strained epitaxial semiconductor film. These islands are commonly observed in thin films undergoing a morphological instability due to the presence of the elasto capillary effect. We first describe both analytically and numerically the formation of an equilibrium island using a two dimensional continuous model. We have found that these equilibrium island-like solutions have a maximum height h0 and they sit on top of a flat wetting layer with a thickness hw. We then consider two islands and we report that they undergo a non-interrupted coarsening that follows a two stage dynamics. The first stage may be depicted by a quasi-static dynamics, where the mass transfers are proportional to the chemical potential difference of the islands. It is associated with a time scale tc that is function of the distance d between the islands, and leads to the shrinkage of the smallest island. Once its height becomes smaller than a minimal equilibrium height h * 0 , its mass spreads over the entire system. Our results pave the way for a future analysis of coarsening of an assembly of islands.PACS numbers: 81.15. Hi, 68.35.Ct, 81.10.Aj, 47.20.Hw Understanding the dynamics of coarsening and its effect on self-organisation is a central question in nonequilibrium physics and solid-state physics since its experimental discovery by Ostwald at the end of the 19th century [1] and the seminal theoretical papers of LishitzSlyosov and Wagner [2,3] in the late 60's, see also [4]. Coarsening is a general phenomena in which the natural size of a pattern increases with time in a continuous manner over a large range of time scales [5][6][7][8]. From a more applied point of view, coarsening has a significant impact on properties of matter such as the size of grains in polycrystalline solids, the hardening of metallic alloys, foam dynamics, sintering, sand dunes, etc. We focus here, on the fundamental aspect of coarsening of strained semiconductor quantum dots, such as the gallium-aluminumnitride or silicon-germanium islands [9][10][11][12][13][14][15][16][17][18][19][20][21]. These islands are extensively under scrutiny both for their present and promising applications in electronics or optics, as for example single photons emitters, and for their insights into the fundamental processes of epitaxial growth. The properties and potential applications of quantum dot assembly are indeed crucially dependent on the amount of coarsening, that may critically affects the size homogeneity of such structures [20]. Moreover, the coarsening of such islands seems to be out of the classical description of Ostwald coarsening and requires more investigation.The formation of self-organized semi-conductor quantum dots results from the Stranski-Krastanov growth mode [22]. In this scheme, growth initially proceeds as planar layers, that transform above a given critical thickness h c , into islands separated by a wetting layer. These islands enable a partial relaxation of the elastic stress of t...

Morphologies, metastability, and coarsening of quantum nanoislands on the surfaces of the annealed Ag(110) and Pb(111) thin films

“…4. As we remarked in the Introduction, the rounded islands are typical for Stranski-Krastanow growth [32], where the lattice-mismatch stress is the dominant effect in the morphology evolution. This explains the prevalence of such islands on the Pb(111) surface.…”

“…Our refined continuum model is based on the following 2D surface diffusion equation [29,30,32,[38][39][40][41][42][43][44][45]:…”

“…is the chemical potential [29,30,32,[39][40][41][42][43]. Here γ(h) is the surface energy density, κ is the surface curvature, and n 3 is the z-component of n. The second term in Eq.…”

“…[23]. Recently, several sophisticated 2D and 3D continuum models of thicker film epitaxial growth and morphological evolution have been published [31][32][33][34][35][36][37].…”

Dynamics of elastically strained islands in presence of an anisotropic surface energy

“…where D is the diffusion coefficient, h x is the slope of the surface height ∂ x h(x, t) and ∂/∂s the surface gradient [31,32].…”