When finite-size effects dictate the growth dynamics on strained freestanding nanomembranes

We investigate the influence of strain-sharing and finite-size effects on the morphological instability of hetero-epitaxial nanomembranes made of a thin film on a thin freestanding substrate. We show that long-range elastic interactions enforce a strong dependence of the surface dynamics on geometry. The instability time-scale τ is found to diverge as (e/H)^- with α = 4 (respectively 8) in thin (resp. thick) membranes, where e (resp. H) is the substrate (resp. nanomembrane) thickness, revealing a huge inhibition of the dynamics as strain sharing decreases the level of strain on the surface. Conversely, τ vanishes as H ² in thin nano-membranes, revealing a counter-intuitive strong acceleration of the instability in thin nanomembranes. Similarly, the instability length-scale displays a power-law dependence as (e/H)^- , with β = α/4 in both the thin and thick membrane limits. These results pave the way not only for experimental investigation, but also, for the dynamical control of the inescapable morphological evolution in epitaxial systems.