On the existence of a critical perturbation amplitude for the Stranski-Krastanov transition

The stability of an elastically strained thin film is investigated within the context of small and large perturbation theories for the case of an anisotropic surface energy which is a function of the film thickness and the surface orientation. Under typical growth conditions (below the roughening transition temperature) the surface energy function has a sharp (cusped) minima at low energy orientations, i.e. the derivative of the surface energy with respect to orientation is discontinuous. This sharp cusp in the surface energy function is treated explicity here without the normal smoothing assumptions. It is found that the smoothed approximation is unphysical in the sharp cusp limit as it predicts that the film is always stable. It is shown here that full treatment of the cusp disagrees with this finding and predicts that minimum energy surfaces are in fact unstable for perturbations above a critical size. A simple linear model for this critical perturbation size is proposed. Off-lattice kinetic Monte Carlo (kMC) simulations are conducted to test these predictions and good agreement is found. It is demonstrated that roughening in highly strained heteroepitaxial systems is possibly beyond the scope of linear perturbation theory, depending on the exact nature of the surface energy function. A non-linear theory for large perturbations is proposed to this effect. It is also found that the wetting layer in InAs/GaAs(0 0 1) and Ge/Si(0 0 1) is effectively stable whilst the surface energy of the wetting layer varies with its thickness.