Growth of the nuclei of a new layer is one of the fundamental aspects of the growth of both bulk crystals and epitaxial layers and plays an important role in the design and fabrication of modern quantum structure-based electronic devices. Our Monte Carlo (MC) simulation results demonstrate that the habit of two-dimensional (2D) growth of Kossel crystals depends on supersaturation and surface diffusion barrier. The simulations show that the shapes of nuclei change from the equilibrium shape, which is edge-rounded square through diamond shape at small supersaturation to square shape. At very small supersaturation, the surface diffusion plays a dominant role, controlling the resulting growth shapes. For higher supersaturation, the growth habit is universal and does not depend on surface diffusion rates. Further increase of the supersaturation leads to unstable growth morphologies. (C) 2006 Elsevier B.V. All rights reserved.