Introduction

    In the following several pages, I will introduce our model in more detail.  For intuitive purpose and simplicity, I will focus on the isotropic case: the phase boundary energy and surface stress is isotropic within the plane of the epilayer, the substrate is elastic isotropic.  Please refer to our papers for anisotropic cases.  It is found that anisotropy in surface stress, phase boundary energy and substrate can considerably change the ordering and patterns.  Please also refer to our papers or contact with me about recent results in the research, such as guided self-assembly.

    First allow me to discuss about the system and coordinates.

    Imagine a monolayer of two atomic species A and B grown on a substrate of atomic species S.  As illustrated in the following figure, the monolayer separates into two phases a  and b.  The substrate occupies the half space , bounded by the  plane.  The in-plane phase size is on the order 10 nm, much larger than the atomic dimension.  Consequently, we will neglect the discreteness of individual atoms. 

    Two situations are considered.  Figure (a) illustrates a cross-sectional view of the a  phase covering a fraction of the substrate surface, the bare substrate surface itself being the second phase.  In Fig. (b) the topmost monolayer comprises two phases a  and b  that both differ from the substrate.  The two situations are described in the same model.

    During deposition, when atoms hit the substrate, their initial positions are random.  To collectively self-assemble into a nanostructure, the individual atoms must be mobile on the surface.  We assume that atoms move by diffusion on the surface.  To maintain a flat monolayer on the solid surface, we further assume that atoms diffuse within the topmost monolayer - that is, atoms neither diffuse into the bulk of the substrate, nor pile up into three dimensional islands.  The system relaxes to an equilibrium state (subject to the constraint of concentration uniformity) by accommodating the misfits among the three kinds of atoms and the free space.  The misfits alter electronic states and the free energy of the system.  The effect is short-ranging in that atoms in the substrate, a few monolayers beneath the epilayer, have the same energy as those in an infinite elemental crystal of S.  We lump the epilayer, together with those adjacent monolayers of the substrate affected by the atomic misfit, into a single superficial object.

    It is clear from the previous discussion that, to account for self-assembling phases in a monolayer, a model should contain the following ingredients: phase separation, phase coarsening, and phase refining.