| In heteroepitaxial growth, the mismatch between the lattice constants in the film and the substrate causes misfit strain in the film, making a flat surface unstable to small perturbations. This morphological instability is called Asaro-Tiller-Grinfeld (ATG) instability, which can drive the film to self-organize into nanostructures such as quantum wires or quantum dots. At low temperature, the surface consists of steps and facets, when the misfit strain causes step bunching, traditional continuum models for ATG instability does not apply directly. In the first part of this thesis, we derive a PDE model for step bunching by taking the continuum limit of the discrete models proposed by Tersoff et al and Duport et al. We study the linear instability of a uniform step train with small perturbations and compare our results with those of discrete models and continuum models for traditional ATG instability. We numerically study the nonlinear evolution of this instability and compare our results with those of discrete models. We also study the equilibrium shapes of step bunches and explain their coalescence. In the second part of this thesis, we derive a nonlinear approximate PDE for the ATG instability. In the ATG instability, the misfit strain is coupled with surface morphology and an elasticity problem must be solved numerically. Linear approximation is made in some cases such as when computing the equilibrium island shapes. Using the exact solution for a cycloid surface obtained by Chiu and Gao, we find that our nonlinear approximation has a wider range of applicability than linear approximation. Numerical simulation using our nonlinear PDE model predicts formation of a cusp-like surface morphology from initially small perturbations of flat surfaces, which agrees well with the result obtained by Spencer and Meiron by solving the elasticity problem numerically. |
