In the first part of this thesis, we study the epitaxial growth on vicinal surfaces, where elasticity effects give rise to step bunching instability and self-organization phenomena, which are widely believed to be important in the fabrication of nanostructures. It is challenging to model and analyze these phenomena due to the nonlocal effects and interactions between different length scales.

We first study a discrete model for step dynamics of epitaxial growth with elastic force monopole and dipole effects of steps. We rigorously identify the minimum energy scaling law and prove the formation and coarsening of step bunching as the number of steps increases. Sharp bounds for the bunch size and the slope of the optimal step bunch profile are obtained.

After that we generalize th...[ Read more ]

In the first part of this thesis, we study the epitaxial growth on vicinal surfaces, where elasticity effects give rise to step bunching instability and self-organization phenomena, which are widely believed to be important in the fabrication of nanostructures. It is challenging to model and analyze these phenomena due to the nonlocal effects and interactions between different length scales.

We first study a discrete model for step dynamics of epitaxial growth with elastic force monopole and dipole effects of steps. We rigorously identify the minimum energy scaling law and prove the formation and coarsening of step bunching as the number of steps increases. Sharp bounds for the bunch size and the slope of the optimal step bunch profile are obtained.

After that we generalize the bunching results to a one dimensional system with Lennard–Jones (LJ) (m, n) interaction. A phase transition from bunching to non-bunching regimes is identified and proved. As a byproduct, we partially recover the crystallization results for one-dimensional LJ system. In the bunching regime, we also obtain the sharp bounds of the minimum energy, the bunch size, and the maximal slope. Our analysis also extends to any critical point of energy, not necessarily the global energy minimizer.

Next, we derive a generalized continuum model for step bunching and prove its well-posedness, energy scaling law, and a sharp maximal slope estimate. The results are consistent with those of the discrete model. For discrete and continuum models, periodic and non-periodic settings are both considered.

In the second part of this thesis, we study dislocation models. The Peierls–Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this thesis, we study the convergence from a full atomistic model to the PN model with γ-surface for the dislocation in a bilayer system (e.g. bilayer graphene). We prove that the displacement field and the total energy of the dislocation solution of the PN model are asymptotically close to those of the full atomistic model. Our work can be considered as a generalization of the analysis of the convergence from atomistic model to Cauchy–Born rule for crystals without defects in the literature.