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 some 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 epitaxial growth with elasticity. We rigorously identify the minimum energy scaling law and prove the formation and appearance of one bunch structure. We also provide sharp bounds for the bunch size and the slope of the optimal step bunch profile.
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 discovered and proved. As a byproduct, we partially recover the crystallization conjecture for LJ system. Our analysis also extends to any critical point of energy, not necessarily the global energy minimizer.
We then derive a generalized continuum model 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 Neumann boundary conditions 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 connection between a full atomistic model and a PN model for the dislocation in a bilayer system (e.g. bilayer graphene). We prove under some stability condition that the displacement field of the atomistic model is asymptotically close to that of the dislocation solution of the PN 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.