Lattice Relaxation, Strain, and Microscopic Continuum Hamiltonian for Twisted Bilayer Graphene

The atomic relaxation in twisted bilayer graphene is commonly understood as the result of a balance between the intralayer elastic energy and interlayer adhesion energy. The elastic energy favors a rigid twist with no distortion in the twisted honeycomb lattices, while the adhesion energy favors Bernal stacking and breaking the relaxation into triangular AB and BA stacked domains. I will present a method for finding a closed form expression for a highly accurate approximation to the solution of an atomic relaxation model down to, and below, the first magic angle. The results compare well with the published Bragg interferometry data. Closed form expressions are obtained in the absence, as well as in the presence, of external heterostrain. The approach also allows determination of the radius of convergence of an attempt at a series solution, finding that the first magic angle is outside the radius of convergence.

Given a lattice relaxation, the continuum electroninc Hamiltonian for a graphene bilayer can be systematically constructed starting from a microscopic lattice theory. The approach allows for an arbitrary inhomogeneous smooth lattice deformation, including a twist. Analysis of such ab initio derived continuum model with the relaxed atomic configuration will also be presented and compared with the commonly used minimal continuum electronic model.