Competition of topological and topologically trivial phases in patterned graphene based heterostructures

We explore the effect of mechanical strain on the electronic spectrum of patterned graphene based heterostructures. We focus on the competition of Kekul\'e-O type distortion favoring a trivial phase and commensurate Kane-Mele type spin-orbit coupling generating a topological phase. We derive a simple low-energy Dirac Hamiltonian incorporating the two gap promoting mechanisms and include terms corresponding to uniaxial strain. The derived effective model explains previous ab initio results through a simple physical picture. We show that while the trivial gap is sensitive to mechanical distortions, the topological gap stays resilient.