Competition of trivial and topological phases in patterned graphene based heterostructures

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