Percolation of topological states in engineered periodic strained graphene

The discovery of correlated phases in twisted bilayer graphene accelerated the search for low-dimensional materials with exotic electronic properties. A promising approach uses engineered substrates to strain the material and modify the electron dynamics. However, designing substrates for tailored properties remains beyond reach due to an incomplete understanding of the relationship between substrate's shapes and electronic properties. By analyzing effective Dirac models for graphene under periodic deformations with generic profiles and distinct symmetries, we identify the critical substrate geometrical parameters for specific electronic properties. We establish the conditions for emerging gaps and isolated quasi-flat bands and obtain spatial charge distributions. We also elucidate specific geometrical features that produce percolating states and associated topological bands and discuss the implications for interpreting local experimental observations.