Periodically strained graphene lattice: flat bands

The conditions for the appearance of flat bands in periodically buckled graphene systems is determined. We use a tight-binding model to calculate the band structure of periodically strained graphene lattice. Three different strain configurations are considered: 1) triangular pseudo-magnetic field (PMF) mode - which is the first two-dimensional buckling mode where flat bands have been observed, 2 ) hexagonal buckling mode - as a common out-of-plane buckling mode in case of stacking different hexagonal lattices, and 3 ) herringbone buckling mode - as the lowest energy configuration in the case of large biaxial strains. We examine the band flattening versus the period of the buckling and the strength of the deformation and give predictions for the necessary conditions to access the regime of correlated phases. Our simulations show that the triangular PMF configuration is the most favourable for the appearance of flat bands due to the PMF-induced electron confinement. Similarly as in the case of energy levels of graphene quantum dots in the magnetic field, we find that flat bands show the same dispersion versus the strength of the deformation with energy separation that is inversely proportional to the buckling period.