Effective Hamiltonian of strained graphene.

Based on the symmetry properties of the graphene lattice, we derive the effective Hamiltonian of graphene under spatially nonuniform acoustic and optical strains. Comparison with the published results of the first-principles calculations allows us to determine the values of some Hamiltonian parameters, and suggests the validity of the derived Hamiltonian for acoustical strain up to 10%. The results are generalized for the case of graphene with broken plane reflection symmetry, which corresponds, for example, to the case of graphene placed on a substrate. Here, essential modifications to the Hamiltonian give rise, in particular, to the gap opening in the spectrum in the presence of the out-of-plane component of optical strain, which is shown to be due to the lifting of the sublattice symmetry. The developed effective Hamiltonian can be used as a convenient tool for analysis of a variety of strain-related effects, including electron-phonon interaction or pseudo-magnetic fields induced by the nonuniform strain.