Velocity renormalization in graphene: The role of trigonal warping and electron-phonon coupling effects

We investigate the combined effects of trigonal warping and electron phonon interactions on the renormalization of the Fermi velocity in graphene. We present an analytical solution to the associated Friihlich Hamiltonian describing the interaction of doubly degenerate-optical phonon modes of graphene with electrons in the presence of trigonal warp within the framework of Lee Low Pines theory. On the basis of our model, it is analytically shown that in addition to its renormalization, Fermi velocity exhibits strong anisotropy due to the trigonal warping. It is also found that in the regime where the trigonal warp starts, distortion of energy bands emerges due to electron phonon coupling, and the bands exhibit strong anisotropy.