Influence of charged impurity screening and surface optical phonons on the n=0 Landau level splitting of graphene with polar substrate

The effect of charged Coulombic impurity screening on the n = 0 Landau level splitting of graphene with a polar substrate under a static magnetic field is investigated theoretically considering the electron-phonon Fröhlich interaction from surface optical phonons of the substrate. The screening is improved using a static dielectric function under the random-phase approximation. A method of linear combination operators is adopted to deal with the influence from a magnetic field and a weak electron-phonon coupling theory is used to discuss the effect from the surface optical phonons. It is found that the splitting due to phonons can be broadened by the charged impurity screening and the broadening increases as increasing of the magnetic field. Our results may provide a possible way to enlarge the energy gap of monolayer graphene.