Comparison of Coulomb Impurity, Longitudinal Acoustic Phonons, and Surface Optical Phonons Affecting the n = 0 Landau Level in Monolayer Graphene

The influences of a charged Coulombic impurity with screened effect and carrier-phonon interaction on the n = 0 Landau level in monolayer graphene with a polar substrate under a high static magnetic field are discussed to compare the competition among the impurities, the longitudinal acoustic phonons in the graphene plane and the surface optical phonons on the substrate. A method of linear combination operators is used to deal with the position and momentum of a carrier in a magnetic field. The method of Lee-Low-Pines variation with an arbitrary carrier-phonon coupling is adopted to derive the effects of phonons. It is found that the energy gap of n = 0 Landau level opened by carrier-longitudinal acoustic phonons cannot be the main mechanism, whereas both the carrier-surface optical phonon interaction and the carrier-impurity interaction play the main roles in determining the energy splitting.