Graphene has many unique properties which have made it a hotbed of scientific research in recent years. However, it is not expected intuitively that the strong effects of the substrate and Coulomb doping in the center of crystal cell on the polaron in monolayer graphene. Here, the interaction energy of surface electron (hole) in the graphene and optical phonons in the substrate, which give rise to weakly coupled polarons, is analyzed in the context of the Coulomb doping. The ground-state energy of the polaron is calculated using the Lee-Low-Pine unitary transformation and linear combination operator method. It is found that the ground-state energy is an increasing function of magnetic field strength, the bound Coulomb potential, and the cutoff wavenumber. Numerical results also reveal that the ground-state energy reduces as the distance between the graphene and the substrate is increased. Moreover, the ground energy level of polaron shows the two (+) and (−) branches and zero-Landau energy (ground) level separation in the graphene-substrate material.
Ground state energy of small electron-hole drops