Dispersion relation of graphene surface plasmon by using a quantum hydrodynamic model

A quantum hydrodynamic model is used to study the properties of surface plasmon at an interface between a monolayer graphene sheet and a Voigt substrate. The substrate is assumed as a semiconductor that applying an external magnetic field to the structure in the Voigt configuration. The dispersion relations of graphene surface plasmon are obtained analytically by solving Maxwell's equations and the quantum hydrodynamic equations. It is found that the quantum effects significantly change the properties of graphene surface plasmon and the features of such plasmon are quite different from those in a classical hydrodynamic model. The results also show that the applied magnetic field and the graphene character greatly affect the graphene surface plasmon. Moreover, The plasmon modes exhibit distinctively different behavior for forward and backward propagating directions, which are in contrast to the cases without the quantum effects. In addition, a one-way plasmon mode is found in the lower band region. Parameter dependence of the effects is examined and discussed.