equivalent voltage and current waves for tm spp in graphene sheets

We present a theoretical model to describe the propagation of a transverse magnetic surface plasmon polariton in graphene based on equivalent voltage and charge current waves, which includes the spatial dispersion effect. electrons and holes in graphene are governed by the boltzmann equation in the particle conserving relaxation time approximation. first, we deduce expressions for the non-equilibrium distributions when there are charge oscillations in graphene as a response to the electromagnetic field applied to it. these distribution functions are used in the boltzmann equations to derive other equations for the following four local macroscopic averages: the oscillating electron and hole densities, and the electron and hole current densities. then, for a specific structure, we solve the wave equations for the electric and vector magnetic potentials to obtain the relations between the charge oscillations and the potentials. so, we reach a homogeneous system of four coupled equations relating the amplitudes of the voltage and the current waves. the non-trivial solutions of the system allow us to compute the dispersion and loss curves for such waves. as it is already known, for a given frequency, we can see that the higher the fermi level is, the lesser the spatial-dispersion effect is. following the analysis, a distributed-element circuit for the equivalent transmission line in which would propagate the waves, is developed. finally, we analyze the dependence of these circuit elements and the impedance on both the frequency and fermi level.