Production of plasmons in two layers of graphene with different doping densities traversed by swift electrons

We apply a fully relativistic formulation to calculate the average number of plasmons excited by an energetic electron traversing two parallel graphene layers separated by a given distance d. We use a Drude model in the non-dissipative limit to describe the conductivity of each layer in the THz frequency range, and address the general case in which the layers have different doping densities of charge carriers by defining an asymmetry parameter. With this, we obtain the probability density of exciting the hybridized Dirac Plasmon-Polariton modes that result from the coupling of the layers, and hence the number of plasmons excited by the passing electron along its trajectory. We analyze the effect of different parameters, such as the interlayer distance, the incident particle velocity, and the asymmetry in doping densities. In particular, we find a cusp-like behaviour for nearly equal conductivities, that indicates a high sensitivity of the system to slight variations in doping densities.