Plasmons in tunnel-coupled graphene layers: backward waves with quantum cascade gain

Plasmons in van der Waals heterostructures comprised of graphene and related layered materials demonstrate deep subwavelength confinement and large propagation length. In this letter, we show that graphene-insulator-graphene tunnel structures can serve as plasmonic gain media. The gain stems from the stimulated electron tunneling accompanied by the emission of coherent plasmons under interlayer population inversion. The probability of tunneling with plasmon emission appears to be resonantly large at certain values of frequency and interlayer voltage corresponding to the transitions between electron states with collinear momenta - a feature unique to the linear band structure of graphene. The dispersion of plasmons undergoes a considerable transformation due to the tunneling as well, demonstrating negative group velocity in several frequency ranges.