Acoustic Graphene Plasmons under a DC bias

It has been shown that in a graphene/insulator/metal structure, the plasmons acquire a linear dispersion whose sound velocity is very near the Fermi velocity in graphene. I show that a DC current causes a redshift of the acoustic plasmons travelling upstream of the direction in which electrons flow. As the redshift of the upstream branch causes acoustic plasmon to approach the particle-hole continuum, the spectral weight of this plasmon branch is substantially reduced, resulting in a near unidirectional plasmon.