Negative dynamic conductivity of a current driven array of graphene nanoribbons

We consider a periodic array of graphene nanoribbons under the action of a strong dc electric field $E_0$ and an external electromagnetic excitation with the frequency $\omega$ and the lateral wave vector $q$. Solving the quasi-classical Boltzmann kinetic equation and calculating the surface dynamic conductivity ${\sigma }_{2D}(q,\omega ,E_0)$ and the absorption coefficient of such a system we show that the real part of the conductivity and the absorption coefficient may become negative under certain conditions. Physically this corresponds to the amplification of the electromagnetic waves at the expense of the energy of the direct current source. The results are discussed in connection with experiments on the surface acoustic waves and on the Smith-Purcell-type graphene-based terahertz emitter.