The effects of the substrate surface roughness on graphene plasmons

We investigate the effects of variation in the gap size between mono-layer graphene and a substrate with a randomly rough surface on the linear response of graphene's π electron bands within the approximation of Dirac fermions. We adopt the electrostatic Green's function developed by Rahman and Maradudin [Phys. Rev. B 21, 2137–2143 (1980)] for the surface of a dielectric medium, which exhibits a Gaussian distributed height profile and combine it with the polarization function of graphene described as a zero-thickness planar layer at a fixed distance from the mean position of the substrate surface. We specifically consider the effects of a random gap size on the two-dimensional sheet plasmon mode in heavily doped graphene, both on its dispersion relation in the long-wavelength limit and its broadening due to Landau damping in the continuum of inter-band electron-hole excitations at shorter wavelengths