Dipole excitation of collective modes in viscous two-dimensional electron systems

We describe the structure of the time-harmonic electromagnetic field of a vertical Hertzian electric dipole source radiating over an infinite, translation invariant two-dimensional electron system. Our model for the electron flow takes into account the effects of shear and Hall viscosities as well as an external static magnetic field perpendicular to the sheet. We identify two wave modes, namely, a surface plasmon and a diffusive mode. In the presence of an external static magnetic field, the diffusive mode combines the features of both the conventional and Hall diffusion and may exhibit a negative group velocity. In our analysis, we solve exactly a boundary value problem for the time-harmonic Maxwell equations coupled with linearized hydrodynamic equations for the flat, two-dimensional material. By numerically evaluating the integrals for the electromagnetic field on the sheet, we find that the plasmon contribution dominates in the intermediate-field region of the dipole source. In contrast, the amplitude of the diffusive mode reaches its maximum value in the near-field region, and quickly decays with the distance from the source. We demonstrate that the diffusive mode can be distinguished from the plasmon in the presence of the static magnetic field, when the highly oscillatory plasmon is gapped and tends to disappear.