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.
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The authors wish to thank D. V. Chichinadze, M. M. Fogler, L. Levitov, A. Lucas, M. Polini, A. Principi, and D. Svintsov for useful discussions. V.A., M.L., and D.M. acknowledge partial support by the ARO MURI Award No. W911NF-14-1-0247 and the Institute for Mathematics and its Applications (NSF Grant No. DMS-1440471) at the University of Minnesota for several visits. The research of V.A. and M.L. was also supported in part by NSF Awards No. DMS-1819220 and No. 1922165. D.A.B. acknowledges support from the MIT Pappalardo Fellowship. The research of D.M. was also partially supported by a Research and Scholarship award by the Graduate School, University of Maryland. Part of this research was carried out when three of the authors (V.A, M.L., and D.M.) were visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by NSF under Grant No. DMS-1440415.
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