  We extend the electrodynamics of two dimensional electron gases to account
for the extrinsic spin Hall effect (SHE). The theory is applied to doped
graphene decorated with a random distribution of absorbates that induce
spin-orbit coupling (SOC) by proximity. The formalism extends previous
semiclassical treatments of the SHE to the non-local dynamical regime. Within a
particle-number conserving approximation, we compute the conductivity,
dielectric function, and spin Hall angle in the small frequency and wave vector
limit. The spin Hall angle is found to decrease with frequency and wave number,
but it remains comparable to its zero-frequency value around the frequency
corresponding to the Drude peak. The plasmon dispersion and linewidth are also
obtained. The extrinsic SHE affects the plasmon dispersion in the long
wavelength limit, but not at large values of the wave number. This result
suggests an explanation for the rather similar plasmonic response measured in
exfoliated graphene, which does not exhibit the SHE, and graphene grown by
chemical vapor deposition, for which a large SHE has been recently reported.
Our theory also lays the foundation for future experimental searches of SOC
effects in the electrodynamic response of two-dimensional electron gases with
SOC disorder.
