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The progress in two-dimensional materials has led to rapid experimentaldevelopments in quantum plasmonics, where light is manipulated using plasmons.Although numerical methods can be used to quantitatively describe plasmons inspatially inhomogeneous systems, they are limited to relatively small setups.Here, we present a novel semi-analytical method to describe plasmons intwo-dimensional inhomogeneous media within the framework of the Random PhaseApproximation (RPA). Our approach is based on the semiclassical approximation,which is formally applicable when the length scale of the inhomogeneity is muchlarger than the plasmon wavelength. We obtain an effective classicalHamiltonian for quantum plasmons by first separating the in-plane andout-of-plane degrees of freedom and subsequently employing the semiclassicalAnsatz for the electrostatic plasmon potential. We illustrate this generaltheory by considering scattering of plasmons by radially symmetricinhomogeneities. We derive a semiclassical expression for the differentialscattering cross section and compute its numerical values for a specific modelof the inhomogeneity.

Semiclassical theory for plasmons in two-dimensional inhomogeneous media