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Abstract

We present a hydrodynamic model, grounded in Madelung's formalism, to describe collective electronic motion in anisotropic materials. This model incorporates nonlocal contributions from the Thomas-Fermi quantum pressure and quantum effects arising from the Bohm potential. We derive analytical expressions for the magnetoplasmon dispersion and nonlocal optical conductivity. To demonstrate the applicability of the model, we examine electrons in the conduction band of monolayer phosphorene, an exemplary anisotropic two-dimensional electron gas. The dispersion of plasmons derived from our hydrodynamic approach is closely aligned with that predicted by ab~initio calculations. Then, we use our model to analyze few-layer black phosphorus, whose measured infrared optical response is hyperbolic. Our results reveal that the incorporation of nonlocal and quantum effects in the optical conductivity prevents black phosphorus from supporting hyperbolic surface plasmon polaritons. We further demonstrate that the predicted wavefront generated by an electric dipole exhibits a significant difference between the local and nonlocal descriptions for the optical conductivity. This study underscores the necessity of moving beyond local approximations when investigating anisotropic systems capable of hosting strongly confined plasmon-polaritons.