Resonance shifts and spill-out effects in self-consistent hydrodynamic nanoplasmonics

The standard hydrodynamic Drude model with hard-wall boundary conditions can give accurate quantitative predictions for the optical response of noble-metal nanoparticles. However, it is less accurate for other metallic nanosystems, where surface effects due to electron density spill-out in free space cannot be neglected. Here we address the fundamental question whether the description of surface effects in plasmonics necessarily requires a fully quantum-mechanical approach, such as time-dependent density-functional theory (TD-DFT), that goes beyond an effective Drude-type model. We present a more general formulation of the hydrodynamic model for the inhomogeneous electron gas, which additionally includes gradients of the electron density in the energy functional. In doing so, we arrive at a Self-Consistent Hydrodynamic Model (SC-HDM), where spill-out emerges naturally. We find a redshift for the optical response of Na nanowires, and a blueshift for Ag nanowires, which are both in quantitative agreement with experiments and more advanced quantum methods. The SC-HDM gives accurate results with modest computational effort, and can be applied to arbitrary nanoplasmonic systems of much larger sizes than accessible with TD-DFT methods. Moreover, while the latter typically neglect retardation effects due to time-varying magnetic fields, our SC-HDM takes retardation fully into account.