Nonlocal formalism for nanoplasmonics: Phenomenological and semi-classical considerations

• Metallic nanostructures are anticipated to exhibit nonlocal dynamics when exploring the true nanoscale (1–10 nm).
• The local-response approximation is extended to account for short-range nonlocal response of the homogeneous electron gas.
• Phenomenological treatment of nonlocal response leads to a Laplacian correction term in the electromagnetic wave equation.
• The Laplacian correction is supported by a detailed semi-classical hydrodynamic model.

The plasmon response of metallic nanostructures is anticipated to exhibit nonlocal dynamics of the electron gas when exploring the true nanoscale. We extend the local-response approximation (based on Ohm's law) to account for a general short-range nonlocal response of the homogeneous electron gas. Without specifying further details of the underlying physical mechanism we show how this leads to a Laplacian correction term in the electromagnetic wave equation. Within the hydrodynamic model we demonstrate this explicitly and we identify the characteristic nonlocal range to be ξNL∼vF/ωξNL∼vF/ω where vFvF is the Fermi velocity and ω is the optical angular frequency. For noble metals this gives significant corrections when characteristic device dimensions approach ∼1–10 nm, whereas at more macroscopic length scales plasmonic phenomena are well accounted for by the local Drude response.