Modelling the Spill-out Effect from a Metallic Nanosphere: A Boundary-based Approach

The Quantum Hydrodynamic (QHD) Model, also known as the Self-consistent Hydrodynamic (SC-HD) Model, is a powerful material response model that can properly address the spill-out effect at the boundary of optical metals [1], and, when combined with an appropriate ground state electron density, is able to reproduce both the plasmon resonances and the spill-out effects, just as in Time-dependent Density Functional Theory (TD-DFT) [2].Although, up to date, the QHD model has been applied to the analysis of the optical response of many geometries, e.g., layered spherical nanoparticles [3], planar interfaces [4], cylindrical bowtie dimers [5], to name a few, its main computational platform is COMSOL, a Finite Element Method (FEM)-base commercial solver.A potential reason for impeding the integration of the model with other more efficient Computational Electromagnetics (CEM) algorithms, e.g., (semi-)analytical approaches [6], integral equation-based approach [7], is that the complicated mathematical formulation (remembering the von Weizsacker term) of the model and the resultant demanding computational burden, blocks its broad application, despite its qualities to investigate extreme nanoscopic optical phenomena.Here, we focus on the simplest geometry, a metallic nanosphere, and propose a very efficient boundary-based numerical algorithm to predict its quantum hydrodynamic response.In detail, the algorithm involves a fictitious boundary that splits the entire space into two regions: an inner and an outer region.The inner region is set large enough to enclose the entire inhomogeneous electron gas.For this region, we exploit the spherical symmetries, use vector spherical harmonics, and reduce the original 3D problem to a 1D one.The reduced problem is then transformed to its weak form by using the Finite Element algorithm.For the outer region, the fields are expressed in terms of spherical wave functions.By matching the fields at the boundary, we find a solvable system of equations and thus recover the fields in space.The developed algorithm does not only provide (semi-)analytical results for future benchmarking coming algorithms and solvers, but is also expected to lay down the very foundation for developing very efficient full-wave solvers for arbitrary 3D nano-geometries.