Projected Dipole Model for Quantum Plasmonics

Quantum effects of plasmonic phenomena have been explored through ab initio studies, but only for exceedingly small metallic nanostructures, leaving most experimentally relevant structures too large to handle. We propose instead an effective description with the computationally appealing features of classical electrodynamics, while quantum properties are described accurately through an infinitely thin layer of dipoles oriented normally to the metal surface. The nonlocal polarizability of the dipole layer—the only introduced parameter—is mapped from the free-electron distribution near the metal surface as obtained with 1D quantum calculations, such as time-dependent density-functional theory (TDDFT), and is determined once and for all. The model can be applied in two and three dimensions to any system size that is tractable within classical electrodynamics, while capturing quantum plasmonic aspects of nonlocal response and a finite work function with TDDFT-level accuracy. Applying the theory to dimers, we find quantum corrections to the hybridization even in mesoscopic dimers, as long as the gap itself is subnanometric.

Figure 1

In the projected dipole model, light-driven quantum-plasmonic response in a metallic nanostructure (left) is represented in classical electrodynamics by an infinitely thin layer of dipoles that point perpendicularly to the surface (right). The loupe (left) highlights the spill-out of the microscopic electron distribution inducing a dipole moment proportional to the Feibelman parameter $d_c$.

Figure 2

(a) TDDFT reflection spectra of a sodium surface for the electric field incidence with different $k_{\parallel }$ (in units of $k_F$) as illustrated in the inset. (b) Long-wavelength polarizability ${\alpha }_L$. (c) Magnitude of polarizability $\alpha$ normalized by ${\alpha }_L$.

Figure 3

Extinction properties of sodium dimers within LRA, PDM, and TDDFT for (a) cylindrical wire with radius 4.9 nm and gap 0.74 nm, (b) cylindrical wire with radius 2 nm and gap 1 nm, (c) spherical particle with radius varying from 3 to 23 nm and with 0.74-nm gap distance, where the extinction cross sections are normalized by the surface area of the sphere, and (d) the 20-nm radius case [dash-dotted line in (c)]. TDDFT results in (a) and (b) are reproduced from Refs. [15] and [16], respectively.

Figure 4

Gap effects for planar sodium-vacuum-sodium gap structures within TDDFT. (a) Effective permittivity for a single interface and for 5 and 0.5 Å gaps. (b) Frequency dependence of the corresponding midgap effective permittivities. Dashed lines indicate the lossless Drude model, with its plasma frequency determined by the midgap equilibrium electron density. (c) Electron-hole backward scattering proportion (BSP) as a function of gap distance and energy.

Figure 5

Extinction cross sections of a cylindrical sodium nanowire dimer with radius 4.9 nm, for gaps decreasing from 5.3 to 0 Å. Comparison of predictions by LRA, TDDFT, PDM, and PDGM. TDDFT spectra originate from Ref. [15].