Plasmonic systems with sub-nanometer gaps exhibit quantum effects that significantly deviate from classical electromagnetic predictions. At these extreme length scales, the continuum approximation of classical electrodynamics breaks down, and quantum mechanical phenomena dominate the optical and electronic responses. Key effects include electron spill-out, tunneling currents, and nonlocal screening, which collectively modify the plasmonic behavior in ways that cannot be captured by classical models alone. Theoretical frameworks such as jellium models and time-dependent density functional theory (TDDFT) provide critical insights into these quantum corrections.

Electron spill-out refers to the extension of electron density beyond the nominal boundary of a metal surface. In classical treatments, the electron density is assumed to terminate abruptly at the metal boundary, but quantum mechanically, electrons tunnel into the vacuum or dielectric gap, creating a smooth decay of charge density. This spill-out effect becomes particularly pronounced in sub-nanometer gaps, where the electron density of adjacent structures overlaps. The spill-out modifies the local dielectric function, leading to a redshift of plasmon resonances and a reduction in field enhancement compared to classical predictions. Jellium models, which approximate the ionic background as a uniform positive charge, demonstrate that spill-out effects are sensitive to the electron density parameter \( r_s \), where lower \( r_s \) values (higher electron densities) exhibit less spill-out due to stronger confinement.

Tunneling currents arise when the gap between plasmonic structures is small enough to allow electrons to quantum mechanically tunnel across the barrier. For gaps below 0.5 nm, direct tunneling becomes significant, leading to conductive pathways that quench the plasmonic response. This effect is particularly relevant in systems like nanoparticle dimers or bowtie antennas, where classical models predict enormous field enhancements that are unrealizable due to tunneling. TDDFT simulations reveal that tunneling introduces a damping mechanism, broadening plasmon resonances and reducing their intensity. The tunneling current is highly dependent on the gap distance and the work function of the metal, with noble metals like gold and silver showing pronounced tunneling effects due to their relatively low work functions.

Nonlocal screening is another quantum correction that arises from the wave-like nature of electrons. In classical electrodynamics, the dielectric response is treated as local, meaning the polarization at a point depends only on the electric field at that point. However, in sub-nanometer gaps, the finite wavelength of electrons leads to nonlocal effects, where the dielectric response at one location depends on the electric field over an extended region. This nonlocality smears out the charge density distribution, further reducing field enhancements and blueshifting plasmon resonances. Hydrodynamic models, which incorporate nonlocal effects through additional terms in the dielectric function, provide a semi-classical approximation, but full quantum treatments like TDDFT are necessary for quantitative accuracy.

Quantum corrections also manifest in the optical conductivity of plasmonic systems. Classical Drude models assume a frequency-dependent but spatially local conductivity, but at sub-nanometer scales, the conductivity becomes spatially dispersive due to electron-electron interactions. TDDFT calculations show that this spatial dispersion leads to a suppression of conductivity at high frequencies, altering the absorption and scattering spectra of plasmonic nanostructures. The quantum corrections are particularly significant for gaps below 1 nm, where the electronic wavefunctions of adjacent structures hybridize, leading to new collective modes that are absent in classical treatments.

The interplay between these quantum effects complicates the design of plasmonic devices for applications like sensing or nanophotonics. For instance, the optimal gap size for maximum field enhancement is not simply the smallest achievable gap but rather a balance between quantum tunneling and spill-out effects. Theoretical studies suggest that gaps around 0.3–0.7 nm exhibit the most pronounced quantum corrections, with larger gaps converging toward classical predictions and smaller gaps dominated by tunneling-induced damping.

Jellium models and TDDFT have been instrumental in quantifying these effects. Jellium models, while simplified, capture the essential physics of electron spill-out and nonlocality without the computational cost of full ab initio methods. TDDFT, on the other hand, provides a more rigorous treatment by explicitly accounting for electron-electron interactions and exchange-correlation effects. Both methods agree that quantum corrections become significant when the gap size is comparable to the Fermi wavelength of the metal, typically around 0.5 nm for noble metals.

In summary, plasmonic systems with sub-nanometer gaps require quantum mechanical descriptions to accurately predict their optical and electronic properties. Electron spill-out, tunneling currents, and nonlocal screening are dominant effects that reshape plasmon resonances and field distributions. Theoretical tools like jellium models and TDDFT offer valuable insights into these phenomena, guiding the development of quantum-aware designs for nanophotonic applications. Future work may explore the role of atomic-scale details, such as crystallographic orientation and surface defects, which could further modulate these quantum effects.