A Generalized Non-local Quantum Theory for Plasmonic Nanostructures

The propagation of light and its interaction with metallic nanostructures at the scale that is smaller than the wavelength of light (subwavelength) is an interesting phenomenon. In recent years, confining light at a subwavelength scale by very small metallic nanoparticles, such that quantum effects cannot be neglected, has generated interest in the scientific community. The light-matter interactions at this level require a careful quantum mechanical treatment to be correctly characterized, hence evolving in a new field named Quantum Plasmonics. In metallic nanostructures with sizes below 10 nm, the collective and coherent oscillations of electrons (plasmon resonance), cannot be described by classical models since the quantum-mechanical effects start dominating and become relevant with changing the plasmons oscillation frequency. Such changes have so far been poorly understood and the experimental measurements that have been carried out, have struggled to be correctly interpreted. Therefore, a quantum model of metal permittivity is required to understand the size-dependent optical properties of very small nanostructures. Here we present the nonlocal quantum model, obtained by applying the Wigner equation with the collision term in the kinetic theory of metals. Our results suggest that the probability of finding electrons at higher energy levels increases in the excitation of quantum plasmons, since their wave functions overlap, and therefore, the quantum tunnelling effect increases. The dispersion relation, damping rate, and decay length of surface and bulk plasmon resonances are investigated in thin metal film slabs and small silver nanoparticles with the various diameter down to atomic size and plasmon wave functions are shown for solutions of infinite quantum well at various quantum levels