Metallic nanoshells, consisting of a metallic layer grown over a solid dielectric core, are unique nanoparticles. These particles exhibit unique optical properties, characterized by their highly tunable plasmon resonances across a wide range of frequencies in both the visible and infrared portions of the spectrum. Such properties have attracted considerable attention and are integral to various devices, such as resonant photooxidation inhibitors, optical triggers for drug delivery implants, and environmental sensors.

In recent years, quantum hydrodynamic theory (QHT) has been applied to compute the absorption spectra of Na nanoshells with varying thicknesses, emphasizing the significant impact of the quantum effect within QHT on its high-energy mode. Notably, the high-energy mode predicted by QHT is significantly redshifted compared to that predicted by the local response approximation (LRA). However, both methods, for the sake of simplicity, assume that in-shell and out-shell media are vacuums, thereby neglecting the influence of these media on its absorption spectrum. Consequently, this study employs QHT to investigate the effects of the in-shell and out-shell media of the Na nanoshell on its absorption spectrum.

Theoretically, the classical Drude model under the LRA for free electrons is typically applied to elucidate the optical response of plasmonic nanostructures. However, when the characteristic size of the nanostructure is less than 10 nm or the gap size of the nanodimers is of the order of subnanometers, the LRA becomes ineffective, and nonlocal corrections must be considered. In the context of free electron gas, the simplest corrections beyond the LRA are achieved by incorporating the Thomas-Fermi (TF) electron pressure into a hydrodynamic-like description, known as the Thomas-Fermi hydrodynamic theory (TFHT). However, both techniques overlook quantum effects, such as electron spillover and Landau damping, essential at nanometer scales. The QHT effectively accounts for the aforementioned quantum effect computationally. Therefore, in this study, we use the LRA, TFHT, and QHT to investigate the influence of quantum effects on the absorption spectrum of Na nanoshells in-shell and out-shell media.

The presence of in-shell and out-shell media induces a redshift in the surface plasmon resonances (Fig. 4). The in-shell medium has a greater effect on the high-energy mode and a smaller effect on the low-energy mode compared to the out-shell medium. Hence, in practical applications, the high-energy mode can be regulated by adjusting the in-shell dielectric constant, whereas the low-energy mode can be adjusted by altering the out-shell dielectric constant. This regulatory approach effectively controls energy modes to meet various application requirements. In addition, the quantum effect exerts a stronger influence on the resonance energy of the high-energy mode of the Na nanoshell compared to the low-energy mode (Fig. 6). Therefore, the influence of the quantum effect can be neglected when investigating the resonance energy of the low-energy mode in Na nanoshells, leading to a reduction in the number of calculations. This streamlining proves advantageous for practical applications.

This study systematically investigates the effects of the in-shell and out-shell media of the Na nanoshell on its absorption spectrum and ground-state conduction electron density. The results show that the extent of conduction electron spillover is related to the in-shell and out-shell media. The static permittivity of both the in-shell and out-shell media directly influences the extent to which the conduction electrons spill out to the outer shell. Additionally, the amplitudes and resonance energies of the surface plasmon resonance modes of the Na nanoshell are significantly affected by the in-shell and out-shell dielectric constants. Specifically, an increase in the in-shell dielectric constant results in a redshift of the surface plasmon resonance modes, accompanied by a reduction in the amplitude of the low-energy mode and an increase in the amplitude of the high-energy mode. As the out-shell dielectric constant increases, the low-energy mode undergoes a redshift, accompanied by an increase in its amplitude, whereas the high-energy mode remains relatively unchanged. In addition, the in-shell medium has a stronger influence on the high-energy mode and a weaker influence on the low-energy mode compared to the out-shell medium. Finally, our investigation into the quantum effect on the resonance modes reveals that with an increase in the in-shell dielectric constant, the quantum effect decreases for the low-energy mode amplitude and increases for the high-energy mode amplitude. With the increase in the dielectric constant of the out-shell, the quantum effect on the low-energy mode amplitude gradually increases, while its impact on the high-energy mode amplitude remains relatively consistent. Moreover, the quantum effect has a stronger influence on the resonance energy of the high-energy mode of the Na nanoshell compared to that of the low-energy mode.