Analytical plasmon dispersion for two-energy level film system

Based on the free electron gas model and the eigen equation method, analytical solutions for the dispersion relationships of both intra-band and inter-band plasmons are derived under the long wavelength approximation in two-energy level thin film system. The analytical solutions aid in understanding how plasmon energies vary with the wave vector and the film thickness. In the long wavelength limit, the dispersion of intra-band plasmons is in agreement with that of pure two-dimensional plasmon. At normal wavelength, the dispersion curve of intra-band plasmon is lower than that of pure two-dimensional plasmon. This is due to the fact that the thickness of the thin film reduces the Fermi energy, thereby introducing a negative correction to the intra-band plasmon energy. The dispersion of inter-band plasmon reveals that the inter-band plasmon energy is larger than the corresponding single particle excitation energy. As the film thickness increases, there is a rapid decrease in the energy of inter-band plasmon due to that the energy interval between two energy bands diminishes.