Analytical and approximation approaches to solve the hybrid plasmonic–plasmonic only transition of the TM₀ mode in a dielectric-semiconductor-insulator-metal four layer structure

An analytical solution to the fundamental TM₀ mode on the model system of a dielectric-semiconductor-insulator-metal four layered planar structure is obtained. A transition from the ‘hybrid plasmonic’ mode to ‘plasmonic only’ mode is ratified by the change from sinusoidal to exponential wave functions in the semiconductor layer as the propagation constant of the TM₀ mode exceeds that of the light in bulk form of the semiconductor. A variational method based on the mode hybridization picture is proposed to approximate the dispersion relation of the fundamental TM₀ mode. It is demonstrated that the variational method can well produce the dispersion relation of the TM₀ mode in the ‘hybrid plasmonic’ region but deviate significantly in ‘plasmonic only’ region if the trial wave function based on mode hybridization is used, which suggests that the mode hybridization idea should only be applied to the ‘hybrid plasmonic’ region.