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. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement