Surface plasmon polaritons at linearly graded semiconductor interfaces

New results are reported on investigation of dispersion curves for surface plasmon polaritons (SPPs) at an inhomogenously doped semiconductor/dielectric interface whereby the dielectric is represented by the same undoped semiconductor. The doped semiconductor is described by its frequency-dependent permittivity that varies with the depth. It is shown that a transition layer (TL) with a linear change in carrier concentration supports one branch dispersion curve regardless of the TL thickness. The obtained dispersion curves reach a maximum at a finite frequency depending on the TL thickness, and subsequently asymptotically approach the zero frequency in the shortwave limit. Therefore two surface plasmon modes are supported at a given frequency: a long-wave mode with a positive group velocity and a short-wave mode with a negative group velocity. A condition of a zero group velocity can be satisfied by tuning the TL layer. It is shown that the conventional dispersion relation for SPPs at a TL with a zero thickness is an asymptotic solution, and the convergence of real dispersion curves is point-wise instead of an expected uniform convergence.