Abstract
The dispersion relation of surface plasmon polaritons in graphene that includes optical losses is often obtained for complex wave vectors while the frequencies are assumed to be real. This approach, however, is not suitable for describing the temporal dynamics of optical excitations and the spectral properties of graphene. Here we propose an alternative approach that calculates the dispersion relation in the complex frequency and real wave vector space. This approach provides a clearer insight into the optical properties of a graphene layer and allows us to find the surface plasmon modes of a graphene sheet in the full frequency range, thus removing the earlier reported limitation $(1.667<\ensuremath{\hbar}\ensuremath{\omega}/\ensuremath{\mu}<2)$ for the transverse-electric mode. We further develop a simple analytic approximation which accurately describes the dispersion of the surface plasmon polariton modes in graphene. Using this approximation, we show that transverse-electric surface plasmon polaritons propagate along the graphene sheet without losses even at finite temperature.
Highlights
Surface plasmon polaritons (SPPs) are collective excitations of charge density coupled to electromagnetic waves that can travel along a conductor-dielectric interface [1]
The existence of each type of SPP mode can be confirmed by solving a dispersion relation between the frequency and the wave vector following from Maxwell’s equations and the conductivity model
We focus on the dispersion of SPP modes in a graphene layer with finite temperature and nonzero chemical potential and show that the complex-frequency analysis developed in this work removes both the upper and the lower limits for the TE SPP mode in graphene
Summary
Surface plasmon polaritons (SPPs) are collective excitations of charge density coupled to electromagnetic waves that can travel along a conductor-dielectric interface [1]. We focus on the dispersion of SPP modes in a graphene layer with finite temperature and nonzero chemical potential and show that the complex-frequency analysis developed in this work removes both the upper and the lower limits for the TE SPP mode in graphene While it is well-known how to calculate the dispersion relations of SPPs in graphene for given optical conductivities [12], this has been done assuming that any SPP mode has a real frequency but complex wave number q = q + iq [12,16,17]. Appendices A–E provide details on derivations of the optical conductivity of graphene and secular equations for the TM and TE SPP modes, and supply an additional material on our study of the TE mode near the lower threshold frequency and on the SPP dispersion at a finite damping
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