Universal behavior of a dispersive Dirac cone in gradient-index plasmonic metamaterials

Abstract

We demonstrate analytically and numerically that the dispersive Dirac cone emulating an epsilonnear-zero (ENZ) behavior is a universal property within a family of plasmonic crystals consisting of two-dimensional (2D) metals. Our starting point is a periodic array of 2D metallic sheets embedded in an inhomogeneous and anisotropic dielectric host that allows for propagation of transversemagnetic (TM) polarized waves. By invoking a systematic bifurcation argument for arbitrary dielectric profiles in one spatial dimension, we show how TM Bloch waves experience an effective dielectric function that averages out microscopic details of the host medium. The corresponding effective dispersion relation reduces to a Dirac cone when the conductivity of the metallic sheet and the period of the array satisfy a critical condition for ENZ behavior. Our analytical findings are in excellent agreement with numerical simulations.

Graphene epsilon-near-zero plasmonic crystals

“…where k z is the wavenumber along z direction. The above assumptions allow us to simplify the system of time dependent Maxwell equations to a one dimensional eigenvalue problem [15] with the Helmholtz type governing equation…”

“…The metallic 2D sheets carry a surface current J s = σ E that acts as a boundary. Maxwell equations dictate that on the boundaries that are defined by the metallic sheets at x = nd, the tangential electric field E must be continuous with discontinuous derivative with a jump discontinuity due to the surface currents on the metallic plates, hence the transmission conditions are [15]:…”

“…± denotes the limit above (+) and below (−) of a metallic boundary. We can get a closed system of equations by using the Bloch-wave ansatz [15,17] in the x direction with k x indicating the real Bloch number:…”

“…In Ref. [15] formula (5) √ ε x ) [15], that occurs at the center of the Brillouin zone [17]. It has been shown that the occurence of the plasmonic Dirac point is a universal behavior of 2D plasmonic crystals with any spatial dependence on the dielectric host along the x direction [15].…”

“…In particular, a plasmonic crystal consisting of periodically stacked graphene layers embedded in a dielectric host leads to negative refraction and ENZ behavior in the THZ and infrared frequencies, and allows to tune optical properties by controlling the operation frequency or the doping amount [17]. It has been furthermore shown that the ENZ condition coincides with a plasmonic Dirac cone in the wavenumber space, where the plasmonic Dirac cone is formed by two asymptotically linear, intersecting dispersion bands [9,13,15,17]. Such a graphene-based plasmonic crystal design, however, has the major shortcoming that ENZ behavior can only be observed for a very small frequency range.…”