Graphene-based hyperbolic metamaterial

We introduce a graphene-based composite multilayer structure that exhibits hyperbolic-like wavevector dispersion at terahertz and mid-infrared frequencies. The multilayer structure comprises graphene sheets separated by dielectric layers. We formulate the effective permittivity tensor of the multilayer using a simple homogenization scheme. In addition, we employ Bloch theory for evaluating the wavevector dispersion for a propagation mode inside the HM, and show that the homogenization scheme is in good agreement with Bloch theory in a very wide spatial spectrum. We report the tunability of transition from elliptic to hyperbolic iso-frequency wavevector dispersion by varying the chemical potential of graphene sheets.


INTRODUCTION
Hyperbolic metamaterials (HMs) are a subcategory of uniaxially anisotropic materials. In general, the isofrequency wavevector dispersion in an uniaxially anisotropic medium can be elliptic for some polarization (extraordinary waves), and under certain conditions it can evolve into a hyperbolic relation which leads to many interesting physical properties [1], [2]. In case of hyperbolic dispersion, ideally, an infinitely wide spatial spectrum of waves can propagate and carry power as opposed to a finite propagating spectrum in common media. However, realistic HMs allow for the propagation of a finite but still very wide spatial spectrum. For this reason, HMs can be employed in novel applications such as enhancing the local density of states [3], super absorbing near-fields at HM surfaces [4], [5], and hyperlensing [6]. In order to gain a physical insight into wave propagation, we represent a HM as an effective medium with a relative effective permittivity tensor xx yy zz where z is the axis of anisotropy.
Hyperbolic dispersion arises when 0 t z ε ε < [7]. A common HM design at optical frequencies is composed of metaldielectric multilayers with subwavelength thicknesses, that lead to 0 t ε < and 0 z ε > (the expressions of t ε and z ε can be found in [7]) in a wide frequency band thanks to the metals' negative permittivity [4].
In this paper we propose a graphene-based HM design for terahertz and low mid-infrared frequencies. In our design, graphene sheets play a similar role to metal layers in a metaldielectric multilayer HM at optical frequencies. Graphene supports plasmonic modes at upper GHz and lower THz range [8], [9] and it can be modeled as a low-loss inductive layer. Moreover, graphene has an extremely subwavelength lattice constant at infrared frequencies, thus spatial dispersion effects are not significant for practical conditions. The tunabilty of graphene's sheet conductivity via electrostatic and/or magnetostatic bias allows extensive control of the graphene response. Figure 1. Graphene-based multilayer HM topology, comprises graphene sheets as inductive surfaces stacked with dielectric spacers leading to hyperbolic dispersion at terahertz and mid-infrared frequencies.

II. GRAPHENE-BASED TUNABLE HYPERBOLIC METAMATERIAL
In this paper we investigate the graphene-based HM structure shown in Fig. 1(a) whose unit cell is composed of a graphene sheet and a dielectric layer with relative permittivity temperature. The multilayer metamaterial in Fig. 1 is modeled via effective medium approximation (EMA), with relative permittivity tensor eff ε By averaging the transverse current and electric field (assumed constant over a unit cell), the "transverse" permittivity t ε is expressed as On the other hand, z d ε ε = due to the continuity of normal displacement field and the assumption of infinitesimally thin graphene sheets. Straightforwardly, when the graphene sheets are sufficiently inductive, i.e., when the effective "transverse" permittivity t ε becomes negative. This gives rise to hyperbolic wavenumber dispersion for extraordinary ( TM z ) wave:  As the frequency increases, t ε′ approaches d ε (not shown here) because graphene conductivity reaches its universal value 2 / (2 ) e h π ∼ with a negligible reactive part. Note also that t ε ′′ exhibits smaller values as the chemical potential is increased ( Fig. 2(b)), thus electrostatic biasing also provides tunability of losses. It is important to assess the limits of EMA and characterization of propagating waves inside the graphene multilayer structure. With this aim, we evaluate the wavevector dispersion using the more accurate Bloch theory. Thus in Fig  (for a deeper discussion, the reader is referred to [4]). In the hyperbolic dispersion case (with c μ = 0.4 eV), z α in Fig. 3(b) is small for a wide spatial spectrum, in which waves can propagate without significant attenuation. In summary, we show that graphene sheets can be used for designing tunable HMs at far and mid-infrared frequencies.
The broad spatial spectrum enables the use of the proposed structure to efficiently absorb near fields generated at the HM surface at terahertz frequencies. Graphene-based HM can be utilized for enhancing the scattered power by particles inside or at its surface, and obtaining tunable decay rate of emitters.