## Abstract

The tunability of graphene-based hyperbolic metamaterial structure operating in SCLU telecom bands is investigated. For the first time it has been shown that for the proper design of a graphene/dielectric multilayer stack, the HMM Type I, Epsilon-Near-Zero and Type II regimes are possible by changing the biasing potential. Numerical results reveal the effect of structure parameters such as the thickness of the dielectric layer as well as a number of graphene sheets in a unit cell (i.e., dielectric/graphene bilayer) on the tunability range and shape of the dispersion characteristics (i.e., Type I/ENZ/Type II) in SCLU telecom bands. This kind of materials could offer a technological platform for novel devices having various applications in optical communications technology.

© 2016 Optical Society of America

## 1. Introduction

Hyperbolic metamaterial (HMM) represents a special class of uniaxially anisotropic metamaterials that can be described by a diagonal permittivity tensor (in Cartesian co-ordinates) with one or two the principal components having the negative value. Such anisotropic behavior results in an unusual hyperbolic dispersion relation for the medium and consequently many interesting phenomena such as hyperlensing [1], sub-wavelength imaging [2], nano-scale waveguiding [3,4], light enhancement [5] etc.

Hyperbolic media are classified as Type I materials that have only one component of the dielectric tensor negative and possess properties of conventional dielectrics, and Type II having two components of the dielectric tensor negative, which behave as metals [6–8]. The artificial HMMs are most commonly realized with two categories of structures such as: sub-wavelength metal-dielectric multilayers (1D-HMM) and metallic nanowires in a dielectric host [6–8].

Recent studies have shown a novel class of HMMs where metallic layers in 1D-HMM are substituted by graphene sheets (1D-GHMM) [9–22]. Graphene, as a two-dimensional, hexagonal lattice of carbon atoms exhibits a wide range of unique electrical and optical properties [23]. In particularly, unlike the metals, the conductivity of graphene can be tuned by shifting the Fermi level with an external electric field [24]. Thus, the great advantage of 1D-GHMM structure, in contrast to “conventional” HMM, is possibility of changing its optical properties in THz, infrared and optical frequency ranges by applying external electric field providing active control over HMMs dispersion characteristics [9,14].

Especially, the possibility of forming the dispersion surface shape for given HMM Type in the wavelength range 0.3 µm ÷ 1 µm has been shown in stack composed of metal/dielectric/TCO triplet layers [10]. Moreover it was reported that the graphene-dielectric composite material allows tunable and controllable transition from hyperbolic (Type II) to elliptic dispersion regime at far- and mid-infrared frequencies [15]. This transition allows to obtain the Epsilon-Near-Zero condition (ENZ) which implies a wide-angle tunable transparency of the graphene-based HMM in contrast to other materials exhibiting ENZ phenomena. A modification of the optical reflection properties of 1D-GHMM structure may have practical applications in tunable optical reflection modulator [13] and photonic switches [16,17] operating at THz frequencies. A controllable elliptic-hyperbolic transition may imply also sharp changes in the Purcell effect which can be enhanced due to increased photonic density of states in both THz and mid-IR frequency range [9,18]. 1D-GHMMs have also a rich potential for designing efficient and innovative tunable absorbers in far-infrared [11,19] and mid-infrared [20,21] frequency region. Recently, a GHMM dual-gated tunable absorber in the near-infrared frequency range has been reported [22].

In this paper we investigate a sub-wavelength graphene/dielectric multilayer stacks operating in SCLU telecom bands (1460 nm ÷ 1675 nm). For the first time it is shown that in this frequency range, by applying external electric field obtaining the HMM Type I, Type II as well as ENZ (ε_{||}~0, i.e. elliptic) regime, is possible. This kind of tunable structures can be applied for instance in optical communications technology (i.e., optical switches, memory devices, etc.) or near-infrared stealth communications. In order to obtain dispersion characteristics of 1D-GHMM structure, the Effective Medium Theory (EMT) and Transfer Matrix Method (TMM) are used. In particular, numerical simulations reveal the effect of refractive index and thickness of dielectric layers as well as a number of graphene sheets on the tunability range and shape of dispersion characteristics of analyzed structure (i.e., Type I/ENZ/Type II) in SCLU telecom bands. Thus, it has been shown that for proper design of graphene/dielectric stack, in contrast to previously discussed GHMM structures [15,16], both HMM types i.e. the Type I and Type II, as well as Epsilon-Near-Zero regime are achievable in the near-infrared range by changing biasing potential.

## 2. Model and theory

Uniaxial anisotropic metamaterial can be identified by the permittivity tensor:

*x*,

*y*-axes) and ⊥ (

*z*-axis) refer to the directions parallel and perpendicular to the interfaces in multilayer realization, respectively (Fig. 1). These uniaxial materials show hyperbolic dispersion if permittivity tensor components satisfy the condition: ${\epsilon}_{\parallel}\cdot {\epsilon}_{\perp}<0$. Under the subwavelength limit (i.e., as long as the thickness of unit cell of periodical multilayer HMM structure is less than the wavelength of light) the Effective Medium Theory can be applied to describe metamaterial structure (so called effective medium homogenization model) [6–8].

In our case, the unit cell of HMM material consists of graphene/dielectric bilayer as depicted in Fig. 1. The graphene sheets are separated by dielectric layers with thickness *t _{d}* and relative permittivity

*ɛ*. Applying the effective medium homogenization model the diagonal components of the relative permittivity tensor are approximated as follows [21]:

_{d}*t*

_{g}is the thickness of graphene layer and

*ε*

_{g}is the graphene’s permittivity. It is assumed that the thickness of dielectric layers are deep subwavelength but thick enough to avoid interaction between graphene layers [25,26]. According to Ref [21]. the graphene’s effective permittivity can be written as:where

*ɛ*is the vacuum permittivity and

_{0}*σ*is the conductivity of a single-layer graphene. Moreover, frequency and chemical potential dependent conductivity of a monolayer graphene can be given by Kubo formula [24]:

*f*is the Fermi-Dirac function:

_{D}(ξ)*f*(

_{D}*ξ*) = [exp(

*ξ*-

*μ*/

_{C}*k*) + 1]

_{B}T^{−1},

*μ*is the chemical potential,

_{c}*T*is temperature,

*k*and

_{B}*h*are Boltzmann and Planck’s constant,

*ω*is the angular frequency of the incident electromagnetic wave, and

*τ*is the phenomenological scattering rate, which we set equal 0.1 meV. It is worth noting that formula [Eq. (4)] is valid in wavelength range up to ~410 nm (i.e., approximately 3 eV). In case of few layer graphene its multilayer conductivity can be described by

*σ*=

_{ml}*σ*∙

*N*, where

_{g}*N*is the number of graphene layers. Considered model can be exploited for

_{g}*N*≤ 6 [16].

_{g}A change of graphene’s chemical potential can be achieved in several ways: chemical or molecular doping, and electrical or thermal stimulation [27]. It turns out that the most effective method of graphene’s conductivity modulation is applying gate voltage *V _{g}*. Relationship between gate voltage and chemical potential can be given by the following formula [16]:

*ħ*is Dirac constant,

*υ*is the Fermi velocity of Dirac fermions in graphene (∼10

_{F}^{6}m/s),

*a*= 9 × 10

_{0}^{16}m

^{−1}V

^{−1},

*V*is offset bias which reflects graphene’s doping and/or its impurities.

_{dirac}In our approach, we assumed that dielectric layer is characterized by constant relative permittivity *ε _{d}*. Since the conductivity of single layer graphene is frequency and chemical potential dependent, the effective tensor components of graphene/dielectric metamaterial structure (see [Eq. (1)]) also gain similar dependence. Thus, by controlling chemical potential of graphene it is possible to change optical properties of graphene-based HMM structure.

In this paper we demonstrate the possibility of shaping of dispersion characteristics of 1D-GHMM structure in SCLU bands by voltage biasing. We show that proper tailoring of parameters (i.e., thickness and refractive index of dielectric layer, number of graphene layers) for given 1D-GHMM structure enables to obtain Type I, ENZ or Type II regime for desired frequency within SCLU bands by proper external electric polarization.

## 3. Results and discussion

The results shown below are obtained by employing EMT model. The structure considered is a periodic stack consisting of *N _{g}* graphene monolayers (thickness of one graphene monolayer is 0.35 nm) and SiO

_{2}dielectric layers with thickness

*t*and

_{d}*ε*= 2.1025 (

_{d}*n*= 1.45). All acquired results were verified by TMM. The both approaches provide approximately the same results for GHMM structure consisted of at least 3 unit cells.

_{d}Figure 2 shows the curves representing both parallel (*ε _{||}*) and perpendicular (ε

_{⊥}) permittivity of graphene/dielectric multilayers versus the wavelength for chemical potential

*μ*= 0 eV (

_{c}*V*= 0 V), Figs. 2(a)-2(c) and

_{g}*μ*= 20 meV (

_{c}*V*~3 mV), Figs. 2(d)-2(f), and for various thicknesses of the dielectric layer

_{g}*t*. In all figures the thickness of graphene is controlled by the number of graphene sheets

_{d}*N*(

_{g}*N*= 1, 3, 6). As we can see, the effective permittivity

_{g}*ε*

_{⊥}exhibits resonant behavior and ε

_{||}almost keeps constant. An increase in the number of graphene layers (for given thickness of the dielectric layer) leads to the shift of the resonant wavelength of

*ε*

_{⊥}towards longer wavelengths. Moreover, the longer wavelength range of Type I hyperbolic dispersion (i.e. negative

*ε*

_{⊥}) is observed. We find, see Figs. 2(a)-2(c) and Figs. 2(d)-2(f), that with the increasing of the thickness of dielectric layer the wavelength range for Type I HMM regime is shifted towards shorter wavelengths covering SCLU telecom bands and simultaneously the wavelength range of Type I HMM is decreased.

The influence of chemical potential, changed by external electric field (described by gate voltage *V _{g}*), on the effective tensor components

*ε*

_{⊥}and ε

_{||}is shown in Figs. 2(d)-2(f). As we can notice, by applying the external electric field, the shift of Type I HMM regime towards shorter wavelengths is possible, whereas the narrower bandwidth for Type I hyperbolic regime is observed.

In Figs. 3(a)-3(d) we show the behavior of the permittivity tensor components *ε*_{⊥} and *ε*_{||} in SCLU telecom bands for different gate biasing (*V _{g}* = 0 V ÷ 2 V) and various unit cells (defined by number of graphene sheets and thickness of dielectric layer). In particular, for

*t*= 6 nm and

_{d}*N*= 1, see Fig. 3(a), the change of the biasing enables obtaining only Type I HMM in wavelength range 1590 nm ÷ 1680 nm (approximately corresponding to LU bands) and elliptic regime in the whole SCLU range. However, increasing the number of graphene sheets up to

_{g}*N*= 6 allows for obtaining Type I, elliptic including ENZ as well as Type II regimes in the whole SCLU bands in considered range of biasing, see Fig. 3(b). The situation is change when we increase the thickness of the dielectric layer. For

_{g}*N*= 1 and

_{g}*t*= 10 nm, see Fig. 3(c), we observe that by changing external polarization (in the same

_{d}*V*range) the elliptic region in the whole SCLU bands is possible. However, in this case Type I regime is obtainable in narrower spectrum range than for the structure having

_{g}*t*= 6 nm as well as Type II is not available. Moreover, the increase of the number of graphene sheets up to

_{d}*N*= 6, see Fig. 3(d), enables again achieving all considered dispersion regimes in the whole SCLU bands by change of biasing (

_{g}*V*= 0 V ÷ 2 V).

_{g}This effect is also observable in Figs. 4(a)-4(d) where the permittivity tensor components *ε*_{⊥} and *ε*_{||} are plotted as a function of the gate voltage *V _{g}* for the wavelength λ = 1.55 µm. Greater number of graphene sheets can provide not only broader Type I HMM, but also existence of Type II as well as ENZ regimes within 0 V ÷ 2 V biasing range (for both 6 and 10 nm dielectric layer thickness).

It is worth noting that for the structure having dielectric layer thickness *t _{d}* = 6 nm (

*N*= 6) the switching between Type I, elliptic also including Epsilon-Near-Zero and Type II could be obtained for smaller range of biasing than for the structure having

_{g}*t*= 10 nm (

_{d}*N*= 6).

_{g}This is illustrated in Figs. 5(a)-5(b), where the permittivity tensor components *ε*_{⊥} and *ε*_{||} are plotted as a function of the gate voltage *V _{g}* for the wavelength λ = 1.55 µm. However for structure having

*t*= 10 nm the broader frequency range of ENZ regime is observed.

_{d}## 4. Conclusions

In this paper we investigate 1D hyperbolic metamaterials in which metallic layers are substituted by graphene sheets (1D-GHMM). The great advantage of 1D-GHMM structure, in contrast to “conventional” HMM, is possibility of tuning its optical properties. For the first time it has been shown that for proper design of graphene/dielectric multilayer stack, the Type I, Epsilon-Near-Zero and Type II regimes in SCLU telecom bands are possible by changing biasing potential. Numerical results reveal the effect of structure parameters such as thickness of dielectric layer as well as a number of graphene sheets in unit cell (i.e., dielectric/graphene bilayer) on the tunability range and shape of dispersion characteristics (i.e., Type I/ENZ/Type II). We believe that this kind of structure offers foundation of new technological platform (compatible with CMOS technology) for novel class of photonic devices (i.e., optical switches, memory devices, etc.) having various applications in optical communications technology or near-infrared stealth communications.

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