## Abstract

A tunable graphene-based hyperbolic metamaterial is designed and numerically investigated in the mid-infrared frequencies. Theoretical analysis proves that by adjusting the chemical potential of graphene from 0.2 eV to 0.8 eV, the reflectance can be blue-shifted up to 2.3 µm. Furthermore, by modifying the number of graphene monolayers in the hyperbolic metamaterial stack, we are able to shift the plasmonic resonance up to 3.6 µm. Elliptic and type II hyperbolic dispersions are shown for three considered structures. Importantly, a blue/red-shift and switching of the reflectance are reported at different incident angles in TE/TM modes. The obtained results clearly show that graphene-based hyperbolic metamaterials with reversibly controlled tunability may be used in the next generation of nonlinear tunable and reversibly switchable devices operating in the mid-IR range.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fascinating effects occurring in artificial electromagnetic metamaterials have inspired scientists to create modern photonic devices, including switches, filters, absorbers, and modulators [1–7]. Indeed, subwavelength-structured media allow researchers to meet the restrictive criteria of advanced multifunctional systems. It is related to the metasurfaces being superior to naturally occurring materials due to variations made at a limited scale that allow manipulation of any incoming signal that comes in to contact. Recently, the properties of plasmonic materials have been broadened by the development of hyperbolic metamaterials (HMMs), which are a new class of uniaxial anisotropic metamaterials that have hyperbolic (or indefinite) dispersion [8–14]. The origin of their unusual electromagnetic behavior is associated with the hyperboloidal isofrequency surface of extraordinary (transverse magnetic polarized) waves, which is given by $\frac{{k_x^2 + k_y^2}}{{{\varepsilon _ \bot }}} + \frac{{k_z^2}}{{{\varepsilon _\textrm{||}}}} = {\left( {\frac{\omega }{c}} \right)^2}$, where *k _{x}*,

*k*and

_{y}*k*are, respectively, the

_{z}*x*,

*y*and

*z*components of the wave vector,

*ω*is the wave frequency and

*c*is the speed of light. The hyperbolic dispersion occurs when different “entrees” of the dielectric permittivity tensor have opposite signs. We distinguish hyperbolic dispersion of type I when a permittivity tensor, out the plane (z), assumes a negative value $({{\varepsilon_\parallel } > 0;{\varepsilon_ \bot } < 0} )$ and type II if the two permittivities’ tensors, in the plane (x-y), possess a negative sign $\left( {{\varepsilon_\parallel } < {0;{\varepsilon_ \bot }} > 0} \right)$ [15–21]. In particular, hyperbolic metastructures with reversibly switchable plasmonic properties are inspiring because of their potential applications in light enhancement, subwavelength imaging, and spontaneous emission [8–14].

The requirements for efficient, low-voltage, and multifunctional infrared modulation devices have pushed these researchers into exploring new materials. Despite this ongoing research, the tuning efficiencies and high power consumption of tunable infrared devices remain major limitations. Therefore, over the past few years, there has been considerable interest in using the conductivity property of graphene to produce tunable infrared components.

Graphene can be exploited in the creation of switchable devices owing to its voltage-dependent sheet conductivity, which can be repeatedly tuned by adjusting the Fermi level via the chemical potential, which thereby allows for ultrafast electrical modulation [22]. These exotic properties have resulted in a significant increase in recent works devoted to studying graphene as a crucial component of metamaterials. For example, Linder and Halterman investigated the absorption properties of graphene-based anisotropic metamaterial structures where the metamaterial layer possesses an electromagnetic response corresponding to a near-zero permittivity [23]. Rufangura and Sabah proposed a graphene-based wideband metamaterial absorber for solar cell application [24]. Zhang et al. designed a dual-band absorber based on graphene formed by combining two cross-shaped metallic resonators of different sizes within a super-unit-cell arranged in mirror symmetry [25]. Liu et al. considered highly tunable hybrid metamaterials employing split-ring resonators strongly coupled to graphene surface plasmons [26]. Meng et al. numerically investigated a graphene plasmonic narrowband perfect absorber based on periodically patterned H-shaped graphene arrays separated from a metallic ground plate by a thick dielectric spacer [27]. The aforementioned works show that graphene has found a wide range of applications in metamaterial structures as an active element that allows for effective switching of their plasmonic resonance [28–30]. However, most of the studies are devoted to the use of graphene in planar metamaterial structures. I. Khromova et al. and M. A. K. Othman et al. demonstrated that using the graphene’s capability to be tuned by control of its conductivity is possible even in stacked structures, which allows construction of waveguide modulators [31], beam steering devices [32], and tunable enhanced near-field absorption [33]. Since publication of their studies, many works on graphene-based hyperbolic metamaterials have been demonstrated [34–41]. Notwithstanding, the development of high-performance and efficient modulators employing the combination of hyperbolic metamaterials with graphene remains a challenge. Furthermore, methods reported so far are insufficient and in most cases focus mainly on showing the property, not designing an effective optoelectronic device, which significantly limits their applications. For this reason, progress in the development of effective, optimized, switchable HMM-based devices employing graphene is still a necessity.

Here, we demonstrate a switchable reflection modulator operating in mid-IR frequencies using hyperbolic metamaterial based on graphene. By modulating its conductivity (via tuning of the chemical potential) or changing the number of graphene monolayers, a modulation of the elliptic/type II topological transition is obtained. Thus, we designed three stacks containing one, three, and six graphene monolayer. The electrical tunability is reported by means of modification of the chemical potential. In addition, the influence of the incidence angle on the reflectance is investigated to obtain a sharp edge filter. Furthermore, an active behavior of the reflection modulator, type II hyperbolic dispersion, and elliptic dispersion has been verified by means of dispersion diagrams for specific TE/TM propagation modes.

## 2. Theoretical model

The finite difference time domain (FDTD) method implemented in Lumerical FDTD Solutions software has been adopted to simulate the active response of a graphene-based reflection modulator. In addition, to ensure the stability and accuracy of the computational algorithm, the Bloch boundary condition, as well as the smallest possible spatial grid size of 1 nm was chosen. The proposed structure is illustrated in Fig. 1. In our approach, the designed device is based on alternating subwavelength layers of graphene and dielectric, which are described by thicknesses *t _{g}*,

*t*, and permittivities ɛ

_{d}_{g}, ɛ

_{d}, respectively. Both thicknesses have been selected in such way that hyperbolic dispersion may occur. It is noteworthy that the thickness of a single graphene monolayer (1MG) was set in the numerical simulations as

*t*= 0.35 nm.

_{g}In our study, the anisotropy of HMM at a subwavelength scale is described on the basis of diagonal components of the permittivity tensor: $\bar{\bar{\varepsilon }} = [{{\varepsilon_{xx}},{\varepsilon_{yy}},{\varepsilon_{zz}}} ]$, where their values are: ${\varepsilon _{xx}} = {\varepsilon _{yy}} = {\varepsilon _\parallel }$, and ${\varepsilon _{zz}} = {\varepsilon _ \bot }$, and they can be determined employing effective medium theory [33,41,42]:

Without taking into account the external magnetic field, the isotropic surface conductivity *σ* of graphene is given by the Kubo formula and can be calculated as the sum of the intraband term *σ _{intra}* and the interband term

*σ*, as discussed in [44]:

_{inter}*T*is the temperature,

*e*is the electron charge, $\hbar $ is the reduced Plank constant, ${k_B}$ is the Boltzmann constant, and ${f_d}(\xi )$ is the Fermi–Dirac distribution written as follows: which gives the probability that a given available electron energy state will be occupied at a given temperature. The value of the Fermi–Dirac distribution function is directly related to the magnitude of the chemical potential (${\mu _c}$), which is of the order of electron volts (eV).

The formula describing the relation between chemical potential and applied voltage (${V_g}$) can be written as follows:

where ${v_F}$ is the Fermi velocity in graphene (∼10^{6}m/s), ${a_0}$ = 9·10

^{16}m

^{−1}V

^{−1}is an empirical constant and ${V_D}$ is the offset bias voltage, which in our study was supposed to be 0 V.

As shown in Eqs. (4)–(7), the graphene surface conductivity is modeled in such a way that it is valid for one graphene monolayer (1MG). However, by scaling the total conductivity by the number of graphene monolayers, the aforementioned model can be used to characterize the graphene-based stacks’ structures with more than one graphene monolayer in the unit cell. Hence, considering the Cartesian coordinate systems shown in Fig. 1, we have used a light with transverse magnetic polarization (with respect to *z*) that strikes the hyperbolic metastructure along the z-direction with the following space–time dependence of fields: $A\exp ({i{k_x}x + i{k_y}y + i{k_z}z - i\omega t} )$.

## 3. Results and discussion

To optimize the performance of a switchable reflection modulator, we first characterized the variation in the reflectance as a function of graphene monolayers (MG) and chemical potential (*µ _{c}*), as shown in Figs. 2(a) and 2(b). In Fig. 2(a), when the number of graphene monolayers (MG) rises from one (1MG) to six (6MG), which corresponds to the increase in the thickness of the graphene whole layer from 0.35 to 2.1 nm, we can observe that the most efficient edge filter characteristic is obtained for six monolayers (6MG). In addition, the reflectance can be reversibly regulated up to 3.6 µm. The frequency shift with respect to the number of graphene sheets is related to the fact, that in our numerical calculations we assume wave propagation that accumulates linear phase inside multilayer graphene. However, so far this is an open issue that requires experimental verification. Also, the increase in the thickness of the graphene results in a change of the period of the unit cell (which is defined in this case as

*t = t*) from 100.35 nm to 102.1 nm. Therefore, by adjusting the number of graphene monolayers, it is possible to effectively control and switch the plasmonic resonance of graphene based hyperbolic metamaterial. It should be noted that these results were obtained for constant values of the chemical potential (

_{g}+ t_{d}*µ*= 0.8 eV for

_{c }*V*= 5 V), number of unit cells (

_{g}*N*= 20) and thickness of the SiO

_{2}layers, which is

*t*= 100 nm. In addition, the resonance can be self-regulated by the voltage-dependent chemical potential, as illustrated in Fig. 2(b). When the chemical potential gradually increases from

_{d}*µ*= 0.2 eV (

_{c }*V*= 0.3 V) to

_{g}*µ*= 0.8 eV (

_{c }*V*= 5 V), the reflectance can be blue-shifted up to 2.3 µm. Note that this process is completely reversible, i.e. when the chemical potential decreases from

_{g}*µ*= 0.8 eV to

_{c }*µ*= 0.2 eV, reflectance is red-shifted up to 2.3 µm. Hence, the type II hyperbolic dispersion can be self-regulated by modifying the number of graphene monolayers as well as by means of the chemical potential.

_{c }In our approach, a graphene-based switchable reflection modulator’s working principle is connected with the transition from elliptic to type II hyperbolic dispersion. Therefore, transitions between elliptic dispersion $({{\varepsilon_\parallel } > and\; {\varepsilon_ \bot } > 0} )\; $ and type II hyperbolic dispersion $\left( {{\varepsilon_\parallel } < {0\; and{\varepsilon_ \bot }} > 0} \right)$ were examined. The real part of the permittivity tensor components as a function of wavelength $({{\varepsilon_\parallel }(\lambda ),{\varepsilon_ \bot }(\lambda )} )$ was determined for different monolayers of graphene, as presented in Fig. 3(a). Note that both chemical potential and dielectric thickness values are constant and are set to be *µ _{c}* = 0.8 eV and

*t*= 100 nm, respectively. Considering the resonance frequencies (i.e., wavelengths for which individual components of the effective diagonal tensor are equal to zero), we are able to determine transition wavelengths, which occur at ${{\lambda }_1}\; $ = 3.8 µm, ${{\lambda }_2}$ = 5.5 µm, and ${{\lambda }_3}$ = 7.3 µm, from elliptic to type II hyperbolic dispersion in the case of 6MG, 3MG, and 1MG, respectively. Note that these wavelengths can be reversibly controlled by a voltage-sensitive chemical potential. Furthermore, transition from high transmission to high reflectance coincides with the change of the isofrequency dispersion regime from elliptic to type II hyperbolic, as depicted in Fig. 3(b).

_{d}Figures 4(a)–4(f) illustrate the angular reflectance characteristics for transverse magnetic (TM) and transverse electric (TE) modes—depending on whether ** E** or

**are considered transverse to the propagation direction—for a metastructure composed of one (1MG), three (3MG), and six (6MG) graphene monolayers. It can be seen in the case of both analyzed structures that the reflectance is more dependent on the incidence angle in case of the TE mode than in the case of the TM mode. This is mainly due to the surface plasmons excited in graphene sheet supporting the transverse magnetic mode (TM).**

*H*The stack structure built based on 1MG behaves completely differently in the case of the TE and TM modes, as evident from Figs. 4(a) and 4(b). In this case, for the TM mode, reflectance takes values below 0.8, which means that we can treat it as transmissive in the analyzed bandwidth. For the TE mode, when the angle of incidence increases from θ = 0° to θ = 60°, the reflectance is blue-shifted by 1.1 µm. Furthermore, the reflectance is more than 80%; therefore, the 1MG stack can be considered a reflective medium in the range of 6.5–8 µm. When the graphene’s thickness increases to 1.05 nm (3MG), we get sharper reflectance characteristics for both the TE and TM modes, as shown in Figs. 4(c) and 4(d). At the same time, the more desirable characteristic is in the case of the TE mode because the reflectance rapidly increases from 4 µm, so that a wavelength of 5.5 µm reaches the maximum value for all incidence angles. In addition, for the TE mode, the blueshift is clearer than for the TM mode, and reaches up to 0.7 µm. Figure 4(e) shows that in the case of the TE mode for a structure made of six graphene monolayers (6MG), when the angle of incidence light increases from θ = 0° to θ = 60°, the reflectance is blue-shifted by 0.5 µm, while in the TM mode, the metastructure seems to behave almost the same for the whole considered angle’s range, as shown in Fig. 4(f). Hence, created hyperbolic metamaterial may work as a selective angled reflector and a polarization dependency reflector.

To further examine the structure and verify the results presented in Figs. 4(a)–4(f), we performed additional numerical simulations to obtain spatial reflectance distributions as a function of incident light (θ) vs. wavelength (λ) for 1MG, 3MG, and 6MG for both the TE and TM modes. The results are presented in Figs. 5(a)–5(f). Please note that the value of the chemical potential is constant and amounts to *µ _{c}* = 0.8 eV. In Fig. 5(a), we see that transition from low to high reflectance takes place at a wavelength of 6.8 µm, and falls again above 9 µm. Moreover, with a larger angle, the band (red area) where the reflectance is high widens. In the wavelength range from 1 µm to 5.5 µm, the stack built on the basis of 1MG, in the case of the TE mode, has transmission properties (blue area). Thus, the spatial distribution visible in Fig. 5(a) perfectly reproduces and confirms the results shown in Fig. 4(a). The same situation occurs if we combine subsequent spatial reflectance distributions with their equivalents in the form of charts obtained for given structures and a given working mode. For instance, in Fig. 5(b), the stack behaves as a transmissive medium up to 6.3 µm, then reflectance gradually increases up to 9 µm. However, for an angle θ = 60°, its decrease is visible for a wavelength of about 8 µm. This is exactly what we see in Fig. 4(b). If we triple the thickness of the graphene layer (3MG), we can see that the area with high transmission is blue-shifted for both the TE and TM modes [see Figs. 5(c) and 5(d)]. In addition, the border between areas with high transmission (blue) and high reflectance (red) becomes thinner. This is because we obtain a sharp low-pass filter characteristic. These effects are most visible for the structure built on the basis of 6MG, as illustrated in Figs. 5(e) and 5(f). Clearly, such properties are characterized by high light transmission (up to 3 µm) and a reflectance greater than 90% (from 3.5 µm to 9 µm), mostly regardless of the wave incidence angle. It is also worth noting that a designed device can be the perfect transmissive medium for a wavelength up to 3 µm, for both the TE and TM modes.

## 4. Summary

The tuning property of graphene was employed for creating an active hyperbolic metamaterial, which can operate as a switchable reflection modulator in mid-IR frequencies. Numerical simulations indicate that reflectance may be blue-shifted/red-shifted in the whole considered range (2–8 µm) by a two-fold approach, namely by increasing the number of graphene monolayers and via changing the chemical potential controlled by tuning of the external voltage. A further blue-shift in reflectance up to 0.5 µm was observed in the case of the 6MG stack’s metastructure in the TE mode for incident light at 60°, while a negligible dependence on the incident light angle was observed for the TM mode. This class of metamaterials is, therefore, dependent on incident light angle/polarization, which is a key point in applications such as angular-selective and polarization-sensitive modulators. This kind of tunable modulator may exhibit elliptic and type II hyperbolic dispersion for both the TE and TM modes, which makes graphene-based hyperbolic metamaterials one of the best candidates as nonlinear reconfigurable optical elements.

Angular reflectance characteristics for both the TE and TM modes are in great agreement with spatial reflectance distributions as a function of incident light (θ) vs. wavelength (λ), which allows us to predict an active behavior of a created graphene-based reflectance modulator. To conclude, the observed features of graphene-based hyperbolic metamaterial as a switchable reflection modulator can find a wide range of potential applications in active optoelectronic systems as an effective edge or narrow-band filter in the mid-IR range [45–48]. Importantly, HMMs support propagating high-k modes and are characterized by enhanced photonic density of states. Therefore, HMMs can be employed to create hyperlenses for far-field super-resolution imaging, meta-cavity lasers, and antennas for second-harmonic generation tomography [49].

## Funding

National Centre for Research and Development (TECHMATSTRATEG1/347012/3/NCBR/2017).

## Acknowledgments

The authors would like to thank Prof. Filippo Capolino from the University of California for fruitful discussion and many valuable comments. This work has been supported by The National Centre for Research and Development under grant No. TECHMATSTRATEG1/347012/3/NCBR/2017 (HYPERMAT) in the course of "Novel technologies of advanced materials - TECHMATSTRATEG."

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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