Quasi-Periodic Hyperbolic Metamaterials Composed of Graphene and Dielectric

Abstract

The hyperbolic properties are demonstrated in Cantor photonic multilayers, which exhibit quasi-periodicity and consist of graphene and dielectric layers, forming a centrosymmetric structure. Different from the traditionally periodic hyperbolic metamaterials, here we introduce the concept of layer weight in quasi-periodic photonic multilayers; the equivalent permittivity tensor presents a hyperbolic dispersion with a specific weight of dielectric thickness. Subsequently, the dispersion relation transforming from hyperbolicity to ellipticity has been explored. The tunability of the hyperbolic properties can be enhanced by varying the Fermi energy of the graphene, the thickness of the dielectric, the relaxation time of graphene, and the number of graphene monolayers. The reflectance and transmittance of light waves in the quasi-periodic hyperbolic metamaterials depend on the hyperbolically dispersive speciality strongly as well. This research provides theoretical guidance for the design and optimization of photonic devices based on hyperbolic metamaterials.

1. Introduction

Unlike naturally occurring materials, metamaterials [1] are a type of macroscopic functional material formed by arranging artificial sub-wavelength structural units in a regular pattern. These units typically range in size from 10 nm to 100 nm, with critical dimensions much smaller than the operating wavelength. By altering the shape, size, arrangement, and material composition of these units, the dielectric constant and magnetic permeability of metamaterials can be adjusted over a broad range, thereby enabling special optical properties hard to achieve with conventional materials and unconventional control of the light field. Hyperbolic Metamaterials (HMMs) are a class of anisotropic metamaterials that exhibit unique electromagnetic properties. In the permittivity or permeability tensor of HMMs, one principal component has an opposite sign compared to the other two principal components, leading to hyperbolic isofrequency contours [2,3]. HMMs are typically fabricated using multilayer film structures or nanowire array structures [4,5]. These materials find applications in various fields, including fundamental research [6,7,8], biological imaging [9,10], optical communication [11,12], and quantum optics [2]. Specifically, the metal–dielectric multilayer HMM can function as optical superlenses by precisely tailoring the optical field [13]. It converts the scattered evanescent waves into propagating waves within the anisotropic medium, thereby magnifying sub-wavelength objects and projecting high-resolution images in the far field. Researchers have also conducted a systematic study on the spectral characteristics of periodically arranged nanostructured HMM absorbers [14]. Simultaneously, they proposed a nanoscale HMM-based prism-coupled waveguide sensor (PCWS) operating in the near-infrared band [15]. Additionally, optical biosensors based on HMMs have been constructed to explore their potential for early cancer cell detection [16]. Another study further proposed a polarization converter assisted by a centrally split S-shaped anisotropic reflective superstructure [17].

In recent years, graphene, as a special two-dimensional material, has received extensive attention due to its excellent electrical and optical properties. The surface conductivity of graphene can be flexibly adjusted by means of external gate voltages, electromagnetic fields, or chemical doping [18,19]. This tunability allows graphene to exhibit excellent optical response characteristics in the near-infrared to terahertz frequency bands [20,21,22,23,24,25,26]. By mixing or stacking graphene with common dielectrics, hyperbolic metamaterials with tunable permittivity can be constructed, and the structure is also called graphene-based hyperbolic metamaterials (GHMMs). This kind of material can provide new ideas for the design of novel functional devices [27,28,29,30]. In the field of metamaterial research, the exploration of graphene-based structures has continuously deepened. Ivan V. Iorsh et al. proposed metamaterials based on the periodic multilayer graphene structure and demonstrated that the tuning from elliptical dispersion to hyperbolic dispersion can be achieved through an external gate voltage in the Terahertz (THz) frequency [29]. Tavana S. et al. proposed a novel structure that sandwiches GHMM between two one-dimensional periodic photonic crystals. Research indicates that this structure can be utilized to fabricate terahertz devices, with its performance tunable by varying the incident angle and the chemical potential of graphene [30]. Previous studies focused on bandgap formation in Cantor multilayers [31,32,33], whereas this work emphasizes tunable hyperbolic dispersion in graphene-based metamaterials. Different from the traditionally periodic hyperbolic metamaterials, here we achieved the hyperbolic characteristics in the Cantor quasi-periodic structure composed of graphene and dielectric. The effective permittivity tensor was calculated through the specific weight distribution of the dielectric thickness. Furthermore, the tunability of the hyperbolic properties can be enhanced by varying the Fermi energy of the graphene, the thickness of the dielectric, and the number of graphene monolayers. Additionally, the impact of hyperbolic dispersion on the reflectance and transmittance of the light beam is characterized. This effect can provide theoretical guidance for the design of the optical switch.

2. Theoretical Model and Method

The structure composed of graphene and dielectric is illustrated in Figure 1. The photonic multilayers are arranged according to the Cantor sequence [34,35], which is a typical quasi-periodic sequence. The quasi-periodic sequence is a special type of non-periodic sequence, but it has stricter long-range order than completely random or disordered non-periodic sequences. Mathematically, the iterative rule of the Cantor sequence is defined as follows: S₀ = A, S₁ = ABA, S₂ = S₁(BBB)S₁, ……, S_N = S_N−1(BBB)^N−1S_N−1, ……, where (BBB)^N−1 represents 3^N−1 instances of B; and N = 2, 3, 4, … denotes the sequence number. Here, A and B represent the graphene monolayer and dielectric, respectively. BaF₂ is selected as the dielectric. BaF2 is a commonly used material in laser optics and is regarded as a lossless dielectric [35]. In the figure, the Cantor sequence number is N = 2, leading to the formation of the structure ABA(BBB)ABA, which exhibits centrosymmetry. The transverse magnetic (TM) wave is incident on the graphene layer along the Z-axis. Specifically, ε_g and ε_d denote the permittivity of the graphene monolayer and dielectric, respectively, while t_g and t_d represent their respective thicknesses.

The surface conductivity of graphene (σ = σ_intra + σ_inter) can be obtained using the Kubo formula [36]. Among them, the intraband conductivity σ_intra and the interband conductivity σ_inter are expressed as follows:

$${\sigma }_{\mathrm{intra}}={\displaystyle \frac{i{e}^2{K}_{B}T}{\pi {\hslash }^2(\omega +i/\tau )}}\left[{\displaystyle \frac{E_F}{K_{B}T}}+2\ln (1+e^{-{\displaystyle \frac{E_F}{K_{B}T}}})\right],$$

(1)

$${\sigma }_{\mathrm{int}\mathrm{er}}={\displaystyle \frac{i{e}^2}{4\pi {\hslash }^2}}\ln \left|{\displaystyle \frac{2E_F-\hslash (\omega +i/\tau )}{2E_F+\hslash (\omega +i/\tau )}}\right|,$$

(2)

where e, $\hslash$, and K_B are the electronic charge, the reduced Planck constant, and the Boltzmann constant, respectively. E_F, τ, and T are the Fermi energy of graphene, the relaxation time, and the ambient temperature, respectively. ω represents the angular frequency of the incident light.

The effective dielectric constant of the graphene [37] can be expressed as

$${\epsilon }_g=1+{\displaystyle \frac{i\sigma }{t_g\omega {\epsilon }_0}},$$

(3)

where ε₀ represents the vacuum permittivity. Since the optical wavelength λ in the near-infrared band satisfies λ >> t_g + t_d, the effective medium theory can be adopted to describe the wave propagation based on graphene [38]. The permittivity tensor of the structure takes a diagonalized form and is expressed as [ε] = [ε_xx, ε_yy, ε_zz], where ε_xx = ε_yy = ε_‖ and ε_zz = ε_⟂. Here, ε_‖ and ε_⟂ represent the parallel and vertical components of the relative permittivity concerning the graphene layers, respectively. These components are formulated as follows:

$${\epsilon }_{xx}={\epsilon }_{yy}={\displaystyle \frac{t_{g1}{\epsilon }_g+t_{d1}{\epsilon }_d}{t_{g1}+t_{d1}}},$$

(4)

$${\epsilon }_{zz}={\displaystyle \frac{{\epsilon }_d{\epsilon }_g(t_{g1}+t_{d1})}{{\epsilon }_d{t}_{g1}+t_{d1}{\epsilon }_g}},$$

(5)

where t_g1 = T_gt_g and t_d1 = T_dt_d. T_g and T_d correspond to the proportional ratios of graphene and dielectric layers to the total layer count, respectively. For N = 2, the structure of Figure 1 consists of nine layers, including four graphene layers and five dielectric layers. The proportional coefficients are T_g = 4/9 and T_d = 5/9, respectively. Based on this, the equivalent thickness of the entire multilayer structure can be derived.

For higher orders (N > 2), the counts of graphene and dielectric layers are determined by the Cantor sequence S_N = S_N−1(BBB)^N−1S_N−1. Correspondingly, the coefficients T_g and T_d are derived. This framework ultimately enables the systematic analysis of hyperbolic dispersion behavior. Additionally, the dielectric constant formula obtained can also be normalized to the periodic structure. This method enhances the robustness of the structure design—even if the local arrangement deviates from the ideal quasi-periodic sequence due to fabrication errors, the overall performance can still be stabilized. When the size of the basic unit is less than one-tenth of the wavelength, this model can be used to perform equivalent modeling of the structure. This model can also be applied to other metal–dielectric structures.

For the propagation of TM-polarized light, the dispersion surface can be expressed as

$${\displaystyle \frac{{k_x}^2}{{\epsilon }_{zz}}}+{\displaystyle \frac{{k_z}^2}{{\epsilon }_{xx}}}={k_0}^2.$$

(6)

k_x and k_z denote the wave vectors along the X-axis and Z-axis, respectively, while k₀ represents the wave vector in free space. When ε_xx ∙ ε_zz < 0, the multilayer structure exhibits hyperbolic dispersion. Conversely, when ε_xx ∙ ε_zz > 0, the multilayer structure demonstrates elliptical dispersion.

For a multilayer structure, the transfer matrix method [39] can be used to analyze the reflection coefficient r and transmission coefficient t, and the matrix elements are functions of the permittivity of the structure. For a specific layer l, when an electromagnetic wave is incident on the structure, the transfer matrix can be expressed as

$$M_l=\left[\begin{array}{cc}\cos {\delta }_l& -{\displaystyle \frac{i}{{\eta }_l}}\sin {\delta }_l\\ -i{\eta }_l\sin {\delta }_l& \cos {\delta }_l\end{array}\right],$$

(7)

where ${\delta }_l=2\pi n_l{t}_l\cos \theta /\lambda$, (l = d,g). η_d and η_g are the effective optical admittances in the dielectric and graphene, respectively. The incident light is denoted as λ, while n_d and n_g are the refractive indices of the dielectric and graphene, respectively. These refractive indices can be derived from ${\mathrm{n}}_d=\sqrt{{\epsilon }_{\mathrm{d}}}$ and $n_g=\sqrt{{\epsilon }_g}$, respectively. The structure consists of nine layers in total in Figure 1. The transfer matrix of the multilayer system is obtained by sequentially multiplying the transfer matrices of each individual layer, described as follows:

$$M={\displaystyle \prod _{l=1}^9{M}_l}=\left[\begin{array}{cc}{A}_o& B_o\\ C_o& D_o\end{array}\right].$$

(8)

Then, the reflection coefficient of the structure can be calculated as

$$r={\displaystyle \frac{A_o{\eta }_0+B_o{\eta }_0{\eta }_{2N+1}-C_o-D_o{\eta }_{2N+1}}{A_o{\eta }_0+B_o{\eta }_0{\eta }_{2N+1}+C_o+D_o{\eta }_{2N+1}}}.$$

(9)

This reflection coefficient can be further decomposed into r = |r|exp(iΦ_r), where |r| represents the amplitude, and Φ_r denotes the complex angle. The reflectance and transmittance of the structure are then determined by R = |r|² and T = |t|².

3. Numerical Results and Discussions

By adjusting the Fermi energy of graphene through electric fields or chemical doping, the carrier concentration in graphene can be significantly increased at terahertz frequencies. Under such conditions, electron energy transitions predominantly occur via “in-band absorption”, and the electron transport behavior becomes more analogous to that of “free electrons”, aligning with the characteristics described by the Drude model [40]. This variation in electron behavior leads to changes in the electrical conductivity of graphene, which subsequently affects the dielectric constant of the structure and further influences its dispersion characteristics.

Figure 2a,b, respectively, represent the changes in the real and imaginary parts of the graphene conductivity in the parameter space composed of the Fermi energy and frequency. In the calculations, the temperature is set as T = 300 K, and the thickness of the dielectric BaF₂ is t_d = 20 nm, and its permittivity is ε_d = 2.2. The number of graphene monolayers in a single graphene layer is N_g = 1. The thickness of the graphene monolayer is t_g = 0.5 nm, and the relaxation time is set as τ = 1 ps in order to take into account the scattering loss from the acoustic phonons [40,41]. The Fermi energy E_F varies from 0.4 eV to 1.0 eV, and the frequency f of the light arises from 10 THz to 100 THz, corresponding to a wavelength range of 3–30 μm. The dimensions of the structure in this manuscript are much smaller than the working wavelength, fully meeting the requirements of the effective medium theory (EMT) for sub-wavelength scales. It can be seen that at low frequencies, the real and imaginary parts of the conductivity increase with the increase in Fermi energy, with intraband transitions dominating, and the conductivity follows a Drude response. Graphene is lossy. At high frequencies, the values of the real and imaginary parts of the conductivity are very small and change slowly with the increase in frequency, indicating that graphene has less loss.

Figure 3 shows the permittivity components in the parameter space composed of the Fermi energy and the frequency. It is assumed that the TM-polarized light is normally incident from the air. Figure 3a represents the real part of the vertical components ε_zz of the relative permittivity. The real part of ε_zz hardly varies with the frequency, and its value is always positive. Figure 3b shows the real part of the vertical components ε_xx of the relative permittivity. The imaginary part of ε_zz is very small at low frequencies, which indicates the low loss at this region.

Figure 3c shows the real part of the parallel components ε_xx of the relative permittivity, and the solid line represents the contour line with a value of 0. In the low-frequency range, the real part of ε_xx is negative, and ε_xx and ε_zz have opposite signs, resulting in ε_xx ∙ ε_zz < 0. Under this condition, the dispersion curve of the structure is hyperbolic. Here, a variable denoted as f_c is introduced to represent the characteristic frequency at which a transition occurs from hyperbolic to elliptical dispersion. When the frequency is further increased to a certain characteristic frequency f_c, with ε_xx > 0, the dispersion changes from hyperbolic to elliptical. Therefore, the sign change in the dielectric constant components is achieved through the epsilon-near-zero (ENZ) point [42]. When E_F increases from 0.4 eV to 1.0 eV, the characteristic frequency f_c increases from 47.72 THz to 76.90 THz. This means that the characteristic frequency of the hyperbolic dispersion shifts to the high-frequency region as E_F increases. The imaginary part of ε_xx in Figure 3d decreases as the frequency increases. When f > 47.72 THz, Im[ε_xx] < 0.001, indicating that graphene has very low loss. Therefore, the hyperbolic properties of the structure can be flexibly adjusted by the Fermi energy of graphene.

Next, calculate the influence of different factors on the dispersion relationship. The dispersion relation formula in Equation (6) is transformed as follows:

$${\displaystyle \frac{{(k_x/k_0)}^2}{{\epsilon }_{zz}}}+{\displaystyle \frac{(k_z/k_0{)}^2}{{\epsilon }_{xx}}}=1.$$

(10)

Set the range of k_x/k₀, solve for k_z/k₀ in the equation, and the dispersion curve showing the variation of k_z/k₀ with k_x/k₀ can be obtained.

Figure 4a displays the dispersion curves at various Fermi energies (E_F = 0.4 eV, 0.6 eV, 0.8 eV, and 1.0 eV for f = 45 THz). All dispersion curves are hyperbolic since ε_xx ∙ ε_zz < 0. As E_F rises from 0.4 eV to 1.0 eV, the position and slope of the curves in the k_y/k₀ direction change. The slope of the curves represents the group velocity, and the group velocities of electrons are different for different Fermi energies.

In Figure 4b, the dispersion contours exhibit an elliptical shape at 58 THz when E_F = 0.4 eV, under which condition ε_xx ∙ ε_zz > 0. For other values of E_F, the dispersion contours remain hyperbolic. In Figure 4c, elliptical dispersion is observed at 68 THz for E_F = 0.4 eV and 0.6 eV, whereas hyperbolic dispersion occurs at E_F = 0.8 eV and 1.0 eV. Specifically, in Figure 4d, the structure retains a hyperbolic dispersion characteristic only at E_F = 1.0 eV for f = 76.89 THz. Consequently, the Fermi energy can significantly alter the dispersion relation at high frequencies.

Figure 5a–d show the variation of the transmittance of the structure with the imaginary part of the conductivity (Im(σ)) of graphene under different Fermi energies (E_F = 0.4 eV, 0.6 eV, 0.8 eV, and 1.0 eV). Im(σ) is a key parameter characterizing the optical loss of graphene, and the larger the imaginary part, the stronger the loss. As shown in the figures, increasing the Fermi energy leads to a rise in Im(σ) and a concurrent decrease in transmittance. Specifically, at E_F = 0.4 eV, the maximum Im(σ) is 0.74 mS, corresponding to a transmittance of 0.75; at E_F = 1.0 eV, the maximum Im(σ) increases to 1.86 mS, with transmittance dropping to 0.33, representing an approximate 2.5-fold increase in loss.

The hyperbolic dispersion characteristics are analyzed at a fixed wavelength. Subsequently, the reflectivity and transmissivity are analyzed through full-wave simulation. Figure 6a displays the reflectance of the structure under different Fermi energies (E_F = 0.4 eV, 0.6 eV, 0.8 eV, and 1.0 eV). The reflectance gradually decreases with increasing frequency f and approaches zero at a characteristic frequency f_c, which shifts toward higher frequencies as E_F increases. Correspondingly, the transmittance in Figure 6b rises with the frequency and peaks near 1 at the same f_c, which also shifts higher as E_F increases. This behavior arises from the near-zero condition of the permittivity (ε_xx ≈ 0) at f_c, where the effective wave impedance of the graphene–dielectric hybrid structure matches that of the surrounding medium. This impedance matching minimizes interfacial reflection losses, leading to maximal transmittance. The transmittance undergoes a change between the “high–low” states by altering the Fermi energy, thereby enabling the on–off function of the optical switch.

Assuming 10 THz is the selected operating frequency, tuning the gate voltage allows control over the Fermi energy of graphene, enabling the transmittance at this frequency to stably switch between two distinct states: the transmittance exceeding 0.75 is defined as the “on state” (for efficient optical signal transmission), while a transmittance below 0.33 is defined as the “off state” (for significant optical signal blocking). Using these values, the modulation depth is calculated as |0.75 − 0.33|/0.75 = 56%.

In addition to being influenced by the Fermi energy, the effective permittivity is also significantly dependent on the thickness of the dielectric. Figure 7 presents the permittivity components in the parameter space defined by the thickness of the dielectric and the frequency. The thickness t_d varies from 20 nm to 50 nm, and the frequency f varies from 10 THz to 100 THz. Other parameters are as follows: T = 300 K, τ = 1 ps, t_g = 0.5 nm, N_g = 1, and E_F = 0.4 eV.

The real part of ε_zz in Figure 7a is always positive and slowly decreases with the increase in t_d at a certain frequency. The imaginary part of ε_zz in Figure 7b is positive and gradually increases with the increase in frequency, but its value is very small. The real part of ε_xx, as shown in Figure 7c, is negative in the low-frequency range. In this case, since ε_xx ∙ ε_zz < 0, the dispersion curve exhibits a hyperbolic characteristic. The characteristic frequency f_c decreases from 47.72 THz to 30.84 THz with t_d increasing from 20 nm to 50 nm, and the hyperbolic region shifts towards lower frequencies. Figure 7d shows the imaginary part of ε_xx, and the value is sensitive to values of low frequencies. Consequently, altering the thickness of the dielectric enables the flexible adjustment of the hyperbolic characteristics of the quasi-periodic structure.

In Figure 8a, all dispersion curves are hyperbolic (open contours) since ε_xx < 0 < ε_zz at 30 THz, allowing propagation of large-k_x waves. In Figure 8b at 47.7 THz, most dispersion contours become elliptical for smaller t_d as ε_xx becomes positive. Only at t_d = 50 nm does the medium remain hyperbolic. The alteration in frequency not only modifies the shape of the curve but also results in distinct regulation patterns of dielectric thickness on dispersion characteristics.

The relaxation time of graphene is a characteristic time scale that describes the time required for electrons to recover from a non-equilibrium state to a thermal equilibrium state after being disturbed by an electric field, light field, etc. Specifically, it is the average free time between two consecutive electron-scattering events. Changing the relaxation time of graphene also exhibits a hyperbolic dispersion characteristic.

Figure 9 shows the variation of the relative dielectric constant components of the structure with frequency under different graphene relaxation times. The graphene relaxation time values are τ = 0.008 ps, 0.01 ps, 0.02 ps, and 1 ps. The frequency range is 40 THz–60 THz. Other parameters are the following: T = 300 K, ε_d = 2.2, E_F = 0.4 eV, td = 20 nm, N_g = 1. Figure 9a represents the real part of the vertical component of the relative dielectric constant ε_zz. The real part of ε_zz increases with the increase in the frequency, but the real part values do not change much under different relaxation times. Figure 9b represents the imaginary part of ε_zz. Its value gradually decreases with the increase in relaxation time. The real part of the parallel component of the relative dielectric constant ε_xx is shown in Figure 9c. The characteristic frequencies fc corresponding to the relaxation times of 0.008 ps, 0.01 ps, 0.02 ps, and 1 ps are 43.39 THz, 44.99 THz, 47.04 THz, and 47.71 THz, respectively. The hyperbolic region moves to the high-frequency region with the increase in relaxation time. When the frequency is greater than fc, the dispersion curve of the structure is elliptical. Figure 9d represents the imaginary part of ε_xx. It decreases with the increase in frequency and gradually decreases in value. Therefore, the hyperbolic characteristics of the structure can be flexibly regulated by changing the frequency and graphene relaxation time.

When the frequency f = 43.38 THz, the dispersion curves of the structure all present a hyperbolic shape for τ = 0.008 ps, 0.01 ps, 0.02 ps, and 1 ps. As shown in Figure 10a, as the graphene relaxation time increases, the opening angle of the isochronous hyperbola becomes larger, which can support propagation modes with high wave vectors. When the frequency is raised to 47.70 THz, as shown in Figure 10b, only when τ = 1 ps, the dispersion curve is a hyperbola, while in other cases, it is an elliptical dispersion. As the graphene relaxation time increases, the minor axis of the elliptical isochronous contour becomes longer, demonstrating strong anisotropy.

The number of graphene monolayers is also an important factor affecting the hyperbolic properties of the structure. Figure 11 shows the permittivity components for different numbers of graphene monolayers. The number of graphene monolayers N_g varies from 1 to 4 in a single graphene layer, and the frequency f varies from 40 THz to 50 THz. Other parameters are as follows: T = 300 K, τ = 1 ps, t_d = 20 nm, and E_F = 0.4 eV. The real and the imaginary parts of ε_zz are shown in Figure 11a,b, respectively. The values are positive and almost independent of low frequencies. The effect of the number of graphene monolayers on the real part of ε_xx is investigated in Figure 11c. The characteristic frequency f_c decreases from 47.71 THz to 47.11 THz with N_g increasing from 1 to 4, and the hyperbolic region shifts towards lower frequencies. The imaginary part of ε_xx in Figure 11d decreases as the frequency increases and has a very small value. Therefore, changing the number of graphene monolayers can also regulate the hyperbolic characteristics.

All dispersion curves exhibit a hyperbolic shape in Figure 12a at 47.10 THz. The opening angle of the isofrequency hyperbola gradually decreases as N_g increases from 1 to 4. At 47.70 THz, as shown in Figure 12b, the dispersion contour becomes elliptical for N_g values of 2, 3, and 4 due to the condition ε_xx ∙ ε_zz > 0. The minor axis of the elliptical isofrequency contour becomes progressively shorter with increasing N_g, demonstrating strong anisotropy. The number of graphene monolayers can effectively regulate the dispersion characteristics of the metamaterial.

Figure 13a presents the electric field distribution within the structure, simulated by the FDTD method implemented in COMSOL Multiphysics 5.5 software. Figure 13b shows the electric field distribution, obtained through the transmission matrix method using MATLAB R2022b software. The horizontal axis corresponds to the direction of the graphene and dielectric layers arrangement, while the vertical axis represents the normalized intensity of the horizontal electric field in the Z direction. The entire structure is much shorter than the wavelength, and the electric field intensity decreases along the Z direction. As shown in the figures, the electric field distributions obtained from both simulation methods are consistent.

For practical device integration, graphene and dielectric layers can be alternately stacked via transfer techniques for constructing the multilayer structure. Furthermore, graphene can also be directly deposited onto the surface of dielectric layers via CVD [43,44]. CVD is a mature technique widely utilized in micro- and nanofabrication, enabling precise control of the thickness and uniformity of dielectric layers. Recent advances in techniques such as PECVD growth of graphene films, dielectric-layer-assisted transfer techniques, gradient surface energy (GSE) modulation methods, and CVD growth of graphene [45,46,47,48] validate the feasibility of fabricating such nanostructures. Regarding dynamic functionality, the optical response of graphene can be changed by electrically regulating the Fermi energy of graphene (via iron doping or external gate voltage [18]).

4. Conclusions

In conclusion, tunable hyperbolic dispersion is achieved in the Cantor-sequence multilayers of graphene and dielectric. The effective medium theory was used to derive the anisotropic permittivity of the structure. With a specific weighting of the dielectric thickness, the equivalent permittivity tensor exhibits hyperbolic dispersion. The frequency range of hyperbolic dispersion can be effectively tuned by adjusting the Fermi energy of graphene, the thickness of the dielectric, and the number of graphene monolayers. The characteristic frequency of hyperbolic dispersion shifts toward higher frequencies with the Fermi energy and the relaxation time of graphene increasing. Conversely, the characteristic frequency shifts toward lower frequencies with the thickness of the dielectric and the number of graphene monolayers increasing. Reflectance and transmittance spectra were analyzed to identify frequency bands supporting low-loss hyperbolic modes. It is hoped that this research can offer valuable theory for designing and optimizing photonic devices based on hyperbolic metamaterials.