Graphene hyperbolic metamaterials: Fundamentals and applications

The deeper understanding and controlling of materials properties are the basis for many important breakthroughs in science and technology. Due to the development of nanofabrication technology, the appearance of metamaterials offers a new route to designing material properties at will. Over the past two decades, metamaterials have been a popular research field due to the fact that they exhibit various extraordinary electromagnetic properties which cannot be realized by natural materials [1]. Among different kinds of metamaterials, hyperbolic metamaterials (HMMs), which are highly anisotropic media with hyperbolic dispersion, have gained plenty of attention recently [2–5]. HMMs have three-dimensional bulk response and are relatively easy to be fabricated by using multilayers or nanowire structures [6,7]. Therefore, HMMs have attracted extensive attention and have potential in many applications such as subwavelength imaging [8,9], spontaneous [10,11], sensing [12,13] and absorber [14].

The design of HMMs covers a wide frequency range from ultraviolet to mid-infrared or even terahertz (THz) frequency ranges [3]. Figure 1(a) shows the materials utilized for designing HMMs in different frequency regions. In earlier researches, most HMMs were designed by metal and dielectric materials. Such conventional metal-dielectric structures show large negative permittivity at THz frequencies which restrict their functions. Differently, graphene, a single sheet of carbon atoms, has gained a lot of attention since it was first fabricated by micromechanical cleavage method in 2004 [15]. Besides, there was also researches reporting that natural graphite shows hyperbolic dispersion in the UV range [16]. Graphene supports transverse magnetic (TM) surface plasmons and behaves like metals/dielectric systems in the THz and far-infrared range [17]. Therefore, a periodic stacking of graphene sheets and dielectric material was expected to behave like an effective hyperbolic medium. What is more, as compared with metal materials, the surface plasmons in graphene has the features of relatively low loss and flexible tunability [18]. The conductivity of graphene can be tuned by external gate voltage, electrostatic biasing, or magnetic field [19]. Hence, graphene is a good candidate for designing HMMs in a wide frequency range from the mid- to far-infrared frequency ranges including THz region. It should be noted that graphene shows strong absorption in visible and near-infrared range, so that it is not a suitable material to construct optical metamaterials in these frequency ranges.

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Graphene HMMs show strong anisotropy which have elliptic or hyperbolic dispersion relation. In the hyperbolic case, graphene HMMs support TM waves, whereas transverse electric (TE) waves are mainly evanescent [20]. Therefore, most researches about graphene HMMs focused on TM waves. The early studies of graphene HMMs included special phenomena such as large Purcell effect [19–21] and negative refraction [22]. Researches also revealed their potential in absorption [23,24], imaging [25], and waveguide [26]. The transition between elliptic and hyperbolic dispersion types in graphene HMMs was discussed in ref. [27]. This transition property allows the access to Epsilon-Near-Zero (ENZ) condition where the dispersion diagram becomes very flat. Besides, graphene HMMs show tunability over a wide bandwidth by adjusting the chemical potential of graphene.

Recent years have witnessed widely explorations of graphene HMMs as tunable devices in THz and infrared ranges. The strong anisotropy makes the HMM become a suitable platform for modulation of light. For example, the propagation angles of unidirectional beam in the graphene-based HMM can be dynamically tuned [28]. Surface electromagnetic waves on the interface of neighbouring graphene-based HMM structures have been investigated [29,30]. The optical properties of graphene HMM with defect layer were also reported [31]. Furthermore, The Goos-Hänchen shift based on one-dimensional graphene HMM photonic crystal [32] and the generation of Cherenkov radiation in graphene hyperbolic grating in THz band [33] were observed. These works enriched the properties of graphene HMMs. Very recently, graphene HMMs have been utilized in plasmon [34–36], microcavity [37], biosensing [38,39], heat transfer [40], and surface-enhanced Raman spectroscopy [41]. The tunability of graphene HMMs have made it much important in modulation of electromagnetic (EM) waves.

The objective of this review is to summarize the so far theoretical and experimental studies in graphene HMMs. Therefore, we will first introduce the common model applied in graphene HMMs studies in the following section. Next, we will mainly discuss the up-to-now fabrication technologies applied in experimental demonstrations of graphene HMMs in the third section. The potential applications in THz and far-infrared frequency range are summarized in the fourth section. Finally, we conclude current progress and provide an outlook for future development.

The optical properties of graphene can be characterized by its surface conductivity ${\sigma}(\omega,\mu_{c},\Gamma,T)$, where ${\omega}$ is the working frequency, ${\mu}_{c}$ is the chemical potential, Γ is the phenomenological scattering rate, and T is the temperature. The conductivity can be modelled as the combination of the interband and intraband contributions, which is expressed by Kubo formula [42,43]

$$\begin{align} &\sigma \left(\omega,\mu_{c},\Gamma,T\right)=\frac{je^{2}k_{B}T}{\pi \hslash^{2}\left(\omega +j2\Gamma \right)}\bigg[\frac{\mu_{c}}{k_{B}T} \nonumber\\ &+2\ln\left(e^{-{\mu_{c}}/\left({k_{B}}T\right)}+1\right)\bigg]+\frac{je^{2}\left(\omega +j2\Gamma \right)}{\pi \hslash^{2}}\!\!\nonumber\\ &\times\int_{0}^{\infty }\frac{f_{d}\left(-E\right)-f_{d}\left(E\right)}{\left(\omega +j2\Gamma \right)^{2}-4\left(E/\hslash\right)^{2}}\mathrm{d}E, \end{align} \tag{ 1 }$$

where e is the charge of the single electron, k_B is the Boltzmann constant, and ${\hslash}$ is the reduced Planck constant. The function f_d (E) is given by the Fermi-Dirac distribution,

$$\begin{equation} f_{d}(E)=\bigg[\exp \frac{(E-\mu_{c})}{k_{B}T}+1\bigg]^{-1}, \end{equation} \tag{ 2 }$$

where E is the energy.

The common structures of graphene HMMs discussed in the literature are similar to metal-dielectric multilayers where the metal layers are replaced by graphene sheets. The schematic of graphene HMMs multilayer structure is depicted in fig. 1(b). It is constructed by stacking graphene sheets and dielectric layers with a thickness d and permittivity ${\varepsilon}_{d}$ alternatively. The wave propagation in multilayers with sub-wavelength period can be modelled by effective medium approximation (EMA) method which is widely utilized in HMM structure analysis [44,45]. The Bloch theory can also be applied in wave propagation model using the transfer matrix of a unit cell (as discussed in chapter 8 in ref. [46]). According to EMA, the periodic multilayer is regarded as a homogeneous anisotropic medium with effective relative permittivity ${\varepsilon}_{\textit{eff}}$ as described below for TM waves [20,27]:

$$\begin{equation} \varepsilon_{\textit{eff}}=\left[\begin{array}{lll} \varepsilon_{t} & 0 & 0\\ 0 & \varepsilon_{t} & 0\\ 0 & 0 & \varepsilon_{z} \end{array}\right], \end{equation} \tag{ 3 }$$

where ${\varepsilon}_{t}$ and ${\varepsilon}_{z}$ are the components parallel and perpendicular to the graphene sheet according to fig. 1(b). Since the graphene sheet is negligibly thin as compared with the dielectric layer, the electric field in the z-direction will not excite any current in graphene. Therefore, the perpendicular permittivity component ${\varepsilon}_{z}$ is equal to ${\varepsilon}_{d}$. Differently, the effective relative "transverse" permittivity is determined by averaging the transverse effective displacement current over the associated electric field in a unit cell which is defined as a dielectric layer with ${\varepsilon}_{d}$ between $z = 0$ and $z = d$ and a graphene sheet at $z = 0$. Hence, the parallel permittivity component can be written as follows:

$$\begin{equation} \varepsilon_{t}=\varepsilon_{d}-j\frac{\sigma (\omega,\mu_{c},\Gamma, T)}{\omega \varepsilon_{0}d}. \end{equation} \tag{ 4 }$$

In eq. (4), the permittivity is highly related to the graphene sheet conductivity, and the conductivity ${\sigma}$ depends on the frequency and chemical potential. The chemical potential ${\mu}_{c}$ can be connected with the carrier density n_s by the equation,

$$\begin{equation} n_{s}=\frac{\varepsilon_{0}\varepsilon_{d}E_{bias}}{e}=\frac{2}{\pi \hslash^{2}\upsilon_{F}^{2}}\int_{0}^{\infty }E\bigg[f_{d}(E)-f_{d}(E+2\mu_{c})\bigg]\mathrm{d}E, \end{equation} \tag{ 5 }$$

where ${\upsilon}_{F}$ is the Fermi velocity with a magnitude around 9.5×10⁵ m/s, and E_bias is the electrostatic field biasing. Therefore, the chemical potential can be modulated by E_bias through electrical gating. By combining eqs. (1) with (5), the relationship between E_bias and ${\mu}_{c}$ are showed in the following formula:

$$\begin{align} E_{bias}&=\frac{2e}{\pi \hslash^{2}\upsilon_{F}^{2}\varepsilon_{0}\varepsilon_{d}}\bigg\{\left(k_{B}T\right)^{2}\int_{-\mu_{c}/\left(k_{B}T\right)}^{\mu_{c}/\left(k_{B}T\right)}\frac{x}{e^{x}+1}\mathrm{d}x\nonumber\\ &\quad+k_{B}T\mu_{c}\ln \left[e^{-{\mu_{c}}/\left({k_{B}}T\right)}+1\right]+k_{B}T\mu_{c}\nonumber\\ &\quad\times\ln\left[e^{{\mu_{c}}/\left({k_{B}}T\right)}+1\right]\bigg\}. \end{align} \tag{ 6 }$$

The effective relative transverse permittivity ${\varepsilon}_{t}$ can be positive or negative when the chemical potential ${\mu}_{c}$ is changed. Figures 2(a) and (b) show the distributions of real part ${\text{Re}}({\varepsilon}_{t})$ and imaginary part ${\text{Im}}({\varepsilon}_{t})$ with different frequencies and chemical potential. A positive ${\text{Im}}({\varepsilon}_{t})$ appears when ${\mu}_{c}$ approaches zero, which would result in absorption loss in propagation. By choosing suitable ${\mu}_{c}$ and f, the imaginary part can be neglected and ${\varepsilon}_{t}$ is simplified to be its real part. The plane waves propagating in a homogeneous uniaxial anisotropic medium can be separated as TE^z (electric field transverse to the z-axis) and TM^z (magnetic field transverse to the z-axis) waves. Accordingly, the wave vector dispersions of TE^z and TM^z waves in graphene HMMs are written as

$$\begin{align} k_{z}^{2}+k_{t}^{2}&=\varepsilon_{t}k_{0}^{2}, \quad \mathrm{TE}^{z}, \end{align} \tag{ 7 }$$

$$\begin{align} \frac{k_{z}^{2}}{\varepsilon_{t}}+\frac{k_{t}^{2}}{\varepsilon_{z}}&=k_{0}^{2}, \qquad \mathrm{TM}^{z}, \end{align} \tag{ 8 }$$

where $k_{0}=\omega \sqrt{\varepsilon_{0}\mu_{0}}$ is the wave number in vacuum and $k_{t}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ is the transverse wave number component.

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The aforementioned analysis shows that iso-frequency contours of the dispersion for TM^z wave can be hyperbolic or elliptical depending on the permittivity component ${\varepsilon}_{t}$. When ${\varepsilon}_{t} < 0$, the TE^z wave is mainly evanescent for any k_t . However, the extraordinary wave (TM^z ) with transverse wave number $k_{t} > k_{0}({\varepsilon}_{d})^{\mathrm{1/2}}$ can propagate in this case which would be otherwise evanescent in homogeneous dielectric medium with permittivity ${\varepsilon}_{d}$ or in a uniaxial anisotropic media with ${\varepsilon}_{t} > 0$. The graphene HMMs have hyperbolic dispersion when ${\varepsilon}_{t} < 0$ and ${\varepsilon}_{z} > 0$; while ${\varepsilon}_{t} > 0$ and ${\varepsilon}_{z} > 0$, they exhibit elliptic dispersion for a limited propagating spectrum with $k_{t} < k_{0}({\varepsilon}_{d})^{\mathrm{1/2}}$.

The Bloch theory for period structures can also be utilized for examining the validity of the EMA method [27]. In this theory, each graphene sheet is modelled with a complex lumped admittance Y_s  = σ. The transfer matrix [T_unit] of a unit cell in graphene HMMs for TE^z /TM^z waves can be written as

$$\begin{equation} \begin{array}{l} \left[\mathrm{T}_{\text{unit}}\right]=\left[\begin{array}{ll} 1 & 0\\ \sigma & 1 \end{array}\right]\left[\begin{array}{ll} \cos \left(\kappa_{d}d\right) & jZ_{d}\sin \left(\kappa_{d}d\right)\\ \dfrac{j}{\mathrm{Z }_{d}}\sin \left(\kappa_{d}d\right) & \cos \left(\kappa_{d}d\right) \end{array}\right]\\ =\left[\begin{array}{ll} \cos \left(\kappa_{d}d\right) & jZ_{d}\sin \left(\kappa_{d}d\right)\\ \dfrac{j}{Z_{d}}\sin \left(\kappa_{d}d\right)\!+\!\sigma \cos \left(\kappa_{d}d\right) & j\sigma Z_{d}\sin \left(\kappa_{d}d\right)\!+\!\cos \left(\kappa_{d}d\right) \end{array}\right]\!\!, \end{array} \end{equation} \tag{ 9 }$$

where $\kappa_{d}=\sqrt{\varepsilon_{d}k_{0}^{2}-k_{t}^{2}}$ is the wave number along the z-axis. For TE^z and TM^z waves, there are $Z_{d}^{\mathrm{TE}}=\omega \mu_{0}/\kappa_{d}$ and $Z_{d}^{\mathrm{TM}}=\kappa_{d}/(\omega \varepsilon_{0}\varepsilon_{d})$, respectively. The solution of the eigenvalue problem $|[T_\mathrm{unit}]{\text{-e}}^{-jk_z^d}{\rm [I]}| = 0$, where [I] is the identity matrix, leads to the simple dispersion relation as follows:

$$\begin{equation} \cos (k_{z}d)=\cos (\kappa_{d}d)+j\frac{1}{2}\sigma Z_{d}\sin (\kappa_{d}d). \end{equation} \tag{ 10 }$$

The relation in eq. (10) is accurate for arbitrary d and k_t . When the dielectric layer's thickness is much smaller than the Bloch wavelength and the working wavelength (which means $|k_{z}d| \ll 1$ and $|{\kappa}_{d}d|\ll 1$), the dispersion relation simplifies to that obtained via EMA method. We can apply the approximations $\cos x \approx 1-x^{2}/2$ and $\sin x \approx x$ so that the dispersion relation has the form similar to eqs. (7) and (8). The specific analysis of the difference between Bloch theory and EMA method is discussed in ref. [20]. Figures 2(c) and (d) show the calculations of k_t via EMA and Bloch theory, respectively. Here, the z-directed wave number is assumed to have complex values as $k_{z} = {\beta}_{z}-j{\alpha}_{z}$. The results show that these two theories are in good agreement when k_t is not too large. For larger k_t , the assumptions of EMA method are no longer valid. The graphene HMM has an upper limitation of propagating spectrum due to the periodicity, which is the first Brillouin zone edge at ${\beta}_{z} = {\pm}{\pi}/d$ with higher k_t .

The fabrication of graphene HMMs was technologically proposed in early works. A single graphene layer can be fabricated by micromechanical cleavage [47], epitaxy growth [48], or chemical vapor deposition (CVD) [49] methods. Besides, the transfer of graphene onto the dielectric layer is significant. There exist several methods for layer transfer, including wet and dry transfer ways. The common wet transfer ways of graphene or other two-dimensional materials are wedging method [51], the polyvinylalcohol (PVA) method [52], and the evalcite method [53]. An all-dry transfer method was proposed in 2014 by using viscoelastic stamps [54] which showed potential in fabricating heterostructures. The dielectric layer can be grown by atomic layer deposition (ALD) and the graphene layer is transferred on the surface of dielectric layer. Figure 3(a) exhibits the fabrication steps of graphene HMMs.

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A five periods graphene-Al₂O₃ multilayer structure on a CaF₂ substrate was realized in 2016. The CVD graphene was grown on copper foil and transferred to the substrate using the poly(methyl-methacrylate) (PMMA) transfer technique [55]. An ammonium persulfate solution was used to etch the copper foil. The Al₂O₃ dielectric layer was deposited by ALD at 150 °C. Trimethylaluminium was the precursor of Al, and H₂O was the oxygen precursor. With the increase of the numbers of hyperbolic metamaterials' periods, the quality of the whole structure will decrease at the same time. A common issue during the transfer progress is the introducing of cracks and defects into the graphene layers. Besides, aligning different layers can be challenging to maintain large number of layers. When graphene is integrated with dielectric layers, gating graphene is difficult since the weak adhesion to most dielectric substrates.

The experimental demonstration of tunable graphene HMMs was proposed in 2019 [56]. As shown in fig. 3(b), the manufactured individual structure was composed of thermally grown SiO₂, deposited Al₂O₃ via plasma-enhanced chemical vapor deposition (PECVD), graphene by CVD and top SiO₂ deposited by PECVD. The 3 nm/100 nm Cr/Au contacts were deposited by lithographically defined patterns on the graphene layer. The contacts were used to gate graphene monolayer against the back silicon substrate. The characterization results showed that a shift in the near-zero response was observed by tuning the Fermi level of graphene.

Recently, another work demonstrated a layer-by-layer graphene HMM deposition method without a layer transfer step [58]. The dielectric material positively charged polyelectrolyte polydiallyldimethylammonium chloride (PDDA) was deposited to the negatively charged substrate. The graphene was converted from graphene oxide by laser-mediated photoreduction. The graphene oxide was uniform dissolution in water and negatively charged to maintain a monolayer structure and prevent aggregation before it attached to positively charged surface. Figure 3(c) shows the process of fabrication. The bandgap would decrease during the convert from graphene oxide to graphene. Such synthesis offers a low-cost novel method of fabricating graphene HMMs.

Due to the extraordinary optical properties, graphene HMMs have shown potential in light modulation in THz and infrared frequency ranges. The past decade has witnessed the exploration of their applications. The tunability is the hallmark of graphene HMMs which is intrinsically important for applications in optical devices.

High-resolution imaging

The diffraction limit of light is caused by exponential decay of evanescent waves. Hyperlens is an anisotropic medium which supports the propagation of evanescent waves due to the hyperbolic dispersion. The evanescent waves can be converted into propagating ones so that they can be captured by traditional imaging system. A usual hyperlens consists of metal-dielectric layers or of metallic wires which could work in infrared and microwave ranges. However, the employment of metal limits the tunability of hyperlens after fabrication.

The first graphene-based hyperlens for the THz range was proposed by Andryieuski and Lavrinenko in 2012 [59]. The hyperlens was made of graphene stripes embedded into a dielectric, as shown in fig. 4(a). The structure was designed as a cylinder and it could resolve two line sources separated by a distance ${\lambda}/5$ according to the simulation result in fig. 4(b). Zhang et al. proposed a hyperlens with cylindrical alternating graphene/dielectric multilayer [25]. The introduction of cylindrical shape aimed to weaking the effect of distorted image brought by different optical paths from different point sources. Moreover, broadband sub-diffraction imaging was achieved by tuning chemical potential ${\mu}_{c}$. Recently, Caligiuri et al. demonstrated a graphene/SiO₂ multilayer HMM with ITO coated glass to ensure the transparency and conductivity for the electrical gating [60]. The unit cell of HMM consisted of a three-sheet graphene layer (1 nm thick) on the top of a 1 nm SiO₂ layer, as displayed in fig. 4(c). Figure 4(d) presents the simulation result. A 3D logo "tehris" could be resolved at the exit layer with a resolution $> {\lambda}/200$. All these simulations show the potential of graphene HMMs in subwavelength imaging.

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Absorption

Single-layer graphene has absorption of 2.3% in the optical range and more in the THz [61]. Besides, graphene supports surface plasmon polaritons at THz frequencies. Therefore, the absorption at resonance frequency will be significantly increased. The periodic structure of graphene-based metamaterials will increase the coupling of energy from incident field. It was shown that graphene/dielectric multilayers can be used as an absorber for near-fields generated at its surface [20]. Then, a perfect absorption of asymmetric graphene HMM structure was proposed where the graphene layers were tilted respect to boundaries [23]. As shown in figs. 5(a) and (b), 100% light absorption with no reflection was achieved for the case of TM waves while the absorption level was low for the TE waves. A similar structure of perfect absorption was discussed by Madani et al. [62]. Critical coupling of two graphene-based HMMs in different stacking direction reported by Xiang et al. also indicated its near-perfect light absorption when the dispersion is hyperbolic with large imaginary part of permittivity [63]. The structure and simulation of absorption field are displayed in figs. 5(c) and (d).

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In recent years, the absorption in different graphene-based HMMs structures has been investigated. Ning et al. proposed a dual-gated tunable absorber graphene-based HMM [64]. A graphene/dielectric multilayer was placed on top of a stacked composed of graphene and two types of dielectrics. The dual-gated bias voltage was added between multilayer or graphene with bottom Cu reflector. The numerical results showed three absorption peaks and an absorption bandwidth of almost 35 THz can be realized at the second peak. Linder et al. investigated the electromagnetic waves incident upon a graphene layer coating an anisotropic epsilon near zero metamaterial. Extremely wide incident angle of absorption was realized in this structure [65]. Wide band absorption has been reported in graphene-based patch resonators [66] and graphene/CsPbCl3 multilayer [67]. Their 90% absorption bandwidth reached 14.59 THz and 14 THz, respectively. Very recently, a graphene-silica multilayer HMM was proposed to realize narrowband near-perfect absorption [68].

The optical properties of graphene HMMs absorbers are associated with polarization and incidence angle of incident light due to the strong anisotropy of HMMs. Perfect absorption has been found to be obtained only in the TM-polarized wave case. The incident angle of light will notably affect the position of absorption peak in spectrum. What is more, the absorption band can be tuned by changing the Fermi energy of graphene. The tunability of graphene HMM absorbers were discussed in detail in these literatures.

Photonic devices

Lasing

In conclusions, graphene HMMs possess unique and strong anisotropic dispersion relation and abundant physical effects. EMA and Bloch theory are two main theoretical study methods for HMMs. The EMA theory is in good agreement with Bloch method when the transverse component of the wave number is not too large (i.e., in long wavelength limit case). The graphene HMMs can be fabricated through CVD methods and transfer processes. Only several layers of graphene HMMs have been experimental realized up to now. The potential applications of graphene HMMs in THz and infrared frequency regions have gained great attention. Various devices including hyperlens, absorbers, optical switches, waveguides and lasers have been proposed.

As an outlook, the up to now graphene HMMs were mainly discussed as simulation models. The experimental fabrications of graphene HMM device still face challenges. Few periods of graphene/dielectric layers lack the ability of modulations. The fabrication of useful multilayers structure and adding external gate voltage between graphene layers with substrates require new techniques. The chemical doping of graphene can modify its chemical potential customized for different applications. Besides, the chemical potential can be tuned by gate voltage, electrostatic field biasing, etc. Due to the flexible tunability of HMMs, the dispersion of graphene HMM can be switched between elliptic and hyperbolic types. An ENZ state can also be realized in it. The tunability of graphene-based HMMs significantly expands the application fields.

This work was supported by National Natural Science Foundation of China (12074127, 11974119); National Key R&D Program of China (2018YFA 0306200); Research project of the Science Education Professional Committee of the Chinese Higher Education Association (21ZSLKJYYB14). Student research program of SCUT in 2022 (x2wl-C9227060).

Data availability statement: The data that support the findings of this study are available upon reasonable request from the authors.