1. Introduction

Hexagonal boron nitride (hBN) is a two-dimensional natural Van der Waals crystal, which attracted recent attention due to its inherent hyperbolic dispersion relation and ability to support hyperbolic phonon polaritons (HPhPs) [1–7]. On the other hand, graphene is an atomically thin plasmonic material that can support surface plasmon polaritons (SPPs) from optical to terahertz frequency range [8–14]. As both hBN HPhPs and graphene SPPs reside in the mid-IR region, a hybrid graphene–hBN film would bring the advantages of electrical tunability of graphene and high quality factor of hBN polariton resonances, providing an effective and viable modulation of hyperbolic polaritons in such van der Waals heterostructure.

Recently, mid IR and optical properties of graphene-hBN heterostructures has been investigated. Kumar et al. [15] explored light-matter interaction within two Reststrahlen bands of hBN. Ye et al. [16] presented graphene-coated hBN nanowires which supported both SPP and HPhP modes. Meanwhile, electromagnetic wave propagation in multilayer graphene-hBN heterostructures were intensively studied [17–20]. Simulations and experiments show that a graphene-hBN hyper crystal exhibits both tunable and hyperbolic characteristics. Moreover, the coupling of plasmon polaritons in graphene and phonons in hBN leads to hybrid plasmon–phonon polaritons modes, which has great potential in IR applications including sensors, filters, thermal management and many others [21–23].

Combining plasmonic graphene and phonon hBN material together, one can enhance and control the HPhPs-SPPs coupling [24,25]. Previous studies focusing on graphene-hBN hybrid platform utilize a continuous graphene sheet, which does not provide significant enhancement due to lack of strong localized plasmonic resonances. A promising way is to use metal gratings covered by hBN to improve the absorption characteristics. Magnetic polaritons coupled with HPhPs in hBN can create hybrid phonon-plasmon polaritons to enhance the absorption [26], although the tunable bandwidth of the structure is limited due to the strong coupling of magnetic polaritons. Therefore, exploring new mechanisms to achieve broadband tunability remains an active research field.

In this paper, we presented a patterned graphene-hBN metasurface, which merit the advantages of strong plasmon-phonon interactions and wideband tunability. To the best of our knowledge, the plasmonic-phononic coupling in the patterned graphene-hBN structure has been seldom reported. Hajian et al. [27] theoretically demonstrated that the nearly perfect absorption by patterned graphene-hBN-graphene in the mid-IR region. However, the patterned hBN with relatively large thickness breeds multiple hybrid resonant modes and brings difficulties for tuning absorption peaks continuously. Here, we used hBN substrate with small thickness to reduce extra hybrid modes. It should be noticed that the reduction of hBN thickness leads to a significant reduction of intensity of hyperbolic phonon resonance. However, we can still expect a strong plasmon-phonon coupling between surface plasmons in graphene and second phonons in the hBN. Based on finite difference time domain (FDTD) method, the absorption and electromagnetic field distribution were analyzed. By varying the Fermi energy of graphene pattern, the absorption peaks can be tuned dynamically and continuously. This study offers new opportunities for the study and the control of plasmon-phonon interactions in a hybrid patterned graphene-hBN structure.

2. Method and analysis

The configuration of the hybrid structure, composed of patterned graphene/hBN heterostructure and lossfree dielectric substrate with a constant refractive index of n = 1.7, is illustrated in Fig. 1. The proposed structure is periodic along both the x and y directions with a period p. Moreover, the length of the square hole patch in graphene is d. The thickness of the hBN and dielectric substrate layer is denoted as t and h, respectively. In this simulation, the geometric parameters are p = 240 nm, d = 100 nm, t = 20 nm and h = 1.6 μm. The gold ground plane is set to be thick enough as there is no transmission through the substrate.

 

Fig. 1 (a) Schematics of the patterned graphene- hBN hybrid structure with structural parameters p = 240 nm, d = 100 nm, t = 20 nm and h = 1.6 μm. (b) Real part of the dielectric function of hBN with perpendicular (black line) and parallel (red line) components.

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hBN is a natural uniaxial crystal in the infrared region. The permittivity of hBN is a tensor, characterized by two parts: ε_||, which is along the crystal symmetry axis, and ε_⊥, which is in the basal plane. The hBN permittivity tensor is given by

In hBN, there are two Reststrahlen bands, the lower frequency Reststrahlen band, corresponds to Type I region (ε_⊥>0, ε_∥<0), and the higher frequency Reststrahlen band, corresponds to Type II region (ε_⊥<0, ε_∥>0). Two dielectric functions representing these two bands can be described by [15,28]

Table 1. Parameters used in Eq. (2) to obtain the permittivity tensors of hBN

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Figure 1(b) shows the calculated real parts of dielectric constants by fitting the perpendicular and parallel components of the hBN dielectric function. As shown in Fig. 1(b), the areas between the dash lines illustrate the two hyperbolicity bands. It should be noticed that besides the main peak near Type I region, there is a minor peak named the second phonon peak for ε_⊥ [28]. For this structure, it is the second phonon resonance enables the plasmon-phonon coupling.

The complex surface conductivity of graphene includes both the interband and intraband transitions and is described by the Kubo formalisms [29,30]. However, at the infrared region, the interband component can be neglected, as the conductivity σ of graphene is given as follows [13,31]:

Here, $E_F$ is the Fermi level, τ is the relaxation time. In simulation, we choose the Fermi level $E_F=0.64eV$. The relaxation time τ can be calculated by

The numerical simulations were conducted using a finite-difference time domain technique (in Lumerical FDTD). In the simulation, periodic boundary conditions in x and y directions are applied to simulate an infinite area, while in z direction the perfectly matched layer (PML) boundary condition is utilized. Moreover, graphene sheet thickness is chosen to be 1 nm, and the mesh with graphene sheet is set to 0.25 nm to ensure the accuracy of the calculation.

3. Results and discussion

Figure 2(a) shows the absorption spectra of the proposed structure. From wavelength of 8 to 18 μm, as shown in Fig. 2(a), there are three resonance peaks, corresponds to mode 1, mode 2, and mode 3, respectively. For comparison, Fig. 2(a) also includes the absorption spectra of the structures which have no graphene sheet and continuous graphene sheet. It is clear that only with patterned graphene can excite such a triple resonance. Moreover, the resonance peak frequency of mode 2 located outside the Type I region of hBN, as we can predict that it is originated from the second phonon resonance.

 

Fig. 2 (a) Absorption spectra of the structures in different cases and (b) contribution of the $P_{abs}$ of graphene and hBN layer within the patterned graphene/hBN heterostructure.

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The absorption of the proposed structure can be further illustrated by power dissipation at every single layer of the structure, as the power absorption per unit volume ($P_{abs}$) is defined as [32]:

Figure 2(b) demonstrates the contribution of the $P_{abs}$ distribution in graphene and hBN layer, as the hBN contributes a fully partial of absorption in mode 2, as well as a larger partial in mode 1 and 3. Hence, we can expect a strong power absorption within hBN layer due to the coupling between hBN phonons and graphene plasmons.

It is known that hybrid plasmon-phonon polaritons can be excited when combining graphene with hBN in hybrid structures. In addition, the hybrid polaritons can be categorized to surface plasmon-phonon polaritons (SP3) and hyperbolic plasmon-phonon polaritons (HP3) which are supported outside and inside of the Reststrahlen bands of hBN, respectively [21,27]. However, from the simulations above, we find that the second phonons can also lead to strong coupling with surface plasmons, exciting hybrid plasmon-phonon polaritons. To further comprehend the absorption mechanism, we calculated the electric field and $P_{abs}$ distributions of the three resonant modes and results are shown in Fig. 3. For the first mode, shown in Fig. 3(a), there are obvious propagating surface plasmon polaritons travelling at the surface of graphene sheet, which indicates that SPP mode was produced within the hybrid structure. Similar phenomenon was observed in Fig. 3(e), as mode 3 is also related to SPP mode. On the other hand, the $P_{abs}$ distributions in Figs. 3(b) and 3(f) suggested that for mode 1 and 3, there is still a certain amount of absorption taking place in hBN, as the surface plasmon polaritons in graphene couple with phonon polaritons in hBN, producing these two SP3 mode. However, for mode 2, the electric field distribution shown in Fig. 3(c) indicates that the intensity of plasmons at the surface of graphene are much weaker than that of SP3 modes. Moreover, from Fig. 3(d), most of the absorption is concentrated in the hBN layer, demonstrating that phonon polaritons in hBN dominate the plasmon-phonon coupling.

 

Fig. 3 Electric field and $P_{abs}$ distribution in a cross-section of the hybrid structure. (a) and (b): mode 1; (c) and (d): mode 2; (e) and (f): mode 3.

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Figure 4(a) illustrates the tunability of the hybrid structure, i.e. the relationship between chemical potentials of graphene and the absorption spectra. For comparison, the tunability of the patterned graphene structure without hBN with same geometric parameters presented in Fig. 4(b) shows that the SPPs in graphene may have a wider range in wavelengths. However, it is the coupling between graphene and hBN that contribute to the hybrid resonance modes. As the chemical potential increases from 0.2 eV to 0.8 eV, all absorption peaks show blue shift. Meanwhile, peak absorption also varies with the change of chemical potential. It is worthy to notice that continuously increasing the chemical potential of graphene could move the peak frequency of mode 3 closer to that of mode 2 (second phonon resonance band). However, just like SP3 and HP3, the resonance frequencies of these two modes have no intersections. Moreover, for mode 1, the resonance frequency shows a redshift with the increase of chemical potential of graphene. It is interesting that the wavelength interval between mode 1 and mode 2 is larger than that between mode 2 and mode 3, which indicates that the SP3 mode cannot be supported within Reststrahlen band of hBN, even though there is no hyperbolic phonon resonance. As shown in Fig. 4(a), for the two SP3 modes, not only the peak absorption frequency but also the absorption intensity can be tuned within a fairly broad wavelength range.

 

Fig. 4 Absorption spectra of the patterned graphene structures for various chemical potentials. (a) With hBN and (b) Without hBN.

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Finally, we analyzed the effect of the structure geometry of the hybrid structure on the absorption spectrum. In the simulation, only one parameter, e.g., the periodicity (p), the length of the pattern in graphene (d), the thickness of the hBN (t), or the thickness of the dielectric substrate (h), was changed, while other parameters were fixed.

The simulation results for different p, d, t, and h are illustrated in Fig. 5. From Fig. 5(a), the increase of p causes the strong redshift of the two SP3 modes, since the increase of the overall length of the patterned graphene causes the increase of the inductance of the resonant structure. However, the reverse trend is observed in Fig. 5(b), as the increase of pattern length d leads to the decrease of the inductance. Hence, the absorption peaks move towards to higher frequencies. Figure 5(c) illustrate the dependence of hBN thickness t on absorption spectra, where the absorption rate of phonon mode is more dependent on t than that of SP3 modes. Moreover, compared with phonon mode, hBN thickness t plays a more important role on the resonant frequency of SP3 mode.

 

Fig. 5 Absorption spectrum dependence on different structural parameters: (a) periodicity, p, (b) length of the pattern in graphene, d, (c) thickness of the hBN, t, and (d) thickness of the dielectric substrate, h.

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Lastly, the effect of the dielectric substrate thicfkness h on the absorption spectra is presented in Fig. 5(d). As seen in Fig. 5(d), reducing h leads to a weaker absorption for all resonant modes. This can be attributed to the reduction of phonon resonance intensity which depends on the thickness of the substrate and thus, further weaken the plasmon-phonon coupling.

4. Conclusion

In summary, by combing patterned graphene with hBN, we have theoretically demonstrated the tunable multi-wavelength absorption within the hybrid structure. The electric field and $P_{abs}$ distributions are investigated to elucidate the intrinsic mechanisms, and it is found that the hybrid plasmon-phonon polaritons modes, originate from the coupling between surface plasmons in patterned graphene and second phonons in hBN, are responsible for the multi-wavelength absorption. Moreover, by applying a voltage to the connected graphene patterns and the gold ground plane, one can easily change the chemical potential of graphene and tune these absorption peaks. This work can broaden the polariton coupling study which leads to the control of the plasmon-phonon interaction within a hybrid graphene-hBN structure, providing new potential application in the mid-IR detection.