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Abstract
In this paper, we utilize a heterostructured graphene/hBN/graphene nanodisk array to implement an electrically tunable absorber in and out of the Reststrahlen band (RSB) region of hBN. Tuning of phonon-type resonance absorption in the RSB region is achieved through phonon-plasmon-polariton hybridization. The hybrid phonon mode enabled a 290 nm shift of the resonant wavelength, and the sensitivity of absorption peak to the electrical control is 362.5 nm/eV. Simultaneously, the nearly perfect absorption is obtained in the condition of high chemical potential of graphene. Moreover, the plasmon polaritons are strongly modified by phonon polaritons of hBN, so the FWHM of absorption peaks out of the RSB region reduce to 45-49 nm, and the maximum Q of absorption reaches 220.44 at EF=0.65 eV, which is paving a way toward coherent emission at the atmospheric transparent band. Importantly, graphene-assisted hyperbolic phonon polaritons of hBN will enable future phonon devices with high optical performance and wide tunability.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hyperbolic materials are anisotropic media for which the in-plane and out of-plane components of the permittivity tensor have opposite signs [1]. Natural hyperbolic materials in which atomic layers are bonded together through van der Waals (vdWs) forces hold the key to unlock the full potential of hyperbolic media in nanophotonics [2,3]. Hexagonal boron nitride (hBN) is a prototypical hyperbolic material that supports volume-confined hyperbolic phonon polaritons (HPP) [4,5]. The recent discovery of strongly confined phonon-polariton modes in hBN enabled a series of major advances in nanophotonics in the mid-IR wavelength region, including sub-diffraction imaging [6–9], negative refraction [10,11] and molecular vibrations sensor [12]. In contrast to the surface plasmon polaritons (SPP)-collective oscillations of free electrons at metal or semiconductor surfaces coupled to electromagnetic fields, HPP is caused by the coupling of electromagnetic fields and lattice vibrations, which exists in the Reststrahlen band (a spectral range between the longitudinal and transverse optical phonons frequencies, defined as LO and TO, respectively). Importantly, the hyperbolic material hBN offers an opportunity to simultaneously achieve sub-diffraction confinement [3], low losses [4], and remarkably lifetimes in the picosecond range [10] through the excitation of HPP modes. Recent advances in the growth and exfoliation of high-quality hBN enabled high propagation momentum of HPP with kp up to 25k0 on a three-monolayer hBN flake [13].
However, HPP in hBN is still weak in active tuning functionalities, as phonon polaritons originate from intrinsic lattice vibrations, which hinders hBN’s application for tunable and reconfigurable and limits its further nanophotonics application. Selecting active materials is most commonly employed to realize indirect tuning. Recent works have demonstrated that the phonon polaritons in hBN can be modified by means of phonon-plasmon-polariton hybridization in a graphene-hBN heterostructure [14–17], which is based on the tailored dispersion of hybridization modes. Also, these heterostructures with phonon-plasmon-polariton hybridization can be effectively applied in absorption devices [18–20], emission cooling [21], and single-photon sources [22]. In addition, Ge3Sb2Te6 (GST) and VO2 as phase-change materials (PCMs) can serve as the versatile platforms to arbitrarily control the propagation of polaritons at the nanoscale. Since the phonon polaritons remain sensitive to local changes in the dielectric function of the ambient environment, the confined HPP in hBN can interact with spatially localized phase transitions of the PCM through the variation of surrounding dielectric environments. Exploiting the PCM to tune the HPP enabled potential applications on refractive elements [23], sub-wavelength focusing [24], and absorber switch [25].
Since large-scale monolayer graphene can be grown easily using the chemical vapor deposition (CVD) compared to other 2D materials, and SPP can be excited on nanopatterned graphene to increase the light-graphene interaction, thereby enhancing the infrared absorption [26–28]. Also, 1D hBN grating are most commonly employed to realize perfect resonance absorption [29]. Depending on the nanostructure design, the tunable mid-infrared absorber with omnidirectional nearly perfect resonance absorption was theoretically demonstrated with respect to the nanopatterned graphene/hBN layer structure [20], whereas lacking the quantitative analysis about the tunable nearly perfect absorption peak in the RSB of hBN. In addition, high-Q resonance with a large modulation is obtained through cascading graphene layers with hBN interlayer, thereby enhancing the sensitivity to ambient environment [30]. It is verified experimentally that the infrared plasmonic response enhanced by exploiting a graphene multilayer stack [31]. In this paper, we investigate the resonant absorption properties of the heterostructured graphene/hBN/graphene nanodisk array. Through illustrating of phonon-plasmon-polariton hybridization in dispersion map, aiming to obtain the tunability absorption caused by modifying HPP mode, and obtain sharp absorption peaks resulted from the modified SPP mode. The resonance absorption in and out of the Reststrahlen band can be well predicted based on phonon-plasmon-polariton hybridization.
2. Design and numerical simulations
To investigate the absorption properties caused by the phonon-plasmon-polariton hybridization in hBN and graphene, we employ the heterostructure that composed of a hBN film sandwiched between two monolayer graphene sheets, and design them into nanodisk arranged in a triangular lattice with center-to-center spacing d, radius r, thickness dg+dh+dg (dg and dh are the thickness of monolayer graphene and hBN film, respectively). As shown in Fig. 1, the graphene/hBN/graphene-layer (G/hBN/G) nanodisk array is placed on the top of a gold reflector that is separated by a CaF2 spacer layer ds. CaF2 is an excellent candidate with lower dispersion in the mid-IR region compared to other dielectric materials, e.g., Al2O3 or SiO2. And the dielectric parameter of CaF2 is taken from the empirical Sellmeier approximation [32]:
Fig. 1. Schematic of the patterned G/hBN/G-layer nanodisk arranged in a triangular lattice on the top of CaF2 spacer and gold reflector. The parameters: center-to-center spacing d=200 nm, radius r=80 nm, thickness of hBN dh=35 nm, thickness of monolayer graphene dg, and thickness of CaF2 spacer ds=270 nm.
hBN is an anisotropic material with two active optical phonon modes in the mid-infrared region, and its relative permittivity can be characterized by a diagonal tensor:
Fig. 2. Relative permittivity of hBN showing the existence of two reststrahlen bands (shaded regions). Type I: ${\varepsilon _\parallel } < 0$, ${\varepsilon _ \bot } > 0$, Type II: ${\varepsilon _ \bot } < 0$, ${\varepsilon _\parallel } > 0$.
Graphene is modeled as surface impedance $Z(\omega ) = 1/\sigma (\omega )$, and the surface conductivity of graphene σ(ω) is calculated using random phase approximation (RPA) [33,34]:
In order to demonstrate the phonon-plasmon-polariton hybridization in hBN and graphene, the dispersion of hybrid polaritons in graphene/hBN/graphene-layered structure can be derived from complex reflectivity ${r_p}$, it can be calculated using the transfer matrix formalism [36]. We use the air/graphene/hBN/graphene/CaF2 structure (Fig. 3) to illustrate the dispersion properties, and assume the first and second monolayer graphene to be located at the two interfaces (i.e., the planes of z = 0 and z = dh) between region 1 (z > 0, air), region 2 (-dh < z <0, hBN), and region 3 (z < -dh, CaF2), where dh is the thickness of hBN. In our case, three layers are included, yielding the following analytical expression for M:
Therefore, according to the Eq. (8), the dispersion of hybrid mode can be calculated for an appropriate choice of dielectric functions of hBN and graphene and hBN’s thicknesses.
We present the dispersion map of HPP of hBN via the above calculations for air/hBN/CaF2 structure, and the dispersion map of SPP of monolayer graphene on the CaF2 layer with EF=0.5 eV, 0.7 eV and 1.0 eV as depicted in Fig. 4(a). The dispersion is visualized using a false-color map of the imaginary part of the reflection coefficient rp (for the case of P-polarization). It may be observed that HPP-SPP coupling takes place in the RSB region.
Fig. 4. (a) Calculated dispersion of the hyperbolic phonon polaritons of h-BN with dh=35 nm, and the surface plasmon polaritons of graphene with EF=0.5 eV, 0.7 eV and 1.0 eV (the arrow indicates the increase of EF). (b) Absorption of the individual hBN nanodisk array in the RSB region (shaded region) and the monolayer graphene nanodisk array out of RSB region.
Moreover, in contrast to the graphene-hBN heterostructure, Fig. 4(b) shows the absorption of individual hBN nanodisk and graphene nanodisk array with identical geometry parameters (r=80 nm, d=200 nm, and ds=370 nm). For the hBN nanodisk array, the absorption peak is located at 6.967 µm, the maximum reaches to 83%, and the FWHM is 69 nm. As for the graphene nanodisk array, with the Fermi energy of graphene varying from 0.6 to 1.0 eV, the absorption peak blue shifts 3.06 µm, and the FWHM decreases from 223 nm to 142 nm. It is obviously to see that the FWHM of resonant absorption in hBN nanodisk array is narrower than that in monolayer graphene nanodisk array, which contributes to the lower loss constants in phononic media that sustaining phonon-polaritons compared with that of the plasmonic materials in the mid-IR spectral range [4].
3. Results and discussions
According to the dispersion calculation above, we can manipulate the phonon polaritons of hBN by means of strong coupling between graphene plasmon and hBN’s phonon polaritons in graphene-hBN heterostructures. The peculiar coupling response is illustrated by the calculated wavelength (λ)/momentum (q) dispersion relations of its polariton modes, which is visualized using a false-color map of Im(rp). Figures 5(a) and 5(c) present the dispersion map of the G/hBN-layer with EF=0.6 and 1 eV, respectively. Clearly, the dispersion curves of phonon polaritons of hBN in RSB are strongly modified compared with that of individual hBN layer [see Fig. 4(a)], and the original HPP of hBN (HP2) is changed to the hyperbolic phonon-plasmon polaritons (HP3). Also, the dispersion branch moves to lower q as the Fermi energy of graphene increases. Following the previous reports [14], we refer to the collective modes existing outside the RSB as surface plasmon-phonon polaritons (SP3). In addition, for the G/hBN/G-layer structure as shown in Figs. 5(b) and 5(d) with EF=0.6 and 1 eV, respectively, two dispersion branches in RSB are modified, and the second dispersion branch possesses higher q, and gives more tuning region as varying the Fermi energy of graphene.
Fig. 5. Calculated dispersion of the hybrid phonon-plasmon-polaritons in (a) G/hBN-layer, and (b) G/hBN/G-layer structure with EF=0.6 eV and dh=35 nm, and in (c) G/hBN-layer, and (d) G/hBN/G-layer structure with EF=1.0 eV and dh=35 nm.
Figures 6(a) and 6(b) provide the absorption spectra in the RSB with respect to the G/hBN and G/hBN/G-layer nanodisk structure, respectively. Since the phonon polaritons of hBN is obviously modified by the plasmons polaritons of graphene, the modification of the hyperbolic phonon response by graphene is clearly manifested in the blueshift of absorption wavelength compared with that of individual HPP in hBN [see Fig. 4(b)]. Importantly, the resonance absorption caused by the HP3 mode in RSB region is continuously tunable under the control of the graphene’s Fermi energy. The absorption peaks reach to above 90%, even are close to 100% at EF=1.0 eV, i.e. near to the perfect absorption, so, high chemical potential of graphene is conducive to perfect absorption in RSB region. And the resonant wavelength blue shifts as the Fermi energy of graphene increases. With the Fermi energy of graphene varying from 0.2 to 1.0 eV, the absorption peaks are shifted from 164 to 290 nm (the sensitivity to the electrical control are 205 nm/eV and 362.5 nm/eV) for G/hBN-layer and G/hBN/G-layer nanodisk structure, respectively [see Fig. 6(a) and 6(b)], which indicates that the tunability of hyperbolic phonon polaritons of hBN have achieved indirectly. Also, the sensitivity to the electrical control for the triple-layer structure is superior to that of the double-layer nanodisk array. To quantitatively evaluate the tunability of the proposed absorber, Figs. 6(c) and 6(d) show the resonance wavelength λ0 with the variation of graphene’s chemical potential and their quadratic fitting as ${\lambda _\textrm{0}}\textrm{ = 6}\textrm{.91 - 0}\textrm{.095}{E_F}\textrm{ - }0.091{E_F}^2$ and${\lambda _\textrm{0}}\textrm{ = 6}\textrm{.91 - 0}\textrm{.147}{E_F}\textrm{ - }0.\textrm{183}{E_F}^2$for G/hBN-layer and G/hBN/G-layer nanodisk array, respectively. The resonant position in RSB region nonlinearly shifts to shorter wavelength as Fermi energy increases, which caused the tunability of hyperbolic phonon polaritons in hBN’ RSB region. Moreover, it is noteworthy to the excellent properties with respect to the graphene-encapsulated hBN structure (i.e. the G/hBN/G-layer nanodisk array), which is because that the dispersion curve is more easily manipulated by the Fermi energy. In this case, the hyperbolic phonon polaritons in hBN is modified strongly and carries the graphene’ tunable attribution.
Fig. 6. Absorption of (a) G/hBN-layer and (b) G/hBN/G-layer nanodisk array in the RSB region. Dependence of HP3 resonation wavelength in the RSB region on the graphene’s chemical potential and their quadratic fitting of (c) G/hBN -layer and (d) G/hBN/G-layer nanodisk array. The parameters: dh=35 nm, r=80 nm, d=200 nm, and ds=370 nm for G/hBN-layer nanodisk array, and ds=270 nm for G/hBN/G-layer nanodisk array.
In addition, except for the HP3-type resonance absorption in the RSB region, SP3-type sharp absorption peaks appear out of the RSB region (8-12 µm). For the G/hBN-layer nanodisk array as shown in Fig. 7(a), with the Fermi energy of graphene varying from 0.75 to 1.0 eV, the absorption peak blue shifts 1.03 µm, and the change of chemical potential from 0.75 to 1.0 eV cause a 1.02 µm wavelength shift from 10.13 to 9.11 µm for the G/hBN/G-layer nanodisk array [Fig. 7(b)]. So, the dependence of absorption peaks resulted from SP3 mode on the chemical potential is almost identical for the proposed double and triple-layer nanodisk array. Although the tunable range is suppressed in comparison with that of monolayer graphene nanodisk [see Fig. 4(b)], the absorption peak blue shifts entirely and move the spectra to 9-11 µm region, and the peaks maximum reach to above 90% mostly. Additionally, compared to the graphene nanodisk array, the FWHM in the absorption spectra of these proposed hybrid structures are getting narrowed as shown in Figs. 7(c) and 7(d). Particularly, the FWHM of SP3-type absorption peaks reduce to 45-49 nm. We present the values of Q for different Fermi energy (Q=λ0/FWHM). The maximum Q of absorption reaches 220.44 at EF=0.65 eV for the G/hBN/G-layer nanodisk array. Also, the FWHM in the RSB region even reach to 37 nm, the maximum Q reaches 184.63 at EF=0.2 eV for the G/hBN/G-layer nanodisk array. The phonon polaritons with longer lifetime enabled stronger electric field confinement, which manifests as higher and sharper absorption (lower FWHM) in the spectral response, then in this case, the SP3 mode carries the attribution of phonon polaritons.
Fig. 7. Absorption of (a) G/hBN-layer and (b) G/hBN/G-layer nanodisk array out of the RSB region. Q value and FWHM of the proposed structure (c) in and (d) out of the RSB region as a function of chemical potential of graphene.
Furthermore, we stress the distinction between the field distributions in HP3 and SP3 spectral regions. The Ez-field distributions at 9.107 µm [ Figs. 8(a) and 8(b)] and 6.582 µm [Fig. 8(c) and 8(d)] are shown to illustrate the difference of SP3 and HP3 mode, respectively. Figure 8(a) shows the x–y-plane of E field distribution of SP3 mode, which is mainly in the vicinity of nanodisks, and both nanodisks oscillate in-phase. In the view of x–z-plane [Fig. 8(b)], the SP3 mode is localized at the graphene/h-BN interface and decays evanescently in the interior of the h-BN, which indicates the attribute of surface wave for the SP3 mode. This SP3 mode has a long lifetime due to the coupling of SPP and HPP mode, so, it exhibits stronger electric field confinement. Whereas the E field distribution of HP3 mode is concentrated inside the hBN, and propagate through the entire nanodisk as a guided mode despite the small thickness, each nanodisk can be considered as a separate resonator as shown in Fig. 8(c), and the collective resonance indicates a higher and sharper absorption in RSB region. Also, the x–z-plane field [Fig. 8(d)] reveal the Fabry–Perot resonances in the h-BN resonators with characteristic of a standing wave [6].
Fig. 8. Ez-field distribution of G/hBN/G-layer nanodisk array in the (a) x–y-plane and (b) x–z-plane at the resonant wavelengths of 9.107 µm, and in the (c) x–y-plane and (d) x–z-plane at the resonant wavelengths of 6.582 µm. The parameters: dh=35 nm, r=80 nm, d=200 nm, ds=270 nm, and EF=1.0 eV.
In addition to considering the gate control using graphene’s chemical potential, the absorption responses are easily affected by relaxation time of graphene as well. The relaxation time of graphene is not only decided by EF, also depends on the carrier mobility of graphene, so, the quality of sample preparation in experiment has a great impact on τ. We show the dependence of absorption FWHM supported by HP3 and SP3 resonance on τ as depicted in Fig. 9(a). In our case, the G/hBN/G-layer nanodisk array with above parameters and EF=0.8 eV is selected. Obviously, for HP3-type resonance absorption, the FWHM change slightly with varying τ, and the FWHM is about 50 nm, which is considered that the spectra responses are almost unaffected by τ in the RSB region. Whereas the FWHM of HP3-type resonance absorption strongly depends on the τ, and is broadened as reduction of τ. Additionally, to establish the feasibility of the proposed absorbers, the dependence of absorption to the angle of incidence and polarization need to be considered. Figures 9(b) and 9(c) illustrate the absorption with varying incident angle in TE and TM polarization, respectively. The peaks location is almost independent of the angle of incidence for G/hBN/G-layer nanodisk array in TE polarization, but have a slight redshift in TM polarization as increasing the incident angle. The SP3-type absorption decreases when θ=60°, but still reaches to 80% in TE polarization, and the perfect HP3-type absorption could operate over a wide-angle range from 0° to 60° in TM polarization.
Fig. 9. (a) Dependence of absorption FWHM supported by HP3 and SP3 resonation on the relaxation time of graphene. Absorption of G/hBN/G-layer nanodisk array with EF = 0.8 eV in (b) TE and (c) TM polarization at θ=0°, 15°, 30°, 45° and 60°.
4. Conclusions
In conclusion, we designed an electrically tunable dual-waveband absorber consisting of G/hBN/G-layer nanodisk array for operating in the mid-IR region. The hybrid HP3 and SP3 modes are presented through calculating dispersion map, for overcoming the limitations of the individual HPP and SPP. Hence, the optimized G/hBN/G-layer nanodisk absorber exhibits a superior tunability to the electrical control in the RSB region. The blueshift of the absorption peak is 290 nm (EF=0.2-1.0 eV), and the sensitivity to the electrical control is 362.5 nm/eV, which indicates that HP3 mode possesses the tunable attribution from graphene. Also, the absorption peaks reach to above 90%, even the nearly perfect absorption is obtained at EF=1.0 eV. Furthermore, the resonance absorption peaks of G/hBN/G-layer nanodisk array out of the RSB region exhibit narrow FWHM (45-49 nm), and the maximum Q of absorption reaches 220.44 at EF=0.65 eV, which is modified by HPP mode and inherit the hBN’s advantages with low losses and long lifetime. Additionally, the proposed absorber can operate over a wide-angle range, and almost independent to the incident angle and polarization. The proposed hybrid structure expands the application of phonon-type devices for mid-infrared nanophotonics and could enable the creation of novel actively tunable, low-loss application at the nanoscale.
Funding
National Natural Science Foundation of China (11675046, 62075048).
Disclosures
The authors declare no conflicts of interest.
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