Results and Discussions In the theoretical model, the vdW heterostructure is modeled as a 2D infinite waveguide (Fig. 1). The thicknesses of top (ε^^[1]) and bottom layers (ε^^[2]) are d₁ and d₂, respectively. It is sandwiched between two semi-infinite plates, which act as the substrate (ε_s=ε^[3]) and cover layer (ε_c=ε^[0]). The electromagnetic modes [Eqs. (S8)-(S10), Supporting materials] and the associated polariton dispersions [Eq. (4)] are obtained by solving the Maxwell’s equations upon the continuities of the electric and magnetic fields at interfaces. With the input of dielectric functions of monolayer graphene [Eq. (2)] and α-MoO₃ [Eq. (1)], the calculated polariton dispersions and contours of the α-MoO₃/Gr heterostructure are shown in Fig. 1 and Fig. S4 in the Supporting materials. In comparison with the pristine monolayer graphene and α-MoO₃ lamina, due to the isotropic plasmon polariton in the graphene, the dispersion contours of the heterostructure are more complex and distorted along the [100] and [001] directions. Therefore, the hybridized plasmon-phonon polaritons can propagate along these directions that are forbidden respectively for the phonon polaritons in Restrahlen Band 1 and Band 2 in α-MoO₃ lamina (Fig. 2). The theoretical results are further corroborated by numerical simulations using FEM (Figs. 1 and 2) and experimental nano-imaging measurements (Figs. 3 and 4). Moreover, the influence of the thickness of α-MoO₃ lamina on the polariton hybridizations in α-MoO₃/Gr heterostructure is also investigated. Because the polariton fields are of evanescent nature, by reducing the α-MoO₃ thickness, the hybridized polaritons converge to the plasmon polariton in monolayer graphene, while with the increase in the α-MoO₃ thickness, the polaritons in the heterostructure evolve into the phonon polaritons in the pristine α-MoO₃ (Fig. 4). The results are also corroborated by the nano-imaging measurements (Fig. 4).

Two-dimensional (2D) van der Waals (vdW) crystals like graphene and α-MoO₃ can support polaritons in the spectral range from terahertz to mid-/far-infrared regime, enabling nanoscale confining, focusing, and controlling of the electromagnetic fields. The hybridization between different polaritons can further enrich the properties of polaritons and bring more degrees of freedom for the regulation of electromagnetic fields at the nanoscale. In this paper, we studied the hybridization of plasmon polaritons and phonon polaritons in a heterostructure composed of an α-MoO₃ vdW thin lamina stacking onto a monolayer graphene. An analytical waveguide model was developed to calculate the polariton propagation characteristics in the vdW heterostructure. The dispersion contours, dispersion relations, and localized electromagnetic field distributions of the hybridized polariton waveguide modes were derived. The theoretical results were then verified by real-space optical nano-imaging and numerical simulations. Our study can provide a quantitative model for the calculation of the hybridized polariton waveguide modes in vdW heterostructures, which can help further exploring the interactions between different types of polaritons in 2D vdW crystals.

In our theoretical model, the vdW heterostructure is treated as a 2D infinite waveguide supported onto SiO₂ substrate, which consists of a monolayer graphene of 0.5-nm thickness and an α-MoO₃ lamina of 115-nm thickness (Fig. 1). The dielectric functions of the graphene and α-MoO₃ are described using the Drude model and Lorentz model, respectively. Because in α-MoO₃ the polaritons can approximately be treated as transverse magnetic (TM) mode, the electromagnetic modes and the associated dispersion relations of the heterostructure can then be obtained by solving the Maxwell’s equations upon the continuities of the electric and magnetic fields at interfaces. Monolayer graphene was grown by the chemical vapor deposition (CVD) method. Microfabrication technique combining electron beam lithography (EBL) and reactive ion etching (RIE) was employed to pattern the graphene into microstructures. The vdW α-MoO₃ laminas were grown using a physical vapor deposition method. Dry transfer method was utilized to prepare the α-MoO₃/graphene (α-MoO₃/Gr) heterostructures, where the monolayer graphene microstructures were covered with α-MoO₃ laminas of different thicknesses. Real-space nano-imaging was conducted using a scattering-type scanning near-field optical microscope (NeaSNOM, Neaspec GmbH, Germany). In a specific measurement, a metal-coated tip (Arrow-IrPt, Nanoworld, Switzerland) was illuminated using a mid-infrared laser (Access Laser, USA) with a wavelength range of 9.20-10.70 μm (934.5-1087.0 cm^-1). The tip was vibrated vertically with a frequency of about 280 kHz. The backscattered light from the tip was detected in a pseudo-heterodyne interferometric manner, where the scattered light was demodulated at the fourth harmonic of the tip vibration frequency. The optical and morphological images of the sample can be simultaneously obtained by scanning the heterostructure underneath the tip. For the numerical study, the real-space polariton waves were manifested as the real-part of the z-component of the electric field, Re(E_z), on the surface of the SiO₂ substrate. They were calculated using the finite element method (FEM) simulations (COMSOL Multiphysics). A vertically-polarized electric dipole source was fixed above the α-MoO₃ with a separation of 50 nm (Fig. S5, Supporting materials). The thicknesses of the air, α-MoO₃, graphene and SiO₂ layers were 500 nm, 115 nm, 0.5 nm and 500 nm, respectively. The permittivities along the three principle axes were calculated according to Eq. (1). The anisotropic dielectric tensors of the α-MoO₃ layer are written as ε^=[ε_x, 0, 0; 0, ε_y, 0; 0, 0, ε_z] and ε^=[ε_xx, ε_xy, 0; ε_yx, ε_yy, 0; 0, 0, ε_zz]. These tensors were imported into the COMSOL package to solve the Maxwell’s equations. For the monolayer graphene and α-MoO₃ lamina, the Re(E_z) was monitored respectively on the planes 5 nm away from their upper surfaces, while for the α-MoO₃/Gr heterostructure, the Re(E_z) was monitored on the plane 4 nm away from the graphene upper surface.

In conclusion, we have established a theoretical model to investigate the hybridizations of plasmon polaritons and phonon polaritons in a heterostructure consisting of an α-MoO₃ lamina covering a monolayer graphene. The propagation characteristics, including the polariton dispersion relation, in-plane dispersion contour, and localized electromagnetic field distribution, were calculated and studied. It is revealed that due to the hybridization effect, the α-MoO₃/Gr heterostructure is able to support polariton propagation along the directions that are forbidden for the phonon polaritons in pristine α-MoO₃ lamina. Additionally, the influence of the α-MoO₃ thickness on the polariton hybridization in the heterostructure was also investigated, indicating that when the α-MoO₃ lamina was thinner/thicker, the hybridized polaritons became more plasmon/phonon polariton-like. The theoretical results were corroborated respectively by the numerical simulations and experimental nano-imaging measurements. We strongly believe that the results obtained in our study can on one hand provide a theoretical model for analytically studying the polariton hybridizations in vdW heterostructures, and on the other hand help further our understanding on the polaritonic physics in low-dimensional materials.