1. Introduction

Polar dielectrics and van der Waals (vdW) materials have recently attracted significant attention in nanophotonics because of their unique physical properties with regards to light-matter interaction such as supporting tunable excitations of half-light and half-matter polaritons which exhibit long lifetimes, low-losses and ultraslow group velocity [1–9]. In particular, polar crystals, such as silicon carbide (SiC), hexagonal boron nitride (h-BN) and quartz, support strongly confined hybrid light-matter waves such as surface phonon polaritons (SPhPs) in their Reststrahlen band (negative permittivities) in the infrared spectrum [7–12]. Since phonons have inherently long lifetimes (~1ps) and low optical loss [3], the SPhPs have lower propagation losses and longer lifetimes as opposed to surface plasmon polaritons (SPPs) (which are hybrid-waves of free charge carriers and photons) typically supported by metals, since the later suffers from high intrinsic optical losses with a lifetime of ~10 femtoseconds [13].These losses primarily due to free charge carrier scattering by impurities or phonons reduces the transmission output, and shortens the propagation lengths of SPPs.

The recent emergence of atomically-thin, two-dimensional (2D) van der Waals materials (vdWs) of varying electronic types enables new avenues for light-matter interactions from ultraviolet-visible to terahertz spectral ranges [14,15], as hosts for ultra-confined polaritonic modes within atomically thin volumes [16,17]. The ability to isolate these materials into atomically-thin, self-passivated with strong in-plane bonding, allows arbitrary stacking of “all-surface” layers into complex van der Waals heterostructures. As a result, optical excitations or hybrid waves supported in these materials are amenable tuning by externally applied electric fields and changes in the dielectric environment. For instance, 2D semimetals such as graphene and small band gap semiconductors such as black phosphorus (BP) exhibit tunable and highly confined SPPs [18–20], which have led to applications such as molecular sensing, tunable THz generation and electro-optical light modulation [21–24]. Likewise, natural hyperbolic materials such as h-BN [10–12,25] and MoO₃ [26,27] support low loss and volume confined hyperbolic phonon polaritons, which exhibit ray-like propagation, high quality factors and hyperlensing effects in the infrared. In the visible frequency regime, transition metal dichalcogenides (TMDCs) support exciton-polaritons which were recently demonstrated in MoSe₂ with a small polariton wavelength (down to 300 nm) and a long propagation length (up to 12 μm) [28].

Among the above cases, the SPPs in graphene [29] and phonon polaritons in h-BN [5,9] have shown the most promise with confinement factors (β = λ₀/λ_p), (defined as the ratio of polariton wavelength to free space photon propagation wavelength), of over 100 have been achieved, which means that infrared-THz light (1-100 ums in free space) can be squeezed down to a wavelength of few hundred nanometers in these polaritonic solid state media. This ability to localize light to achieve deeply subwavelength control is of great technological significance as it allows the integration of the merits of both electronics and photonics at high device density into a single technology justifying the great interest in vdW crystals for nanophotonics. Bulk polar dielectrics and metals can also support SPhPs and SPPs respectively. However, they are limited by two disadvantages as compared with 2D van der Waals layers for prospective applications: first, with SiC’s bulk SPhPs one cannot achieve arbitrary confinement at any frequency within the Reststrahlen band. These small and fixed confinement factors therefore limit the control of light squeezing capabilities using bulk SPHPs in SiC and second, the intrinsic lattice vibrations of polar crystals are bosonic and charge neutral in nature limiting their active tunability with external electric fields. More recently, a new strategy for highly confined SPhPs has been proposed by employing ultrathin dielectric environment on polar crystals [7,8,30]. For instance, a recent experiment reported 190-times squeezed SPhP in a heterostructure of atomic-thin vdW dielectrics on SiC [8]. Likewise, reversible optical switching of squeezed SPhPs (~70 times free space wavelength) has also been recently demonstrated by controlling the structural phase of a phase-change material (PCM) employed on quartz substrates [7], which has later been reproduced on SiC substrates [30].

In this work, we propose a mixed-dimensional heterostructure [31] of 2D monolayer graphene and few layer vdW dielectrics (MoS₂ and h-BN) on bulk 3D SiC. Due to the strong coupling between graphene (g-) plasmon and vdW dielectrics tailored (d-) SPhP, such an heterostructure forms a hybrid between the two excitations referred to as g-SPhP, which exhibits high confinement factors (>100) and electrical tunability. By laterally patterning the ultrathin vdW dielectrics between the 2D graphene and 3D SiC, these propagating g-SPhPs can be confined to form hybrid localized surface phonon-plasmon polariton resonance (LSPh-SPR). This nanopatterning of the vdW spacer dielectric to support highly confined LSPh-SPRs is feasible in terms of fabrication since the vdW dielectric is barely few nm in terms of thickness and advantageous in terms of performance as opposed to high aspect ratio dielectric resonators (width/height~250/1000nm) created from bulk SiC [2,32–34], here the height of vdW dielectric antennas are within only a few tens of nanometers, which is capable for extreme near-field enhancement and ultra-compact devices (λ₀³/v_a³~100³, the antenna valume is defined as v_a), despite operating at free-space wavelengths of 10-13 um. Due to the relatively low-loss in SiC’s phonon mode, the LSPh-SPRs retain a narrow resonance linewidth (△ω_res~10cm⁻¹) and correspondingly high-quality factor (Q = ω_res/△ω_res~85), which is nearly an order of magnitude larger than plasmonic modes. Furthermore, the collective resonance behaviors with graphene plasmon, enable such LSPh-SPRs outstanding tunability. By tuning the Fermi Level of graphene with a gate voltage, the LSPh-SPRs result strong free-space modulation of reflection in infrared region with modulation depths > 95%. Such tunable LSPhRs with ultra-compact volume and high-quality factors, are suitable for applications such as infrared biosensors, electro-optic modulators and photodetectors at room temperature.

2. Hybrid phonon-polaritons at 2D multilayer and SiC heterointerfaces

2.1 Highly confined d-SPhP and tunable g-SPhP at 2D/SiC heterointerfaces

The schematics of bulk SPhP on SiC, ultra-thin dielectric tailored d-SPhP in heterostructure of MoS₂/SiC, and graphene plasmon hybridized g-SPhP in heterostructure of graphene/h-BN/SiC are depicted in Fig. 1. To compare their light confining capabilities, we first analytically derive a transfer matrix formalism (see Appendix C.) to trace their dispersions, as shown in Fig. 1(b, e, h), respectively. Here, k_p is normalized by free-space wave-vector k₀, and the confinement factor is defined by β = k_p/k₀, which also represented by the reduced polariton wavelength λ_p relative to the free-space value λ₀. Such polaritons can be supported over a spectral range referred to as the ‘Reststrahlen’ band, between the longitudinal (LO) and transverse optic (TO) phonon frequencies of SiC, where the permittivity is negative (ε<0) (see Appendix A.) [3]. The confinement factor of bulk SPhP (Fig. 1(b)) is smaller than 4. For instance, as shown by the inset z-components of the electric field distributions (E_z), the wavelength of λ_SPhP = 4.9 μm is only 2.15 times shorter than free space wavelength λ₀ = 10.58 um. The corresponding mode profile (Fig. 1(c)) and electric field distribution (Fig. 1(b) inset) suggest that the mode largely extends out into free space and bulk SiC on either side of the surface for up to micron scale thicknesses thereby minimizing confinement.

 

Fig. 1 (a) Schematic, (b) dispersion, and (c) electric E_z field mode profile for SiC based bulk SPhP, inset (b) show the E_z fields of the propagating SPhP mode. (d-f) The results for vdW dielectric tailored d-SPhP in multilayer MoS₂ (5nm) on SiC. (g-i) The results for electric tunable graphene plasmon hybridized g-SPhP in heterostructure of graphene/h-BN (5nm) on SiC, the Fermi level (μ_c) of graphene is 0.3ev.

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Transition metal dichalcogenides (TMDCs) are attractive materials for optoelectronic applications in the visible and near-infrared range. Thus far, it has been overlooked that multilayer TMDCs also have high in-plane refractive indices. For instance, the real part of index of MoS₂ is ~3.8 at infrared frequencies (see Appendix A.). Owing to this, atomically thin layers of MoS₂ on SiC can permit the study of extreme limits of confinement in aforementioned d-SPhPs, as has been reported experimentally in [8]. The concept is similar to d-SPPs on a thin dielectric layer on a metal [35], the d-SPhP mode also exists in our air/MoS₂/SiC stack (Fig. 1(d)): in the limit of large k_, the mode does not “see” much of the outside material (air), but exhibits subwavelength modal size within the dielectric slab and very slow group velocity over an large frequency bandwidth. As shown in Fig. 1(e), the confinement factor of the d-SPhP in 5nm MoS₂ on SiC is greatly enhanced, exceeding 100, the corresponding polariton wavelength (Fig. 1(e) inset) of d-SPhP squeezed down to a scale of ~λ₀ = 11.25 μm to λ_p = 110 nm. Upon further inspection of electric field (Fig. 1(e) inset) and mode profiles (Fig. 1(f)) of d-SPhP, we observe that the mode is primarily confined at the upper and bottom interfaces of MoS₂, with the fields extending out to ~100 nm height in z-direction on either side and only a small part of the electric field is confined in the ultrathin high-index dielectric slab, indicating that it is character is still similar to that of a the bulk SiC based SPhP shown in Figs. 1(b) and 1(c). Furthermore, for d values smaller than a certain threshold, the mode exhibits a regime of negative group velocity (v_g = ∂ω/∂k_p), as observed by the decreasing slope in the dispersion shown in Fig. 1(e).

The natural negative v_g of d-SPhP has not been effectively utilized in prior works. If such d-SPhP travels into another slab that supports polaritons with positive v_g, the refracted polariton may stay on the same side of the normal breaking Right-hand’s law of refraction, resulting in negative refraction in-plane (Fig. 3(a)). A prominent application of negative refraction is the superlens [36], a planar device capable of focusing all the spatial Fourier components of a source, realizing a perfect image in the 2D plane, its realization in this vdW materials based polaritons is attractive and has realistic advantages in terms of applications [37], particularly since the low-loss and highly confined nature of the polaritons presents new opportunities for applications such as near-field nano-imaging, and tunable, integrated optics. To obtain a positive v_g polariton with minimum perturbation, we can cover monolayer graphene on the dielectric (such as h-BN, Fig. 1(g)) to form a typical metal-insulator-metal (MIM) gap waveguide mode (g-SPhP) [38]. This MIM stack can further flip the sign of g-SPhP group velocity back to positive, as shown in Fig. 1(h). Since both graphene and SiC feature negative permittivities in this infrared frequency regime, we find an ultraconfined out-of-plane E_z electric field (Fig. 1(i)) in the gap between the graphene and the SiC, this is typical for acoustic graphene plasmon [39,40], characterized by a nearly linear dispersion at small wavevectors (white dot line in Fig. 1(h)). Here, h-BN was chosen as the middle layer dielectric, both graphene and h-BN have honeycomb lattice with hexagonal symmetry, and h-BN has been demonstrated as the best encapsulation layer for graphene. Thus, the technology of graphene/h-BN/SiC heterostructure is feasible in practical fabrication. Further, g-SPhP originates from the hybridization of graphene plasmon and the d-SPhP, and can thus be actively tuned with the Fermi level of graphene (Fig. 1(h)).

2.2 The confine factor and figure of merit of d-SPhP and g-SPhP

As mentioned above, the confinement factors (β = k_p/k₀) of the d-SPhP and g-SPhP are strongly dependent on the thickness of the dielectrics (MoS₂ and h-BN). To investigate and illustrate this further, we perform thickness dependent calculations of dispersion relations as plotted by the blue and red curves for d-SPhP and g-SPhP in Fig. 2(a), respectively. For such triple-layer structures with dielectric thickness d, the confinement factors can be simplified [8,11,41] and expressed under the large momentum limit, i.e., k ≫ k₀ as shown in Eq. (1) below. Here, ε_i is the permittivity of i layer (air, vdW dielectrics, and SiC) from up to bottom side. Since h-BN is an anisotropic material, i.e., ε̂ = diag{ε_x, ε_y, ε_z}, ε_x = ε_y≠ε_z, the components of its permittivity have different value in basal plane (ε_x = ε_y) and in normal plane (ε_z). Z₀ ≈377Ω is the free-space impedance and σ is the surface conductivity of graphene.

 

Fig. 2 Simulated dispersions of d-SPhP (green curves) and g-SPhP (red curves) for (a) different MoS₂ and h-BN thicknesses, and (b) different dielectric index layers (5nm). (c) The figure of merit (γ⁻¹) for different polaritons, the thickness of MoS₂ (green curve) and h-BN (red curve) is 5nm. The Fermi level (μ_c) of graphene is 0.3ev.

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For the case of bilayer stack d-SPhP comprising of SiC and an ultrathin high-index dielectric (MoS₂), setting σ = 0 in Eq. (1) will result in the correct confinement factor. The first proportional term in Eq. (1) explains that β increases with reducing d both for d-SPhP and g-SPhP for a fixed wavenumber which agrees well with the simulated results in Fig. 2(a). If the h-BN thickness is fixed at d = 5nm, then changing the graphene doping from 0.3ev to 0.2ev, results in an even larger confinement factor of g-SPhPs (black open circle line in Fig. 2(a)). As vdW dielectrics can be easily scaled down to atomic layer thickness, this is a practically realizable system which suggests that ultra-confined SPhPs with polariton wavelengths reduced down to a few tens of nanometers as possible to create and observe in simple heterostacks. The logarithmic term in Eq. (1) solely depends on the permittivity (or refractive index) of each layer. Figure 2(b) demonstrates the dependence of dispersion relation on the refractive index of the middle layer. The key observation is β variation with refractive index has opposite trends for the case of d-SPhP vs g-SPhP due to the opposite sign of group velocity and curvature in their dispersion relations. Therefore, while higher index spacer can help confine the d-SPhP mode to a greater degree, the g-SPhP mode is better confined with a comparatively lower index dielectric.

The figure of merit (FOM) for propagating polaritons is the losses incurred during propagation. This can be represented by the magnitude of the ratio γ^-1 = Re(k_p)/Im(k_p). We compare this ratio for the polariton modes under consideration here. For h-BN in its first Reststrahlen band (ε_x/y>0, ε_z<0) from 780 to 830 cm⁻¹, the γ^-1 of HP is smaller than 15 [5,9]. However, in our case the h-BN is behaving as a dielectric since the frequency ω >830 cm⁻¹. Hence the γ^-1 of graphene plasmon on h-BN substrate (black line in Fig. 2(c)) shows a large value that extends to 40, which coincides with the experimental results at the same frequencies in [29]. Thus, these results guarantee the prediction of γ^-1 of g-SPhP in our heterostructure of graphene/h-BN/SiC (red line in Fig. 2(c)). In addition, we also checked the γ^-1 of d-SPhP (MoS₂/SiC) in [8], the experimental γ^-1 demonstrated a lower value of 8, this proves that d-SPhP on MoS₂/SiC has higher loss again per our prediction in Fig. 2(c).

2.3 In-plane negative refraction between d-SPhP and g-SPhP

The schematic of the negative refraction between d-SPhP and g-SPhP is shown in Fig. 3(a). Here, both the thicknesses of h-BN and MoS₂ slabs are set as 10nm, and a monolayer graphene is coved on h-BN. Practically, it would be challenging to fabricate the proposed structure. However, we would also like to point out that recent advances in selective etching of h-BN and MoS₂ and use of fluorinated graphene as an etch stop [42] can allow fabrication of these structures with relative ease. In addition, according to Eq. (3) and the dispersion curves in Fig. 2(b), a negative refraction lens cannot be designed using the same 2D dielectric layer on both left and right regions. This is because the negative d-SPhP and positive g-SPhP have no intersection points in the frequency or energy domain (Fig. 2(b)) with the same dielectric layer (e.g. n_MoS2~3.8) which indicates that the momentum (k_p) and energy (ω) are mismatched. However, for our lateral junction design (Fig. 3(a)) between MoS₂ and graphene/h-BN, both the momentum and the energy can be selected from their dispersions in Fig. 3(b) to be perfectly matched. For instance, at the frequency of ω = 883cm⁻¹, corresponds to a confine factor β = 80 for d-SPhP in left MoS₂/SiC, we can tune the right graphene Fermi level μ_c = 0.3ev to realize momentum match. Namely, both the polariton wavelengths are 80 times smaller than the incident wavelength. FDTD simulations (Lumerical FDTD 3D solver) are implemented to illustrate the superlens effect, a dipole source is placed 400 nm away from the MoS₂-hBN interface at the left MoS₂ side to launch the d-SPhP mode. Experimentally, such a mode can be launched using a metallic tip such as the one used in near-field scanning optical microscopy (NSOM). The top-view of the in-plane |E|² and H_y fields are shown in Figs. 3(c) and 3(d). As can be clearly seen, all angle negative refraction occurs at theMoS₂-Graphene interface, resulting in the g-SPhP mode in the right side graphene region showing a mirror image of the dipole (H_y of Fig. 3(d)). The normalized H_y field (Fig. 3(e)) at the image plane (vertical cut by the dashed line in Fig. 3(d)) shows the resolution of the superlens. With a FWHM of the peak equaling 120 nm for a free space wavelength of ~11300 nm (883 cm⁻¹) the superlens is highly effective and suggests deep-subwavelength focusing.

 

Fig. 3 (a) Schematic of the in-plane superlens, the thickness of MoS₂ and h-BN is 10nm. (b) The dispersions of d-SPhP and g-SPhP, particularly g-SPhP is electric tunable by controlling graphene’s Fermi level. (c, d) The distributions of in-plan |E|² and H_y field (5nm above SiC substrate) at ω = 883 cm⁻¹ and μ_c = 0.3 ev, corresponds to a momentum matched n_eff = 80 for both polaritons. (e) The normalized H_y field at the image plane (white dashed line in (d)). (f-h) The corresponding results at ω = 883 cm⁻¹and μ_c = 0.5 ev.

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An interesting aspect of using graphene and polaritons mode in graphene for deep sub-wavelength focusing is the ability to electrically tune graphene. We demonstrate this in our in-plane superlens by changing the Fermi-level of graphene to tune the lens properties. For instance, if keep the frequency fixed at ω = 883 cm⁻¹, but change the graphene Fermi level (from 0.3 ev to 0.5 ev), we can electrically tune the focal length of the superlens, as can be compared in Figs. 3(c) and 3(f). Changing (increasing) the Fermi level changes (decreases) the effective index of graphene and because of this the transmission from high-index (n_g = 80) to low-index region (n_g = 42), the refract angle θ_t increasing leads to the decrease of focal length f. This is exactly opposite to the case of conventional lenses in which the larger index material results in smaller focal length of the lens. This phenomenon demonstrates the opposite working rules of negative refraction. The totally focal intensity (Fig. 3(f)) is lower than that in Fig. 3(c), due to the momentum (index) mismatch at the interface which results in loss of some light via reflection. Further, the lower effective index of graphene (n_gra = 62) also increases the refracted wave period (H_y in Fig. 3(g)), and correspondingly decreases the resolution of the superlens (FWHM = 155 nm).

3. Ultra-compact localized SPh-SPR optical modulator

3.1 Hybrid localized phonon resonance with high quality factor

While propagating polaritons are of great interest for applications such as waveguiding and lensing, their subwavelength resonator counterparts-localized surface polariton resonances (LSPR)-are also equally useful and conceptually rich as a platform for strong light-matter interaction for sensing [43,44], modulation [45,46], and tunable antennas [47,48]. Below we demonstrate that due to the highly thickness or z-confined nature of above hybrid d-SPhP (Fig. 1(d)) and g-SPhP (Fig. 1(g)) at 2D/SiC interfaces, it is possible to laterally confine the polaritons into localized phonon resonances (LSPhR) by subwavelength patterning the vdW dielectrics on SiC. For instance, we observed dielectric tailored localized surface phonon resonance (d-LSPhR) within MoS₂-grating on SiC, and obtain electric tunable LSPh-SPR resonance in heterostructure of graphene/MoS₂-grating/SiC (Fig. 4(a)).

 

Fig. 4 (a) Schematic of the optical modulator with graphene-hybrid localized surface phonon resonance (LSPh-SPR), the thickness, width and period of MoS₂ grating are 40nm, 150nm and 300nm, respectively. (b) Simulated reflectance spectra for graphene nano-ribbons (w/p = 150/300nm) based g-LSPR (black line), non-graphene coved d-LSPhR (green line) hybrid LSPh-SPR (red line), and graphene on SiC gratings (d/w/p = 2/1.5/3μm) hybrid Fabry Pérot cavity-graphene plasmon model (FP-SPR), respectively. (c, e, g and i) Electric filed intensity enhancements |E/E₀|² and (d, f, h and j) the real part of E_z field distributions for four different types of localized polariton resonances. The Fermi level (μ_c) of graphene is 0.45ev.

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In fact, nanostructured polar dielectrics (such as SiC) resonators have been proposed previously to offer low intrinsic loss and provide extremely high Q LSPhR [2]. But owing to the chargeless nature of the phonon-polariton, it is difficult to tune LSPhPs directly using an electric field. However, they can be tuned via carrier injection [49] and statically changing the geometric parameters of the antenna or the refractive index of the environment [2,34]. More recently, graphene plasmonic resonators have been employed for the above described applications in the mid-IR range, One issue with graphene resonators is the weak intrinsic optical absorption (~2.3%) in single-layer [50,51] and double-layer [52–54], which limits modulation depths of graphene-based modulators [55–57], and another is the low Q factor of graphene plasmon resonance [45,46]. While, the weak absorption issue can be resolved using clever light trapping techniques [58,59], the issue of low Q remains challenging to solve. Here, our proposed heterostructure (Fig. 4(a)) combine 2D resonator with polar SiC, the novel coupling of 2D and SiC phonon polariton, allows us to achieve electric tunable and high Q LSPhR resonance.

To make a comparative assessment, in Fig. 4(b) we simulate g-LSPR in graphene nano-ribbons, d-LSPhR in MoS₂-grating/SiC, the hybrid LSPh-SPR in graphene/MoS₂-grating/SiC, as well hybrid FP-SPRs in a graphene/SiC-grating. Unlike previously reported bulk SiC resonators with high aspect ratios (height/width~1000nm/250nm), the height (width) of MoS₂ gratings we report (Fig. 4(a)) is only 40nm (150nm), and the period is 300nm. This device geometry presents mode confinement in extremely small volumes (~λ₀³/100³) as opposed to bulk SiC resonators [32,33]. Since the Q factor (ω_res/△ω_res) describes the energy stored to energy loss rate, the electronic losses from the scattering of free carriers by phonons and impurities limits the Q values of graphene g-LSPR to ~28 (black line, Fig. 4(b)). In comparison, due to the polar nature of SiC’s phonon, such losses are largely absent in d-LSPhR, resulting in Q factors up to 165 (green line, Fig. 4(b)), with a narrow resonance linewidth (△ω_res~5cm⁻¹), an improvement of more than one order of magnitude over noble metallic antennas [60,61].

The most interesting structure among the four cases discussed above is our proposed graphene/MoS₂-grating/SiC heterostructure. The collective resonance behaviours of localized phonon and graphene plasmons, enable the hybrid LSPh-SPR to inherit the high quality factor of d-LSPhR concurrently with the electric tunability of graphene plasmons resulting in superior electrostatic modulation of light from the overall structure. While the hybrid LSPh-SPR has a Q lower that the purely dielectric based d-LSPhR, it still retains a high Q factor of 85 (red line in Fig. 4(b)) roughly thrice that of graphene g-LSPR. High Q factors represent high confinement and mode quality which is easily perturbed by small variations in dielectric environments. This makes high Q resonances apt for sensing devices. When the resonance is spectrally overlapped with the molecule absorption spectrum, the enhanced molecule-resonator coupling can lead to a change in either the frequency or the strength of the resonance, from which the molecular fingerprints can be extracted. The purple line in Fig. 4(b) shows that Fabry Pérot cavity-graphene plasmon hybrid model (FP-SPR) which also shows a high Q factor (~83) [62], due to the strong coupling between SiC-grating phonon and graphene plasmon. However, the SiC grating has a larger device volume (~λ₀³/5³) of d/w/p = 2/1.5/3um as compared to LSPh-SPR.

Figures 4(c)-4(j) show the electric field intensity enhancements |E/E₀|² and E_z field distributions of four types of antennas, all of their maximum electric field enhancements achieve a value of 10⁴. Upon closer inspection at the nature of the (E_z) electric field profile; Fig. 4(d) shows the field enhancement at the edge of graphene nano-ribbons, suggesting it is an electric dipole mode, which has been widely studied and observed before [21,63]. For the case of d-LSPhR, the electric field (Fig. 4(f)) again shows two hotspots at the bottom edges of the antenna, suggesting that the d-LSPhR resonance is also an electric dipole. Finally, for the LSPh-SPR in hybrid graphene/MoS₂-grating/SiC heterostructure (Fig. 4(a)), the electric field hotspots are distributed along the suspended portion of graphene monolayer as well as the bottom MoS₂-SiC interface. We believe that such a geometry is practically possible and reasonable to achieve considering the recent progress on making angstrom size capillary channels in van der Waals heterostructures [64,65]. This geometry allows more space for effective light matter interaction, especially considering the field enhancement above the graphene which may be easily perturbed by a change in dielectric environment and therefore benefit chemical and biological sensing applications. Figure 4(g) shows that the hybrid LSPh-SPR resonance is dominated by the d-LSPhR mode since the localized resonance at the edges of MoS₂ antenna excites the propagating graphene SPP mode, more details can be seen from Figs. 6(e) and 6(g). This coupling of d-LSPhP and SPP can be modulated by changing the carrier density (Fermi level) of graphene. Therefore, this coupling allows the modulation depth of LSPh-SPR to tremendously improve (~95%) due to the strong near-field enhancement of d-LSPhR on the MoS₂ grating, despite weak intrinsic optical absorption (∼2.3%) of the graphene monolayer as discussed below in detail.

3.2 Electric tunable LSPh-SPR in graphene/MoS₂-grating/SiC

Firstly, we can statically tune the local resonance by means of changing the grating geometry. This is illustrated in Fig. 5(a) for the d-LSPhR resonances wherein, by increasing the widths (w) of MoS₂ antenna, the resonant frequencies of d-LSPhR blue-shift, and the reflectance dips decrease to near zero. The same phenomenon is observed in graphene/MoS₂-grating/SiC heterostructure which shows two different LSPh-SPRs (Fig. 5(b)), both of which can be tuned with the increment of MoS₂ antenna heights (d). We observe that the resonant frequencies of LSPh-SPR-1 red-shifts, the reflectance dips at first and then increases and show a maximum at a height of d = 30nm of the MoS₂ antennas. Furthermore, the Q factors of LSPh-SPR-1 decrease drastically, with a very low value (Q~34) at the height of d = 100nm due to reduced coupling between the d-LSPhR and g-SPP modes. However, for LSPh-SPR-2, the resonance frequency stays constant at 952 cm⁻¹, but the reflectance dips reduces in amplitude.

 

Fig. 5 (a) The modulation of d-LSPhR with the heights of MoS₂ antenna (d/p = 40/300nm), each y-tick represents the reflectance from 0 to 1. (b) The modulation of graphene hybrid LSPh-SPRs with the widths of MoS₂ antenna (w/p = 150/300nm). (c) The modulation of LSPh-SPRs with different graphene Fermi levels. The E_z electric distributions of g-LSPhR-1 with (d) weak coupling (μ_c = 0ev, light red circles in (c)) and (e) strong coupling (μ_c = 0.65ev, dark red circles in (c)). (f) The reflectance differences (△R) and (g) resonant frequencies ω as a function of graphene’s Fermi-levels μ_c for LSPh-SPR-1 and LSPh-SPR-2.

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Secondly, the most important aspect of this hybrid heterostructure device (Fig. 4(a)) is the electrical tunability of the LSPh-SPR. As seen from Fig. 5(c), both two LSPh-SPR resonances are actively tuned by controlling the Fermi level (electrostatic gate bias) of graphene. Figures 5(d) and 5(e) show the E_z electric fields of the weakly coupled resonance (μ_c = 0ev) and strongly coupled resonance (μ_c = 0.65ev). In the strong-coupling regime, relatively larger near-field enhancement occurs at the graphene surface, which then leads to zero reflectance at the dip (dark red circle in Fig. 5). Figures 5(f) and 5(g) show the maximum differential reflectance (△R) plots and the resonant frequencies of two LSPh-SPR resonances as a function of graphene’s Fermi-levels. The △R is defined as the normalized reflectance minus the un-gated (μ_c = 0ev) reflectance value (top plot in Fig. 5(c). For LSPh-SPR-1, both the resonance frequencies and the △Rs increase with the increment of graphene’s Fermi level. For instance, at frequency of ω = 938cm⁻¹ in Fig. 5(c), changing μ_c of the graphene from 0ev to 0.65 eV results in 95% reflectance modulation as shown in Fig. 5(f) red plot. But for LSPh-SPR-2, the resonant frequencies are almost stable at 952 cm⁻¹, and with more than 70% reflectance modulation. The difference between LSPh-SPR-1 and LSPh-SPR-2 resonances also arises from the d-LSPhR component. As seen in Fig. 6, the d-LSPhR part of LSPh-SPhR-1 (Figs. 6(a)-6(d)) shows two hotspots at the bottom edges of MoS₂ antenna, but for LSPh-SPR-2 (Figs. 6(e)-6(h)), the E_z electric fields of d-LSPhR are located in the air-gap between graphene and SiC suggesting that the dipole momentum lies along the interface between air-gap and SiC. Increasing the Fermi level of graphene would result strong coupling between d-LSPhR and the propagated g-SPP. Further, larger μ_c also results longer polariton wavelength of g-SPP, as clearly demonstrated in Figs. 6(c) and 6(g), which are coincidence with the propagating properties of graphene plasmon.

 

Fig. 6 The E_z electric distribution and Poynting vector of the weak coupling resonance (a, b) and strong coupling resonance (c, d) for LSPh-SPR-1; the weak coupling resonance (e, f) and strong coupling resonance (g, h) for LSPh-SPR-2, d/w/p = 40/150/300nm. The corresponding reflectance spectra are shown in Fig. 5(c).

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Finally, we can also use 3D semiconductors (such as Si and Ge) instead of MoS₂ to obtain the LSPh-SPR, we predict it may provide new route for future nanophotonics at the extremely nanoscale. Furthermore, the heterostructure points to an interesting direction for future works wherein combining the concept of nanostructured dielectric resonators on polar materials demonstrated here, with other optical modulation approaches (such as optical carrier injection [49] or thermal control) could enable the realization of more diverse and functional optical devices including switches, modulators and sensors, with high quality factor LSPhP resonances.

4. Conclusion

In summary we computationally demonstrate a new strategy of using ultrathin 2D dielectrics (or resonators) on polar SiC for generating ultra-confined propagating (or localized) d-SPhPs. Further, when monolayer graphene suspended over these vdW resonators, the d-LSPhPs can strongly couple to propagating graphene SPPs. The coupled LSPh-SPR modes have high-quality factors (~85) roughly 3 times higher as compared to localized surface plasmons in graphene resonators (~28) and are highly tunable with gate-voltage or Fermi level of graphene. The high quality factors are predominantly due to the low-loss SiC phonon resonance which dominated the character of the hybrid mode. Taking advantage of this hybridization and tunability, we demonstrate strong free-space modulation of g-LSPhRs, with modulation depth of reflectance exceeding 95%. Our results can open the door to many other applications including tunable metasurfaces, biosensing and ultrafast THz modulator devices.

Appendix A Dielectric functions of polar dielectric SiC and h-BN, MoS₂

As shown in Fig. 7(a), the dielectric function for polar dielectrics such as SiC and h-BN can be described by a Lorentz oscillator model [3]:

 

Fig. 7 (a) Real part permittivity of h-BN and SiC. h-BN’s first reststrahlen band (ε_x/y>0, ε_z<0) is from 780 to 830 cm⁻¹, and h-BN work as a hyperbolic dielectric at frequencies ω>830 cm⁻¹; SiC’s reststrahlen band (ε<0) is from 797 to 973 cm⁻¹. (b) In-plane real and imaginary permittivity of MoS₂.

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(2)$$\epsilon (\omega )={\epsilon }_{\infty }(1+\frac{{\omega }_{LO}{}^2-{\omega }_{TO}{}^2}{{\omega }_{TO}{}^2-{\omega }^2-i\omega \gamma })$$

with a pole at ω_TO, and a zero-point crossing at ω_LO, which are the TO and LO phonon frequencies, respectively. For SiC, ε_∞= 6.56, ω_LO = 973 cm⁻¹, ω_TO = 797 cm⁻¹, and γ = 4.76 cm⁻¹. h-BN is an anisotropic material in two wavelength regions (reststrahlen bands), i.e., ε̂ =diag{ε_x, ε_y, ε_z}, ε_x=ε_y≠ε_z, both the in-plane (x, y) and the out-of-plane (z) dielectric function can be described by Eq. (2). For the in-plane permittivity, ω_TOx = 1370cm⁻¹, ω_LOx = 1610cm⁻¹, and γ_x = 4.87 cm⁻¹; for the out-plane permittivity, ω_TOz = 780cm⁻¹, ω_LOz = 830cm⁻¹, and γ_z = 2.95 cm⁻¹_. The dielectric function of MoS₂ (Fig. 7(b)) is obtained from the fitting of the experimental data in Supplementary of [8].

Appendix B Optical conductivity of graphene

In this work, we use random phase approximation (RPA) to model the optical conductivity of graphene. The conductivity is given by [1,66]:

(3)$$\sigma (\omega )=\frac{e^2{E}_F}{\pi {\hslash }^2}\frac{i}{\omega +i{T}^{-1}}+\frac{e^2}{4{\hslash }^2}\left(\theta (\hslash \omega -2E_F)+\frac{i}{\pi }\log \left(\left|\frac{\hslash \omega -2E_F}{\hslash \omega +2E_F}\right|\right)\right)$$

where e is the unit electric charge, ℏ is the reduced Planck constant, θ denotes a step function, E_F is the graphene Fermi level, and T=300 K. The electron relaxation time τ is determined by τ=μE_F ∕ev_F², wherein the carrier mobility is assumed as μ=10⁴cm²V⁻¹s⁻¹ and the v_F ≈ 10⁶m/s is the Fermi velocity.

Appendix C Dispersions of the hybrid phonon-plasmon-polaritons

We follow an approach similar to the one used by Huber [20] to trace the dispersions of interface polaritons in black phosphorus heterostructures by the imaginary part of the Fresnel reflection coefficient r_p. Because the modes of polaritons are the singularity poles in the coefficient of r_p, we can visualize the dispersion of these hybridized polaritons via a false-color plot of Im(r_p) as a function of k_x and ω. The coefficient r_p for a multilayered system can be calculated using the transfer matrix formalism. In our case, as shown in Fig. 8, three layers are included, layer 1 (z>d, air), layer 2 (0 <z<d, MoS₂ or h-BN), and layer 3 (-H<z<0, SiC substrate), where d is the thickness of layer-2 slabs.

 

Fig. 8 The schematic of three-layer heterostructure, layer 1 (z>d, air), layer 2 (0 <z<d, MoS₂ or h-BN), and layer 3 (-H<z<0, SiC substrate), where d is the thickness of layer-2 slabs. The following derivations give the dispersions of bulk SPhP, d-SPhP and g-SPhP, respectively.

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A matrix M can be used to describe the interaction of electromagnetic waves in the heterostructure:

(4)$$M=\left[\begin{array}{cc}{M}_{aa}& M_{ab}\\ M_{ba}& M_{bb}\end{array}\right]=R_{1,2}\cdot T_2\cdot R_{2,3}$$

Where matrices R_i,j describe the reflection at each interface between layers i and j, and the matrix T₂ describes the propagation of the electromagnetics wave through layer 2. The matrices R_1,2, R_2,3 and T₂ are given by

(5)$$R_{1,2}=\frac{1}{t_{1,2}}\left[\begin{array}{cc}1& r_{1,2}\\ r_{1,2}& 1\end{array}\right],R_{2,3}=\frac{1}{t_{2,3}}\left[\begin{array}{cc}1& r_{2,3}\\ r_{2,3}& 1\end{array}\right],T_2=\left[\begin{array}{cc}{e}^{i{k}_{z2}d}& 0\\ 0& e^{-i{k}_{z2}d}\end{array}\right]$$

Where, r_i,j (t_i,j) are the Fresnel reflection (transmission) coefficients for a single interface between two infinite half spaces; k_zi is the out-of-plane k-vector of the electromagnetic wave in layer i, such as:

(6)$$k_{z1}=\sqrt{\frac{{\omega }^2}{c^2}{\epsilon }_1-k_x{}^2},k_{z2}=\sqrt{\frac{{\omega }^2}{c^2}{\epsilon }_{2x}-k_x{}^2\frac{{\epsilon }_{2x}}{{\epsilon }_{2z}}},k_{z3}=\sqrt{\frac{{\omega }^2}{c^2}{\epsilon }_3-k_x{}^2}$$

Where ε₁ and ε₃ are the relative permittivity of layer 1 and 3, respectively. Since h-BN is a hyperbolic material, its relative permittivity is characterized by an anisotropic diagonal tensor [ε_2x, ε_2y, ε_2z], and the reflection and transmission coefficients are expressed as,

(7)$$r_{i,j}=\frac{{\epsilon }_{x\text{j}}{k}_{zi}-{\epsilon }_{xi}{k}_{zj}}{{\epsilon }_{xj}{k}_{zi}+{\epsilon }_{xi}{k}_{zj}},t_{i,j}=\frac{2{\epsilon }_{xj}{k}_{zi}}{{\epsilon }_{xj}{k}_{zi}+{\epsilon }_{xi}{k}_{zj}}$$

The reflection coefficient r_p for the whole heterostructure is then given as a ratio of two matrix components of M in Eq. (4),

(8)$$r_p=\frac{M_{ba}}{M_{aa}}$$

Finally, the dispersions of d-SPhP mode in MoS₂/SiC region are visualized by using a false-color map of r_p, as shown in Fig. 1(e). Similarly, if layer 2 is set as air, we can solve the dispersion of bulk SPhP mode on the SiC substrate, as shown in Fig. 1(a). For g-SPhP in graphene/h-BN/SiC heterostructure, the surface conductivity (σ) of graphene would largely change the reflection and transmission coefficients at the interface between layer-1 and layer-2, thus the corresponding derivations for r_1,2 and t_1,2 should be replaced by,

(9)$$r_{1,2}=\frac{{\epsilon }_{x2}{k}_{z1}-{\epsilon }_{x1}{k}_{z2}\text{+}\sigma k_{z1}{k}_{z2}/\omega {\epsilon }_0}{{\epsilon }_{x2}{k}_{z1}+{\epsilon }_{x1}{k}_{z2}\text{+}\sigma k_{z1}{k}_{z2}/\omega {\epsilon }_0},t_{1,2}=\frac{2{\epsilon }_{x2}{k}_{z1}}{{\epsilon }_{x2}{k}_{z1}+{\epsilon }_{x1}{k}_{z2}\text{+}\sigma k_{z1}{k}_{z2}/\omega {\epsilon }_0}$$

The dispersions of g-SPhP are electric tunable by controlling the graphene’s Fermi level, as shown in Fig. 1(h) of the main text.

Funding

China Academy of Engineering Physics; U. S. Army Research Office (W911NF1910109); University of Pennsylvania (Penn Engineering).

Acknowledgments

DJ acknowledges support in part by, the U. S. Army Research Office under contract number W911NF1910109 and from startup funds of Penn Engineering.

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