Tunable plasmon-phonon polaritons in anisotropic 2D materials on hexagonal boron nitride

3.1 Modal characteristics of SPPs supported by a suspended A2DM

We proceed to investigate the impact of substrate on the modal properties of A2DM SPPs. In order to study the properties of surface modes in materials and structures with in-plane anisotropy, we analyze EFCs ω(βx, βy)=const rather than plasmon dispersion β(ω). The reason is that the direction of the mode wavefront propagation in anisotropic systems, defined by wavevector β, does not generally coincide with the direction of energy flow, defined by the group velocity ∇βω(β) [15]. From the property of the gradient operator it follows that the surface mode in anisotropic materials carries energy in the directions orthogonal to the EFCs.

The EFCs, density plots, and electric fields associated with SPPs supported by a suspended A2DM sheet are shown in Figure 2. The subscripts 1, 2, and 3 of the letters marking the panels of the figure distinguish between the results obtained for λ=5, 6.5, and 8 μm, respectively (and similar in Figure 3, Figures S2, and S3). In Figures 2, 3 and Figure S2 the concentration of electrons in the A2DM is 1014 cm−2. Moreover, for all numerical calculations in this paper, the z-polarized electric dipole is located at x = y = 0, 10 nm away from surface of the structures and the top-view mode profiles are calculated at a plane 5 nm way from the structures surface.

Figure 2:

(a_i) EFCs, (b_i) transmission coefficients, and [(c_i), (e_i)] density plots of electric fields associated with SPPs supported by a suspended A2DM sheet [Figure 1(a)] with n=1014cm−2; (c_i) and (d_i) are the top-view spatial distributions of Re(Ez) and |Ez|, respectively, and (e_i) are the side-view of the field. The panels denoted by subscripts 1, 2, and 3 correspond to λ=5μm, 6.5μm and 8μm, respectively; dashed horizontal line shows the location of the A2DM sheet.

In agreement with Figure 1(d), at 5 μm(2000 cm−1), σ″xx=3.56ℒ10−4 S and σ″yy=−5.22ℒ10−5S, thus |σ″xx/σ″yy|=6.82 and σ″xxσ″yy<0. In this case, only one component of the conductivity tensor is of metallic type and, consequently, A2DM supports SPPs with hyperbolic topology at this wavelength [Figure 2(a₁)]. For the suspended A2DM sheet, the hyperbola’s asymptotes can be obtained from Eq. (14) by assuming that βx, βy≫β0 and neglecting the losses [10], [11], [15],

The density plot of the transmission coefficient of the suspended A2DM sheet for λ=5 μm is shown in Figure 2(b₁). A good agreement between the EFC and the trend of the density plot can be seen. It is also observed that the wavenumbers that satisfy the condition (|βx|, |βy|)<(65β0, 125β0) are mostly responsible for the strong excitation of the SPPs. The top views of the Re(Ez) and |Ez| field distributions, shown in Figure 2(c₁) and (d₁), clearly demonstrate the hyperbolic directional guiding of the SPPs. More specifically, the white dashed line in Figure 2(d₁) indicates that the SPP ray travels along the suspended A2DM sheet at an angle of φ=24.7° with respect to the x axis. This value is in good agreement with the one predicted by the asymptotic behavior

which gives the normals to the asymptotes and thus a good estimation of the group velocity direction of the SPPs supported by a suspended A2DM sheet. The angle that the SPP propagation direction makes with the x axis is seen to increase as the operation wavelength becomes smaller (for example, φ=36.87° for λ=4 μm). As explained in Section C of the Supporting Information, this angle can be actively tuned by changing the concentration of electrons in the A2DM.

An even more interesting scenario arises in the purely anisotropic regime (i. e., for σ″xx>0 and σ″xx>0) when one of the imaginary components of A2DM conductivity tensor dominates over the other (e. g., σ″xx≫σ″yy). This regime is characterized by the support of SPPs with an anisotropic elliptic or figure-eight-like topology which favors propagation in specific directions. With this topology, A2DM can canalize most of the plasmon energy toward the x axis due to low values of σ″yy. The first example of the considered topology for λ=6.5 μm (σ″xx≈7.16σ″yy) is shown in Figure 2(a₂)–(e₂). Due to relatively small σ″yy, this case is also called a σ-near-zero regime [6]. The comparison of Figure 2(a₂) and (a₁) shows that the change of dispersion from hyperbolic to σ-near-zero leads to a noticeable modification of the EFC. According to Figure 2(b₂), all the wavenumbers in the EFC contour with (|βx|, |βy|)<(67β0, 127β0) correspond to high densities in the transmission plot.

The top-view profiles of Re(Ez) and |Ez| are shown in Figure 2(c₂) and (d₂). The value of φ corresponding to the dashed white line in the |Ez| profile is 11.86°. In this instance the group velocity points predominantly along the directions that make smaller angles with the horizontal axis as compared to the case of λ=6.5 μm. Thus, SPPs carry energy in the form of narrow rays propagating almost parallel to the x axis [15].

Figure 2(a₃)–(e₃) correspond to λ=8 μm when σ″xx≫σ″yy but σ″yy is less smaller than σ″xx compare to the previous case (σ″xx≈3.39 σ″yy), thus this case is called as anisotropic regime. Now the material is less anisotropic than for λ=6.5μm, and the figure-eight-like shapes appear in the respective EFC [Figure 2(a₃)] and density plot [Figure 2(b₃)]. Notice that since the ratio σ″xx/σ″yy is now smaller than before, the figure-eight-like EFC correspond to smaller values of βy/β0 [10], [15]. Much like for λ=6.5 μm, all the wavenumbers in the EFC contour for λ=8μm take high density values indicating that they all can be involved in the mode excitation. Moreover, the top-view of the Re(Ez) field [Figure 2(c₃)] suggests that the spatial distribution of the electric field at this wavelength is significantly modified as compared to the previous cases. The latter observation is supported by the |Ez| field distribution in Figure 2(d₃). For the dashed white line, the direction of the group velocity with respect to the x axis is given by the angle φ=1.43°. This value indicates that the SPPs carry energy in the form of subdiffractional rays propagating along the x axis, which in the present case is the crystallographic axis of A2DM.

It should be noted that Figures 2(e_j), 3(e_j), and 4(e_j) for j=1, 2, 3 show Re(Ez(x, z)) in the plane y=0. As discussed earlier, SPPs of a A2DM sheet propagate in the x and y directions with hyperbolic, σ-near-zero, and anisotropic dispersions. The side-view mode profiles in panels (e_j) allow one to estimate the strengths of the SPPs localization in the structure in all these cases.

We note that the effect of an underlying trivial substrate is minimal, and we defer their analysis to the Supporting Information. Hereinafter, the SPPs of the suspended A2DM sheet and the A2DM/sub structure are simply denoted as SP²s.

3.2 Hybrid guided modes supported by a A2DM/hBN/substrate heterostructure

Besides being compatible with a A2DM, hBN has a very useful phononic property in the MIR range — it behaves as a natural hyperbolic material within its RS bands and as a uniaxially anisotropic dielectric with low-losses outside of them [33], [34], [35], [36], [37], [38], [39]. In this section we analyze a A2DM/hBN/sub heterostructure [Figure 1(c)] which is more functional than the suspended A2DM sheet and the A2DM/sub structure considered earlier. The comparison of shaded regions in Figure 1(e) and (f) shows that the wavelength domain from 5 to 8 µm contains both hyperbolic regions of the A2DM with n=3ℒ1013 cm−2 and the type-II RS band of hBN. Note that these hyperbolic regions of A2DM and hBN do not overlap for n=1014cm−2, as seen from Figure 1(d) and (f). In the following calculations thickness of the hBN film, l, is taken as it is assumed that 20 nm unless stated otherwise.

It is well-known that once hBN is combined with a 2D plasmonic material such as graphene that supports SPPs or SP²s, the resulting structure supports hybrid plasmon-phonon modes which are combinations of SP²s and HP²s and may be referred to as SP³s and HP³s outside of the RS band and inside this band, respectively [35], [37], [38]. Here we use the same notations for the modes supported by our structure. The EFC diagram and the transmission coefficient associated with the A2DM/hBN/sub heterostructure at λ=5 μm are shown in Figure 3(a₁) and (b₁). The dispersion of SP²s supported by the suspended A2DM sheet (solid blue curve) and the A2DM/sub structure (solid red curve) are also shown for reference. One can see that when hBN acts as a uniaxially anisotropic dielectric, the SP³s dispersion is almost the same as the dispersion of SP²s in the A2DM/sub heterostructure. Similarly, the wavenumbers that are mostly involved in the excitation of the mode satisfy the condition (|βx|, |βy|)<(80β0, 120β0).

Figure 3:

(a_i) EFCs, (b_i) transmission coefficients, and [(c_i), (e_i)] density plots of electric field associated with the guided modes supported by the A2DM/hBN/sub structure [Figure 1(c)] with n=1014cm−2. The meaning of panels (c_i), (e_i) is the same as in Figure 2. Black curves in (a_i) are the guided modes of the A2DM/hBN/sub structure whereas blue and red curves correspond to the SP²s of the suspended A2DM sheet and the A2DM/sub structure, respectively. Dotted horizontal line in (e_i) shows the interface between the hBN layer and the substrate.

The analysis of EFCs at other wavelengths shows that when hBN is used as a buffer layer, the guided plasmon-phonons possess modal features that are typical for the modes supported by the A2DM/sub structure at wavelengths below the edge wavelength of the type-II RS band of hBN, λ≤6.2 μm (as explained below, topological transitions occur for the guided modes within the type-II RS band of hBN). This claim is supported by the observation from the Re(Ez) field distribution [Figure 3(c₁)] that the SP³ mode travels in the plane of the A2DM sheet with a profile similar to the one shown in Figure S2(c₁). In particular, in agreement with Figure S2(a₁), the angle of ray propagation with respect to the x axis, φ=25.64°, calculated from the dashed white line in Figure 3(d₁), is exactly the same as that for the A2DM/sub structure. The comparison of |Ez| in Figure 3(d₁) and Figure S2(d₁) also shows that for λ=5 μm the SP³ mode propagates along the A2DM/hBN/sub heterostructure as far as the SP² mode along the A2DM/sub structure, which means that the decay lengths in both cases are almost the same.

The topological features of HP³s within the type-II RS band of hBN at λ=6.5 μm differ significantly from the features of SP²s in the suspended A2DM sheet and the A2DM/sub structure. The EFC of the guided mode at this wavelength [Figure 3(a₂)] is a mixture of the figure-eight-like dispersion of the A2DM SP²s and elliptic/circular dispersion of the hBN HP²s; therefore, we label it as HP³. The density plot in Figure 3(b₂) shows that all the wavenumbers in the EFC diagram are responsible for the excitation of the guided mode. This point was observed in the density plots of SP²s of the suspended A2DM sheet [Figure 2(b₃)] at λ=8 μm, confirming the change in the topology of the HP3 mode in the type-II RS band due to the presence of hBN. In fact, the topology of HP³ is anisotropic (not σ-near-zero) at this wavelength. This is further evidenced by the fact that Re(Ez) and |Ez| of the HP³ mode at λ=6.5 μm [Figure 3(c₂) and (d₂)] resemble similar field distributions of the suspended A2DM sheet SP²s at λ=8 μm. Besides, the angle of ray propagation with respect to the x axis calculated from Figure 3(d₂), φ=1.38°, is very close to the value 1.43° found from Figure 2(d₃).

More investigations reveal that at the lower boundary of the type-II RS band, λ=6.2 μm, the dispersion topology of the guided mode supported by the A2DM/hBN/sub structure is σ-near-zero, similar to the one of the A2DM SP² at 6.5 μm [Figure 2(a₂)–(e₂)]. At the boundary frequency, the SP³s of the A2DM/hBN/sub structure are characterized by φ=11.86°, which is the same as for the suspended A2DM sheet SP²s at λ=6.5 μm.

Similarly, at other wavelengths outside of the type-II RS band of hBN, e. g., λ=8 μm, the topology of the SP³ mode of the A2DM/hBN/sub structure differs from the topologies of the SP² modes of the suspended A2DM sheet and the A2DM/sub structure. From Figure 3(a₃) it is seen that the EFC of the SP³ mode (black curve) is different from the EFCs of SP²s of the suspended A2DM sheet (blue curve) and the A2DM/sub structure (red curve) but resembles the red curves in Figure S2(a₂). It can therefore be concluded that the SP³s of the A2DM/hBN/sub structure at λ=8 μm have a σ-near-zero topology. The density plot [Figure 3(b₃)] shows that in this case all the wavenumbers in the EFC are involved in the mode excitation. The comparison of Figure 3(c₃) with Figure S2(c₃) (anisotropic topology) and Figure S2(c₂) (σ-near-zero topology) gives further evidence that the dispersion topology of the SP³s of the A2DM/hBN/sub structure at λ=8 μm is predominantly σ-near-zero rather than anisotropic. The direction of energy propagation corresponding to the white dashed line in Figure 3(d₃) is given by the angle φ=14.57°, which coincides with the value obtained from Figure 2(d₂) and confirms the same topology of the modes.

The performed analysis shows that the presence of hBN in the case of n=1014cm−2 leads to: (i) a change in the dispersion topology of the modes within the type-II RS band and outside of it for λ>7.3 μm; (ii) the appearance of the modal features of the SP²s of the suspended A2DM sheet and the A2DM/sub structure [c.f. Figures 2, 3, and Figure S2]; and (iii) the support of HP³ modes inside the type-II RS band of hBN that possess the features of both the SP²s of A2DM and the HP²s of hBN. These findings may enhance the interaction of A2DM/hBN heterostructures with light and improve their performance in asymmetric guiding, beaming, and unidirectional propagation, as will be discussed in the next section.

3.3 Light–matter interaction, asymmetric guiding, beaming, and unidirectionality

As the Purcell effect explains [49], when an excited emitter — e. g., an atom, molecule, or quantum dot — is placed near a system that supports photonic modes, its lifetime changes as compared to the emission in free space. The strength of this effect, determined by the ratio of the quality factor to the volume of the available photonic modes, can be characterized by the spontaneous emission rate (SER) [50]

where P and P0 are the powers generated by the emitter with and without the structure, μ→p and r→0 define the orientation and position of the emitter, and G¯¯s(r→0, r→0, ω) is the scattering term of the Green function of the system.

Owing to the support of subwavelength mode volumes by nanophotonic structures (e. g., by graphene-based structures), the light–matter interaction and thus the SER in their vicinity are enhanced. It has been reported that BP [7] and hBN [36] exhibit high SERs for materials with in-plane and out-of-plane hyperbolic dispersions. The enhancement in spontaneous emission of a quantum emitter is observed due to the considerable increase in the local density of states (LDOS) which is a result of the support of high-β modes by hyperbolic materials. Much like graphene, which is an isotropic 2D plasmonic material [51], [52], BP can also considerably enhance spontaneous emission of a quantum emitter outside of its hyperbolic region due to the support of surface plasmons [7].

Figure 4 illustrates the impact of an hBN buffer layer between the A2DM sheet and the substrate on the SER of a z-oriented point dipole located 10 nm above the A2DM sheet. Solid blue and red spectra are the SERs of the emitter for n=3ℒ1013 cm−2 and 1014cm−2, respectively. The lower electron densities are seen to result in higher SERs. The black dash-dotted spectra show that the SER of the source placed above the hBN/sub structure is considerably enhanced inside the type-II RS band of hBN, due to the excitation of the HP² modes (hyperbolic phonon polaritons). The presence of a 20-nm-thick hBN layer in the A2DM/hBN/sub structure is seen to decrease the SER of the source for both values of the electron density [c.f. the dashed and solid spectra in Figure 4(a)]. The increase of the hBN layer thickness to 100 nm considerably enhances the SER, as evidenced by the comparison of the dashed spectra in the two panels. For this thickness the SERs in the vicinity of the A2DM/hBN/sub structure are comparable to or even higher than the SERs near the A2DM/sub structure. Therefore, in case broadband enhancement of spontaneous emission is concerned, the thickness of the hBN layer is an important parameter. This broadband enhancement may not be achieved by graphene/hBN heterostructures [36]. Consequently, as far as, both enhanced light–matter interactions and guiding functionalities are concerned, A2DM/hBN/sub heterostructure can be a better candidate than the A2DM/sub structure.

Figure 4:

SERs of quantum emitter located 10nm above A2DM/sub structure (solid curves) and A2DM/hBN/sub structure (dashed curves) with 20- and 100-nm-thick hBN layers for n=3ℒ1013cm−2 (blue curves) and n=1014cm−2 (red curves). Dash-dotted curves show SERs of emitter above similar hBN/sub structures; dotted vertical lines mark the boundaries of the type-II RS band of hBN.

Using a uniform graphene sheet modulated by a closely located corrugated ground plane [8], a patterned graphene sheet or a patterned BP layer [4], [5], [9], one can obtain a uniaxially anisotropic surface conductivity i. e., metasurface. This uniaxially anisotropic conductivity enables the realization of hyperlensing [4], [5], [8] and canalization [9] regimes. The hyperlensing regime allows transferring subwavelength images from a source point to an image point without diffraction while canalization makes possible diffractionless beaming of single subwavelength-scale confined rays. Similar functionalities have been reported in the visible range for periodic metallic gratings [12], [13]. It is also known that gapped Dirac materials such as bilayer graphene and transition metal dichalcogenides behave as chiral optical media under the illumination with circularly polarized light [14]. For these materials SP²s are reciprocal while the edge modes are non-reciprocal. The nonreciprocal behavior becomes more pronounced with the increase of the off-diagonal component of the conductivity tensor of the material [14], [15].

Consider a double heterostructure composed of a A2DM/sub structure for x<0 and a A2DM/hBN/sub structure for x>0, which is shown at the top of Figure 5. The A2DM sheet covers the entire heterostructure and is located on a 20-nm-thick hBN layer. The two point dipoles exciting the heterostructure are separated by distance d and located in plane x=0 (white dashed line) above the A2DM sheet.

Figure 5:

(top) Double heterostructure composed of A2DM/sub structure for x<0, A2DM/hBN/sub structure with l=20 nm for x>0, and a uniform A2DM sheet on top of them, and (bottom) density plots of electric field |E| at different wavelengths for [(a₁)–(e₁)] n=1014cm−2 and [(a₂)–(e₂)] n=3ℒ1013cm−2. A pair of z-oriented electric dipoles (red arrows) are 30 nm apart and 10 nm above the A2DM sheet. Vertical dashed lines show the boundary between the A2DM/sub and A2DM/hBN/sub structures.

The electric field distributions plotted in Figure 5 for different wavelengths and electron densities in A2DM illustrate how the hBN buffer layer leads to the asymmetry in the excitation of the modes. This effect enables asymmetric guiding, beaming, and unidirectionality of the guided modes supported by a uniform sheet of A2DM. The field distributions are shown for the most illustrative wavelengths and two electron concentrations. The mode profiles in Figure 5(a₁)–(e₁) agree well with the mode profiles in Figure 3 for x>0 and in Figuree S2 for x<0. It is seen from Figure 5(a₁) that at λ=5μm the mode profiles on both sides of the white dashed line are similar, featuring hyperbolic rays that show symmetric guiding in the A2DM/hBN/sub and A2DM/sub structures.

From the mode profiles in Figure 5(b₁) one can see that the topology of the guided modes in both parts of the heterostructure is σ-near-zero. At this wavelength, SP² modes are supported by the A2DM/sub structure and HP³ modes are supported by the A2DM/hBN/sub structure. Therefore, the guided modes traveling in the plane of the A2DM sheet at λ=6.4 μm are asymmetric with respect to plane x=0. This asymmetry, caused by the presence of the hBN layer for x>0, will be referred to as the asymmetric guiding of the modes. A similar asymmetric guiding can be observed for the modes of a truncated chiral 2D material [14], [15].

The asymmetric guiding becomes more pronounced for λ=6.6 μm [Figure 5(c₁)]. At this wavelength the SP² guided mode travels in the region x<0 with the σ-near-zero topology while the anisotropic HP³ mode in the region x>0 exhibits strong beaming in the +x direction. A similar effect can be observed for a pair of different adjacent 2D anisotropic media [9].

The most extreme asymmetry in the propagation of the two modes is illustrated for λ=7 μm by Figure 5(d₁). One can see that the propagation of the HP³ mode in this case is blocked for x>0, whereas the SP² mode can still unidirectionally propagate in the region x<0. The unidirectionality of propagation can be also observed for a uniform sheet of A2DM excited by an appropriately driven elliptically polarized electric dipole [16], [17]. The effect of asymmetric guiding is also present in Figure 5(e₁), because the dispersion topologies of the SP³ and SP² modes propagating for x>0 and x<0, respectively, are the same as in Figure 5(c₁).

The same conclusions regarding the asymmetric guiding, beaming, and unidirectionality can be drawn from the mode profiles in Figure 5(a₂)–(e₂). In this case a special consideration should be given to the symmetric edge mode of the uniform A2DM sheet that is confined to the y axis [white dashed line in Figure 5(c₂)]. The symmetric edge modes are usually confined to the truncated edge of a A2DM, e. g., a truncated layer of BP [15], [28] or graphene-based heterostructures [53], [54]. However, Figure 5(c₂) gives strong evidence that the edge modes with symmetric features can be supported by a uniform A2DM sheet of the considered double heterostructure as well. It is worth mentioning that in addition to the in-plane nanophotonics applications, similar to the graphene-based structures [55], [56], [57], metasurfaces and metamaterials based on A2DMs can also find practical applications. However, their investigations is out of the scope of this study.