### Nanofabrication of the gadgets

Excessive-quality α-MoO_{3} flakes had been mechanically exfoliated from bulk crystals synthesized by the chemical vapour deposition (CVD) technique^{19} after which transferred onto both business 300 nm SiO_{2}/500 μm Si wafers (SVM) or gold substrates utilizing a deterministic dry switch course of with a polydimethylsiloxane (PDMS) stamp. CVD-grown monolayer graphene on copper foil was transferred onto the α-MoO_{3} samples utilizing the poly(methyl methacrylate) (PMMA)-assisted technique following our earlier report^{44}.

The launching effectivity of the resonant antenna is especially decided by its geometry, along with a trade-off between the optimum dimension and illumination frequency^{45,46}. We designed the gold antenna with a size of three μm and a thickness of fifty nm, which supplied a excessive launching effectivity over the spectral vary from 890 to 950 cm^{−1} inside the α-MoO_{3} reststrahlen band. Alternatively, a thicker antenna with a stronger *z* part of the electrical area may very well be used to launch the polaritons extra effectively in future research. Observe that slender antennas (50-nm width) had been used to stop their shapes from affecting the polariton wavefronts, particularly when their propagation is canalized (akin to in Figs. 2 and 3), whereas wider antennas (250-nm width) had been used to acquire the next launching effectivity and observe polaritons propagating throughout the SiO_{2}–Au interface in our experiments (for instance, Fig. 4).

Gold antenna arrays had been patterned on chosen α-MoO_{3} flakes utilizing 100 kV electron-beam lithography (Vistec 5000+ES) on an roughly 350 nm PMMA950K lithography resist. Electron-beam evaporation was subsequently used to deposit 5 nm Ti and 50 nm Au in a vacuum chamber at a stress of <5 × 10^{−6} torr to manufacture the Au antennas. Electron-beam evaporation was additionally used to deposit a 60-nm-thick gold movie onto a low-doped Si substrate. To take away any residual natural materials, samples had been immersed in a sizzling acetone bathtub at 80 °C for 25 min after which subjected to light rinsing with isopropyl alcohol (IPA) for 3 min, adopted by drying with nitrogen fuel and thermal baking (for extra particulars on the fabrication and characterization of the Au–SiO_{2}–Au in-plane sandwich construction, see Supplementary Figs. 26 and 27).

The samples had been annealed in a vacuum to take away a lot of the dopants from the moist switch course of after which transferred to a chamber crammed with NO_{2} fuel to attain completely different doping ranges by floor adsorption of fuel molecules^{47}. The graphene Fermi power may very well be managed by various the fuel focus and doping time, attaining values as excessive as ~0.7 eV (Supplementary Fig. 9). This gas-doping technique offers wonderful uniformity, reversibility and stability. Certainly, Raman mapping of a gas-doped graphene pattern demonstrated the excessive uniformity of the tactic (Supplementary Fig. 9). Because the deposition of NO_{2} fuel molecules on the graphene floor happens by bodily adsorption, the topological transition of hybrid polaritons in graphene/α-MoO_{3} heterostructures could be reversed by controlling the fuel doping. For instance, after fuel doping, the Fermi power of graphene may very well be lowered from 0.7 to 0 eV by vacuum annealing at 150 °C for two h. The pattern might subsequently be re-doped to achieve one other on-demand Fermi power (Supplementary Fig. 28). It ought to be famous that the graphene Fermi power solely decreases from 0.7 to 0.6 eV after being left for two weeks beneath ambient situations, which demonstrates the excessive stability of the doping impact (Supplementary Fig. 29). Observe that chemical doping has been demonstrated to be an efficient solution to tune the traits of polaritons, akin to their energy and in-plane wavelength^{48,49,50,51}.

### Scanning near-field optical microscopy measurements

A scattering scanning near-field optical microscope (Neaspec) geared up with a wavelength-tunable quantum cascade laser (890–2,000 cm^{−1}) was used to picture optical close to fields. The atomic pressure microscopy (AFM) tip of the microscope was coated with gold, leading to an apex radius of ~25 nm (NanoWorld), and the tip-tapping frequency and amplitudes had been set to ~270 kHz and ~30–50 nm, respectively. The laser beam was directed in direction of the AFM tip, with lateral spot sizes of ~25 μm beneath the tip, which had been adequate to cowl the antennas in addition to a big space of the graphene/α-MoO_{3} samples. Third-order harmonic demodulation was utilized to the near-field amplitude photographs to strongly suppress background noise.

In our experiments, the p-polarized plane-wave illumination (electrical area **E**_{inc}) impinged at an angle of 60° relative to the tip axis^{52}. To avert any results attributable to the sunshine polarization path relative to the crystallographic orientation of α-MoO_{3}, which is optically anisotropic, the in-plane projection of the polarization vector coincided with the *x* path ([100] crystal axis) of α-MoO_{3} (Supplementary Fig. 6). Supplementary Fig. 30 reveals the tactic used to extract antenna-launched hybrid polaritons from the complicated background alerts noticed when the polaritons propagate throughout a Au–SiO_{2}–Au in-plane construction to understand partial focusing.

### Calculation of polariton dispersion and isofrequency dispersion contours (IFCs) of hybrid plasmon–phonon polaritons

The switch matrix technique was adopted to calculate the dispersion and IFCs of hybrid plasmon–phonon polaritons in graphene/α-MoO_{3} heterostructures. Our theoretical mannequin was based mostly on a three-layer construction: layer 1 (*z* > 0, air) is a canopy layer, layer 2 (0 > *z* > –*d*_{h}, graphene/α-MoO_{3}) is an intermediate layer and layer 3 (*z* < –*d*_{h}, SiO_{2} or Au) is a substrate the place *z* is the worth of the vertical axis and *d*_{h} is the thickness of α-MoO_{3} (Supplementary Fig. 31). Every layer was considered a homogeneous materials represented by the corresponding dielectric tensor. The air and substrate layers had been modelled by isotropic tensors diag{*ε*_{a,s}} (ref. ^{53}). The α-MoO_{3} movie was modelled by an anisotropic diagonal tensor diag{*ε*_{x}, *ε*_{y}, *ε*_{z}}, the place *ε*_{x}, *ε*_{y} and *ε*_{z} are the permittivity elements alongside the *x*, *y* and *z* axes, respectively. Additionally, monolayer graphene was situated on high of α-MoO_{3} at *z* = 0 and described as a zero-thickness present layer characterised by a frequency-dependent floor conductivity taken from the native random-phase approximation mannequin^{54,55}:

$$start{array}{rcl} {sigma left( omega proper)}& = &{frac{{i{{rm{e}}^2}{k_{rm{B}}}T}}{{{{uppi}}{hbar ^2}left( {omega + frac i tau} proper)}}left[ {frac{{{E_{rm{F}}}}}{{{k_{rm{B}}}T}} + 2ln left( {{{rm{e}}^{ – frac{{{E_{rm{F}}}}}{{{k_{rm{B}}}T}}}} + 1} right)} right]}{}&{}&{ + ifrac{{{{rm{e}}^2}}}{{4{{uppi}}hbar }}ln left[ {frac{{2left| {{E_{rm{F}}}} right| – hbar left( {omega + frac i tau} right)}}{{2left| {{E_{rm{F}}}} right| + hbar left( {omega + frac i tau} right)}}} right]}finish{array}$$

(1)

which will depend on the Fermi power *E*_{F}, the inelastic rest time *τ* and the temperature T; the comfort time is expressed by way of the graphene Fermi velocity *v*_{F} = *c*/300 and the service mobility *μ*, with (tau = mu E_{mathrm{F}}/ev_{mathrm{F}}^2); *e* is the elementary cost; *okay*_{B} is the Boltzmann fixed; *ℏ* is the diminished Planck fixed; and *ω* is the illumination frequency.

Given the robust area confinement produced by the construction into account, we solely wanted to think about transverse magnetic (TM) modes, as a result of transverse electrical (TE) elements contribute negligibly. The corresponding p-polarization Fresnel reflection coefficient *r*_{p} of the three-layer system admits the analytical expression

$$start{array}{*{20}{c}} {r_{mathrm{p}} = frac{{r_{12} + r_{23}left( {1 – r_{12} – r_{21}} proper){mathrm{e}}^{i2k_z^{left( 2 proper)}d_{mathrm{h}}}}}{{1 + r_{12}r_{23}{mathrm{e}}^{i2k_z^{left( 2 proper)}d_{mathrm{h}}}}},} finish{array}$$

(2)

$$start{array}{*{20}{c}} {r_{12} = frac{{{{Q}}_1 – {{Q}}_2 + SQ_1Q_2}}{{{{Q}}_1 + Q_2 + SQ_1Q_2}},} finish{array}$$

(3)

$$start{array}{*{20}{c}} {r_{21} = frac{{{{Q}}_2 – {{Q}}_1 + SQ_1Q_2}}{{{{Q}}_2 + Q_1 + SQ_1Q_2}},} finish{array}$$

(4)

$$start{array}{*{20}{c}} {r_{23} = frac{{Q_2 – Q_3}}{{Q_2 + Q_3}},} finish{array}$$

(5)

$$start{array}{*{20}{c}}the place {Q_j = frac{{k_z^{left( j proper)}}}{{{it{epsilon }}_t^{(j)}}},} finish{array}$$

(6)

$$start{array}{*{20}{c}} {S = frac{{sigma Z_0}}{omega }.} finish{array}$$

(7)

Right here, *r*_{jk} denotes the reflection coefficient of the *j*–*okay* interface for illumination from medium *j*, with *j*,*okay* = 1–3; ({it{epsilon }}_t^{(j)}) is the tangential part of the in-plane dielectric perform of layer *j* for a propagation wave vector *okay*_{p}(*θ*) (the place *θ* is the angle relative to the *x* axis), which could be expressed as ({it{epsilon }}_t^{(j)} = {it{epsilon }}_x^{(j)}mathop {{cos }}nolimits^2 theta + {it{epsilon }}_y^{(j)}mathop {{sin }}nolimits^2 theta) (the place ({it{epsilon }}_x^{(j)}) and ({it{epsilon }}_y^{(j)}) are the diagonal dielectric tensor elements of layer *j* alongside the *x* and *y* axes, respectively); (k_z^{left( j proper)} = sqrt {varepsilon _t^{left( j proper)}frac{{omega ^2}}{{c^2}} – frac{{varepsilon _t^{left( j proper)}}}{{varepsilon _z^{left( j proper)}}}q^2}) is the out-of-plane wave vector, with ({it{epsilon }}_z^{(j)}) being the dielectric perform of layer *j* alongside the *z* axis; and *Z*_{0} is the vacuum impedance.

We discover the polariton dispersion relation *q*(*ω*,*θ*) when the denominator of equation (2) is zero:

$$start{array}{*{20}{c}} {1 + r_{12}r_{23}{mathrm{e}}^{i2k_z^{left( 2 proper)}d_{mathrm{h}}} = 0.} finish{array}$$

(8)

For simplicity, we thought of a system with small dissipation, in order that the maxima of Im{*r*_{p}} (see color plots in Supplementary Determine 10) roughly resolve the situation given by equation (8), and due to this fact produce the sought-after dispersion relation *q*(*ω*,*θ*) (see further dialogue in Supplementary Observe 1).

### Electromagnetic simulations

The electromagnetic fields across the antennas had been calculated by a finite-elements technique utilizing the COMSOL bundle. In our experiments, each tip and antenna launching had been investigated. For the previous, the sharp metallic tip was illuminated by an incident laser beam. The tip acted as a vertical optical antenna, changing the incident gentle right into a strongly confined close to area under the tip apex, which could be considered a vertically oriented level dipole situated on the tip apex. This localized close to area supplied the mandatory momentum to excite polaritons. Consequently, we modelled the tip as a vertical *z*-oriented oscillating dipole in our simulations (Fig. 1c,f), a process that has been extensively used for tip-launched polaritons in vdW supplies^{56}. For the antenna launching, the gold antenna can present robust close to fields of reverse polarity on the two endpoints, thus delivering high-momentum near-field elements that match the wave vector of the polaritons and excite propagating modes within the graphene/α-MoO_{3} heterostructure^{45,46}. Our simulations of polariton excitation via antennas, akin to in Fig. 3b,d, integrated the identical geometrical design as within the experimental constructions.

We additionally used a dipole polarized alongside the *z* path to launch polaritons, and the gap between the dipole and the uppermost floor of the pattern was set to 100 nm. We obtained the distribution of the actual a part of the out-of-plane electrical area (Re{*E*_{z}}) over a airplane 20 nm above the floor of graphene. The boundary situations had been set to completely matching layers. Graphene was modelled as a transition interface with a conductivity described by the native random-phase approximation mannequin (see above)^{55,57}. We assumed a graphene service mobility of two,000 cm^{2} V^{–1} s^{–1}. Supplementary Fig. 1c,d reveals the permittivity of SiO_{2} and Au, respectively, on the mid-infrared wavelengths used.