### Nanofabrication of the gadgets

Excessive-quality α-MoO_{3} flakes have been mechanically exfoliated from bulk crystals synthesized by the chemical vapour deposition (CVD) technique^{19} after which transferred onto both industrial 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 principally 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 offered 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. Notice that slender antennas (50-nm width) have been used to forestall their shapes from affecting the polariton wavefronts, particularly when their propagation is canalized (equivalent to in Figs. 2 and 3), whereas wider antennas (250-nm width) have 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 have 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 strain 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 have been immersed in a scorching acetone bathtub at 80 °C for 25 min after which subjected to mild rinsing with isopropyl alcohol (IPA) for 3 min, adopted by drying with nitrogen gasoline 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 have 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} gasoline to attain completely different doping ranges by floor adsorption of gasoline molecules^{47}. The graphene Fermi vitality may very well be managed by various the gasoline focus and doping time, reaching values as excessive as ~0.7 eV (Supplementary Fig. 9). This gas-doping technique gives wonderful uniformity, reversibility and stability. Certainly, Raman mapping of a gas-doped graphene pattern demonstrated the excessive uniformity of the strategy (Supplementary Fig. 9). Because the deposition of NO_{2} gasoline molecules on the graphene floor happens by bodily adsorption, the topological transition of hybrid polaritons in graphene/α-MoO_{3} heterostructures may be reversed by controlling the gasoline doping. For instance, after gasoline doping, the Fermi vitality of graphene may very well be lowered from 0.7 to 0 eV by vacuum annealing at 150 °C for two h. The pattern may subsequently be re-doped to succeed in one other on-demand Fermi vitality (Supplementary Fig. 28). It ought to be famous that the graphene Fermi vitality solely decreases from 0.7 to 0.6 eV after being left for two weeks underneath ambient situations, which demonstrates the excessive stability of the doping impact (Supplementary Fig. 29). Notice that chemical doping has been demonstrated to be an efficient technique to tune the traits of polaritons, equivalent 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 have been set to ~270 kHz and ~30–50 nm, respectively. The laser beam was directed in the direction of the AFM tip, with lateral spot sizes of ~25 μm underneath the tip, which have been enough 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 pictures 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 brought on by the sunshine polarization course 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* course ([100] crystal axis) of α-MoO_{3} (Supplementary Fig. 6). Supplementary Fig. 30 exhibits the strategy 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 appreciate 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 have 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 parts alongside the *x*, *y* and *z* axes, respectively. Additionally, monolayer graphene was positioned on prime 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 is determined by the Fermi vitality *E*_{F}, the inelastic leisure time *τ* and the temperature T; the relief time is expressed by way of the graphene Fermi velocity *v*_{F} = *c*/300 and the provider mobility *μ*, with (tau = mu E_{mathrm{F}}/ev_{mathrm{F}}^2); *e* is the elementary cost; *ok*_{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 consideration, we solely wanted to contemplate transverse magnetic (TM) modes, as a result of transverse electrical (TE) parts 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*–*ok* interface for illumination from medium *j*, with *j*,*ok* = 1–3; ({it{epsilon }}_t^{(j)}) is the tangential part of the in-plane dielectric perform of layer *j* for a propagation wave vector *ok*_{p}(*θ*) (the place *θ* is the angle relative to the *x* axis), which may 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 parts 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-about a system with small dissipation, in order that the maxima of Im{*r*_{p}} (see color plots in Supplementary Determine 10) roughly clear up the situation given by equation (8), and due to this fact produce the sought-after dispersion relation *q*(*ω*,*θ*) (see extra dialogue in Supplementary Notice 1).

### Electromagnetic simulations

The electromagnetic fields across the antennas have been calculated by a finite-elements technique utilizing the COMSOL package deal. In our experiments, each tip and antenna launching have 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 may be considered a vertically oriented level dipole positioned on the tip apex. This localized close to area offered the required 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 parts 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 by the use of antennas, equivalent to in Fig. 3b,d, integrated the identical geometrical design as within the experimental buildings.

We additionally used a dipole polarized alongside the *z* course 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 true 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 have 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 provider mobility of two,000 cm^{2} V^{–1} s^{–1}. Supplementary Fig. 1c,d exhibits the permittivity of SiO_{2} and Au, respectively, on the mid-infrared wavelengths used.