H exagonal boron nitride (hBN) is a representative van der Waals (vdW) crystal, 1 in which atomic layers are bonded together by weak vdW forces. Thanks to its high chemical stability, 2 insulating nature, and clean and atomically flat surface, 3 hBN has been widely used in nanoelectronics as an excellent substrate and dielectric spacer. 4 For electronic applications, heterostructures combining hBN and other vdW crystals have been shown to support ultrahigh electron mobility 3,5 and can protect sensitive materials in atmosphere. 6,7 In addition to electronics, hBN also offers interesting properties for photonics and plasmonics. It emits ultraviolet light 8,9 and can host a variety of defects functioning as single photon emitters. 10 In the mid-infrared spectrum, hBN is a natural hyperbolic material, and it supports hyperbolic phonon polaritons. [11][12][13][14][15][16][17][18] While phonon polaritons and plasmon polaritons are both hybrid light-matter propagating modes, 19,20 the former provide larger optical confinement and lower damping. Hyperbolic phonon polaritons in hBN have therefore led to a series of advances in nanophotonics, including subdiffractional focusing, 21,22 biochemical sensing, 23 flat optics, 17,18 internal structure analysis, 24 and a natural platform for metamaterials and metasurfaces. 25 In order to achieve a tunable hyperbolic response and to expand the functionality of hBN polaritonic devices, the modification of hBN polaritons combining hBN with other materials has also been investigated. 4 Representative previous results have reported the tuning of polaritons in hBN with a plasmonic graphene layer 26 and with phase change materials. 27 However, these effects are often accompanied by deteriorative damping in hyperbolic phonon polaritons due to added material loss intrinsic in the surrounding materials. A recent work has demonstrated loss compensation 28 of hyperbolic phonon polaritons in hBN via isotropic enrichment, in order to reduce the phonon-defect scattering for propagating polariton waves.

While it appears promising to explore low-loss and flat substrates, such as silicon nitride 29 in order to improve the figures of merits for propagating polaritons, in this work we investigate the modification of hyperbolic phonon polaritons by suspending hBN crystals. The modification is directly observed with infrared nanoimaging and confirmed in theory and simulations. This modification of polariton properties can be attributed to a change of dielectric environment for the supported hyperbolic phonon polaritons. While we achieve tuning of the polariton wavelength, we are also able to reduce the polariton damping because of the elimination of substrate loss in the suspended geometry. Thus, this technique opens interesting opportunities to improve the performance of polaritonic nanodevices.

The schematic of our infrared nanoimaging experiments on suspended hBN is shown in Figure 1a. Our substrate comprises two layers: a doped Si wafer (gray) capped with a 285 nm thick SiO 2 (orange). The capped SiO 2 layer was patterned to contain 1.5 μm wide air trenches by reactive ion etching. hBN crystals were then mechanically exfoliated from bulk samples and transferred to the substrate, such that part of the hBN crystal remained suspended (Figure 1a). The infrared nanoimaging experiment was performed using a scattering-type scanning near-field optical microscope (s-SNOM, www.neaspec.com) and quantum cascade lasers (QCLs, www.daylightsolutions. com). In the s-SNOM nanoimaging, the PtIr coated atomic force microscope (AFM) tip (radius ∼20 nm) was set to tapping mode with a frequency ∼270 kHz and amplitude ∼70 nm. When the sample was scanned underneath, we illuminated the AFM tip with the QCL and collected the backscattered optical signal. The scattered signal was then collected in a pseudoheterodyne interferometric detection module, 30 such that the AFM topography (Figure 1b), s-SNOM amplitude (s(ω), Figure 1c,d), and s-SNOM phase could be recorded simultaneously. In order to obtain the genuine near-field signal, we then demodulated the s-SNOM output at the third harmonics of the tip tapping frequency.

In the experiment, once illuminated with the QCL, the AFM tip acts as an optical antenna that conveys the energy of the incident photons and launches hyperbolic phonon polaritons (Figure 1a) with a wavelength λ p much smaller than the one of free-space photons λ 0 . Here λ 0 = 1/ω, where ω is the QCL emission frequency in wavenumbers (cm -1 ). Hyperbolic phonon polaritons are propagating waves in hBN, and they can be recognized in the observed s-SNOM images (Figure 1c,d) as periodic oscillations (fringes). Similar to previous works, 28,31 the polariton fringes usually exhibit a period Δ = λ p /2 close to the hBN edge (white dashed line). The s-SNOM line profiles, taken along dashed lines in the s-SNOM image (Figure 1c), are plotted in Figure 2a. The polariton fringes appear as oscillation peaks and show the strongest oscillations next to the edge (x = 0), followed by a series of weakly damped ones. These fringes originate from the standing wave interference between tip-launched polaritons and edgereflected polaritons. The Δ = λ p /2 fringes are actually superpositioned with weak Δ = λ p fringes in the images, as confirmed by the appearance of α (Δ = λ p ) and β (Δ = λ p /2) peaks in the corresponding Fourier transform (FT) spectra in Figure 2b. As studied in previous works, the Δ = λ p fringes are attributed to polaritons launched by the crystal edge. 31,32 Because the Δ = λ p /2 fringes dominate in the s-SNOM images, the following analysis will rely mainly on the λ p /2 fringes.

On the basis of this experimental technique, we now analyze the properties of hyperbolic phonon polaritons observed in suspended and supported hBN. Polariton fringes are observed in both suspended and supported hBN (Figure 1c,d). They appear to support similar oscillations, as displayed in the s-SNOM line profiles (Figure 2a). Interestingly, although the suspended and supported hBN are in the same crystal and show no topographic difference (Figure 1b), the polariton wavelength in the suspended hBN is longer than the one in the supported hBN: λ sus > λ sup . This wavelength difference is unambiguously revealed in the s-SNOM line profiles and the corresponding FT spectra, as seen in the difference between Δ sus (blue arrow, Figure 2a) and Δ sup (red arrows) and the difference between β sus and β sup (Figure 2b). A direct implication of the fact that λ sus > λ sup , observable in the s-SNOM images (Figure 1c,d), is that polariton fringes in the suspended part are bent toward the crystal interior compared with those in the supported part.

We attribute the difference between λ sus and λ sup to the change of permittivity in the polariton substrate: from SiO 2 ε SiO 2 = 1.22 + 0.02i to air ε air = 1 at this frequency ω = 1450 cm -1 . The dispersion of hyperbolic phonon polaritons in hBN can be derived using the Fabry-Perot quantization condition, which yields 11 Ä

where d is the hBN thickness, l = 0, 1, 2, 3, ... is the branch index of hyperbolic phonon polaritons. ε s is the permittivity of the substrate, and ε x = ε y = ε t and ε z are principal components of the hBN permittivity tensor. k = k 1 + ik 2 is the complex inplane momentum of polaritons and relates to the polariton wavelength by k 1 = 2π/λ p . As described by eq 1, the polariton wavelength λ p depends on the substrate permittivity ε s . Specifically, in the upper Reststrahlen band Im ε z ∼ 0, one obtains

For ε SiO 2 = 1.22 + 0.02i and air

In our experiment, the modification of polariton wavelength by sample suspending can also be observed at other frequencies (e.g., ω = 1440 cm -1 in Figure 1d) and spans over the whole upper Reststrahlen band of hBN. We outline the dispersion of λ sus and λ sup from the same crystal by plotting the frequency (ω)wavelength (λ p /λ 0 ) dispersion in Figure 3b. The simulation results (blue and red curves, using ε air and ε SiO 2 ) excellently match our experimental data (blue dots and red squares).

We now demonstrate that not only the polariton wavelength λ p is modified by the sample suspension but also the polariton damping γ is reduced in this geometry. A reasonable figure of merit for the polariton damping, γ = k 2 /k 1 , describes the decay of oscillation amplitude of propagating polariton waves normalized to the wavelength. In the experiment, the polariton damping γ can be obtained by fitting the s-SNOM line profile with the sinusoidal wave function s(ω) = Ax -1/2 e 2kx . 31,32 To remove the influence of Δ = λ p fringes, we performed the inverse FT of the β peak (Figure 2b), such that the output data (Figure 2c) contain the genuine Δ = λ p /2 fringes. At a representative frequency ω = 1450 cm -1 , we extracted γ sus = 0.061 whereas γ sup = 0.070, as shown by the fitting envelop es in Figure 2c. Following the same procedure, we observe a reduction of polariton damping over a broad range in the Reststrahlen band (Figure 3a), with excellent agreement between simulation results (blue and red curve) from eq 1 and experimental data (blue dots and red squares). This polariton loss reduction γ sus < γ sup is attributed to the removal of dielectric loss of the substrate in the suspended geometry. Having demonstrated the modification of wavelength and damping of hyperbolic phonon polaritons by sample suspension, we quantify these modifications by defining the coefficient for wavelength elongation Δλ = |λ sup -λ sus |/λ sup and damping reduction Δγ = |γ sup -γ sus |/γ sup . On the basis of wavelength λ and damping γ, we further extract the polariton propagation length L p = (2k 2 ) -1 = λ/4πγ and the propagation length elongation coefficient ΔL p = |L p sup -L p sus |/ L p sup . In Figure 4, we plot both the experimental data (black squares) extracted from Figures 1-3 and simulations from eq 1 for Δλ, Δγ, and ΔL p . We provide three important results. First is the modification coefficient Δλ, where Δγ ranges from 0 to 20% and ΔL p ranges from 0 to 30% and they all vary with frequency in the Reststrahlen band. Second is the damping reduction, where wavelength and propagation length elongation via sample suspension are most effective for the l = 0 branch of the hyperbolic phonon polaritons: as the branch index l (see l = 1, 2, and 3 curves in Figure 4a,b) increases, the polariton modification effects (Δλ, Δγ, and ΔL p ) decreases. The different modification effects for different polariton branches are related to their spatial field distribution: 26,33 as l increases, the electromagnetic field of polaritons is better confined inside the hBN slab; therefore, it is less sensitive to the environmental permittivity change (e.g., sample suspending in this work). Third, once the frequency ω is fixed, the modified coefficients Δλ, Δγ, and ΔL p stay universal for hBN crystals of different thicknesses, as verified in our experiments and simulations (Figure 4c,d).

In summary, using infrared nanoimaging and detailed electromagnetic simulations, we have investigated hyperbolic phonon polaritons in suspended hBN. We demonstrated that sample suspension can effectively reduce the polariton damping and elongate the wavelength and propagation length of the polaritons by eliminating the substrate loss and changing the environment permittivity. In our experiment, the damping reduction Δγ reaches 18% in the suspended geometry, thus, providing better figures of merits for hyperbolic phonon polaritons. Sample suspension to modify polariton wavelength and reduce damping, as introduced in this work, is expected to be generalizable to other polaritonic media, such as lowdimensional materials 20,34 including graphene, 35,36 transition metal dichalcogenides, 37,38 topological insulators, 39 black phosphorus, 40 and nanowires 41,42 as well as conventional plasmonic systems including metals and metamaterials. 43 The modification of polaritons by changing the dielectric environment, such as sample suspension, may be further engineered and programmed in order to achieve sophisticated nanophotonic functionalities, including propagation steering and transformation polaritonics. 44

Corresponding Author *A. Alu. E-mail: aalu@gc.cuny.edu.