Long-lived phonon polaritons in hyperbolic materials

Natural hyperbolic materials with dielectric permittivities of opposite sign along different principal axes can confine long-wavelength electromagnetic waves down to the nanoscale, well below the diffraction limit. This has been demonstrated using hyperbolic phonon polaritons (HPP) in hexagonal boron nitride (hBN) and  -MoO 3 , among other materials. However, HPP dissipation at ambient conditions is substantial and its fundamental limits remain unexplored 1 , 2 . Here, we exploit cryogenic nano-infrared imaging to investigate propagating HPP

We investigate hyperbolic phonon polariton (HPP) propagation and dissipation in isotopically pure hexagonal boron nitride (h 10 BN, h 11 BN), along with naturally abundant hBN and in -MoO3 van der Waals crystals using near-field infrared (IR) microscopy. Nano-IR methods allow one to directly visualize polaritonic standing waves on the surface of these hyperbolic materials 1-6 . Similar to graphene plasmon polariton investigations [7][8][9][10][11][12] , both the wavelength of the HPP ( ) and its dissipation  can be readily obtained from nano-IR images (Figs. [1][2], with these two quantities allowing one to extract the complex dielectric function () =  ′ +  ′′ of a phononic medium [3][4][5][6] . Earlier attempts have characterized HPP in isotopically pure hBN (h 10 BN, h 11 BN) and in -MoO3 crystals at ambient conditions 6,[13][14][15][16][17][18] . However, the fundamental limits of HPP dissipation and lifetime remain to be determined as this necessarily relies on temperaturedependent nano-imaging of polaritonic waves. In this work, we report cryogenic nano-imaging results for these van der Waals materials for the first time, demonstrating record-long HPP lifetimes. We studied isotopically pure hBN and biaxial hyperbolic -MoO3 crystals, and compares these results with naturally abundant hBN crystals. Combined with theoretical models, we examined the physics governing HPP dissipation in isotopically pure hyperbolic crystals.
To perform cryogenic nano-imaging, we utilized a home-built scanning near-field IR microscope operating at variable temperatures 7,[19][20][21] . In this setup, the incident light with IR frequency  is focused onto the metallized tip of an atomic force microscope (AFM). As the tip approaches the sample, a concentrated evanescent field excites polaritonic modes with a ( ) that is much shorter than the free-space wavelength IR = 2π / of the incident photons [22][23][24][25][26] . This unique nano-IR apparatus has been routinely employed in studies of plasmon and phonon polaritons as well as for visualizing inhomogeneities in complex oxides [19][20][21] . In the previous HPP nano-imaging studies, a physical sample edge was chosen as the HPP wave reflector 6   We begin with a survey of a large-area (25  20 m 2 ) image of HPP standing waves in h 11 BN obtained at T = 50 K (Fig. 1c) with an IR laser operating at IR = 6.6 m. Here we display raw data in the form of the scattered near-field amplitude s normalized to the corresponding signal detected from the gold disks, whose optical response provides a convenient temperature (T)independent reference. The most prominent aspect of the image in Fig. 1c is that the entire field of view is filled with HPP fringes. As expected, the p-fringes dominate the field of view, emanating from the Au antenna and propagating radially outwards. Even a cursory inspection of Fig. 1c reveals that HPP remain highly confined with IR/p > 20 and yet they travel over tens of microns, far exceeding previous measurements at ambient temperature 6,13 .
Nano-IR data in Fig. 2 attest to a clear reduction of HPP losses in monoisotopic hBN specimens at lower temperatures with the incident frequency of 1522 cm -1 (6.57 m). HPP in h 11 BN films (thickness of 180 nm) at room temperature exhibit 15 µm propagation lengths, corresponding to quality factor Qp ~ 35 as defined below (Fig 2d&e). As the temperature is reduced, we observe a systematic increase in quality factor and propagation length. Specifically, at T = 45 K, Qp exceeds ~ 60 which represents the highest quality factor achieved to date. To elucidate the underlying polariton scattering mechanisms, we have also performed T-dependent studies in h 10 BN and naturally abundant samples for systematic comparisons (Fig. 2a&c). Two observations can be drawn. First, we observed that the monoisotopic specimens of h 10 BN and h 11 BN share nearly identical propagation lengths at all temperatures. These HPP oscillations exhibit scattering lifetimes 2-3 times longer than those observed in naturally abundant hBN crystals, consistent with prior results 6 . Second, HPP propagation lengths at a given frequency generally increase with reduced temperatures for all hBN specimens, as documented in Fig 2d&e. Notably, we found that both h 10 BN and h 11 BN exhibit a different temperature dependences compared with naturally abundant hBN crystals of similar thickness, as manifested by their HPP oscillations and the quality factor Qp (Fig 2d&e). These overall trends suggest different underlying mechanisms of phonon scattering between isotopically-pure and natural abundant hBN, as discussed in the following sections in detail. For contrast, we have also investigated temperature dependence of hyperbolic HPP in exfoliated crystals of -MoO3, an emerging natural material that exhibits in-plane hyperbolicity at mid-IR frequencies, to determine if the same temperature dependent scattering is observed as in hBN 13,14,16 . In contrast to the convex wavefronts in isotropic materials, a concave wavefront of the HPP modes in thin slabs of -MoO3 have been observed in ambient conditions 13,14,16,17 . In Fig. 3, we show raster scans of an -MoO3 crystals (~ 200 nm thickness) exfoliated onto an Au antenna obtained at temperatures ranging from 50 to 300 K and at the frequency of 928 cm -1 . At room temperature, HPP interference fringes with concave shapes along the (100) direction were observed, consistent with the isofrequency curves as expected from the in-plane hyperbolic responses 13,14,16 . As the temperature is reduced, we observe a systematic increase in both the overall propagation distance and the number of detectable HPP oscillations, consistent with hBN results. At T = 50 K, HPP oscillations approach propagation lengths of 20 µm (Fig. 3a). Similar temperature-dependence of HPP was also detected in the elliptical response regime (see SI). These overall trends are similar to our results on hBN isotopes, indicating the uniform characteristics of phonon propagation and dissipation emerging at low temperatures in hyperbolic crystals. As shown by solid curves superimposed on polariton line-profiles in Fig. 2d & Fig. 3b, the fitted results match well with the experimental data across variable temperatures. This analysis allowed us to extract the temperature-dependent quality factors Qp (T) for polaritons in all isotopes of hBN and -MoO3 crystals for the first time, as presented in Fig. 2e & Fig. 3c. Specifically, we found that the highest quality factor Qp (T) stems from the near-monoisotopic hBN flakes at cryogenic temperatures with Qp exceeding 60.
We now outline the damping rate analysis based on the temperature-dependent results extracted from hBN crystals. In Fig. 4, we plot the temperature-dependent HPP damping rates . Several pieces of information arise from Fig. 4. Firstly, the HPP damping rates scale quasi-linearly with temperature for all samples. This is different from the nonlinear temperature dependence of plasmon damping rates observed in pristine graphene 7 .
Secondly, compared to h 11 BN and h 10 BN, the polariton damping rate in naturally abundant hBN displays a higher residual damping in the zero-Kelvin limit and a steeper slope to the temperature dependence.
The temperature dependent studies offer insights into the damping mechanisms of HPP in both hBN and -MoO3 crystals. We posit that the temperature dependence can only be described by the participation of acoustic phonons in the scattering process, as the optical phonons are not thermally occupied in the temperature range of < 300 = 208 −1 (see SI). In isotopically pure hBN, we attribute the leading damping channel to momentum relaxation caused by acoustic phonons, that is, the polariton being scattered from its intial state with momentum to the final state with momentum = + by absorbing/emiting an acoustic phonon with momentum ± (Fig. 4b). Under reasonable approximations (see SI Sec. 4.1), the resulting scattering rate reads where 0 is a temperature-independent scattering rate due to other decay processes 27,28 . The dashed fitting lines in Figure 4 correspond to 0 = 0.4 −1 , 0 = 0.6 −1 , 0 = 0, = 0.5 −1 , i = 2.5 and Λ = 450 . Note that these fitting parameters are obtained for our simplified analytical model. Further dedicated first-principle calculations are possibly needed for fine tuning, which is beyond the scope of the current work. On top of the above intrinsic damping channels, the dielectric loss from the SiO2 substrate contributes < 20% to the total damping rate at the experimental frequency of 1522 cm -1 . Finally, the damping rate in -MoO3 is smaller than natural hBN, while slightly larger than monoisotopic hBN. The illustration of scattering of a hyperbolic polariton from an initial state (blue dot) to a final state (red dot) on the iso-frequency surface by either emitting/obsorbing an acoustic phonon or by impurity potential.
In summary, our results account for HPP dynamics in high-quality hyperbolic crystals.
Quality factors and scattering rates of hyperbolic polaritons in naturally abundant and monoisotopic hBN, as well as -MoO3 have been extracted from cryogenic nano-imaging studies.
Though limited by acoustic phonon scattering and other decay channels, the HPP propagation length and lifetime in monoisotopic hBN at low-T exceed 25 m and 5 ps, a new record among all natural hyperbolic materials reported so far. Our first report of HPP nano-imaging studies of high-quality hyperbolic crystals at cryogenic temperatures demonstrates promising opportunities for the exploration of advanced phononic switching 29 , polariton periodic orbits in cavities 30 and nonlinear phenomena 31 in ultra-pure polar samples at technologically relevant liquid nitrogen temperatures.