Substrate-mediated hyperbolic phonon polaritons in MoO₃

1 Introduction

The ability to confine and to guide light at length scales smaller than its free-space wavelength (λ₀) enables numerous nanophotonic applications, including surface-enhanced absorption and scattering [1], [2], [3], [4], [5], nanoscale waveguides [6], [7], [8], and non-linear optics [9], [10]. Polaritons, hybrid excitations of light and coherent charge oscillations in materials, offer a means to achieve such optical control at the nanoscale. For example, several spectroscopic techniques utilize surface plasmon polaritons (SPPs) in metals to boost their sensitivities [2], [11], [12]. However, the fast carrier scattering times in metals (≈fs) results in high losses that generally hinder many nanophotonic applications, especially in the mid-infrared (mid-IR) [13], [14].

Phonon polaritons (PhPs), by contrast, couple light with optical phonons in a polar crystal. These excitations can exist only within spectral ranges, known as reststrahlen bands, delimited by transverse optical (TO) and longitudinal optical (LO) phonon pairs [14], [15]. Within these bands, at least one component of the material’s electric permittivity tensor is negative, resulting in high optical reflectivity while supporting surface-bound PhP modes. Strongly anisotropic materials, in which the real permittivity differs in sign along orthogonal principal axes, support hyperbolic PhPs (HPhPs) within the material’s volume, propagating at angles determined by the wavelength-dependent permittivity [14], [16], [17]. Such hyperbolic materials, therefore, enable high optical confinements (≈1% λ₀) in the mid-IR, with longer lifetimes (≈ps) and lower losses compared with SPPs [14], [17], [18], [19]. Additionally, HPhPs provide the potential for negative refraction, superlensing, and other effects [16], [20], [21].

A number of natural hyperbolic materials supporting HPhPs have been discovered, including h-BN [22], [23] and MoO₃ [24], [25], [26], which have been the subject of recent reviews [27], [28]. Among these materials, MoO₃ is of particular interest due to its natural in-plane anisotropic dispersion [25]. To date, the optical dispersion relations in these materials have been predominantly determined by using scattering-type scanning near-field optical microscopy (s-SNOM) thanks to its ability to image HPhP propagation in real space with nanoscale resolution [20], [24], [29], [30]. Recently, similar investigations were accomplished with photoinduced force microscopy (PiFM) [26], [31], [32] and photothermal induced resonance (PTIR) [32], [33], [34], two nanoscale-capable IR techniques that do not require a spectrally sensitive detector in the far-field. In particular, PTIR was shown to detect a more complete set of theory-predicted HPhP modes in h-BN frustum structures [35], [36], revealing excitations not observed by s-SNOM. With PTIR, light absorption in a sample is measured by transducing the photo-induced thermal expansion of the sample directly beneath the probe tip of an atomic force microscope [37]. Notably, PTIR can sense sample composition far below the exposed surface [38], [39], [40], at depths exceeding 1 µm [33], [41], with its signal proportional to the local sample absorption coefficient [42], enabling easy comparison of PTIR spectra with far-field IR databases [40], [42], [43]. These characteristics enable broad applications in materials science [44], [45], [46] and biology [47], [48], [49], [50], [51]. In nanophotonics, PTIR has been used to characterize integrated waveguides [52], plasmonic modes [53], [54], and near-field absorption enhancement of plasmonic nanostructures [55], [56], [57]. Here, we leverage PTIR to visualize HPhPs propagating in MoO₃ single-crystalline thin layers exfoliated on gold-coated glass or on SiO₂/Si substrates patterned into one-dimensional (1D) gratings. Fourier analyses of PTIR absorption maps and real-space parameterized models of HPhP propagation were used to determine the dispersion relations and to quantify lifetimes, propagation lengths, and optical compressions of HPhPs in MoO₃ single crystals. Increased HPhP propagation lengths and lifetimes of about 2× were observed in suspended regions of MoO₃ flakes compared with those in direct contact with SiO₂. In addition to demonstrating similar capabilities to s-SNOM, here we highlight the ability of PTIR to detect and to discern intrinsic (within MoO₃) and extrinsic (contaminants) subsurface defects. Understanding and controlling the factors that influence HPhP propagation, as examined here, enhances the ability to engineer light–matter interactions at dimensions below the diffraction limit, which is critical for a variety of materials and nanophotonic applications.

2 Results and discussion

The structure of orthorhombic α-MoO₃ (Figure 1a) consists of stacked bilayers composed of distorted MoO₆ octahedra with three crystallographically inequivalent oxygen atoms [58]. Thanks to its layered, van der Waals type structure, single crystals of α-MoO₃ can be exfoliated into flakes of various thicknesses and sizes, with edges predominantly aligned with crystallographically defined [100] and [001] directions. Additionally, MoO₃ is characterized by numerous IR-active optical phonon modes, resulting in multiple reststrahlen bands that can support HPhP propagation [59]. Two of these bands, between about 820 and 972 cm⁻¹ (“lower band”), and between about 958 and 1004 cm⁻¹ (“upper band”) [24], [25], [26], [59], were probed in this work (for further reststrahlen band details see discussion S1 in Supplementary material).

Figure 1:

Photothermal induced resonance (PTIR) detects hyperbolic phonon polaritons (HPhPs) in a MoO₃ single crystal.

(a) Crystal structure of orthorhombic α-MoO₃ (space group D2h16-Pbnm), depicting Mo (in gray) at the center of octahedra composed of three crystallographically inequivalent oxygen atoms (labeled “O₁,” “O₂,” and “O₃”) in red. (b) Real components of the complex electric permittivity tensor (ε), relative to that of free space (ε₀), along the [001] (blue line) and [010] (red line) crystallographic directions in MoO₃. The spectral regions highlighted with pale red and blue backgrounds represent the two, partially overlapping, reststrahlen bands probed in this work. Real-space PTIR absorption maps at (c) 910 cm⁻¹ and at (d) 988 cm⁻¹ of a MoO₃ flake (≈315 nm thick) deposited on a uniform, gold-coated glass substrate. Dashed lines depict the approximate locations of the flake edges. Polaritons in MoO₃ manifest as a superposition of periodic fringes of varying absorption intensity. The [100] and [001] crystallographic directions are inferred from the fringes observed in each reststrahlen band. Elliptic in-plane propagation (type I) is observed in the upper band, while hyperbolic in-plane propagation (type II) is observed in the lower band. Note that non-linear color scales are used, with bands of approximately equal change in absorption intensity indicated by tick marks along the right sides of the color bars. The topography map acquired simultaneously with (c) is available in the Supplementary material (Figure S2).

The excitation of HPhPs typically requires the use of evanescent illumination geometries or light-scattering sites to bridge the momentum mismatch between incident photons in free space and polaritons within a material [14], [30]. Intrinsic features, such as crystal edges, or extrinsic ones, such as plasmonic nanostructures or the tip of a scanning probe microscope, commonly fulfill this role [30], [36]. For materials confined in at least one dimension, as in the case of the thin flakes studied here, HPhPs are permitted only with specific (quantized) momenta [35], [60], meaning that they propagate only at selected angles determined by the permittivity tensor of the material. Consequently, in s-SNOM and PTIR real-space maps, HPhPs appear as periodic fringes with spacing proportional to the flake thickness (Figure S1). The polariton fringes observed in PTIR measurements result from especially intense local heating and thermal expansion of the sample that occurs in regions where the strong tip-enhanced near-field overlaps with the polariton field [36]. Polaritons launched at crystal edges (“edge-launched”) propagate through the material with characteristic wavenumbers and are detected at the probe with corresponding spatial frequencies determined by the material dispersion. Similarly, light scattered by the probe tip generates “tip-launched” polaritons that emanate outward and are detected after reflecting back to the probe from another interface such as a flake edge. Both edge-launched and tip-launched HPhPs can exist and be detected simultaneously in PTIR absorption maps, which capture a superposition of all excited modes. However, tip-launched polaritons are characterized by fringe periods that are approximately half that of edge-launched modes of the same order (see below) [61]. Depending on the relative excitation efficiencies of a particular sample, either tip- or edge-launched (or both) HPhPs may be prominently generated and observed. Nevertheless, tip- and edge-launched HPhPs are fundamentally the same phenomenon and their approximate 2:1 wavenumber ratio is just a consequence of the scanning probe measurement scheme (see below).

Representative PTIR absorption maps measured in two MoO₃ reststrahlen bands (Figure 1) reveal significant differences in HPhP propagation. In the lower band (Figure 1c), absorption fringes are aligned parallel to the left and right edges of the MoO₃ flake (nearly vertical) but no fringes are observed along the top (horizontal) edge of the flake. By contrast, in the upper reststrahlen band (Figure 1d), fringes parallel to all visible flake edges are observed, though with lower spatial frequencies than in the lower band. Additionally, in this case, interference between cross-propagating polaritons is evident, especially near the corners of the flake (e.g., Figure 1d, top-right). Such differences reflect the optical anisotropy of MoO₃ and the distinct hyperbolic characters of the two reststrahlen bands (upper band, type I; lower band type II), as observed previously in s-SNOM studies on SiO₂ substrates [24], [25], [26]. Therefore, PTIR images of HPhP propagation along flake edges can be used to identify the crystallographic orientation of the MoO₃ flake, which is indicated in Figure 1.

Real-space PTIR absorption maps encode information related to the superposition of all HPhP modes propagating in the hyperbolic material as variations (e.g., fringes) in the PTIR signal intensity. Discrete Fourier transforms (DFTs) of these real-space maps separate, in frequency space, the polaritons (peaks in the computed power spectra; see, e.g., Figure 2c) from the background absorption and other artifacts. Polaritons observed in this manner can be attributed either to edge- or tip-launched HPhPs propagating in the hyperbolic media, which we model here as damped harmonic oscillations.

Figure 2:

Real-space photothermal induced resonance (PTIR) imaging enables characterization of hyperbolic phonon polariton (HPhP) propagation.

(a) Topography (top) and PTIR absorption map (bottom) of a MoO₃ flake (approximately 279 nm thick) on a uniform, gold-coated glass substrate, illuminated at 910 cm⁻¹. (b) Average height (top) and absorption (bottom) line profiles from (a), scaled to arbitrary units. The dashed lines in (a) and (b) denote the position of the topographically identified flake edge; profile segments inside and outside the flake edge are shown in purple and in black, respectively. (c) Magnitude of the discrete Fourier transform (DFT) of the absorption line profile, within the interior of the flake only, showing a prominent peak representing a tip-launched HPhP mode. (d) Dispersion relation of HPhPs in the MoO₃ flake shown in (a). Dashed lines represent the theoretical HPhP dispersion (see Supplementary material S1.2) for edge-launched modes of order n. (e) Group velocities and the corresponding (f) lifetimes of the edge-launched HPhP modes from (d). Dashed line in (e) represents the derivative of the theoretical dispersion (n = 1) from (d). Error bars in (d–f) represent uncertainties in the mean values, propagated from least-squares fit covariance matrices.

For HPhPs propagating in one dimension along an x-coordinate axis, with angular wavenumber k and experiencing damping γ, we fit Equation (1) to the measured real-space absorption profiles.

In this model, a phase parameter (φ) accounts for offsets in the relative positions of the measurement origin (taken to be the topographically identified flake edge) and the fringe pattern, while A scales the model amplitude to match the measured intensities. Since, in general, PTIR absorption profiles represent superpositions of HPhPs, the model sums over i independent modes, which are identified by peaks (k) in their power spectra (see below and Figure 2c). By fitting this model to the PTIR absorption profiles we obtain estimates of the parameters (A_i, γ_i, k_i, φ_i) that characterize each mode. These parameters can be used to quantify the HPhP damping rates and lifetimes. This analysis is illustrated in Figure 2 for a MoO₃ flake with its edge aligned nearly perpendicular (vertical) to the fast-scan direction of the probe (horizontal). In this configuration, sequentially measured scan lines possess an approximate translational symmetry along the slow-scan direction of the probe (vertical). Consequently, column-wise (vertical) averaging of pixels in the resulting PTIR images enables 1D analyses with enhanced signal-to-noise ratios. The DFT power spectrum (Figure 2c) of the averaged, 1D absorption profile reveals a prominent polariton peak (≈16.7 µm⁻¹). By plotting the angular wavenumbers of the detected DFT peaks (blue circles in Figure 2d) obtained from the Fourier analysis of a series of PTIR images, measured at different IR wavelengths, we constructed the HPhP dispersion curve of this MoO₃ crystal. Clearly, the measured polariton dispersion does not match the trend line predicted by theory for the first (n = 1) or second (n = 2) order edge-launched HPhP modes (dashed lines in Figure 2d); see theory details in Supplementary material S1.2. However, the measured angular wavenumbers are approximately double the values predicted for the first order edge-launched mode. Therefore, we attribute the prominent peaks in the DFT plots to HPhPs launched by the gold-coated probe tip [61]. For tip-launched HPhPs, movement of the probe toward or away from a flake edge changes the distance traversed by tip-launched HPhPs by twice that of the corresponding edge-launched HPhPs (see Equation (1)), leading to an apparent doubling of the spatial frequencies of HPhPs launched by the tip [61]. Scaling the observed spatial frequencies by a factor of 1/2 enables reconstruction of the complementary edge-launched HPhP dispersion curve (red circles in Figure 2d), which coincides closely with theoretical predictions for edge-launched polaritons. The relative amplitudes of the tip- and edge-launched peaks in the DFT power spectra depend on the relative launching efficiencies and can vary between samples and scattering sites [30]. Typically, tip-launched HPhPs are assumed to originate from an azimuthally symmetric source (probe tip), and propagate outward with a circular wave front that decays in intensity proportional to an additional x prefactor not shown in Equation (1) but accounted for in our fitting and subsequent analyses. For the remainder of this work, HPhP propagation characteristics are reported for the cases of edge-launched HPhPs, either observed directly or from scaled analyses of more prominently detected tip-launched HPhPs.

Additional HPhP characteristics can be derived from the model-fit parameters. Significantly, the damping factors describe the rates at which HPhPs decay in the material and can be related to their propagation lengths (L), as shown in Figure S3, using Equation (2).

Additionally, the speed (group velocity, v_g) of HPhPs propagation through the material can be determined from the slope of the dispersion curve,

which is shown in Figure 2e for HPhPs in the lower reststrahlen band. We note that HPhPs in MoO₃ [25], and other hyperbolic materials [14], [19], propagate with high optical compressions (≈30×, λ_HPhP ≈ 3% λ₀) and at speeds much slower than the corresponding free-space photons; here, v_g ≈ 0.1% c, among the slowest reported values for such materials [25], [62]. Furthermore, HPhP lifetimes (τ) can be estimated from the approximate relationship between the propagation length and group velocity shown in Equation (4).

These characteristics enable quantitative comparisons of HPhP propagation in different materials and samples, thus permitting a comparison of their effectiveness in various nanophotonic applications. To this end, HPhPs in the lower MoO₃ reststrahlen band were found to have lifetimes up to about 9 ps (Figure 2f), which are about two orders of magnitude longer than SPP in Au at near-visible wavelengths [63], [64]. The lifetimes measured here, for HPhPs propagating in MoO₃ on a gold-coated substrate, are longer than the lifetimes reported previously on SiO₂ surfaces (≈1.9 ps) [25]. This difference, in large part, is due to the long propagation lengths of the HPhPs (determined to be tip-launched) measured in these crystals.

Despite traveling within the volume of a host material, HPhPs are also sensitive to the surrounding media through their evanescently decaying external fields. These evanescent fields limit the range of HPhP interactions outside of the hyperbolic material, typically to a few tens of nm, similar to the decay range of SPPs [65], [66]. Consequently, variations in the local complex refractive index, such as near material discontinuities, can modify the dispersion, damping, and optical compression of HPhPs [67], [68], [69], [70], [71]. Measurements of a MoO₃ crystal deposited onto a linearly patterned SiO₂ substrate grating (alternating SiO₂ lines and 95-nm-deep trenches of nearly equal widths; see Figure 3c and Section 4 for details) enabled side-by-side examination of tip-launched HPhPs propagating in both substrate-supported and suspended regions, revealing clear differences (Figure 3b and d). In suspended regions, HPhP fringes extend farther from the crystal edges and have lower optical compressions than HPhPs propagating in regions of the MoO₃ crystal in direct contact with the SiO₂ substrate. Quantitative comparison of PTIR absorption profiles measured at 910 cm⁻¹, averaged along narrow bands centered within each region, reveals that HPhPs in suspended regions propagate with wavelengths ≈1.5× longer than in regions in direct contact with the substrate (Figure 3f). Extension of this analysis to other excitation wavelengths yields the dispersion relations and propagation parameters characterizing HPhPs in each of these regions (Figure S4), including lifetimes of up to ≈12 ps in suspended regions of the MoO₃ crystal (Figure 3h). We hypothesize that the shortened lifetimes for HPhPs propagating in the supported regions (≈5 ps) are a result of dissipative interactions with the SiO₂ substrate, which is characterized by a broad and strong (i.e., large imaginary refractive index) phonon resonance near 1075 cm⁻¹. These interactions are not experienced as strongly within suspended MoO₃ regions. Unlike the sharp, step-like changes in the grating topography, the HPhP fringes in the PTIR absorption maps exhibit gradual transitions from a compact, high spatial frequency arrangements in the SiO₂-contacted regions of the flake, to more diffuse, low spatial frequency arrangements in suspended regions. The extent over which these transitions occur illustrates the lateral range of the gradually weakening influence of the supporting substrate on HPhPs propagation, which is proportional to the polaritonic wavelength. As expected, these transitions become sharper for HPhPs with higher optical compressions (i.e., shorter polariton wavelengths) [67]. These results demonstrate that substrate engineering is a viable strategy for controlling HPhP propagation with direct, localized effects on HPhP lifetimes and optical compressions, in agreement with previous studies [69], [70]. Since we detect no significant deformations of the MoO₃ crystal due to uneven surface contact or mechanical loading by the probe tip in the suspended regions, we expect lattice strain to play a negligible role in the variations in HPhP propagation observed here.

Figure 3:

Topographically patterned substrates enable control over hyperbolic phonon polariton (HPhP) propagation.

(a) Topography, (b) photothermal induced resonance (PTIR) absorption map at 910 cm⁻¹, and (c) schematic of a MoO₃ flake (thickness t ≈ 185 nm) deposited on one-dimensional SiO₂ grating lines (width w ≈ 1.4 µm, pitch p ≈ 3 µm) separated by trenches of depth d ≈ 95 nm. (d) Magnified PTIR absorption map at 910 cm⁻¹ of the region enclosed by the dashed rectangles in (a) and (b). (e) Averaged (165-nm-wide) absorption line profiles measured along the indicated paths in (d). (f) Magnitudes of the discrete Fourier transforms (DFTs) of the absorption profiles shown in (e); cubic spline interpolation plotted. Peaks in the respective DFT power spectra indicate that HPhPs propagate with higher optical compressions (≈1.5×) in regions of the MoO₃ flake in direct contact with the SiO₂ substrate compared to suspended regions, above the trenches. (g) Group velocities and corresponding (h) HPhP lifetimes. Dashed lines in (g) represent the theoretical group velocity of first order, edge-launched HPhPs. Error bars in (g) and (h) represent uncertainties in the mean values, propagated from least-squares fit covariance matrices. Note that a non-linear color scale is used in (d) with bands of approximately equal changes in absorption intensity indicated with tick marks along the right side of the color bar.

The PTIR measurements presented here also reveal crystal defects and buried contaminants that are imperceptible to examinations of only the exposed sample topography or that would be difficult to detect with primarily surface-sensitive methods. Unlike s-SNOM, for which the sensitivity of the tip-scattered light decreases rapidly as a function of the depth [33], [72], PTIR is capable of detecting subsurface features at depths up to a few µm, comparable with the IR light penetration depth [33], [34], [73]. Figure 4 shows PTIR absorption maps at 985 and 910 cm⁻¹ of the same MoO₃ flake seen in Figure 3 (different location), revealing relatively strong absorption in regions localized over trenches in the substrate. Absorption spectra measured on these features exhibit a distinctive peak at ≈1264 cm⁻¹ that we attribute to the Si–CH₃ asymmetric deformation of polydimethylsiloxane (PDMS) residue [40]. Since no significant topographic features were observed on the exposed surface, we conclude that these features represent PDMS contaminants trapped primarily in the trenches beneath the crystal, derived from the polymer stamp used to prepare the sample (see Section 4). We observe HPhPs propagating in the MoO₃ crystal directly above the PDMS contaminants (Figure 4b). However, due to the PDMS background absorption, the effect of PDMS on HPhP propagation could not be quantified properly here. In addition to extrinsic contamination, seemingly intrinsic defects present in some MoO₃ crystals were also observed in PTIR absorption maps that do not correspond to obvious topographic features and with apparent negligible affect HPhP propagation (see Figure 1 and Supplementary materials Section S6.1).

Figure 4:

Photothermal induced resonance (PTIR) reveals trapped contaminants that can influence hyperbolic phonon polariton (HPhP) propagation and imaging. Absorption maps at (a) 985 cm⁻¹ and at (b) 910 cm⁻¹, magnified region from (a), of the same MoO₃ crystal shown in Figure 3 (but at a different location), supported by a SiO₂/Si grating. Relatively strong, localized absorption features (absent in topography) reveal contaminants trapped beneath the MoO₃ flake. (c) Absorption spectra measured at locations one to six indicated in (a) identify the contaminants as polydimethylsiloxane (PDMS), derived from sample preparation. The contaminants underneath the MoO₃ flake affect the HPhP propagation and change the apparent spacing between successive fringes, indicated by arrows in (b), compared with those observed in uncontaminated regions (e.g., Figure 3b and d). Topography maps acquired simultaneously with (a) and (b) are available in the Supplementary material (Figure S6).

3 Conclusions

In summary, PTIR absorption maps visualize HPhPs propagating in exfoliated α-MoO₃ single crystals. By combining real-space PTIR images and Fourier analyses, we determined the polariton dispersion relations, propagation lengths, lifetimes, and group velocities in MoO₃ crystals. Long HPhP lifetimes, in excess of 10 ps, were measured for HPhPs propagating in MoO₃ with optical compressions >30×. The effects of substrate morphology were tested by comparing HPhP propagation in regions of a MoO₃ flake, deposited on a grating substrate, that were either suspended or in direct contact with SiO₂. Polaritons in suspended regions of the flake were found to propagate farther, with longer lifetimes (≈2×) and lower optical compressions (≈1.5×), than those in regions in direct contact with the substrate. Additionally, by leveraging the ability of PTIR to sense subsurface features, we detected and identified the composition of contaminants (PDMS) beneath regions of a MoO₃ flake, which originated from the sample preparation procedure. We also observed that some intrinsic crystal defects in MoO₃ had negligible impacts on HPhP propagation. This work demonstrates the versatility of PTIR for nanophotonic measurements of HPhPs, which provides a useful complement to s-SNOM studies. Furthermore, leveraging substrate morphology and material composition, as demonstrated here, represents a viable technique that enhances the ability to engineer and to control HPhP propagation in other nanophotonic systems.

4 Methods