Light interaction with excitons, bound states of electrons and holes in crystals, is a rich subject that has been studied extensively in semiconductor quantum wells (1–11) and – more recently – in van der Waals (vdW) semiconductors (12–21). When excitons and light couple strongly, they form bosonic quasiparticles termed exciton polaritons (22). Exciton polaritons display intriguing physics: they have been made into Bose-Einstein condensates (3–4) demonstrating superfluidity (5–6) with quantized vortices (7–8). Furthermore, exciton polaritons have been used to realize unconventional topological insulators (9), low-threshold switching devices (10), lasers without population inversion (11), etc. Hyperbolic vdW materials, where permittivities have opposite signs along different crystal axes, are predicted to host exotic new kinds of exciton polaritons called hyperbolic exciton polaritons (HEPs) (23–24). HEPs may confine light to deeper subdiffractional wavelengths and provide enhanced polaritonic density of states relative to conventional exciton polaritons (20–21). Hyperbolic polaritons can be composed of any polar excitation: they have been observed with phonons (25–26), plasmons (27), and even transient excitonic transitions (28), but never with steady-state excitons. HEP confinement can reach length scales comparable to the exciton Bohr radius, leading to unique nonlocal and quantum effects (23). An experimental realization of HEPs will thus open new pathways to manipulate excitons and light at the nanoscale.
Exciton polaritons are commonly achieved within Fabry-Pérot microcavities, where highly-reflective mirrors on either side of a semiconducting well ensure strong coupling of excitons and photons. Energy oscillates between trapped photons and excitons (Rabi oscillations), leading to Rabi splitting of mode frequencies and coalescence of scattering rates (1–2). Slabs of vdW semiconductors without a closed cavity can themselves serve as low-quality resonators that support propagating waveguide modes interacting with excitons (14), in a process sometimes referred to as self-hybridization (15). This geometry lends itself to scanning probe nano-optical techniques since the material surface is exposed, permitting direct nano-imaging of exciton polaritons (Fig. 1a). At room temperature, waveguide modes reveal negative phase velocity or “backbending” dispersion and increased dissipation around exciton resonances (16–18). Upon cooling, exciton resonances sharpen, causing an anti-crossing in the waveguide or cavity mode dispersion (19). In hyperbolic materials, we will show HEPs appear inside the anti-crossing with multimode dispersion.
HEP imaging is technically challenging since light must acquire a large momentum to couple to HEPs, and exciton resonances must be strong enough to drive the permittivity to negative values while keeping damping low – requiring cryogenic temperatures in the case of most vdW materials (29). In this work, we developed the first-ever cryogenic near-infrared near-field microscope to satisfy these experimental requirements and present the first direct images of steady-state HEPs in the vdW semiconductor chromium sulfide bromide (CrSBr). Excitons in CrSBr have large oscillator strength and small scattering rate even in bulk crystals, enabling observation of exciton polaritons without the need for a closed cavity (21). Using our home-built nano-optics apparatus, we establish the existence of HEPs through a combination of energy-, temperature- and thickness-dependent measurements. Furthermore, CrSBr is known to be an A-type antiferromagnet (AFM), where individual ferromagnetic vdW layers order antiferromagnetically below the bulk Néel temperature TN=132 K (30–31). Excitons in CrSBr have been found to couple to magnetic order (32–33) and additional unidentified optical transitions appear below TN (20, 34). We observe HEPs in CrSBr coupling to these optical transitions, indicating that these transitions persist in the high-momentum response. Finally, we note that the onset of hyperbolicity is concurrent with the appearance of intralayer ferromagnetic order at 160 K (31, 35), suggesting magneto-electronic coupling may be at play. Calculations within a self-consistent excitonic-vertex-corrected many-body perturbative approach (QS\(G\widehat{W}\)) (36) reveal exciton delocalization upon magnetic ordering, which contributes to increased exciton spectral weight and robust hyperbolicity in CrSBr.
We first extract in-plane optical constants of bulk CrSBr at different temperatures from broadband reflectance measurements by fitting a multilayer model (Materials and Methods). CrSBr is orthorhombic with Pmmn (D2h) space group and lattice parameters a = 0.4767 nm, b = 0.3506 nm, and c = 0.7965 nm (30); it has a diagonal biaxial dielectric tensor in the principal basis. The b-axis dielectric permittivity is dominated by exciton resonances below the semiconducting bandgap at 1.5 eV (37). In this work, we focus on the exciton resonance (X) peaked at 1.343 eV (295 K)-1.3675 eV (20 K). The X linewidth is broad at room temperature but narrows significantly at cryogenic temperatures (Fig. 1b), likely due to suppression of phonon-induced scattering (38). Along the a- and c-axes, the dielectric functions are effectively constant in the near-infrared range (\({\epsilon }^{a}=12\) at 295 K; \({\epsilon }^{c}=3.7\) at 20 K, 4.5 at 50 K, and 8 at 295 K). c-axis values were extracted from near-field microscopy (Fig. S3). Because X appears only along the b-axis and is sharp enough at low temperature to allow for negative \({\epsilon }_{1}\) values, we observe an energy band around 1.4 eV (orange region in Fig. 1b) where the CrSBr isofrequency surface is hyperboloidal. Further, we note the existence of sidebands of X, labeled X* and X**, at 1.3830 eV and 1.3935 eV in the low-temperature b-axis dielectric function (Fig. 1b, right panel).
While strong coupling between excitons and photons causes an anti-crossing of the waveguide mode dispersion, the highly anisotropic nature of the excitons in CrSBr also admits HEPs between the split waveguide modes (Fig. 1c). HEPs can be either surface or bulk modes. Surface modes disperse from inside the material light cone and have evanescent surface fields. Bulk modes, on the other hand, exist only to the right of the material light cone. In CrSBr, HEPs have in-plane hyperbolic wavefronts with oscillatory out-of-plane electric fields propagating through the bulk like waveguide modes. The fundamental (n=0) HEP mode is a surface mode inside the material light cone and becomes a bulk mode outside. On the other hand, higher-order HEP modes (n=1, 2…) always exist as bulk modes (Supplementary Note S1). A finite difference time domain simulation (Materials and Methods) of the n = 1 bulk HEP propagating through CrSBr is shown in Fig. 1d. Furthermore, the sidebands noted earlier in Fig. 1b may couple to HEPs. The backbending features in the HEP dispersion (Fig. 1c) correspond exactly to X* and X** energies.
To access HEPs experimentally, we performed cryogenic scattering-type scanning near-field optical microscopy (s-SNOM) on a home-built system illuminated by a tunable continuous-wave near-infrared laser (Fig. 1e). We also use a commercial s-SNOM system to characterize CrSBr waveguide modes at room temperature (Materials and Methods). In s-SNOM experiments, a tapping metallized tip is illuminated by laser light and the backscattered signal is demodulated at the higher harmonics of the tip-tapping frequency. The tip can launch modes, including high-momentum polaritons that cannot be excited with free-space light, which propagate to sample edges, transmit or reflect, and interfere with light from the tip. As the tip scans, interference fringes appear, corresponding to various light modes inside the material (e.g. waveguide modes and polaritons) or air modes, which are free-space standing waves between tip and sample that disperse like the vacuum light cone.
We verify at room temperature that excitons and photons couple directionally in CrSBr. Figure 2 shows select room-temperature s-SNOM images and line profiles from corners of CrSBr crystals at photon energies in the range 1.26–1.55 eV. Fringes in Figs. 2a and 2b are superpositions of the fundamental transverse magnetic waveguide mode (TM0) and air mode interference fringes. Near-field amplitude line profiles along the b-axis (Fig. 2c) show a region of negative or backbending dispersion near X (follow dashed lines). This backbending in real-(ω, k) space corresponds to a Rabi splitting in complex-\(\omega\) electrodynamics of \({\Delta }\omega\)=226 meV (Fig. S6). A crude estimate of the coupling strength \(\kappa \sim 114\) meV (Supplementary Note S2) tells us that exciton-photon coupling along the b-axis is well into the strong coupling regime, even by a more stringent definition \({\kappa }>\frac{{{\gamma }}_{ex}+{{\gamma }}_{ph}}{2}\sim 96\) meV. In contrast, line profiles along the a-axis (Fig. 2d) maintain a positive linear dispersion throughout the investigated energy range, indicating that a-axis TM0 modes do not interact with excitons. In Fig. 2e, after applying a standard geometrical correction (Supplementary Note S3), Fourier peaks of both a- and b-axis profiles are overlaid on the Im \({r}_{p}\) loss function based on the dielectric functions obtained from far-field reflectometry. Fast Fourier transforms (FFTs) of line profiles in Figs. 2c and 2d can be found in Fig. S7. We note excellent agreement between the calculated dispersion and momenta extracted from near-field profiles, affirming that exciton-waveguide mode coupling occurs only about the b-axis.
Next, we perform near-field nano-imaging of CrSBr at cryogenic temperatures. The waveguide mode dispersion now splits about the hyperbolic band with a complex-ω Rabi splitting energy of \({\Delta }\omega =\)163 meV at 20 K (Fig. S6d). \({\Delta }\omega\) is smaller at 20 K than at room temperature because of the reduced \({\epsilon }^{c}\) (Fig. S3), which was overlooked by other studies (20). We now expect HEPs to appear within the splitting. Figure 3a shows s-SNOM images at E = 1.378 eV of a 200 nm CrSBr crystal at T = 295 K and 50 K in the same region. Both images are normalized to the substrate and plotted on the same color scale. The low-temperature image is noticeably brighter, consistent with a negative \({\epsilon }_{1}\) and the appearance of a hyperbolic band. Moreover, an additional subdiffractional fringe with different periodicity appears in the 50 K image (black arrow). After a geometrical correction (Supplementary Note S3), its corresponding momentum is ~ 1.74\(\times\)105 cm− 1 at 20 K; which is beyond the material light cone with k(E = 1.376 eV, T = 20 K) \(\approx\) 1.32\(\times\)105 cm−1. This momentum is consistent with the n = 1 bulk HEP mode.
In real-space (Fig. 3b, left panel), HEP fringes (orange diamonds) are partially obscured by air modes (green circles). In FFT spectra (right panel), however, HEPs (orange diamond) and air modes (green circle) can be readily distinguished. We fit two Lorentzian lineshapes (dashed blue lines) to the measured FFT spectrum and show the corresponding real-space inverse FFT in the left panel. If only the HEP peak is filtered in, then the corresponding real-space fringes are shown in the bottom left of Fig. 3b. We see that the HEP propagates for ~ 1 µm with a confinement factor \(k/{k}_{0}\) of ~ 2.5. Similarly, we can use a two-dimensional FFT filter to remove air mode fringes from Fig. 3a. The filtered near-field image at 50 K is shown in Fig. 3c and the filtered regions of Fourier space are circled in red in the bottom inset. The HEP fringes are now unobscured and we note that the averaged line profile (black line) is comparable to the one-dimensional Fourier-filtered line profile from Fig. 3b.
Line profiles along the white dashed b-axis line in Fig. 3a are shown in Fig. 3d at T = 295, 100, 50, and 20 K in black, gray, cyan, and purple, respectively. At room temperature, we observe TM0 waveguide modes and air modes with similar periodicities. At 100 K, TM0 modes disappear as the waveguide mode dispersion splits and only air modes remain. New peaks appear between the air mode fringes at lower temperatures, changing position from 50 to 20 K (follow black dashed lines). Calculations using far-field optical constants suggest that the HEP wavelength should increase with decreasing temperature (Fig. S9a) as \({\epsilon }_{1}\) becomes less negative – in agreement with the experimental line profiles. Figure 3e shows a 50 K near-field image outside the hyperbolic band, at E = 1.304 eV, with corresponding FFT spectrum below. Figure 3e looks qualitatively different from Fig. 3a at 50 K. We see the TM0 waveguide mode since the probe energy is far from the anti-crossing. Unlike the HEP in Fig. 3b, the corrected momentum of the TM0 peak in Fig. 3e is ~ 1.2\(\times\)105 cm− 1, less than the momentum of the material light cone (blue dashed line). Also, the TM0 mode is much less lossy than the HEP, as seen by either comparing propagation lengths or FFT peak intensities relative to air modes.
To further confirm our assignment of HEPs, we investigated another CrSBr crystal of different thickness. Figure 4a shows a near-field image of the corner of a 107 nm CrSBr crystal at T = 20 K and E = 1.376 eV. We again see a subdiffractional fringe propagating along the b-axis. No subdiffractional fringe was observed along the a-axis – as expected from the in-plane hyperbolic isofrequency contour (left panel). The large anisotropy between observed a- and b-axis air modes, on the other hand, is primarily due to the different interference paths along the two directions; which are also different from Fig. 2 (Fig. S8). In Fig. 4b, the averaged b-axis line profile in the 107 nm sample (blue) is compared to a line profile from a 200 nm sample (purple) at the same temperature and photon energy. The fringe wavelength shortens with decreasing thickness, which is consistent with the well-established property that hyperbolic modes have higher momenta in thinner samples (25–26). By contrast, waveguide modes follow the opposite trend (Fig. S9b): allowing us to readily distinguish HEPs from waveguide modes and conventional exciton polaritons.
Next, we analyze HEP momenta quantitatively and assess the experimental evidence for coupling between HEPs and exciton sidebands. Figure 4c shows experimental HEP momenta (blue squares) from near-field data taken at T = 20 K for a crystal with thickness d = 107 nm with probe energies E = 1.367–1.39 eV. FFTs of line profiles can be found in Fig. S7. Error bars represent full-width half-maxima of FFT peaks. Data are overlaid on the Im \({r}_{p}\) loss function (maxima in orange) and calculated dispersion (white line). Note that \(\text{max}\text{I}\text{m} {r}_{p}\) is not always a good indicator of poles for lossy modes near the light cone (Supplementary Note S1). Our data agree with the calculated dispersion (white line) and support the existence of backbending near X* - indicating that HEPs couple to exciton sidebands. Furthermore, near-field images taken with E > 1.381 eV do not have noticeable HEP fringes, suggesting that X* may be enhancing polariton dissipation. Figure 4d shows near-field images at E = 1.387 eV (top) and 1.376 eV (middle). The black arrow indicates the HEP fringe in the 1.376 eV image, which is absent in the 1.387 eV image. The bottom panel of Fig. 4d shows calculated HEP propagation lengths with and without X*. The X* sideband causes a significant reduction in propagation length at higher energies, which explains the absence of noticeable HEP fringes in the 1.387 eV image.
Recall that bulk CrSBr is an A-type AFM below TN=132 K. Intralayer ferromagnetic interactions aligning in-plane spins of Cr orbitals (Fig. 5a, top) actually appear at a slightly higher temperature of TC=160 K (31). Between TN and TC, short-range ferromagnetic domains form (35, 39) before giving way to AFM order when interlayer magnetic coupling dominates below TN (Fig. 5a, bottom). In Fig. 5b, experimental b-axis \({\epsilon }_{1}\)from far-field spectroscopy is shown near TN and TC. The hyperbolic band of CrSBr caused by X (white region, \({\epsilon }_{1}<0\)) first emerges immediately below TC and broadens with decreasing temperature. In Fig. 5c, we plot the experimental spectral weight (black crosses) of X (Equation M4) as a function of temperature. All optical conductivities are shown in Fig. S11. The spectral weight increases rapidly below TC, then plateaus below TN. Similar behavior has been observed in other transition metal magnets and was attributed to magneto-electronic coupling (40–42). QS\(G\widehat{W}\) calculations on CrSBr (Materials and Methods) indeed predict a significant increase in exciton oscillator strength going from paramagnetic (PM) to AFM states (gray lines) and show remarkable agreement with angle-resolved photoemission spectroscopy (43). Electrons taking part in exciton formation gain kinetic energy and become more dispersive along the \({\Gamma }\)-Y direction and so less localized in the AFM phase (Fig. 5d). Considering electrons hopping in the crystal lattice roughly as atomic transitions, such that hopping is forbidden between atoms of antiparallel spin, then electrons should indeed be more itinerant in ferromagnetically polarized monolayers than in a PM lattice with random spins. Simultaneously, the onsite d-d components of the exciton wavefunction are reduced in the ordered AFM phase (Fig. 5e), leading to a larger oscillator strength (44–45). Together with reduced scattering rates at low temperature, the increased spectral weight from magnetically-induced exciton delocalization allows for robust HEPs in CrSBr.
In summary, we have observed HEPs for the first time in a steady-state experiment. The temperature-, thickness-, and energy-dependence of subdiffractional fringes in our near-field experiments establish expected HEP properties. Further, we demonstrated coupling between HEPs and exciton sidebands in CrSBr, as evidenced by backbending and enhanced HEP losses above 1.381 eV. Lastly, by measuring exciton spectral weight near magnetic transitions, we proposed that hyperbolicity in CrSBr is partly driven by magneto-electronic coupling. Future work may integrate CrSBr into an open cavity photonic crystal to improve HEP propagation lengths while still permitting direct imaging (46–47). Improving quality may also enable imaging of the n = 0 surface mode – observations of surface exciton polaritons are rare (48–49) and near-field imaging could provide direct proof of their existence. Finally, measuring CrSBr in an s-SNOM capable of applying magnetic fields (50–51) would allow for studying the interplay between HEP propagation and magnetic order.