Charting the Exciton-Polariton Landscape of WSe 2 Thin Flakes by Cathodoluminescence Spectroscopy

Semiconducting transition-metal dichalcogenides (TMDCs) provide a fascinating discovery platform for strong light-matter interaction effects in the visible spectrum at ambient conditions. While most of the work has focused on hybridizing excitons with resonant photonic modes of external mirrors, cavities, or nanostructures, intriguingly, TMDC flakes of sub-wavelength thickness can themselves act as nanocavities. Here, we determine the optical response of such freestanding planar waveguides of WSe 2 , by means of cathodoluminescence spectroscopy. We reveal strong exciton-photon interaction effects that foster long-range propagating exciton-polaritons and enable direct imaging of the energy transfer dynamics originating from cavity-like Fabry-Pérot resonances. Furthermore, confinement effects due to discontinuities in the flakes are demonstrated as an efficient means to tailor mode energies, spin-momentum couplings, and the exciton-photon coupling strength, as well as to promote photon-mediated exciton-exciton interactions. Our combined experimental and theoretical results provide a deeper understanding of exciton-photon self-hybridization in semiconducting TMDCs and may pave the way to optoelectronic nanocircuits exploiting exciton-photon interaction beyond the routinely employed two-oscillator coupling effects. The condensed matter quantum systems,

increasing interest [1][2][3][4][5] . Whereas temperature is the typical control knob to drive a system of collective excitations to its ground state, quasiparticle density in solid-state systems, such as the density of plasmons 6 or excitons 7,8 , has recently emerged as a versatile control parameter 9 , toward high-temperature condensation. In general, excitons are promising for condensation due to their light mass and bosonic nature. More specifically, excitons in semiconducting group VI transition-metal dichalcogenides (TMDCs), are advantageous because of their exceptionally high binding energies of a few 100 meV giving rise to stable and robust exciton excitations at room temperature [10][11][12] . In addition, the high oscillator strengths associated with the excitonic resonances result in exciton lifetimes longer than 100 fs, enabling effective light-mater coupling 13 . For these reasons, TMDCs are now a particularly fruitful platform for the observation of strong coupling effects between excitons and photons 14 including the formation of excitonpolariton quasiparticles with effective masses reduced by several orders of magnitude 15 .
In order to realize strong exciton-photon couplings in practice, high-quality microcavities [16][17][18][19] , plasmonic nanoantennas 6 , or plasmon polaritons 20,21 typically need to be used. The effective interaction between cavity photons and excitons generally promotes the creation of excitonpolaritons, with spontaneous coherence enabling coherent energy transfer and the possible use in optoelectronic devices that, e.g., combine ultrafast optical routing with electronics functionalities 22, 23 . However, the application of microcavities often involves complicated manufacturing processes limiting achievable coupling strengths, and microcavities also typically sustain only a limited operational bandwidth and are usually large in size, which hinders their use in hybrid electronic-optics nanocircuits. Plasmons, on the other hand, themselves suffer from large dissipative losses 24 . Therefore, simple effective platforms for strong exciton-photon coupling are in principle needed. Recently, thin TMDC waveguides were investigated along these lines by optical microspectroscopy and scanning near-field microscopy, where exciton-photon anticrossing behavior in individual flakes was observed 25,26 and exciton-polariton propagation was directly imaged using monochromatic photons 27,28 , respectively. Yet, the underlying coupling mechanisms remain largely unclear because a spectroscopic investigation at high spatial resolution and at a broad energy range is missing. Moreover, spectroscopy techniques utilizing spontaneous interactions, in contrast with laser-based techniques, could be used to unambiguously unravel the mechanism of spontaneous coherence compared to lasing.
Here, we investigate the spontaneous optical properties of WSe2 thin flakes by means of cathodoluminescence (CL) spectroscopy 29,30 . Using an electron beam-based probe, we rule out the possibility of lasing as the underlying coherence mechanism. Transversal one-dimensional optical confinement within the thin film and the propagation of the optical waves along the longitudinal orientation result in strong exciton-photon coupling as manifested in a Rabi energy splitting of 0.24 eV (corresponding to a wavelength splitting of approximately 110 nm) and the formation of exciton-polaritons whose spatial coherence and propagation mechanisms are investigated at high spatial resolution. Moreover, dispersion control of the optical modes via film thickness, stress engineering, two-dimensional and ultimately three-dimensional confinement is used to tailor the exciton-photon coupling strength. Finally, multi-oscillator coupling physics is explored on the basis of photon-mediated exciton-exciton couplings between A and B excitons, i.e., polariton-polariton interactions.

Results
Cathodoluminescence spectroscopy of WSe2 waveguides. Electron microscopy has continuously been advanced as a probe of both structural and optical properties of materials, with arguably the highest spatial, temporal and energy resolutions 31 . Regarding the optical properties, typically either electron energy-loss spectroscopy (EELS) 32 or CL spectroscopy 30,33 are used. CL spectroscopy, in particular, has proven powerful in analyzing exciton and bandgap excitations in semiconductors and defects 34 . It has only recently been revolutionized to probe not only incoherent processes, but also spontaneous coherence, particularly of surface plasmon polaritons 35 , localized plasmons 36 , and optical modes of photonic crystals 37,38 . However, to the best of our knowledge, CL spectroscopy has not yet been applied to probe exciton-polariton coherence and spatial correlations, which is partially due to the competing incoherent and coherent radiation channels in semiconductors 39 . Below, we will show that CL can probe exciton polaritons and Rabi splitting in atomically flat single-crystalline TMDC flakes of sub-wavelength thickness.
Specifically, we investigate the optical response of 80 nm thick flakes of the prototypical excitonactive TMDC semiconductor WSe2, using 30-keV electrons in a field-emission SEM system equipped with a parabolic mirror and optical detectors (see Methods Section for details). Excitons are excited upon electron irradiation and exciton-photon interaction, exciton-polariton formation, as well as the propagation dynamics are probed by CL spectroscopy (Fig. 1a). Whereas in semiconductor optics weak exciton-photon interaction is omnipresent, strong interaction requires standing-wave-like patterns of light with increased interaction time. This is normally realized by optical cavities. However, the atomically flat interfaces of a material can also act as mirrors exploiting total internal reflection of the light. Thus, photonic modes of a slab waveguide 40 , propagating within the xy-plane and forming standing-wave patterns along the transverse z-axis, facilitate both strong exciton-photon interactions and exciton-polariton transfer dynamics (Fig. 1b). The corresponding WSe2 slab waveguides were fabricated by liquid ) and are given by  Experimental probing of exciton-polaritons using CL spectroscopy. In the experiments, we first look at the CL response of a WSe2 flake with a thickness of 80 nm (Fig. 2). CL spectra, acquired at selected electron impact positions, demonstrate a wavelength splitting on the order of 120 nm, comparable to the predicted values, and a dip at the A exciton wavelength of 751 nm (Fig. 2c).
This splitting is fully reproduced in the calculated CL spectra (Fig. 2g). Upon scanning the electron beam vertically to the edge of the flake, we observe several maxima with their relative distance depending on the wavelength (as indicated by the dashed arrows in Fig. 2b), and a dip at the A exciton energy splitting the excited polaritons into energy intervals above and below the exciton A energy associated with UP and LP branches, respectively. The energy of the photonic mode depends on the thickness of the slab waveguide. Hence, the exciton-photon coupling strength and the observed energy splitting strongly depend on the thickness of the slab waveguide (Fig.   2d). For a slab waveguide at a thickness of 65 nm, the splitting in the wavelength is on the order of ∆ = 250 nm, whereas by increasing the thickness to 80 nm, the wavelength splitting becomes ∆ = 120 nm. The increase in the wavelength splitting is understood by generating more resonant conditions between photons and excitons ( Supplementary Fig. 4). Nevertheless, for larger thicknesses, the excitation of higher-order modes and the Cherenkov radiation result in exciton-photon interactions as well ( Supplementary Fig. 6).
When comparing to the results of numerical calculations (using the finite-difference time-domain method, see Methods section), we indeed observe peaks at the predicted spectral and spatial positions in the two-dimensional CL spectrum. However, the maximum-minimum contrast, in the experimental data is less pronounced, which we attribute to structural imperfections, potentially caused by our liquid exfoliation and transfer methods, as well as excitation of incoherent channels, such as non-radiative electron-hole pairs. The occurrence of spatial interference fringes is well known in the formation of localized plasmonic modes in nanoscopic and mesoscopic plasmonic nanoantennas, such as nanorods 47 and microplatelets 48 . Based on these similarities between propagating surface plasmon excitations 49-51 and the interference fringes observed in Fig. 2, we first hypothesize that reflection from the apex of the triangular flake might cause the interference fringes (Fig. 2e). The fringe periods would then be observable as a result of the constructive near-field interference and the ability of CL to probe such near-field patterns. For this to happen, constructive interference between the excited and reflected exciton-polaritons from the apex would imply between the spatial interference fringes and the calculated exciton-polariton dispersion (Fig. 1d) rules out this hypothesis (since the resulting phase constant will be much smaller than the calculated phase constant of the TMx mode) as a possible reason for the observation of the interference fringes, and points to differences between EELS and CL, as the former is a probe of the near-field excitations, and the latter is a measure of far-field contributions. Our second hypothesis, thus, assumes that the observed interferences reflect interferences between the transition radiation and the scattering of exciton-polaritons from anomalies or edges, implying a far-field constructive interference pattern in the form of EP Lm   . The transition radiation is the radiation caused when an electron crosses a surface so that the dipole formed by the moving electron and its image inside the material is sharply annihilated. Although this type of radiation mostly occurs for metals, we notice that indeed it can be observed in dielectrics 52 . Moreover, when an electron traverses the edge of the material, but not directly propagating through the material, the diffraction radiation can occur due to the interaction of the electron with truncated edges. Based on our second hypothesis, the periodicity of the spatial interference fringes should be equal to the exciton-polariton wavelength of = 2 EP ⁄ .
We provide a direct test for the proposed hypothesis, by acquiring the spatial interference fringes in a higher quality triangular flake with a thickness of 95 nm (shown in Fig. 3a and b). Our CL spectrometer is equipped with a dispersive grating, which allows for projecting the spectrally dispersed optical rays onto a CCD camera. By tilting the grating, and hence changing the central wavelength, we focus more on the UP excitonic branch, where the observed numerous interference fringes in the wavelength-distance CL map are a clear signature of the spontaneous coherence caused by the excitation of exciton-polaritons (Fig. 3b). Hyperspectral images at selected photon wavelengths capture the spatially resolved standing wave patterns, parallel to the edges (Fig. 3c). This can be understood as the constructive interference between the TR and the diffraction of the excited exciton-polaritons from the edges of the flakes at the far field.  Fig. 3d for details). A direct comparison between the calculated surface exciton-polariton dispersions and the measured spectral and spatial fringes reveals a good agreement with the TMx mode, supporting our second hypothesis (Fig. 3e). This places CL spectroscopy parallel to scanning near-field optical microscopy, as a direct measure of spatially and spectrally resolved polaritonic transport mechanisms 27 . However, a better agreement can only be provided by considering edge exciton-polaritons as an individual transport mechanism parallel to surface polaritons, as will be discussed in the following. Edge exciton-polaritons. Edge polaritons, in contrast to surface polaritons, are spatially confined to and propagate along the pristine edges of the flakes 11,32 . The spatial confinement and different screening mechanism of edge exciton-polaritons compared to bulk excitons lead to lower attenuation constants and shifted exciton energies. By scanning the edge of a WSe2 flake with a trapezoidal geometry (Fig. 4a), spatial interference fringes of up to several orders are observed (Fig. 4b), due to lower attenuation constants compared to surface polaritons. The Fourier-transformed CL map allows for acquiring the dispersion of the edge exciton-polaritons ( Fig. 4c). A bright intensity at || = 0 is caused by the background noise, and particularly nonpropagating evanescent modes with high attenuation constants. Those higher-order modes strongly contribute to the dissipation of the energy delivered by the excitation source without contributing to the desired spatial coherence associated with the propagating optical waves. In addition, however, a pronounced signal with the momenta ( = || ) lying outside the light cone is observed. Theoretical modelling of the propagating edge exciton-polaritons are performed by calculating the optical modes supported by a rectangular rib waveguide, with the width and height of 500 nm and 100 nm, respectively. The fundamental mode of this rib waveguide has optical mode profiles confined to the edges of the waveguide (Fig. 4e). Several higher-order optical modes exist with field profiles confined to the edges and surfaces of the flakes. However, only the fundamental mode is experimentally captured, as understood by comparing the calculated dispersion diagram of the fundamental mode and the experimental results ( Fig. 4c; solid lines). Hyperspectral images as well support the presumed excitation of edge excitonpolaritons and also unravel the ultra-confined mode volumes associated with these modes (Fig.   4d).

Photon-mediated exciton-exciton interactions.
Natural flakes with anomalies, allowing for lateral confinements of the photonic modes, provide a natural mechanism for further tuning the energy and modal configurations of the photonic excitations (see Fig. 5a). Particularly in those flakes, we indeed observe not only an omnipresent dip in the CL spectra at the wavelength associated with the A exciton, but also a series of peaks and dips in the wavelength range of the UP branch (Figs. 5b and c). These spectral fringes are revealed in numerical simulations as well ( Fig. 5 d), in a good agreement with experimental results.
Higher-order photonics modes of a slab waveguide, though can cause oscillations in the LP range, they cannot result in the spectral fringes we observe in the UP range. Moreover, Cherenkov radiation (CR) caused by an electron moving inside a bulk material, can strongly interact with the excitons, similar to the photonic modes of the slab waveguide (see Supplementary Note 2 and Supplementary Fig. 4). CR can be excited and captured inside thick films with > 400 , and can be further released to the far-field zone by virtue of its interaction with discontinuities. However, our theoretical and experimental CL analysis rules out the possibility of observing spectral fringes in the UP branch that can be related to either photonic modes or CR. In order to better understand the mechanisms underlying the oscillations we observe for the UP branch, we propose a model based on photon-mediated exciton-exciton interactions.
Equation (1) can be modified to include photon-mediated couplings between the A and B excitons (Fig. 6a) where a system of three coupled harmonic damped oscillators is considered (Fig. 6b). The timeharmonic response of this coupled system of equations represents three eigenvalue solutions to the system, the values of which can be tuned by ph  . We express these eigenvalues by i  where 1, 2,3 i  (Fig. 6c). We indeed assume that excitons A and B can only indirectly interact with each other, by virtue of exchanging photons. Therefore, only two coupling coefficients between the photonic modes and the exciton peaks are considered. By considering the coupling coefficients 1 = 26.3 meV and 2 = 65 meV, we observe a strong shift of the second eigenwavelength from 610 nm to 730 nm, by reducing the frequency of the photonic mode from 2.2 eV to 1.6 eV (blue line, Fig. 6c). For ph ex,A   , we observe a strong red shift of the third eigenvalue by reducing further the photon energy (Fig. 6c, red line). By assuming a Lorentzian line shape centered at each calculated eigenvalue, we reveal an overall spectral shape that is in good agreement with the measured results represented in Figs. 5c and d. Moreover, though in general four spectral peaks are apparent in the acquired CL data (Fig. 4), we notice that the system of three coupled oscillators is sufficient to lay open the underlying mechanism, since the additional middle peak at the wavelength of = 660 nm can be understood by the overlapping tails of adjacent oscillators (Fig. 6d).

Discussion
The Cathodoluminescence response of semiconductors is naturally understood per excitation of many electron-hole pairs, where the statistical distributions of the generated photons do not resemble coherent photon statistics. We indeed ascertain that the presence of spontaneous coherence supported by the excitation of exciton-polaritons, provides evidence for CL spectroscopy to be able to probe Rabi oscillations and nonlinear exciton-photon interactions. In addition and in an inverse approach, semiconductors with room-temperature exciton excitations like TMDCs can be used to generate both X-ray photons and coherent visible-range photons upon electron irradiation, and thus provide a means for designing electron-driven photon sources of higher photon yield compared to plasmonic structures 53 .
Although our results demonstrate evidence for photon-mediated exciton-exciton interactions in natural flakes, related to the energy relaxation between excitonic states, further control of the coupling mechanisms by nanostructuring and controlling the temperature will provide a toolbox for better understanding the formation of long-range spatial correlations associated with Bose-Einstein condensation. The CL response of thinner few-layered WSe2 structures is more intriguing. We observe higher photon yields in general in thinner materials, due to the transition from indirect to direct bandgap 54 . Moreover, when a thin flake is transformed on top of a holy-carbon structure, the strain can heavily manipulate the spatial correlations reported so far 55 (see supplementary Fig.   3). We indeed observe competing strain-induced and exciton-polariton mechanisms in thin films of these materials.
The Stokes shift between the luminescence and absorption peaks will result in a shift of the CL peaks compared to EELS spectra 56 . However, this effect has been shown to cause only a minor effect compared to the strong-coupling effects we observe here, for nominally undoped materials, whereas it can be significantly increased for n-doped materials 57 . The short interaction of the electron beam with our structure does not cause any significant sample heating or doping, as the positions of the peaks do not depend on the acquisition time, the acceleration voltage within the range of 5 keV to 30 keV, and also not on the probe current within the range of 10 nA to 20 nA (see Supplementary Fig. 6). We therefore conclude that the possible Stokes shift has a negligible effect compared to the effect of strong exciton-photon interactions on the spectra. In