Charting the Exciton-Polariton Landscape in WSe2 Thin Flakes by Cathodoluminescence Spectroscopy
Masoud Taleb, Fatemeh Davoodi, Florian Diekmann, Kai Rossnagel, Nahid Talebi
11 Charting the Exciton-Polariton Landscape in WSe Thin Flakes by Cathodoluminescence Spectroscopy
Masoud Taleb , Fatemeh Davoodi , Florian Diekmann , Kai Rossnagel , Nahid Talebi Institute of Experimental and Applied Physics, Kiel University, 24098 Kiel, Germany Ruprecht Haensel Laboratory, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany E-mail: [email protected]
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 , 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 the exciton-photon coupling strength, along the edges of natural flakes. 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. The phenomenon of spontaneous collective coherence in condensed matter quantum systems, as most notably manifested by Bose-Einstein condensation, continues to be a subject of increasing interest . 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 or excitons , has recently emerged as a versatile control parameter , 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 . In addition, the high oscillator strengths associated with the excitonic resonances result in exciton lifetimes longer than 100 fs, enabling effective light-mater coupling . For these reasons, TMDCs are now a particularly fruitful platform for the observation of strong coupling effects between excitons and photons including the formation of exciton-polariton quasiparticles with effective masses reduced by several orders of magnitude . In order to realize strong exciton-photon couplings in practice, high-quality microcavities , plasmonic nanoantennas , or plasmon-polariton quantum and nanosystems typically need to be used. The effective interaction between cavity photons and excitons generally promotes the creation of exciton-polaritons, with spontaneous coherence enabling coherent energy transfer and the possible use in optoelectronic devices that, e.g., combine ultrafast optical routing with electronics functionalities . 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 optoelectronic nanocircuits. Plasmons, on the other hand, themselves suffer from large dissipative losses . 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 and exciton-polariton propagation was directly imaged using monochromatic photons , respectively. Yet, the underlying coupling mechanisms remain largely unclear because a spectroscopic investigation at high spatial resolution over a broad energy range is missing. Moreover, spectroscopy techniques utilizing spontaneous interactions could, in contrast to laser-based techniques, be used to unambiguously unravel the mechanism of spontaneous coherence compared to lasing. Here, we investigate the spontaneous optical properties of WSe thin flakes by means of cathodoluminescence (CL) spectroscopy . 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 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 dynamics are investigated at high spatial resolution. Moreover, dispersion control of the optical modes via film thickness, stress engineering, two-dimensional and ultimately one-dimensional confinement is used to tailor the exciton-photon coupling strength along the edges of natural flakes. Results
Cathodoluminescence spectroscopy of WSe 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 . Regarding the optical properties, typically either electron energy-loss spectroscopy (EELS) or CL spectroscopy are used. CL spectroscopy, in particular, has proven powerful in analyzing exciton and bandgap excitations in semiconductors and defects . It has only recently been revolutionized to probe not only incoherent processes, but also spontaneous coherence, particularly of surface plasmon-polaritons and optical modes of photonic crystals . 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 . Below, we will show that CL spectroscopy can probe exciton-polaritons and Rabi splitting in atomically flat single-crystalline TMDC flakes of sub-wavelength thickness. Figure 1. Strong exciton-photon interactions in WSe slabs. (a) Schematic of the experimental setup where exciton-polaritons in a WSe thin film are excited by an electron beam and probed using cathodoluminescence (CL) spectroscopy. (b) Schematic of a thin film supporting quasi-propagating optical excitations that are confined in the transverse direction and can strongly interact with excitons. (c) High-magnification image of thin-film WSe flakes as deposited on a holey carbon grid. (d) Schematic dispersion diagram where strong interaction of quasi-propagating photonic modes with A excitons causes an energy splitting and anticrossing in the energy-momentum relation of the emergent exciton-polaritons. Exciton binding energies and photon dispersions are indicated by solid and dashed lines, respectively. (e) Simulated energy-momentum EELS map, where the anticrossing is resolved for a 80 nm thick WSe film. (f) Comparison of EELS and CL spectra. The EELS spectrum is governed by exciton absorption peaks. In contrast, strong-coupling effects and a resulting energy splitting are apparent in the CL spectrum (see Methods section). In (e) and (f) exciton energies are indicated by dotted lines. Specifically, we investigate the optical response of 80 nm thick flakes of the prototypical exciton-active TMDC semiconductor WSe , 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 , propagating within the xy -plane and forming standing-wave patterns along the transverse z -axis, facilitate both strong exciton-photon coupling and exciton-polariton transfer dynamics (Fig. 1b). The corresponding WSe slab waveguides were fabricated by liquid exfoliation and placed on top of holey carbon transmission electron microscopy (TEM) grids (Fig. 1c) or carbon discs. Theoretical description of strong exciton-photon coupling in WSe thin films. The coupled harmonic oscillators, underlying the strong exciton-photon interactions, are modelled by virtue of decomposing the optical responses (for example the permittivity of the material) versus photons captured in a slab waveguide and the Lorentzian responses associated with certain excitons; here the A and B excitons in WSe . To better account for inhomogeneous broadening, the generalized model reported elsewhere for modelling the permittivity of gold is adopted here as well (see Supplementary Note 1 and Supplementary Fig. S1 for details). A dielectric slab waveguide can support optical modes of the so-called transverse electric (TE x ) and transverse magnetic (TM x ) polarizations . Within the thickness ranges considered here, the fundamental TE x and TM x modes are both excited; however, only the TM x mode can strongly interact with the excitons and be detected by electron spectroscopy techniques. The latter is due to the fact that electrons can only interact with those photonic modes that support an electric-field component oriented along the electron trajectory . The photonic modes supported by a dielectric slab waveguide with r can strongly interact with the A excitons in the WSe thin films, leading to an energy splitting and level repulsion (Fig. 1d): a phenomenon often referred to as anticrossing . The resulting polaritonic branches, referred to as Lower Polaritons (LP) and Upper Polaritons (UP), emerge in the energy ranges below and above the A exciton energy, respectively, and exhibit rather similar group velocities. For a slab waveguide with a thickness of (cid:1) = 80 (cid:2)(cid:3) , the energy gap opening is on the order of (cid:4)(cid:5) = 0.24 eV , (cid:4)(cid:6) = 110 nm . Whereas for all thicknesses (cid:1) > 60 (cid:2)(cid:3) , the photonic modes can strongly interact with the A excitons, we do observe only weak interactions with the higher-energy B excitons. This behavior does not depend on the thickness of the slab waveguide, but indeed lateral confinements along the y -axis can lead to the modification of the dispersion of the photonic modes, thus allowing for observing strong interactions with both A and B excitons, as will be shown later. A momentum-resolved electron energy-loss spectroscopy map, calculated for an electron with a kinetic energy of 30 keV interacting with a slab waveguide of thickness (cid:1) = 80 (cid:2)(cid:3) , can unravel the dispersion of propagating optical modes (Fig. 1e), and shows the same energy splitting and anticrossing as our model analysis predicted (see Supplementary Note 2). This demonstrates in principle the ability of swift electrons with kinetic energy as low as 30 keV (and even lower; see Supplementary Fig. 2) to efficiently couple to exciton-polaritons in WSe thin films and to probe their coherent dynamics. However, the required momentum resolution at lower energy-loss ranges imposes a challenge for current instruments, since the polariton-induced angular recoil experienced by the electrons covers only a few microradians for relativistic electron beams of the sort used in transmission electron microscopes. Moreover, since both radiative and loss channels contribute to EELS , probing coherent exciton-photon interactions and analysis of transport properties are less straightforward using EELS compared to CL spectroscopy (Fig. 1f). In fact, the signal integrated over all the momentum components reveals that exciton absorption peaks are the dominant contributions to the overall electron energy-loss signal, for the thicknesses of interest of our thin films. In contrast, the observation of a dip – instead of a peak – at the A exciton wavelength in the CL spectra as well as a wavelength splitting as large as 110 nm demonstrates the possibility of using CL spectroscopy for probing exciton-polaritons in thin TMDC flakes. A deeper understanding of the level repulsion phenomenon in quantum electrodynamics systems, and the originating Rabi oscillations, demands a more rigorous treatment of light-matter coupling. Here, we comply with a simple semi-classical model that perfectly captures the main mechanisms, defining the coupling between the electric field ( E (cid:1) ) of the photonic modes and the macroscopic exciton polarization ( P (cid:1) ) as exph iP i P gEiE i E gP (cid:1) (cid:1) (cid:1)ɺ(cid:1) (cid:1) (cid:1)ɺ (1) where ex and ph are the angular frequencies of the exciton and photon oscillators, and are their damping constants, g is the coupling strength, and time derivative is indicated by a dot on top. The stationary solution of Eq. (1), derived by assuming harmonic fields, is given by ph ex k i , (2) with k k i k , (3) and
12 ph ex g . Equation (2) defines two solutions for the system associated with the LP and UP branches. Going beyond previous works, we allow for nonzero damping terms, thus the complex-valued parameter can be used to model both the Rabi oscillation and its damping. Intriguingly, for the resonance condition ( ph ex k ), the damping of the Rabi oscillation is only determined by the damping of its underlying oscillators as Im 2 i . In contrast, detuning of the energy ( ℏ ℏ ℏ ) can represent another mechanism for the relaxation of the Rabi oscillation. We have assumed a dispersive-dependent model for the photonic mode, allowing for efficient modeling of the anticrossing behavior similar to the complete treatment in Fig. 1d. Using the extracted parameters for the A exciton energy and photonic modes (see Supplementary Note 1), and under resonance condition, we indeed observe an energy splitting of for a coupling strength of (cid:7) = 182 meV . Experimental probing of exciton-polaritons using CL spectroscopy.
In the experiments, we first look at the CL response of a triangular WSe 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 100 nm, comparable to the predicted values, and a dip at the A exciton wavelength of 751 nm. Upon scanning the electron beam along the edge of the triangular flake, we observe two maxima Figure 2. CL spectroscopy and hyperspectral imaging of a triangular WSe flake. (a) SEM image of the flake. (b) Measured CL spectra (solid lines) at selected electron excitation impact positions, as indicated in panel (a), compared to calculated spectra (dashed lines). (c) CL wavelength-distance map, measured along the line depicted in panel (a), showing the spatial distribution of the luminescence signal. Dashed arrows indicate a change in the spectral peak position versus the electron impact position. (d) Corresponding simulated CL wavelength-distance map. (e) Schematic illustration of far-field (A) versus near-field (B) interference effects as possible hypotheses for describing the energy-distance CL peaks. Shown is a snapshot of the electron-induced z-component of the electric field at a given time. with their relative distance depending on the wavelength (as indicated by the dashed arrows in Fig. 2b), and a rather diffuse bright excitation at longer wavelengths, associated with UP and LP branches, respectively. 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. The occurrence of spatial interference fringes is well known in the formation of localized plasmonic modes in nanoscopic and mesoscopic plasmonic nanoantennas. Based on these similarities between propagating surface plasmon excitations 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. 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 EP L m , where EP is the phase constant of the exciton-polaritons, L is the distance between the electron and the apex, and is the phase shift due to reflection from the apex. Thus, the period of the fringes would correspond to approximately one half of the wavelength of the excited polaritons. A comparison 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 TM x mode) as a possible reason for the observation of the interference fringes, and points to differences between EELS and CL, as the former probes near-field excitations, and the latter measures far-field contributions. Our second hypothesis, thus, assumes that the observed interferences reflect interferences between the transition or diffraction radiation and the scattering of exciton-polaritons from anomalies or edges, implying a far-field constructive interference pattern in the form of EP L m . The transition (diffraction) radiation is the radiation caused when an electron crosses a surface (an edge) 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 it can indeed happen in dielectrics . Based on our second hypothesis, the periodicity of the spatial interference fringes should be equal to an exciton-polariton wavelength of (cid:6) (cid:1)(cid:2) = 2 (cid:8) (cid:9) (cid:3)(cid:4) ⁄ . We provide a direct test for the second 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 Figure 3.
Propagation mechanisms of exciton-polaritons of the upper branch. (a) SEM dark-field image of the investigated WSe flake. (b) CL wavelength-distance map. Dashed arrows indicate the dispersion of spatial peaks versus wavelength, where the wavelength-dependent distances between them is (cid:1)(cid:2)(cid:3) (cid:1) (cid:4) . (c) Hyperspectral images at selected wavelengths, where the interference fringes caused by the reflection from the edges of the triangle are apparent. (d) Sketch of the proposed mechanism for the observed spatio-spectral interference fringes. TR: transition radiation. EP: exciton-polaritons contributing to the radiation due to their scattering from the edges. (e) Measured (scattered data) and calculated phase constant ( (cid:5) (cid:2)(cid:3) ; blue solid line) and attenuation constant ( (cid:6) (cid:2)(cid:3) ; dashed solid line) of the surface exciton-polaritons. Photon dispersion in free space is indicated by a dashed red line. 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 in the far field. Therefore, a constructive interference pattern can be formed by the excitation of the exciton-polaritons with phase fronts parallel to the edges (Fig. 3d). The observed peaks in the CL energy- distance maps are broad, due to dissipative losses associated with the TE mode. The propagation constant of the slab modes can be decomposed into its real ( EP ) and imaginary part ( EP ), corresponding to the phase and attenuation constant, respectively (Fig. 3e). The relatively large attenuation constant for the TE mode at the measured photon energies is the reason for the large broadening of the peaks in the distance-wavelength CL map. To extract the phase constant from the energy-distance CL map, the distance between the peaks is measured at selected wavelengths as D and related to the phase constant by EP 0 0 D (see Fig. 3d for details). Direct comparison of calculated thin-film exciton-polariton dispersions and measured spectral and spatial fringes reveals good agreement with the TM x mode, supporting our second hypothesis (Fig. 3e). These results place CL spectroscopy parallel to scanning near-field optical microscopy, for direct measurement of spatially and spectrally resolved polaritonic transport mechanisms . However, even better agreement can be achieved by considering edge exciton-polaritons as an individual transport mechanism parallel to the thin-film exciton-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 . The spatial confinement and different screening mechanism for edge exciton-polaritons compared to bulk excitons lead to lower attenuation constants and shifted exciton energies. Along the edge of a WSe flake with a trapezoidal shape (Fig. 4a), spatial interference fringes up to several orders are easily observed (Fig. 4b), due to the lower attenuation constant compared to surface polaritons. The Fourier-transformed CL map reveals the dispersion of the edge exciton-polaritons (Fig. 4c). A bright intensity strip at (cid:11) || = 0 is caused by background noise and, particularly, by non-propagating evanescent modes with high attenuation constants. These 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. At momenta lying outside the light cone ( (cid:12) = (cid:11) || (cid:13) ), however, another clear signal is observed. The Figure 4. Edge exciton-polaritons. (a) The SEM dark-field image of the investigated WSe flake. (b) Distance-wavelength CL map along the edge marked by the yellow arrow in panel (a). (c) Fourier-Transformed CL map, revealing the dispersion associated with the edge exciton-polaritons. Calculated dispersion curves of a rib WSe waveguide with a width and height of 500 nm and 100 nm, respectively, are indicated by white solid lines, for the fundamental mode. Dashed white lines represent photon dispersion in free space ( k c ). (d) Hyperspectral images at selected wavelengths, where the distribution of the optical modes confined to the edge of the flakes are spatially resolved. (e) Calculated spatial distribution of the y - and z -components of the electric field at the cross section plane ( yz -plane) for the fundamental guided mode propagating along the x -axis, at E = 1.9 eV ( (cid:3) (cid:7) 653 nm ). intensity of this CL signal peaks at E = 2.0 eV. The propagating edge exciton-polaritons are modelled on the basis of optical modes supported by a rectangular rib waveguide with a 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 (Fig. 4f). However, only the fundamental mode is experimentally captured, which can be understood by comparing the calculated dispersion of the fundamental mode with the experimental results (Fig. 4c; solid lines). Hyperspectral imaging provides additional support for the presumed excitation of edge exciton-polaritons and also unravels the ultra-confined mode volumes associated with these modes (Fig. 4d).
Discussion
The CL response of semiconductors is naturally understood by excitation of many electron-hole pairs, where the statistical distributions of the generated photons do not resemble coherent photonic statistics. Still, we can 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, such as semiconducting TMDCs, can be used to generate 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 . The CL response of thinner few-layer WSe structures is more intriguing. We observe higher photon yields in general in thinner materials, possibly due to the reduced dielectric screening. Moreover, when a thin flake is transformed on top of a holy-carbon structure, the strain can heavily influence the spatial correlations reported so far (see Supplementary Fig. 3). We indeed observe competing strain-induced and exciton-polariton mechanisms in thin films of these materials. In summary, we have demonstrated strong coupling between excitons of WSe and the propagating photons of freestanding thin WSe flakes with a Rabi splitting on the order of 0.24 eV. The presented energy-momentum EELS map and CL spectroscopy results show anticrossing of upper and lower polariton branches and long-range propagation of exciton-polaritons in high-quality atomically flat flakes. The possibility of controlling exciton-polariton dispersions along the edges of natural flakes is another exciting and potentially useful feature of these thin-film van der Waals materials. Overall, our results demonstrate that CL spectroscopy can be used to probe the mechanisms of spontaneous coherence in atomically flat single-crystalline TMDC flakes. Methods
Synthesis and exfoliation of single-crystalline WSe flakes. Single crystals of 2 H -WSe were grown by the standard chemical vapor transport method: A near-stoichiometric mixture of high-purity W and Se with a slight Se excess (4 mg/cm ) was placed in a quartz ampoule together with iodine (5 mg/cm ) as transport agent; the ampoule was sealed and heated in a four-zone furnace under a temperature gradient of 920-860 ◦ C. The samples were grown within 900 h. Thin nanosheets were prepared by applying liquid phase exfoliation (LPE) from bulk WSe crystals in isopropanol (Merck, ≥99.8%) (Supplementary Fig. 5). Exfoliation was performed with an ultrasonicator (320 W, Bandelin Sonorex, RK100H) equipped with a timer and heat controller to avoid solvent evaporation. Sonication was conducted in an ice bath by applying a cycle program of 5 min on, followed by 1 min off for a total duration of 60 min. The resulting suspension was drop casted either on a high-purity graphite planchet or on a holey carbon mesh grid for further characterization. Cathodoluminescence imaging.
High-resolution SEM observations were performed using an optical field emission microscope (Zeiss SIGMA) operated at 30 kV. For the CL measurements, a beam current of 14 nA was utilized to scan the specimen and excite their surfaces. The generated CL radiation was collected and analyzed using a CL detector (Delmic B.V) equipped with an off-axis silver parabolic mirror (focal distance: 0.5 mm). The acceptance angle of the mirror and dwell time were 1.46π sr and 400 µs, respectively. The mirror was positioned on the upper side of the specimen and a hole with a diameter of 600 µm above the focal point was supporting the electron excitation. Numerical simulations.
We have employed both a home-built finite-difference time-domain method and the COMSOL Multiphysics software to gain insight into the temporal distribution of the electron-induced radiation and exciton-photon interactions in our WSe sample. The obtained results are generally in good agreement with each other and with the experimental results. In order to simulate CL spectra, we have employed a classical method to model an electron beam by a current density distribution corresponding to a swift electron charge at a kinetic energy of (cid:5) = 30 keV . The mesh size was 2 nm, and a higher-order absorbing boundary conditions was used. The dielectric function of WSe was modeled by the two critical point functions shown in the supporting information. The CL spectra are calculated by employing the discrete Fourier transformation on the field distributions at planes circumventing the structure at distances of from the structure, and calculating the Poynting vector. Convergence was achieved after 8,000 time iterations. To calculate EELS and CL data in the frequency domain, we have employed the radiofrequency (RF) toolbox of COMSOL in a 3D simulation domain, where the Maxwell equations are solved in real space and in the frequency domain. We have utilized an oscillating “edge current” as a fast electron beam along a straight line representing the electron beam. The current was expressed by (cid:1) = (cid:1) (cid:1) exp (cid:2)(cid:3)(cid:4) (cid:5) (cid:6) (cid:2) ⁄ (cid:8) . The CL signal was calculated by using a boundary probe at a plane normal to the electron beam. The dielectric function of WSe was defined by using the interpolation function of COMSOL. Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program, Grant Agreements No. 802130 (Kiel, NanoBeam) and Grant Agreements No. 101017720 (EBEAM). Financial support from Deutsche Forschungsgemeinschaft under the Art. 91 b GG Grant Agreement No. 447330010 and Grant Agreement No. 440395346 is acknowledged.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Author Contributions
MT has performed the CL measurements, data analysis, and fabricated the thin-film samples. FD and NT have performed the simulations. FD and KR have provided the bulk crystals. NT and KR have written the manuscript. All the coauthors have contributed to discussions. NT has supervised the work.
Ethics Declarations
Competing interests
The authors declare no competing interests.
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0 Supporting Information
Charting the Exciton-Polariton Landscape in WSe Thin Flakes by Cathodoluminescence Spectroscopy
Masoud Taleb , Fatemeh Davoodi , Florian Diekmann , Kai Rossnagel , Nahid Talebi Institute of Experimental and Applied Physics, Kiel University, 24098 Kiel, Germany Ruprecht Haensel Laboratory, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany E-mail: [email protected]
Supplementary Fig. 1. Decomposed permittivity components for WSe . Supplementary Fig. 2. Measured CL Spectra of a thin WSe2 flake with (cid:14) (cid:7) (cid:15)(cid:16) (cid:17)(cid:18) , at various electron energies versus the electron kinetic energies.
For almost all electron energies, we observe a dip at the exciton A wavelength, signifying strong exciton-photon interactions.
Supplementary Fig. 3 CL Spectra from a thin WSe2 flake positioned on a holy carbon TEM grid along edges. (a) Dark-field SEM image of the structure. (b) CL spectra versus the electron kinetic energies along scanning paths (i) and (ii), as depicted in panel (a). Supplementary Fig. 4 Calculated MREELS for thin films and bulk materials, for the depicted thicknesses.
Supplementary Fig. 5 Images of Single Crystalline WSe2 flakes.
Before (a) and after (b, c) liquid exfoliation.
Supplementary Note 1. Modeling the WSe permittivity We adopted the method described elsewhere to model the optical response of gold and silver, to analyze the oscillator strength of the excitonic structures in WSe
2 1 . Based on this model, the permittivity component of gold can be decomposed into to two analytical functions, as
01, 2 0 1 0 1 i i i ir r i ii i i e eA i i , (S1) where the first term on the RHS, is a constant (in contrast with the original model where a metallic Drude model was considered), whereas the last terms are associated with exciton excitations. In order to obtain a good agreement between the experimental data and our analytical and numerical results, we observe that neither the calculated permittivity of WSe monolayers nor the reported data for the measured bulk permittivity represent a reasonable solution. The latter in particular does not allow for the excitation of
3 exciton polaritons in the whole range of thicknesses and structural morphologies associated with our grown atomically flat WSe flakes and particularly underrepresent the oscillator strengths associated with the excitons. We propose a slightly modified version for the permittivity with the parameters given by r , A , , , , A , , , and (See Supplementary Fig. S1), for the individual terms associated with the background dielectric photonic modes, exciton A, and exciton B oscillators. Supplementary Note 1. EELS and CL calculations
We have adopted a vector potential approach reported elsewhere to calculate the EELS and CL responses of bulk WSe and WSe thin films . The electron charge density distribution is modeled as , z J r t (cid:1) el el ev x y z v t and its Fourier transformed function is inserted inside the Helmholtz equation as , , , z r r z r z A r k A r J r (cid:1) (cid:1) (cid:1) (cid:1)ɶ . Having the solutions to the vector potential, the field components are calculated as ˆ, , , r r E r i A r A ri (cid:1) (cid:1) (cid:1) (cid:1)(cid:1) (cid:1) (cid:1) (cid:1) (S2) for the electric field, and
1, , r H r A r (cid:1)(cid:1) (cid:1) (cid:1) , (S3) For the magnetic field respectively. These equations are then used to calculate the field components and satisfying the interface boundary conditions. We then calculate the Ponting vector at a plane normal to the electron trajectory positioned at far-field for modelling the CL signal. For calculating the EELS signal, we adopted the generally used procedure based on the decelerating force acting on the electron and expanding that versus the probability loss . In order to better illustrate the effect of thin films on the excitation of propagating photonic modes and thus the strong coupling effect, we have calculated the MREELS signal from both thin films with different thicknesses and also the bulk contribution (no boundary included) (See supplementary Fig. 5). We observe that in addition to the photonic modes, the Cherenkov radiation (CR) can also strongly interact with excitons. Nevertheless, the excitation CR angle is larger than the critical angle for the total internal reflection; thus CR cannot contribute to the CL signal. Moreover, in thin films, the so-called Begrenzungseffekt is indeed a competing effect compensating for the excitation of CR in thin films . References
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