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Abstract
Transition metal dichalcogenides (TMDs) have attracted great attention in valleytronics. Owing to the giant valley coherence at room temperature, valley pseudospin of TMDs open a new degree of freedom to encode and process binary information. The valley pseudospin only exists in non-centrosymmetric TMDs (e.g., monolayer or 3R-stacked multilayer), which is prohibited in conventional centrosymmetric 2H-stacked crystals. Here, we propose a general recipe to generate valley-dependent vortex beams by using a mix-dimensional TMD metasurface composed of nanostructured 2H-stacked TMD crystals and monolayer TMDs. Such an ultrathin TMD metasurface involves a momentum-space polarization vortex around bound states in the continuum (BICs), which can simultaneously achieve strong coupling (i.e., form exciton polaritons) and valley-locked vortex emission. Moreover, we report that a full 3R-stacked TMD metasurface can also reveal the strong-coupling regime with an anti-crossing pattern and a Rabi splitting of 95 meV. The Rabi splitting can be precisely controlled by geometrically shaping the TMD metasurface. Our results provide an ultra-compact TMD platform for controlling and structuring valley exciton polariton, in which the valley information is linked with the topological charge of vortex emission, which may advance valleytronic, polaritonic, and optoelectronic applications.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recently, the emergence of two-dimensional transition metal dichalcogenides (TMDs) has attached growing interest in optoelectronic [1,2] and valleytronic [3–5] applications thanks to their visible to near-infrared and layer-dependent bandgaps [6–9]. Especially, owing to the intrinsic broken inversion and time reversal symmetry of two-dimensional (2D) monolayer TMDs, where a pair of degenerate direct bands located at the corners of the Brillouin zone [4,6,7,10], provide perfect playgrounds for valleytronics. Beyond monolayer TMDs, bulky three-dimensional (3D) 3R-stacked TMDs, which also has a non-centrosymmetric structure, and has been reported to support robust valley coherent excitons at room temperature [11–15].
When photons strongly bond with those TMDs excitons, brand-new bosons called exciton-polariton would arise, with a large exciton binding energy that promises abundant fascinating fundamental physics and applications [16–18]. As hybrid exciton-photon quasiparticles, the TMDs exciton-polariton have both the advantages of the photon and exciton such as the light velocity, strong correlation, low effective mass of photons, and spin-selective valley excitations and light emission associated with the optical circular polarization both in linear and nonlinear regime [19–22]. Thus, exciton-polariton in TMDs systems provide an ideal platform for investigating the phenomenon of interaction between light and materials such as Bose–Einstein Condensation [23,24], polariton biostability [25], superfluidity [26], virtually threshold less polariton lasers [17,27], vortex state and energy effective all-optical logic gates [28].
Most current literature studies focused on either the intralayer exciton in monolayer and interlayer exaction in bilayer TMDs or the moiré excitons in twisted van der Waals heterostructures [29–35]. Besides, there are still missing scenarios to explore excitonic behaviors in mixed-dimensional TMDs (i.e., 2D/3D heterostructure) and 3R-stacked layered TMDs. The bulky TMDs could be very interesting, since layered TMDs have high refractive index (n > 4) in the visible range, which allows the strong interaction and confinement of light even with deeply subwavelength nanostructures [36,37].
On the other hand, the current work focuses on how to generate exciton polariton by coupling TMDs monolayers with nanophotonic cavities [38–43], such as distributed Bragg reflectors (DBRs) [39,40,42], photonic crystals (PhCs) [18,43], metasurface [44] and topological cavities [18]. Fundamental and application research of structural beams is still in the frontier areas [45–48]. The next developing subject towards functional and structured exciton-polaritons (e.g., vortex emission), which is strongly demanded but remains unexplored.
Here, we proposed a generalizable framework to realize valley dependent vortex emission from exciton-polariton in non-centrosymmetric TMDs. The ultrathin (∼25 nm) TMDs layers are directly patterned to symmetric protected BIC metasurface, which hosts two-fold advantages: (1) the exciton coupling to a photonic BIC of ideally infinite Q factor; and (2) the exciton polariton inheriting the fundamental topological properties of BIC (i.e., spin dependent topological vortex), which results valley-locked vortex emission from exciton-polariton regarding to the spin–valley locked selection rule of TMD’s excitons. This synthetic TMDs–metasurface platform opens a new route to further explore the polaritonic physics in low dimensional materials and paves the way towards valley dependent light-emitting diodes and vortex lasers, as well as integrated polariton condensation devices on-chip.
2. Results and discussion
The schematics and selection rules of our proposed valley-dependent vortex emission from exciton-polariton in mix-dimensional and 3R-stacked TMDs are shown in Fig. 1. In particular, monolayer TMDs (Fig. 1(a)) have direct bands at the K and K′ points of their hexagonal Brillouin zones (Fig. 1(c)), the so-called valleys. Owing to the intrinsic inversion symmetry breaking and strong spin-orbit interaction in TMDs lead to spin–valley locking at K and K′ valleys. Consequently, the excitons at different valleys exhibit distinct responses to light helicity depending on their valley pseudospin [49]. With the increase of the stacking number of 3R-TMDs, the character of the valence bands is unexchanged between two energy valleys as shown in Fig. 1 (b). Thus, interlayer hopping is strongly restrained in 3R-TMDs stackings, which means that the valley exciton is robustly protected with the increase of 3R-TMDs stacking layer. It means that the optical transition selection rules for both the monolayer TMDs and 3R-TMDs can be depicted by Fig. 1 (c). Accordingly, a generalizable framework to realize valley dependent vortex emission from exciton-polariton based on BIC and non-centrosymmetric TMDs is proposed in Fig. 1(d-f). As shown in Fig.1d, in the strong coupling regime, TMDs exciton modes collectively coupled to the photonic mode (dashed black lines in Fig. 1(d)), leading to two polariton branches, namely the upper and lower polariton bands. At the lower branch near BIC, the photonic BIC transfers to polaritonic BIC. Hence, the polaritonic BICs inherent in both the valley pseudospin of TMD’s exciton and the topological properties (polarization vortex at momentum space) of the photonic BIC. As shown in Fig. 1(e), the bottom bulk TMDs metasurface designed with BIC, and the top monolayer TMDs support exciton, the strong coupling between aforementioned two modes, leading to the spin-valley-locked vortex emission from exciton-polariton. When a σ-/σ+ beam incident on the mix-dimensional structure (Fig. 1(e)), a vortex emission carrying cross-polarization component (σ+/σ-) would be generated in the far-field. The similar design principle can be easily transplanted to the 3R-stacked TMDs, as shown in Fig. 1(f), where both the photonic BIC and exciton modes are provided by the 3R-stacked TMDs metasurface.
Fig. 1. Valley-dependent vortex emission from exciton-polariton in non-centrosymmetric transition metal dichalcogenide metasurfaces. (a),(b) The schematic of atom structures for monolayer TMDs and 3R-TMDs. (c) Schematics of band structures and optical transition selection rules of excitons in monolayer and 3R-TMDs. σ-(σ+) denotes left (right) circularly polarized (circular arrows). (d) The schematic of strong coupling between symmetric protected BICs and exciton of TMDs. Upper (blue) and lower (red) polariton branches arise from the collective coupling between a photon band (black dashed line) and TMDs exciton states. (e), (f) The valley dependent vortex emission from exciton-polariton in mix-dimensional (2D/3D) TMDs. (e) and 3R-TMDs (f) photonic crystals. The purple arrows indicate the incident light, and the red and blue lights indicate the LCP and RCP emitted vortex beam.
To fully demonstrate our proposed approach, we start by introducing the strong coupling between exciton resonances in a WSe2 monolayer and photonic BIC modes in an ultra-thin 2H-WS2 photonic crystal slab, the schematic as shown in Fig. 2(a). The periodicity a and thickness t of C4 symmetric WS2 photonic crystal slab is 340 nm and 25 nm respectively. Circular air etch hole, whose radius r is equal to 90 nm, is located at the center of each lattice. Such a C4 symmetric photonic crystal slab supports symmetry-protected BIC, of which the eigenfrequency ω = ω0 - iγ can be calculated using the finite-element method. And the Q-factor can be calculated as Q = ω0 / 2γ, where ω0 and γ are the real and imaginary parts of the intrinsic frequency, respectively. TE band structure near the Γ-point and the corresponding radiative Q factor without monolayer WSe2 are shown in Fig. 2(b) and (c) respectively. Here, the fundamental BIC mode (TE1, specially marked with yellow in Fig. 2(b) is selected for spin valley locked emission owing to its electric field distribution exhibits an annular circulation around the nanohole, which is important to match with the near-field circular polarized exciton dipoles. Such a TE1 band supports a BIC around 690 nm and possesses infinite Q factors, of which the energy location of BIC matches the exciton energy of monolayer WSe2 exciton (1.74 eV).
Fig. 2. The formation of exciton-photon and vortex emission from exciton-polariton in mix-dimensional TMDs metasurface. (a) The schematic of mix-dimensional TMDs metasurfaces. An ultrathin multilayer WS2 photonic crystal covered by a monolayer WSe2 on quartz substrate (nsub = 1.42). The C4 symmetric lattice constant a and thickness t are 340 nm and 25 nm respectively, and the radius of circular air hole is 90 nm. (b) The bandstructure of bar WS2 photonic crystal without monolayer WSe2. The color of the TE1 band corresponds to the value of Q factor. Inset shows the electric field and displacement vector around BIC. (c) The Q factors of four resonant bands. The TE1 band (color line) reaches the photonic BIC at Γ point of ideally infinite Q factor. (d) Angle-resolved absorption spectra of WS2 photonic crystal without covering monolayer WSe2. (e) Angle-resolved absorption spectra of mix-dimensional TMDs heterostructure in (a). The line fits respond to the upper (red) and lower (blue) polariton branches arise from the collective coupling between a BIC band (yellow) and WSe2 exciton states (green). (f) A cross-section of the absorption spectrum at the anti-crossing point along the red solid line in (e), the corresponding Rabi-splitting value is 22.03 meV that comprehensively represents the energy difference between the UP and LP.
We performed full wave finite-difference time-domain simulation to further study the mix-dimensional system. For pure BIC, unit cell WS2 photonic crystal with SiO2 substrate is set up for simulation, with PML boundary conditions along z direction, and periodic boundary conditions along x and y directions. The light source is set with plane wave at the top of unit cell WS2 photonic crystal. The reflection spectrum (R) is calculated on the same side of the source region, and the transmission spectrum (T) is calculated below the PhC layers. The absorption spectrum is calculated by A = 1-T-R. The absorption spectrum as a function of incident angle θ of bar-WS2 photonic crystal without covering monolayer WSe2 is shown in Fig. 2(d). The disappearance of resonance in angle-resolved absorption spectra at θ = 0°, which confirms the presence of symmetric protected BIC at the Г-point as predicted from the band diagram in Fig. 2(b). The simulation settings for the exciton coupling part are consistent with the BIC part. Single layer WSe2 is set to a thickness of 0.7 nm. After covering monolayer WSe2 on WS2 photonic crystal slab, as shown in Fig. 2(e), the mixed-dimension system demonstrated a distinct absorption Rabi-splitting at the wavelength around 710 nm, which generates the strong coupling between the BIC and the WSe2 exciton.
To confirm the strong coupling regime between the BIC and the WSe2 exciton, we fit the calculated dispersion with that of coupled harmonic oscillators model [50] in the Hamiltonian representation as
At zero detuning, EBIC - Eexciton = 0, the coupling strength g of the coupled system can be calculated as:
where Ω is the value of the vacuum Rabi splitting. By fitting the coupling strength, UP’s, and LP’s dispersion relationship in Eq. (2), the coupling strength and the corresponding Rabi splitting can be determined as g = 11.02 meV and Ω = 22.03 meV, respectively. This Rabi splitting energy satisfies the strong coupling criteria, i.e., Ω > (γBIC + γexciton)/2, indicating that our system reaches the strong coupling regime, and manifests the generation of exciton polaritons. The line fits in Fig. 2(e) corresponding to the upper (red) and lower (blue) polariton branches matched well with the dispersion relationship in the calculated angle-resolved absorption spectra. Next, the valley-dependent vortex emission characteristics of the above proposed mix-dimensional TMDs system are explored. The schematic is shown in Fig. 3(a). In order to better understand how this type of photonic crystal slab structure with periodicity in real space generates vortex emission, it is necessary to study the topological properties of the lower polariton branch by investigating the far-field polarization states. Fundamentally, BICs are topological defects in the far-field polarization, which carry integer topological charges: here ϕ(k) is the angle between the polarization major axis of radiation from the mode k and the x-axis. C is a path performed counterclockwise along any closed curve around the BIC. As shown in Fig. 3(b), the pseudo-color map is the phase distribution around the Γ point, and the line distribution means the polarization states in momentum space. Forced by symmetry, they will form vortex topologies, where q equals to +1. The phase change along the vanes of anticlockwise around the center can be calculated is 2π, and the polarization distribution near the center of the momentum space is azimuthal polarization. Both the phase and polarization are undefined at kx = ky = 0. When a circularly-polarized beam is normally illuminated on the PhC slab, the winding polarization states around the Γ points can induce geometric phase [Pancharatnam–Berry (PB) phases] vortices in the outgoing cross-polarized beam in momentum space. The transmitted light would gain a spiral phase front, of which the topological charge is l = ${\pm} $2q, where q is the charge of the Γ points and the sign is determined by the handedness of the incident light. Such phase vortices exist in both momentum and real spaces.Fig. 3. The valley dependent far-field vortex emission properties in mix-dimensional TMDs metasurface. (a) The schematic of proposed vortex-generating from strongly coupled exciton-polariton. The metasurface is illuminated by an LCP/ RCP Gaussian beam, which generates σ- (σ+) exciton polaritons by resonances and then produces RCP/LCP vortex emission in the vertical direction. (b) The far field polarization state (line distribution) and phase (pseudo-color distribution) distribution in the momentum space. (c),(d) The cross-section of far-field patterns and (e),(f) phase distributions for emitted vortex beams.
When a right-handed circularly-polarized (RCP) beam shines onto the PhC slab, the transmitted left-handed circularly-polarized (LCP) beam gains the spiral phase front, of which the topological charge is m = -2q. To illustrate the far-field pattern of such a polaritonic BIC, the photonic crystal slab is set as 50 × 50-unit cells in simulation. The incident beam is a Gaussian beam whose waist radius w0 = 0.86um and divergence angle θ0 = 10° respectively and the beam is focusing. Figure 3(c, d) and (e, f) are the far-field intensity distributions and phase distributions of cross-polarization and co-polarization components when incident with RCP and LCP, respectively. The field distribution of cross-section shows a hollow distribution, and the phase distribution shows a phase singularity, the topological of cross-section is the same as the prediction of l = ${\pm} $2q. Note that here, the polarization winding of BIC realizes the spin-to-orbit angular momentum conversion. At the same time, owing to the strong coupling between BIC and WSe2 exciton, the valley pseudospin of exciton would also lock with the spin-orbit interaction process, and hence led to our proposed valley-dependent vortex emission in the far-field.
As proposed in Fig. 1(f), the multilayer 3R-stacked TMDs provide a more compact platform for studying exciton polaritons. Figure 4(a) schematically shows the valley dependent far-field vortex emission properties in 3R-stacked TMDs metasurface. The 3R-WS2 photonic crystal is designed with similar C4 symmetry. The thickness of 3R-stacking WS2 PhC t is 9 nm, and the lattice constant a and the radius r of etch hole satisfied a2 = fπr2, where f = 0.08 is the duty cycle. Figure 4(b) shows the absorption spectrum with incident angle ranging from -20° to 20° with lattice constant a = 350 nm. The upper and lower polariton branches are fitted with red and blue solid lines. To ensure such coupled system is still in the strong coupling regime, the anti-crossing of its hybrid resonances needs to be analyzed as that in Eq. (1–3). The fitting parameter of exciton resonance Eexciton = 2.013 eV (green dashed line). The yellow dashed line denotes the theoretically calculated PhC mode with coupled wave theory [51]. After similar line fitting at zero detuning EBIC - Eexciton = 0, we get the dissipation rate of 3R-stacked WS2 exciton and BIC are γexciton = 59 meV and γBIC = 1.4 meV respectively, the coupling strength of g = 55.54 meV, and the Rabi splitting energy of Ω = 95 meV.
Fig. 4. The exciton-photon in 3R-stacked TMDs metasurface. (a) The schematic of exciton-polariton relevant far-field vortex emission properties in 3R-stacked TMDs metasurface. (b) Coupled wave theory calculated angle-resolved absorption spectra of 3R-stacked TMDs metasurface in (a). The line fits respond to the upper (red) and lower (blue) polariton branches arising from the collective coupling between a BIC band (yellow) and WSe2 exciton states (green). (c) Absorption spectra for different periodicity a with the area of etch equaling to 0.08a2. (d) 2D contour plot of the calculated absorption spectrum as a function of periodicity a. The line fits respond to the upper (red) and lower (blue) polariton branches arising from the collective coupling between a BIC band (yellow) and WS2 exciton states (green).
In order to further prove the strong coupling interaction, we next tune the geometric parameter (different lattice constant a with constant f = 0.08) of the q-BIC metasurface to observe the anti-crossing behavior and the Rabi splitting of the exciton polariton. The relationship between absorption spectrum and lattice constant a is shown in Fig. 4(d). Line fitted curves of upper branch (red solid line), lower branch (blue solid line), q-BIC (yellow dashed line), and WS2 exciton energy (green dashed line) are also depicted in Fig.4d. As the yellow curve shows, the wavelength of q-BIC resonant mode increases with the raising of lattice constant a. The q-BIC resonant line crosses with exciton energy near a = 354 nm and splits to upper branch and lower branch. The Rabi splitting Ω can be calculated Ω = 88.9 meV, γBIC = 1.4 meV can be fit by fano curve and γexciton equals 59 meV. Thus Ω > (γBIC + γexciton)/2 also satisfies the strong coupling criterion.
Taking into account that our proposed PhCs only use the C4 symmetric PhC, the topological charge of BIC is limited to ${\pm} $1. The wide choice of other lattice structures of PhCs would enable multiple and versatile vortex generation relevant to exciton-polariton based on our proposed TMDs metasurface platform. More generally, TMDs have rich materials types (such as XS2, XSe2, and XTe2) with different bandgaps. Thus, flexibly choosing the BIC cavity and excitonic roles of different TMDs, our proposed approach can be scaled to work in different wavelength ranges from visible or near-infrared. Moreover, the novel stacking of van der Waals heterostructure (e.g., mix-dimensional TMDs or 3R-TMDs) may result in the interlayer exciton hopping between the 3D and 2D-TMDs based on their different natures (indirect versus direct) or the resultant accessibility of the excitonic response. The future studies are promising to explore new and rich polaritonic characteristics and functionalities in mixed-dimensional or 3R-stacked TMD systems.
A recently published work reports a polariton Bose-Einstein condensate from a bound state in the continuum in optical grating [24]. Non-radiation nature of BIC is beneficial for the accumulation of exciton-polaritons. Thus Bose-Einstein condensate can be reached with a low threshold energy density. Transferring fundamental topological properties to polariton condensates of BECs in BICs is shown in this work as well. In our work, we focus on the polariton exciton in two systems of TMDs photonic crystals. Owing to the high reflective index of TMDs, ultra-thin vortex emission photonic crystal devices can be realized based on BIC. Moreover, inheriting of BIC topological properties entails valley-locked exciton-polariton associated vortex emission regarding to the spin–valley locked selection rule of TMD’s excitons. The results open new avenues to the realization of polariton Bose-Einstein condensate and potential applications to ultra-compact coherent light emitters and low threshold vortex laser devices at room temperature.
3. Conclusion
In summary, we have exploited BICs to achieve strongly coupled exciton polariton in an ultrathin mix-dimensional (monolayer/2H-stacked) metasurface and a full 3R-stacked TMDs metasurface. The Rabi splitting in our work achieves 22.03 meV for mix-dimensional photonic crystals and 95 meV for 3R-stacked WS2 photonic crystals respectively. Owing to the topological nature of BIC, the valley pseudospin of non-centrosymmetric TMDs would link with the winding polarization pattern of BIC in the momentum space, which hence enabled spin-valley-locked vortex emission from exciton-polariton at the far-field.
Funding
National Natural Science Foundation of China (12174047, 62205049); Sichuan Province Science and Technology Support Program (2022YFSY0023, 2022YFH0082).
Acknowledgment
The support from the start-up funding of University of Electronic Science and Technology of China.
Disclosures
The authors declare no conflicts of interests
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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