Routing valley exciton emission of a WS2 monolayer via delocalized Bloch modes of in-plane inversion-symmetry-broken photonic crystal slabs

The valleys of two-dimensional transition metal dichalcogenides (TMDCs) offer a new degree of freedom for information processing. To take advantage of this valley degree of freedom, on the one hand, it is feasible to control valleys by utilizing different external stimuli, such as optical and electric fields. On the other hand, nanostructures are also used to separate the valleys by near-field coupling. However, for both of the above methods, either the required low-temperature environment or low degree of coherence properties limit their further applications. Here, we demonstrate that all-dielectric photonic crystal (PhC) slabs without in-plane inversion symmetry (C2 symmetry) can separate and route valley exciton emission of a WS2 monolayer at room temperature. Coupling with circularly polarized photonic Bloch modes of such PhC slabs, valley photons emitted by a WS2 monolayer are routed directionally and are efficiently separated in the far field. In addition, far-field emissions are directionally enhanced and have long-distance spatial coherence properties.

To develop valleytronic devices based on TMDCs, effective approaches to separate valleys in the near or far field are indispensable. One feasible way is to selectively excite valleys by utilizing different external stimuli such as optical and electric fields [14][15][16][17][18] , while the usually required low-temperature environment makes it difficult for practical applications. Due to the powerful ability of manipulating light, nanostructures [19][20][21] are also proposed to separate valleys [22][23][24][25][26][27][28][29][30][31][32] . For example, based on either the transverse spin momentum of surface plasmons 27,28 or the variable geometric phase of metasurfaces 31 , valley separation was reported to be achieved in the near or far field at room temperature. However, both the intrinsic loss of metal materials and the localized spatial distribution of resonant modes of nanoantennas limit efficient valley separation, leading to a low degree of valley polarization [24][25][26][27][28][29][30] . As a counterpart of metasurfaces, photonic crystals (PhCs) eliminate all these disadvantages due to delocalized photonic Bloch modes and low-intrinsic-loss dielectric constituents. In addition, these Bloch modes are found to have peculiar polarization properties. With these attractive properties, PhCs have been widely applied in various studies, such as bound states in the continuum [33][34][35][36][37] , topological valley photonics 38,39 , PhC lasers [40][41][42] , and spontaneous emission control of TMDCs 43 . However, to date, there are no reports of effective valley separation in TMDCs by using PhCs.
In this article, we demonstrate that two-dimensional alldielectric PhC slabs without in-plane inversion symmetry can be used to efficiently separate valley exciton emission of a WS 2 monolayer in the far field at room temperature. The valley exciton emission is routed with high directionality and a high degree of valley polarization, as shown in Fig. 1d. For this type of PhC slab, paired delocalized Bloch modes with different circular polarizations not only play a critical role in separating and enhancing directional valley exciton emission but also lead to spatial coherence properties of the emission field, which have not been discussed in past studies. Experimentally, the angleresolved photoluminescence (PL) results directly show efficient valley separation in the far field, with a degree of valley polarization up to 88%. Time-resolved PL measurements indicate a 75% enhancement of the exciton radiative rate. In addition, the double-slit interference results reveal that the spatial coherence length of the emission field for a WS 2 monolayer on a PhC slab without in-plane inversion symmetry is longer than 6 microns (29 microns in theory).

Principle of separating and routing valley exciton emission
Analogous to electronic band structures in solids, Bloch scattering by periodic artificial atoms of PhC slabs alters the dispersion relation of light in the slab, resulting in photonic bands 44 . Each optical state in photonic bands corresponds to a delocalized Bloch mode with well-defined energy and momentum. Modes above the light cone are radiative due to coupling to the free space 44 . For these radiative modes, their polarization states in the far field are strictly defined. The corresponding polarization states of radiative Bloch modes in an arbitrary photonic band can be further projected into the structure plane and mapped onto the Brillouin zone, defining a polarization field in the momentum space 33,34 . These polarization properties in principle could be used to control the radiation of luminescent materials. However, owing to high rotation symmetry, the polarization field is nearly linear in most PhC slabs 45 . As a consequence, the polarization states of those PhC slabs can only cover a belt near the equator of the Poincaré sphere 46 (a space to describe all polarization states, shown in Fig. 1b). With a large area including two poles not covered, it is useless for us to utilize these Bloch modes of PhC slabs to separate valley exciton emission of TMDCs. In contrast, as is known, broken inversion symmetry is of vital importance in the appearance of inequivalent valley excitons in TMDCs. Similarly, we recently reported that paired circularly polarized states with different chiralities emerge from vortex singularities after breaking the in-plane inversion symmetry of PhC slabs 46 , as shown in Fig. 1c. Then, in addition to the areas near the equator, the polarization states cover the whole sphere, including two poles of the Poincaré sphere, corresponding to polarization states with a high degree of circular polarization in momentum space. Therefore, this type of PhC slab with circularly polarized radiative states could be an ideal platform for us to separate valley exciton emission of TMDCs in the far field, as illustrated in Fig. 1d.
First, valley photons could couple to circularly polarized states with corresponding chirality and become separated in the momentum space. Second, these Bloch modes are delocalized and could be used in coherent emission 47,48 . The spatial coherence properties of the emission field lay the foundation for the directionality and highly efficient separation of valley exciton emission of a WS 2 monolayer. A detailed discussion is provided in Supplementary Material section 1.

Results and discussion
To demonstrate the existence of opposite circularly polarized states in momentum space, we designed an inplane inversion-symmetry-broken PhC slab and studied the transmittance spectra in theory and experiment, as shown in Fig. 2. The slabs here are made of silicon nitride (Si 3 N 4 , refractive index ∼2) and silicon dioxide (SiO 2 , refractive index ∼1.5). The thickness of the Si 3 N 4 layer is 150 nm. The thickness of the SiO 2 layer is 500 microns, which could be considered infinite compared to the wavelength of visible light. Square lattices of holes with a period a = 390 nm are etched in the Si 3 N 4 layer. To break the in-plane inversion symmetry, the shape of the etched hole in a unit cell is set as an isosceles triangle, with the height h and baseline length b of the triangle being equal (h = d = 250 nm), as shown in Fig. 2a. More details about the sample design can be found in Supplementary Material section 3.
We first simulated the angle-resolved transmittance spectra under σ + -polarized incidence by Rigorous Coupled Wave Analysis (RCWA), with the incidence plane along the Γ-X direction. The spectra are asymmetric, and there are some diminished regions on the photonic bands, indicated by blue arrows in Fig. 2b. These diminished regions correspond to the nonexcited states under σ +polarized incidence. Hence, those states in the diminished regions are σ − polarized. Changing the incident light to σ − polarization, the diminished regions switch to the other side (Fig. S1b). To show this effect experimentally, we fabricated samples using electron-beam lithography and reactive ion etching (for more details, see Methods). By using a homemade polarization-resolved momentumspace imaging spectroscopy system (Fig. S4), angleresolved transmittance spectra are measured (Fig. 2c), in accordance with the simulation. Both the simulated and experimentally measured results confirmed the appearance of optical modes with a high degree of circular polarization in our designed PhC slab. For comparison, we also researched the angle-resolved transmittance spectra of the PhC slab with in-plane inversion symmetry. Shown in Fig. 2d, the designed shape of the etched hole in the unit is a circle (diameter d = 210 nm). As expected, we did not observe asymmetric spectra under σ + -polarized incidence in either the simulation or experiment, as shown in Fig. 2e, f. When changing the incidence to σ − polarization, the transmittance spectra are the same as those in the case of σ + polarization (Fig. S1c). The results demonstrate that by breaking the in-plane inversion symmetry of PhC slabs, circularly polarized states emerge in photonic bands.
The large area of the WS 2 monolayer is grown on a Si/ SiO 2 substrate by the CVD process and then transferred onto PhC slabs. Both PhC slabs and part of the unstructured flat Si 3 N 4 substrate are covered (Fig. S10). To study the PL distribution in the far field, angle-resolved PL spectra are measured (Supplementary Material section 5), as shown in Fig. 3a-f. The detection plane is along the Γ-X direction, in accordance with the transmittance spectra measurement in Fig. 2. We selected σ + (σ − ) PL by placing a quarter-wave plate and a linear polarizer in the detection path (Fig. S4). Figure 3e, f shows the asymmetric σ + (σ − ) PL spectra of the WS 2 monolayer on the PhC slab without in-plane inversion symmetry. The σ + (σ − ) PL enhanced regions correspond to regions with a high degree of σ + (σ − ) polarization in photonic bands. Figure  3a, b shows σ + (σ − ) PL spectra of the WS 2 monolayer on a flat substrate. Figure 3c, d shows σ + (σ − ) PL spectra of the WS 2 monolayer on the PhC slab with in-plane inversion symmetry. Different from those in Fig. 3e, f, all spectra in Fig. 3a-d are symmetric for both σ + and σ − detection. From the abovementioned experimental results, we can draw the conclusion that, as shown in the asymmetric spectra, valley photons emitted by the WS 2 monolayer have been separated in the far field by PhC slabs without in-plane inversion symmetry. In addition, we performed time-resolved PL measurements at room temperature (Supplementary Material section 10). Compared with that for the WS 2 monolayer on a flat substrate, the exciton radiative rate, namely, the reciprocal of radiative lifetime, is enhanced by 75% when the WS 2 monolayer is on a PhC slab without in-plane inversion symmetry.
To further study the degree of separation in Fig. 3e, f, we plotted the angle-resolved σ + (σ − ) PL spectra for a single wavelength, as shown in Fig. 3g, h. The dotted line refers to 615 nm, and the solid line refers to 628 nm, which are also marked in Fig. 3e, f. We observed that σ + (red) and σ − (blue) PL maximums separately appear at different angles. The σ + and σ − PL peaks are separated by nearly 6 degrees at 615 nm and 3 degrees at 628 nm. For comparison, PL spectra on a PhC with in-plane inversion symmetry for corresponding wavelengths are shown in Fig. S5, with the σ + and σ − PL maximums overlapping at the same angle. We also show that the photoluminescence of the WS 2 monolayer on this PhC slab without in-plane inversion symmetry is highly directional. As shown in Fig.  3g, h, the full width at half maximum of the PL peaks (Δθ) is less than 3 degrees at 615 nm and 2 degrees at 628 nm. This result is due to the delocalized property of Bloch modes, leading to the long-distance spatial coherence property of the far-field emission by the WS 2 monolayer on PhC slabs. According to the Fourier relation between momentum and position, a wide distribution in the real space means that the mode is localized inside a small area in the momentum space. This effect corresponds to the small angle distribution of the far-field emission, i.e., the directional emission, and will be further discussed later in this article. For this reason, although the separation of σ + and σ − PL peaks is small, the valley exciton emission could still be efficiently separated in the far field. Further, we quantify the degree of valley polarization by P θ ð Þ ¼ I þ θ ð Þ À I À ðθÞ I þ θ ð Þ þ I À ðθÞ where I + (I − ) refers to the PL intensity with σ + (σ − ) polarization for a single wavelength, and θ is the radiation angle. The degree of valley polarization is plotted in Fig.  S7, with the maximum degree of valley polarization calculated up to 84%. These results indicate that the PL of the WS 2 monolayer on the PhC slab without in-plane inversion symmetry is highly directional and has a high degree of valley polarization. Based on the measured angle-resolved σ + (σ − ) PL spectra of the WS 2 monolayer on the PhC slab without in-plane inversion symmetry, we mapped the PL intensity distribution of a single wavelength in momentum space, as shown in Fig. 4a-d. The upper (lower) row corresponds to 615 (628) nm. The PL spectra along different directions in momentum space were measured by rotating the sample in-plane relative to the entrance slit of the imaging spectrometer. The projected momentum k is calculated by k = k 0 sinθ (k 0 = 2π/λ is the wavevector of light in the free space, θ is the emission angle Then, we used P(k) to qualify the degree of valley polarization in momentum space, which is similarly defined by P k ð Þ ¼ I þ k ð ÞÀI À ðkÞ I þ k ð ÞþI À ðkÞ , as shown in Fig. 4e, f. Here, I + (I − ) refers to the PL intensity with σ + (σ − ) polarization for a single  Fig. 4 Experimental measurement of PL spectra and valley polarization in momentum space. a-d σ + and σ − PL intensity distribution in momentum space at 615 nm (upper) and 628 nm (lower). These are for the WS 2 monolayer on the PhC slab without in-plane inversion symmetry. e, f Images of valley polarization P(k) in momentum space. The maximum calculated P reaches 88% wavelength. Experimentally, the maximum calculated P reaches 88%, as shown in Fig. 4f. Note that the maximum P did not appear along the Γ-X direction in momentum space. The result is as expected because the circular polarized states of the designed PhC slab without in-plane inversion symmetry are slightly shifted from the Γ-X direction in momentum space 40 . The sign of P(k) reverses at opposite sides of the momentum space, demonstrating the separation of valley exciton emission with different chiralities. In contrast, we also measured and calculated P(k) of the emission by a WS 2 monolayer placed on a flat substrate, and P(k) was negligible (Fig. S12).
In addition to valley-related directional emission in momentum space, we expected the spatial coherence property of emission by the WS 2 monolayer on the PhC slab without in-plane inversion symmetry. Young's double-slit experiments were performed, as shown in Fig.  5. The experimental setup is illustrated in Fig. 5a, and the working principle is based on Fourier transformation. The double slit is mounted on the real image plane inside the optical measurement setup to select radiation fields from two different positions on the sample. The radiation fields from these two positions intersect with each other on Fourier image 2 at the entrance of the spectrometer. Therefore, the spatial coherence properties on the surface of the sample could be directly detected in the far field. Changing the etched depth of the PhC slab, we were able to overlap the measured photonic band with the PL spectra of the WS 2 monolayer to obtain enough signal intensity. Interference fringes are observed in the angleresolved PL spectra along the Γ-X direction, as shown in Fig. 5b. The red-marked line is further plotted in Fig. 5c, showing the interference intensity distribution at 621 nm.
The fringe visibility V is calculated to be~50%, defined by V ¼ I max ÀI min I max þI min , where I max and I min are the intensities of adjacent maximums and minimums 49 . In this measurement, the real double-slit distance d is 120 microns. The scanning electron microscopy image of the double slit is presented in Fig. S13. The magnification of the real image is 20, so the effective double-slit distance on the sample is 6 microns. The 6-micron effective double-slit distance is almost ten times the emission wavelength, demonstrating that the measured spatial coherence length is larger than 6 microns. Moreover, the spatial coherence length could be calculated by λ Δθ in theory, which is widely used in optical coherence theory 50 . Here, Δθ is~0.0215 (1.23 degrees) at 621 nm (Fig. S14), and the calculated spatial coherence length is~29 microns. In comparison, no interference fringes are observed when the WS 2 monolayer is placed on a flat substrate, as shown in Fig. 5b, c. This result means that the far-field emission of the WS 2 monolayer on a flat substrate has no long-distance spatial coherence property. Hence, we reveal that the far-field emission by the WS 2 monolayer on the PhC slab without in-plane inversion symmetry has a long-distance spatial coherence property. This property of the PhC slab extends the coherence control on the PL of the WS 2 monolayer from temporal coherence to spatial coherence.
In summary, we proposed in-plane inversionsymmetry-broken all-dielectric photonic crystal slabs to route valley exciton emission of a WS 2 monolayer in the far field at room temperature. By breaking the in-plane inversion symmetry of the PhC slab, we observed paired circularly polarized states with different chiralities emerge from vortex singularities. Via coupling with those delo- The experimental results for the case of a 6-micron effective double-slit distance. The real double-slit distance d is 120 microns. The magnification of the real image is 20, so the effective double-slit distance on the sample is 6 microns. The upper panel shows the WS 2 monolayer on the PhC slab without in-plane inversion symmetry. The lower panel is for the WS 2 monolayer on a flat substrate, with the signal intensity shown at two-fold magnification. The detection plane is along the Γ-X direction. c The interference intensity distribution in b at 621 nm. The far-field emission of the WS 2 monolayer on the PhC slab without in-plane inversion symmetry has a long-distance spatial coherence property monolayer were separated in momentum space, and the exciton radiative rate was significantly enhanced. In addition, both the directional emission and the longdistance spatial coherence property benefit the applications of in-plane inversion-symmetry-broken PhC slabs to route valley exciton emission. In addition, our method could be extended to manipulate valley exciton emission of other TMDC monolayers. The ability of these PhC slabs to transport valley information from the near field to the far field would help to develop photonic devices based on valleytronics.

Sample fabrication
The fabrication of a photonic crystal slab The sample structure was two slab layers, with a thin silicon nitride layer on the silicon dioxide substrate. The silicon dioxide substrate was cut from a 500-micron-thick quartz wafer. Then, a silicon nitride layer was grown on a silicon dioxide substrate by plasma-enhanced chemical vapour deposition (PECVD). The thickness of the grown silicon nitride layer was nearly 150 nm, and the thickness could be tuned by controlling the deposition time. To fabricate the designed structure, the raw sample was spincoated with a layer of positive electron-beam resist (PMMA950K A4) and an additional layer of conductive polymer (AR-PC 5090.02). Then, a hole array mask pattern was fabricated onto the PMMA layer using electronbeam lithography (ZEISS sigma 300). The sample was further processed by reactive ion etching (RIE). Anisotropic etching was achieved by RIE using CHF 3 and O 2 . The patterned PMMA layer acted as a mask and was eventually removed by RIE using O 2 . The size of every designed structure is~80 × 80 microns.

Transfer process for the WS 2 monolayer
The CVD WS 2 monolayer on the Si/SiO 2 substrate was spin-coated with poly(L-lactic acid) (PLLA) before baking for 5 minutes at 70°C. Afterwards, a PDMS elastomer was placed on top of the PLLA film and then torn off. The composite was then attached to a glass slide and put under a microscope on a transfer stage. The PhC slab placed under the glass slide was aligned carefully using the microscope, and the glass slide was lowered to contact the PhC slab. The stage was heated to 70°C to improve the adhesion, and then, the glass slide was lifted with PDMS, leaving a WS 2 monolayer on the PhC slabs. After dissolving PLLA in dichloromethane, the WS 2 monolayer was finally transferred to the designed photonic crystal slabs.

Optical measurements Experimental measurements of time-resolved PL
Please see Supplementary Material section 4 for the schematics and discussions.
Measurement setup of the polarization-resolved momentumspace imaging spectroscopy system and double-slit experiment Please see Supplementary Material section 5 for the schematics and discussions.

Simulations
The transmittance spectra were simulated by Rigorous Coupled Wave Analysis (RCWA). The periodic boundary conditions were applied in the x and y directions. The polarization angle was set to π/4, and the phase difference was set to π/2 or 3π/2 to obtain circularly polarized incidence (the polarization angle 0 (π/2) corresponds to p(s) polarization). The Si 3 N 4 refractive index was set to 2, and the SiO 2 refractive index was set to 1.5. All the materials were considered to have no loss in visible light.