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## Creator

Xuezhi Ma, Kaushik Kudtarkar, Yixin Chen, Preston Cunha, Yuan Ma, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Xiaofeng Qian, M. Cynthia Hipwell, Zi Jing Wong, Shoufeng Lan

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[Coherent momentum control of forbidden excitons](https://mdr.nims.go.jp/datasets/b6005dfc-c129-4d05-8bbf-80a19dcfbede)

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Coherent momentum control of forbidden excitonsArticle https://doi.org/10.1038/s41467-022-34740-5Coherent momentum control of forbiddenexcitonsXuezhi Ma1,2, Kaushik Kudtarkar1, Yixin Chen3,4, Preston Cunha1, Yuan Ma1,5,Kenji Watanabe 6, Takashi Taniguchi 7, Xiaofeng Qian 4,8,9,M. Cynthia Hipwell1, Zi Jing Wong3,4 & Shoufeng Lan 1,4A double-edged sword in two-dimensional material science and technology isoptically forbidden dark exciton. On the one hand, it is fascinating for con-densedmatter physics, quantum information processing, and optoelectronicsdue to its long lifetime. On the other hand, it is notorious for being opticallyinaccessible fromboth excitation and detection standpoints. Here, we providean efficient and low-loss solution to the dilemma by reintroducing photonicsbound states in the continuum (BICs) to manipulate dark excitons in themomentum space. In a monolayer tungsten diselenide under normal inci-dence, we demonstrated a giant enhancement (~1400) for dark excitonsenabled by transverse magnetic BICs with intrinsic out-of-plane electric fields.By further employing widely tunable Friedrich-Wintgen BICs, we demon-strated highly directional emission from the dark excitons with a divergenceangle ofmerely 7°. We found that the directional emission is coherent at roomtemperature, unambiguously shown in polarization analyses and interferencemeasurements. Therefore, the BICs reintroduced as a momentum-spacephotonic environment could be an intriguing platform to reshape and redefinelight-matter interactions in nearby quantum materials, such as low-dimensional materials, otherwise challenging or even impossible to achieve.Dark excitons (XD) in semiconductors have a long lifetime due to theirdecoupling from radiative channels and spin-flip processes1–5, makingthem perfect to serve as the quantum bits (qubits) for the ongoingdevelopment of quantum computing6. Dark excitons in two-dimensional (2D) semiconductors, such as transition metal dichalco-genide (TMD) monolayers, have also attracted broad research inter-ests thanks to the atomically thin structure of 2D materials that madethem suitable for compact planar devices7. However, since the exci-tonic transitionofdark excitons does not satisfy the selection rules andpossesses zero in-plane dipole moments3,8,9, achieving dark excitonoptical brightening with conventional far-field optical techniques hasbeen a challenging task10. Additionally, collecting XD emission requiresa large-numerical-aperture (NA) objective lens due to their out-of-plane dipole-like radiation pattern, which made it challenging toachieve information read-out using conventional far-field opticaltechniques. To demonstrate a robust dark exciton qubit read-outsystem, two tasks need to be satisfied, namely dark exciton opticalbrightening, and efficient collection of the XD emission.Received: 19 October 2022Accepted: 4 November 2022Check for updates1Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA. 2Institute of Materials Research and Engineering, Agency forScience, Technology and Research (A*STAR), Singapore, Singapore. 3Department of Aerospace Engineering, Texas A&M University, College Station, TX77843, USA. 4Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA. 5Department of Mechanical Engi-neering and Research Institute for Intelligent Wearable Systems, The Hong Kong Polytechnic University, Hong Kong, China. 6Research Center for FunctionalMaterials, National Institute for Materials Science, Tsukuba, Japan. 7International Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, Tsukuba, Japan. 8Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA. 9Department of Electrical andComputer Engineering, Texas A&M University, College Station, TX 77843, USA. e-mail: shoufeng@tamu.eduNature Communications |         (2022) 13:6916 11234567890():,;1234567890():,;http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-1627-288Xhttp://orcid.org/0000-0003-1627-288Xhttp://orcid.org/0000-0003-1627-288Xhttp://orcid.org/0000-0003-1627-288Xhttp://orcid.org/0000-0003-1627-288Xhttp://orcid.org/0000-0003-2108-6774http://orcid.org/0000-0003-2108-6774http://orcid.org/0000-0003-2108-6774http://orcid.org/0000-0003-2108-6774http://orcid.org/0000-0003-2108-6774http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34740-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34740-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34740-5&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-022-34740-5&domain=pdfmailto:shoufeng@tamu.eduFor dark exciton brightening, firstly, the current techniques toexcite the XD emission usually require an ultra-strong in-plane mag-netic field (>14 T)2 or out-of-plane polarized surface plasmon polar-itons (SPPs)3. The former brightening method uses an ultra-strong in-plane magnetic field to tilt the effective internal magnetic field in theconductionband (CB) to gain an in-plane component from the originalout-of-plane electron spin. The lattermethod uses the SPP structure toconvert the in-plane polarized incident light into an out-of-plane SPPmode to efficiently couple with the out-of-plane transition dipolemoment of dark excitons. In addition, out-of-plane transition dipolemoments of dark excitons in WSe2 monolayer encapsulated betweenthin h-BN flakes can couple with out-of-plane polarized incident lightprovided by a 90° rotated objective lens, i.e., the dark excitons can beexcited by the horizontally oriented objective lens10. These methods,however, are intrinsically restricted to cryogenic temperature condi-tions (T < 30K). Alternatively, a tip-enhanced photoluminescence(TEPL) system manifested itself as capable of brightening dark exci-tons at room temperature4. The oblique incident laser can be coupledin, and its out-of-plane polarized component can be selectivelyenhanced by the scanning probe microscope (STM) driven gap modebetween the gold tip and gold substrate, yielding a greater couplingefficiency with the out-of-plane transition dipole moment of darkexcitons. However, its complicated setup limits its application, espe-cially when integrating the dark exciton read-out system with an on-chip systems11.Secondly, efficient XD emission collection is another bottleneckfor the qubits read-out system. A nanoscale trench coupler3,waveguide-based method9, or oblique collecting setups4 are usuallyused to boost the collection efficiency of XD emissions. However, thecomplicated setup, relatively low collection efficiency, or the absenceof the enhancement of XD emission limit their applications. On theother hand, optical resonators such as photonic crystal cavities12,whispering gallery mode (WGM) resonators13, antenna array Mieresonators14, and metamaterials15 are used to enhance anddirectionally emit the electroluminescence (EL) or photoluminescence(PL) signals from bright excitons. To combine those advantages,designing a structure that can achieve the directional XD emissionwithsuitable deflection angles can break the bottleneck for the XD read-outsystem. In this way, XD emission can be easily detected by either aconventional microscope or a single detector. Practically, recentlyemerged optical bound states in the continuum (BICs) with infinitequality factors (Q-factors) or quasi-BICs with finite Q-factors stand outfrom optical resonators for this mission, thanks to their high Q-factorand compatibility with planar optical platforms. These BICs can besupported in various photonic systems such asphotonic crystal slabs16,plasmonic structures17, metasurfaces18, and fiber Bragg gratings19. BICscan be assigned to three categories: (i) symmetry-protected BICs atGamma-points (Γ-points, the center of the Brillouin zone), (ii) off-Γaccidental BICs, and (iii) Friedrich-Wintgen BICs20,21. In 1985, Friedrichand Wintgen suggested that BIC can occur due to the interference ofresonances belonging to different channels that cause an avoidedcrossing22. The avoided crossing was then extended to acoustic23,quantum24 and optical systems25. Compared with accidental BICs,which rely on carefully tuning the geometric parameters, Friedrich-Wintgen BICs are stable and efficient, making them perfect candidatesfor XD directional emission26,27.In the present work, we designed a suspended photonic crystal(PhC) slabmade of lossless silicon nitride (Si3N4) which simultaneouslysupports on-Γ symmetry-protected BICs and off-Γ Friedrich-WintgenBICs (Fig. 1a). In this design, a transverse magnetic (TM) like BIC at theΓ-point canefficiently convert the in-plane polarized normally-incidentpump laser into out-of-plane polarized near-field energy to gain sig-nificant efficiency to couple with the out-of-plane transition dipolemoment of dark excitons11,16,28. In this way, the spin-forbidden elec-trons transition from the valence band (VB) to the conduction band(CB) with opposite spin direction can be allowed, resulting in darkexciton brightening (Fig. 1b, the XD transitions show this process).Meanwhile, the off-Γ Friedrich-Wintgen BIC can selectively couple with++142 23 3� MomentumEnergyK' Kbf20μm1μmacCBVBXD XD Wavevector, kBICPosition, xCavitydeE Fig. 1 | Dark exciton brightening and directional emission. a Schematic ofdirectional emissionof dark excitons in theWSe2monolayerwith normally incidentpumping light. The dark excitons PL signal can then be directionally emittedthrough the Friedrich-Wintgen bound states in the continuum (BIC) supported bythe samePhC slab.b Split-band configuration of bright and dark exciton states. Thespin-forbidden optical transition of the dark exciton (XD) is brightened by theconverted E with enhancement on the top surface of the PhC slab. CB, conductionband; VB, valence band. c A scanning electron microscope (SEM) image of the PhCslab made of silicon nitride (Si3N4). d and e BIC, and cavity modes obtain opticalconfinement in momentum (k) and real (x) space, respectively. f A sketch of theoptical band structure of the PhC slab with three types of the BICs: ① and ④ are theon-Γ symmetry-protected BICs, ② is the off-Γ accidental BIC, and ③ is the Friedrich-Wintgen BIC due to the destructive interference of resonances belonging to dif-ferent bands (red and blue).Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 2the out-of-plane dipole moment of dark excitons and directionallyemit the XD-PL signals29,30. Unlike the regular bound states, which areconfined in real space and rely on the well-like structure to achieve theconfinement (Fig. 1d), the BICs are confined in the momentum spacewith infinite Q-factors and can get rid of the well-like structures,making the on-chip applications possible20.ResultsIn the experiments, we transferred exfoliated monolayers of WSe2onto the suspended Si3N4 PhC slab device (Supplementary Fig. 2, theoptical microscope picture of the device and Fig. 1c, the scanningelectronmicroscope image of the PhC slab, and the transfer details seeMethod). The sketch of the band structure of the device, as shown inFig. 1f, was extracted from the numerically simulated angle-resolvedreflection spectroscopy mapping to highlight the BICs modes sup-ported by the designed PhC slab. It clearly shows three types of BICs:①and ④ are on-Γ symmetry-protected BICs, ② is an off-Γ accidental BIC,and ③ is an off-Γ Friedrich-Wintgen BIC due to the destructive inter-ference of resonances belonging to different TM-like bands (red andblue). In our device, the PhC slab fulfills two important roles simulta-neously: it converts polarization and enhances the incident light, and itselectively enhances and directionally emits the XD-PL signal for higherXD collection efficiency. Onlywith these two roles achieved at the sametime can theWSe2monolayer integrate with photonic chips for robustdark exciton qubits read-out system. In light of the relatively smallemission energy difference between dark and bright excitons(~40–50meV), a large enhancement factor of XD emission, typically, isnecessary to increase thedark-to-bright excitons contrast, especially atroom temperature4. However, the lifetime of dark excitons would belargely decreased due to the Purcell effect, hindering the excitonsapplications, especially in a quantum system. Alternatively, we usedthe off-Γ Friedrich-Wintgen BIC to provide a dark-exciton emissionchannel to separate the dark exciton signal in momentum space for anincreased dark-to-bright excitons contrast. With the PhC slab, thedouble BICs mediated far-field-to-near-field-to-far-field mode trans-formation, giving rise to a ~1400-fold XD-PL enhancement demon-strated in the WSe2 monolayer with a relatively small Purcell factor of26.7. Furthermore, the Friedrich-Wintgen BIC can be tuned to emit theXD signal in various angles from 31.4° to 59.5° by tuning the deviceparameters, demonstrating an angle-tunable dark exciton read-outsystem. Finally, we also demonstrate the XD directional emissions arecoherent at room temperature.Dark exciton brightening using on-Γ symmetry protected BIC atroom temperatureTo demonstrate the mechanism of dark exciton brightening by on-Γsymmetry protected BIC, we first designed a suspended Si3N4 PhC slab(n = 2.23, near 700 nm in wavelength, thickness is 265 nm) with asquare array of cylindrical holes (periodicity is 450nm, hole radius is140 nm). The slab material, Si3N4, is deposited using low-pressurechemical vapor deposition (LPCVD) on top of an <100> oriented sili-con wafer, providing low absorption and the ability to guide theelectromagnetic energy inside. The optimized fabrication protocol(see Method) was applied to yield a PhC slab with minimum surfaceroughness to minimize scattering and improve the quality of theinterface between the WSe2 monolayer and the PhC slab. The devicewas 100-by-100 periods (45 ×45 μm2), which was sufficiently large toguarantee the C4 symmetry and periodic boundary condition of thelattice in the central region of the device. Supplementary Fig. 2a showsthe transferred WSe2 monolayer with a few-layer h-BN flake coveringthe suspended PhC slab. It is worthmentioning that the h-BN flake canintroduce a red-shift in spectra to the PhC slab31,32. To compensate forthis redshift, we reduced the thickness of the PhC device by severalnanometers in our experiment to minimize any spectral differencesbetween the simulation and the real device. This compensationstrategy is effective, thanks to the small difference in refraction indexbetween our Si3N4 slab (n = 2.23, near 700nm in wavelength, moredispersive data see Supplementary Section 15) and the exfoliated h-BNflake (n = ~2.2, near 700nm inwavelength)33. Thedemonstrationof thiscompensation strategy can be found in Supplementary Section 17.To better understand the optical properties of the PhC device, weperformed numerical simulations and characterizations of the device.Figure 2a shows the dispersion curves of the eight lowest energy bands(optical band structures, by the MPB band solver34) along the Γ-X line[kðΓÞ= ð0,0Þ*ð2π=aÞ,kðXÞ= ð0:5,0Þ*ð2π=aÞ,k= ðkx +kyÞ andkx = ðω=cÞsinðθÞ]. The four blue bands are TM-like, whereas the four red bandsare transverse electric (TE) like. Three TM-like bands and one TE-likeband can be excited by the p-polarized incident light as shown in themapping of the reflection spectra as a function of the incident angle inFig. 2b. The comparison between the numerical simulation (left) andthe angle-resolved spectrometric measurement (right) shows mini-mum differences, demonstrating the quality of the fabricated device.Two symmetry-protected TM-like BIC modes at the Γ-point at 694 nmand 755nmareclearly visible in Fig. 2b.Wedesigned aBICat694 nmasthe incident light polarization converter because the BIC can convertthe in-plane polarized incident light into the out-of-plane polarizedcomponent with a large enhancement ratio, as shown in Fig. 2c (theout-of-plane electric field component, Ez distribution on theX-Y plane(upper) and theX-Zplane (lower) in theWSe2monolayer). A numericalsimulation using COMSOL Multiphysics shows that Ez can be createdand enhanced with an enhancement ratio reaching ~3 × 104-fold at themaximum point of resonance (BIC) when compared with the originalincident electrical field. In the real experiment, the tightly focusedmono-color incident laser spot still has a relatively large divergentincident angle and a wide width of wavelength in the spectra. As aresult, only a small part of the energy of the incident laser can couplewith the BIC and be converted and enhanced, even if the 5× beamexpander (Thorlabs, BE05-10-A was used to compress the beam size,more details see Supplementary Fig. 8a) and the laser line filter(Thorlabs, FL694.3-10 was placed with a small twisted angle, moredetails see Supplementary Figs. 8b and c) were applied to compressthe laser energy into the BIC mode. Figure 2d shows the reflectionspectrum with an oblique incident angle of 3° (indicated by the blackdashed line) and the maximum local Ez enhancement ratio over theincident light electricfieldon the top surfaceof the PhC slab,where theWSe2monolayer is seated.However, it is clearly seen thatonly the laserenergy that is close to the resonance peak benefits from the largeenhancement factor. To quantitatively estimate the Ez conversionand enhancement efficiency, we calculated the average Ez=E0� �2as theenhancement factor (more details see Supplementary Section 8) andfound that the average Ez=E0� �2 is 216 when the full-width-half-maximum (FWHM) of the wavelength is 3 nm and the FWHM of theincident laser divergence angle is 4° (± 2°), which provides sufficientlylarge enhancement of the out-of-plane component of the near fieldenergy for dark exciton brightening.The PL spectra of the WSe2 monolayer device were excited bylaser with a wavelength of 694 nm (blue, on-BIC) and 647 nm (red, off-BIC) respectively, which are shown in Fig. 2e. We used a 40× largenumerical aperture objective lens (40× Nikon S Flour high NA objec-tive, NA =0.9) to excite the incident laser and collect the PL signal. Astrong XD emission peak was observed (blue spectrum) when theincident lasermatched theBICmode,whereas theXD emissionchannelwas closed when the incident laser was off-resonance (off-BIC, redspectrum). From the PL spectra for XD and bright excitons (X0), weobtained ~52meV of intravalley energy splitting between the XD andX0. This result is in good agreement with other XD emission observa-tions, namely 47meV obtained by in-plane magnetic field2, 42meV bySPP coupling3, and 46meV by TEPL setup4. To eliminate the possibilitythat emission stemmed from bi-excitons, which have similar emissionenergy compared to dark excitons in WSe235, we measured the PLArticle https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 3signal intensity as a function of excitation power. The intensity of thePL emission and the excitation power are related, i.e., IPL / Iαexcited ,where α is the order of the excitation intensity. From this equation, thePL emission can be recognized as bi-excitons when α = 2 and as brightor dark excitonswhenα= 1, respectively35. In a realistic experiment, thefactor α would be expected in the range of 1.2–1.9 for bi-excitons,which have been extensively studied35,36. In our experiment, the loga-rithmic plot of the dark excitons PL peakwith respect to the excitationpower, as shown in Fig. 2f, shows the fitted exponent of α = 0.9, indi-cating the emission stemmed from dark excitons.Directional and coherent emissionof dark excitons by Friedrich-Wintgen BICBecause of their out-of-plane dipole radiation nature, XD emits pri-marily towards the in-plane direction, making the PL signal difficult tobe collected with conventional microscopic objective lenses10. Sup-plementary Fig. 10 shows the collection efficiency of PL emissionradiated by dark excitons and bright excitons, respectively. We alsoprovide the collection efficiency comparison of this work and othermethods (Supplementary Fig. 19). It is clear to see barely any of the XDemission can be collected by an objective lens with a sufficiently largenumerical aperture (NA = 0.9). To gain higher dark exciton read-outefficiency, we demonstrated a directional and polarized emissionchannel with tunable angles for dark excitons using the Friedrich-Wintgen BICs. This directional emission channel can selectively couplewith the out-of-plane dipole moment of dark excitons and enhancetheir directional emission. The enhanced directional emission has asmall full width at half-maximum (FWHM)of ~7° divergence angle in airand a ~1400-fold total enhancement factor with linear polarization. Bytuning the geometric parameters of the PhC slab, the Friedrich-Wintgen BIC can be tuned to provide different deflection angles fordark exciton emission, making dark exciton read-out possible by asingle detector placed in the proper position.In this experiment, the Friedrich-Wintgen BIC serves as a stableresonator with a high Q-factor and provides an efficient radiationchannel for XD directional emission30,37. We designed a suspendedSi3N4 PhC slab (thickness is 233 nm) with a square array of cylind-rical holes (periodicity is 510 nm, hole radius is 102 nm) that sup-ports a TM-like symmetry-protected BIC at the Γ-point of 596 nm fordark exciton brightening (Supplementary Fig. 5) and another TM-like Friedrich-Wintgen BIC at kx = 0.74 (47.85°) and 770 nm for theXD emission (Fig. 3a and b). Figure 3a and Supplementary Fig. 4show the simulated andmeasuredmapping of the reflection spectraas a function of incident angle (optical band structure), respec-tively. It is clear to observe the avoided crossing (with white dashedline labels) in Fig. 3a. Figure 3b shows the zoomed-in optical bandstructure in the Friedrich-Wintgen BIC region, indicating that theFriedrich-Wintgen BIC occurs due to the destructive interferencebetween two TM-like bands (Mode analyses see SupplementaryFig. 7). In the reflection spectra (Fig. 3c), each mode can be descri-bed by an asymmetrical Fano line shape. Vanishing of the Fanolineshape occurred at the lower branch at 47.85° indicating that theBIC emerged with an infinite Q-factor via the destructive inter-ference of the two resonances. We used a numerical simulationFig. 2 | Brightening of dark excitonswith BICs. aThe band structure of a PhC slabthat supports a symmetry-protected on-Γ BIC with a zero-degree incident angle.The blue bands are the transversemagneticmode-like (TM-like) bandswhereas thered bands are the transverse electric mode-like (TE-like) bands. Only four of thebands (solid lines) can be observed under p-polarized incident light. The PhC onlysupports the on-Γ BIC to illustrate the dark exciton brightening. b The simulated(left) and the measured (right) angle-resolved reflection spectra mapping of thePhC slab. It is clear to see the two on-Γ BICs at wavelengths 694 nm and 750nm,respectively. c Electric-field profile Ez=E0 of the on-Γ BICs, plotted on the topsurface of the PhC slab (top) and the y = −r/2 slice (bottom). d The reflectionspectrum with an oblique incident angle of 3° is shown by the black dashed line,while the maximum local electric field amplitude enhancement ratio on the top ofthe PhC slab is plotted in red.WSe2 monolayer was considered. e The PL spectra ofdark excitons and bright excitons. The blue spectrum was taken when the pumplaser matched the on-Γ BICs at the wavelength of 694 nm (on-BIC) whereas the redspectrum was taken when the pump laser was at the wavelength of 647 nm (off-BIC). f A log plot of the power dependence of PL intensity of dark excitons. Theblack line is a fit of the dark exciton emissions exhibiting a linear power depen-dence. The fitted slop α is 0.9 indicating the PL stem fromdark excitons rather thanbi-excitons.Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 4(COMSOL Multiphysics), eigenfrequency model, to extract theQ-factors along each of the two bands as shown in Fig. 3d. TheQ-factor of the upper branch remains relatively low (~103) whereasthe Q-factor of the lower branch has an infinite peak ataround 47.85°.In addition, the Friedrich-Wintgen BICs are momentum tunable,and thus we can control the XD emission towards different deflectionangles. We designed a series of the Friedrich-Wintgen BICs at thewavelength of 770 nm in Fig. 3e. Thismomentum tunability of the darkexciton’s directional emission channel, i.e., the Friedrich-Wintgen BIC,can also benefit dark exciton information read-out using a singledetector. The Friedrich-Wintgen BICs stem from the full destructiveinterference of two TM bands and thus are stable and easier to designin comparisonwith the off-Γ accidental BIC.We found that the avoidedcrossing of the twoTMbands, exactlywhere the Friedrich-WintgenBICoccurs, can be tuned by varying the PhC slab’s thickness. It is clear tosee, as shown in Fig. 3e, the avoided crossing moved towards a higherangle in the momentum space as the slab thickness increased. Tomatch the dark exciton energy (770 nm in wavelength), we changedthe periodicity and as well as the radius of holes as compensation foreach PhC design. The continuum tunable angle ranges from 31.4° to59.5° as we demonstrated for some discrete cases. The white dashedline traces the avoided crossing to show the tuning trend of theFriedrich-Wintgen BICs. We further plotted the deflection angle byFriedrich-Wintgen BICs as the function of the PhC slab thickness. Therelationship was quasi-linear, implying that we can extend the tunabledeflection angle to a broader range.The directional and polarized emission of dark excitonsthrough the Friedrich-Wintgen BIC has been analyzed in Fig. 4. Thefew-layer h-BN flake and monolayer WSe2 heterostructure weresequentially transferred onto the PhC slab that supported theFriedrich-Wintgen BIC and pumped by a wavelength-tunablefemtosecond-ultrafast-laser with the power of 50 μW withp-polarization. Figure 4a shows the PL emission momentum dis-tribution mapped with kx and ky axis. Four shining emission spots,located at the Γ-X line with kx=k or ky=k = 0.74 and with C4 sym-metry, are clearly visible. The PL emission momentum distributionwas collected by a sensitive silicon camera (Princeton InstrumentsPIXIS 400) that was placed in the k-plane (see Method). It is worthnoting that both X0 and XD emissions were collected by the cameraat the same time. To elucidate the PL emission momentum dis-tribution in further depth, the polarization-resolved images arepresented in Fig. 4b. The four shining spots were radially polarizedand exhibited similar intensities. Three pieces of evidence are pre-sent to explain why the origin of these four shining spots comesfrom the dark excitons. 1) Supplementary Fig. 6 shows the PLemission polarization analysis of the bright or dark excitons, and itshows that only dark excitons’ out-of-plane dipole radiation naturecan provide the emission with C4 symmetry (radial polarization); 2)The four shinning spots have similar emission intensity. If thosespots came from bright excitons with p-polarized (along with the y-direction) in-plane momentum, the spots along the x-directionshould be brighter than those along the y-direction. This argumenthas already considered the depolarization effect of bright excitonat room temperature (more details see Supplementary Section 12);3) Only dark excitons with out-of-plane dipole momentumhave high coupling efficiency with the Friedrich-Wintgen BICthat comes from the interference between two TM-like bands.Fig. 3 | Tunable Friedrich-Wintgen BICs. a The dispersion spectra of the PhC thatsupports BICs. The Friedrich-Wintgen BIC due to the interference of two TM-likebands is highlighted by the white dashed box. b A close look of the Friedrich-Wintgen BIC at a wavelength of 770 nm and oblique incident angle of 47.85° (redline). The white dashed line indicates the origin of the two modes. c Spectra at aseries of different incident angles. Avoided crossing and linewidth vanishing of thelower branch band at 47.85° are observed due to the interference between the twomodes. d Quality factors (Q-factors) of two bands that form the Friedrich-WintgenBIC as a function of the oblique incident angle. The blue circles and the red circlesrepresent the Q-factors of the upper branch and the lower branch of the avoidedcrossing bands, respectively. Q-factors of the lower branch are rapidly increasingwhen the oblique incident angle approaches 47.85° where the Friedrich-WintgenBIC occurred. e Friedrich-Wintgen BIC modes for XD directional emission are tun-able for different deflection angles. The thicker the PhC slab, the higher thedeflection angle of the Friedrich-Wintgen BIC emission channel. f Quasi-linearrelations (blue) between the deflection angle by Friedrich-Wintgen BICs and thePhC slab thickness. The Friedrich-Wintgen BIC can be tuned in the momentumspace from 31.4° to 59.5° and maintain the wavelength close to ~770 nm (red).Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 5(For the E-field analysis of the BIC mode see Supplementary Fig. 7).Furthermore, the polarization for each directional emission spot islinear and the extinction ratio is relatively high (shining spot can befully blocked under cross-polarizer), proving its coherence at roomtemperature38.The angle-resolved PL emission spectra mapping extracted fromthe PL emission momentum distribution with p-polarization (y-direc-tion) by the diffraction grating of the spectrometer is shown in Fig. 4c.The enhanced XD emission located at ky=k = ± 0.74 at 772 nm with asmall divergence angle is clearly visible (~7°, Fig. 4d). We furtherextracted the spectra from the angle-solved PL emission as shown inFig. 4e, the blue line of which shows the XD emission spectrumextracted from ky=k = 0.74 and the red line shows the bright excitonsemission spectrum from ky=k = 0.39. This is clear evidence that thePhC slab can separate the emission from bright and dark excitons andselectively enhance the XD emission at a specified angle with a smalldivergence angle.To further understand the coupling mechanism between the out-of-plane dipole and the Friedrich-Wintgen BIC supported by the PhCslab, a Finite-difference time-domain (FDTD) simulation was per-formed using commercial software (Lumerical FDTD Solutions, ANSYSInc.) to show how the PhC slab selectively couples with the out-of-plane dipole and enhances their emission. A series of out-of-planedipoles with excited wavelengths near 770 nm (on-resonance ofthe Friedrich-Wintgen BIC) were randomly placed on the top surfaceof the PhC slab (30-by-30 periods with scattering boundary condi-tions). Figure 4f shows the simulated out-of-plane dipole emissionmomentum distribution, which perfectly matched the XD emissionpart of the PL emission momentum distribution (Fig. 4a). The broad-ening of the emission features in the azimuthal direction of the XDemission can be attributed to the thermal expansion at roomtemperature.We also quantitatively estimate the enhancement factor (EF) ofthe XD emission by the Friedrich-Wintgen BIC performed by the FDTDsimulation (more details see Method). The enhancement factor equa-tion is EF = ∣ EzE0∣2× γPF ×η, where γPF is the Purcell factor of theFriedrich-Wintgen BIC, and the η is the modal overlap (also known asthemode-matching factor) between the on-ΓBIC and FW-BIC.We usedthe FDTDmethod to calculate the Purcell factor the highest of which is26.7 at a wavelength of 769 nm (more details see SupplementarySection 9). Because both themonolayerWSe2 flake size and the tightlyfocused incident light spot are larger than one periodof the PhC lattice(510 nm), we used the average ∣Ez=E0∣2 enhancement ratio instead ofthe near-field E-field enhancement ratio at any local point to estimatethe EF. The average ∣Ez=E0∣2 enhancement ratio by the on-Γ BIC at596 nm is 116 (more calculation details see Supplementary Section 8)and the modal overlap is 45.73% (more calculation details see Sup-plementary Section 18). The EF is estimated to be as high as ~1400 at awavelength of 769 nm.Coherence demonstration of directional emission from darkexcitonsDue to the long lifetime of the dark exciton and the high Purcell factorof the Friedrich-Wintgen BIC cavity, the directional emission of theFig. 4 | Directionality control of dark excitons. a PL emission momentum dis-tributionmappingwithkx andky axis. Four shining emission spots, locatedat the Γ-X line and kx or ky= 0.74 with C4 symmetry, are clearly visible. b Polarizationanalysis of the PL emission momentum distribution in a. The four shining emissionspots show radial polarization indicating they are from dark excitons. c Angle-resolved PL emission spectra mapping extracted from y-polarized PL emissionmomentum distributionmapping in b. It is clear to see the dark exciton directionalemissions have small divergence angles at wavelengths of around 772 nm andtowards oblique emission angles of around48°.dThemeasured (red solid line) andFDTD simulated (dark green dashed line) PL intensity as a function of the in-planemomentum (ky=k) along the y-direction. The full-width-half-maximum (FWHM) ofthe measured XD emission lobes is 7° indicating the ultra-low divergence angle ofthe directional emission. e Spectra extracted fromoblique angles of 48° and 23° forthe dark exciton emission and the bright exciton emission, respectively. f Thesimulated PL emission momentum distribution by the Lumerical FDTD. Fourshining emission spots show high correspondence to the measured PL emissionpattern in a.Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 6dark excitons inmonolayerWSe2maintains their coherent nature39. Todemonstrate this, we used a cylindrical lens to focus the emissionpattern to observe its interference pattern40,41(more details see Sup-plementary Section 12). Before the coherence demonstration, we useda laser spot of Gaussian shape, reflected by a silicon wafer to observethe bright exciton emission pattern in k-space from monolayer WSe2on a SiO2/Si substrate to demonstrate this method’s reliability. Fig-ure 5a, c, and e respectively show the k-space images (left) and theintensity profiles (right) of the XD directional emission, laser spot, andbright exciton emission, whereas Fig. 5b, d, and f show their corre-sponding cylindrically focused patterns (left) and intensity profiles(right). Because the bright exciton emission would lose its valleycoherence at room temperature and shows nearly random polariza-tion, the intensity profile (labeled by a dashed line) of the k-spacepattern (Fig. 5e, right) and the cylindrically focused pattern (Fig. 5f,right) should be Gaussian-like4. On the other hand, the laser spot hasnaturally good coherence and as a result, its intensity profile of thecylindrically focused pattern (Fig. 5c, right) should have a narrowerFWHMbecauseof the constructive interferenceof the commonphase.Finally, we cylindrically focused the dark exciton directional emissionpattern into one dimension, meaning the left and right emission spotsoverlap. Because they (the left and right spots) have the same polar-ization and a phase difference of π (C4 symmetry of this system) as wediscussed in the manuscript, the destructive interference pattern inthe middle is expected (Fig. 5b).DiscussionWedesigned a suspended Si3N4 PhC slab that simultaneously supportsa TM-like on-Γ symmetry-protected BIC and a tunable off-Γ Friedrich-Wintgen BIC. The supported on-Γ BIC can efficiently convert the in-plane polarized normal incident pump laser into out-of-plane polar-ized near-field energy to gain large efficiency as high as 116 to couplewith the out-of-plane transition dipole moment, thus brightening thespin-forbidden dark excitons inmonolayerWSe2 at room temperature.We also demonstrated that the pump laser could regulate the XDemission by switching frequencies on- or off- the BIC mode. Further,the off-Γ Friedrich-Wintgen BIC can selectively couple with the out-of-plane transition dipole moment of dark excitons and provide anefficient directional and polarized emission channel for the darkexcitons withminimized divergence angle (FWHMof ~7° in the air) and~1400-fold enhancement factor. By tuning the thickness of the PhCslab, the Friedrich-Wintgen BIC can be tuned to emit the dark excitonssignal at various angles from 31.4° to 59.5°. Furthermore, we demon-strated the coherence of the XD directional emission at room tem-perature by the constructive interference of the directional emissionsthat towards opposite directions. The dark exciton brightening anddirectional and coherent emission by a planar device demonstrate arobust dark exciton information read-out system, paving the way foron-chip computing and communications.MethodsSample fabricationA300 nm thick layer of Si3N4wasdeposited onSi substrates using low-pressure chemical vapor deposition (LPCVD). The periodic PhC pat-ternwith a periodicity of 465 nm (or 510 nm) and hole radius of 140 nm(or 102 nm) was written on ZEP-520A photoresist using the TESCANMIRA3 E-beam lithography system. Then the RIE-ICP cyclic dry etchstrategywas applied to transfer the pattern from the photoresist to theSi3N4 layer. After removing the ZEP-520 photoresist residual with hotacetone (90 °C) and an oxygen plasma cleaning procedure, the silicon(<100> oriented) beneath the PhC pattern was undercut using KOHsolution (30wt%) at 120 °C to suspend the PhC slab. Finally, modifiedRIE etching was applied without a mask to reduce the thickness of theslab to 265 nm (or 233 nm).Few-layers h-BN flakes and monolayer WSe2 samples weremechanically exfoliated from bulk crystals onto 295 nm thick SiO2/Siwafers. WSe2 monolayers were identified under an optical microscopeand verified using photoluminescence measurements. Few-layer h-BNflakes and WSe2 monolayer samples were transferred onto suspendedphotonic crystal devices using the capillary-force-assisted clean-stamptransfer method42.Optical measurementThe Fourier-optics-based spectroscopy has three operating modes: animaging mode, an optical band-structure (angle-resolved reflectionspectroscopy) mode, and a spectra analysis mode. For the imagingmode, a broadband emission halogen lamp (QTH10, Thorlabs) with awavelength range of 400–2200nm was used to provide white light. ACCD camera (16MP high-speed USB 3.0 digital camera, AmScope) wasplaced at the image plane after the tube lens to show the imageFig. 5 | Room-temperature coherence of the directional emission. Momentumdistribution and spatial interference enabledby a cylindrical lens for the directionalemission of dark excitons supports an interference pattern b. The destructiveinterference in the middle of b is due to a π-phase shift between the left and rightparts of the light field in a, showing that the directional emission is stronglycoherent at room temperature. Momentum distribution and spatial interferencefor a coherent laser beam. The center of d is brighter than that of c, which is causedby constructive interference. Momentum distribution and spatial interference forincoherent bright exciton emission of monolayer WSe2 on SiO2/Si substrate. Theintensity profile of f is similar to that of e because the bright exciton lots its valleycoherence at room temperature.Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 7(Supplementary Fig. 3). For the optical band-structure (angle-resolvedreflection spectroscopy)mode, the back focal plane of an objective lens(20× Mitutoyo Plan Apo Infinity Corrected Long WD objective, NA =0.42 or 40× Nikon S Flour high NA objective, NA=0.9) was projectedby a Fourier transform system. A pinhole was placed at the image planeafter the tube lens (f = 200mm) to regulate the field of view for the lightreflected from the photonic crystal slab. A slit spectrometer (PrincetonInstruments, spectrometer SP300, and silicon camera PIXIS 400) wasplaced at the K-plane and switched to spectrometer mode for theoptical band-structure or imaging mode for k-space imaging. For thespectra analysis mode, the Fourier lens L4 (f = 100mm) was replacedwith an imaging lens L3 (f= 50mm) to Fourier transform the k-planeinto an image plane. The spectrometer can be used to collect thespectra in the image plane (Supplementary Fig. 3b).Photoluminescence spectroscopy can be achieved by switchingflip mirror to the laser mode. The sample was pumped by awavelength-tunable mode-locked Ti: Sapphire laser (Chameleon UltraII, Coherent) and a cascade of OPO (optical parametric oscillator). Ahigh numerical aperture (NA = 0.9) 40× objective lens (40× Nikon SFlour high NA objective) was used to collect the PL signal at roomtemperature. The PL signal was then sent to the k-plane by the Fouriertransform system and imaged at the spectrometer for either k-planeimaging or the PL emissionmomentum distribution. A short pass filter(FESH0700, Thorlabs) was used to clean the undesired wavelengthbefore the pump laser reached the objective length and a long passfilter (FELH0700, Thorlabs) was used to block the pump power tomake the PL signal stand out.Numerical simulationNumerical simulations of the far field emission patterns and Purcellenhancement factors are carried out using three-dimensional full-wavefinite-difference time-domain methods (Lumerical FDTD Solutions,ANSYS Inc.). The simulation spans inx-andy-directionsencompassing asample area of 30-by-30 periods. Perfectly matched layers (PMLs) areused as boundary conditions in all directions. The XD emission is simu-lated by placing a z-oriented dipole array on the upper surface of thephotonic crystal slab. The radiationphaseof eachdipole is set randomly.The minimum mesh size is set as 2 nm around the emitting region. Tocollect the emission profile, a far field pattern monitor is placed 1μmabove the sample surface. To calculate the Purcell enhancement, acontrol simulation with the same Si3N4 slab without the holes is carriedout. The Purcell enhancement factor is calculated by extracting theemission intensity at 48° for both cases with and without lattices.Numerical simulations of the angle-resolved reflection spectramapping are carried out using the finite element method (COMSOLMultiphysics). We selected one unit cell as the simulation domain anduse the periodic (Bloch) boundary condition in x and y directions andthe periodic type of port to excite the horizontal magnetic field withvarious oblique incident angles for the input light. The S-parameter,S11, was used for the reflection spectra.Data availabilityAll data that support the findings of this study are available in themaintext, figures, and Supplementary Information. They are also availablefrom the corresponding author upon reasonable request.References1. Ye, Z. et al. Probing excitonic dark states in single-layer tungstendisulfide. 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Nano Lett. 17, 6961–6967 (2017).AcknowledgementsS.L. acknowledges the seed grant from the Department of MechanicalEngineering, the financial support from Texas A&M Triads for Transfor-mation (T3), and the start-up funding from Texas A&M University. Z.J.W.acknowledges financial support from the President’s Excellence Fund(X-Grant). M.C.H. acknowledges the Governor’s University ResearchInitiative (#2018-01), Texas A&M University and Texas A&M EngineeringExperiment Station (TEES) departmental start-up, and Mr. Holly Frost.X.Q. acknowledges the support from the U.S. National Science Foun-dation under Award Number DMR-2103842. K.W. and T.T. acknowledgesupport from the Elemental Strategy Initiative conducted by the MEXT,Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (GrantNumbers 19H05790, 20H00354 and 21H05233).Author contributionsX.M. and S.L. conceived the idea and initiated the project. X.M. per-formed the numerical simulation with Y.C., Z.J.W., Y.M., and M.C.H.helped. X.M. fabricated devices and performed optical measurementswith P.C. and K.K. helped. X.M. analyzed the data. K.W. and T.T. grew thebulk boron nitride crystals. X.M. and S.L. wrote themanuscript with P.C.and K.K. helped. All authors discussed the results and manuscript. S.L.supervised the project.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-022-34740-5.Correspondence and requests for materials should be addressed toShoufeng Lan.Peer review information Nature Communications thanks the anon-ymous reviewer(s) for their contribution to the peer review of this work.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons license, and indicate ifchanges were made. 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To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2022Article https://doi.org/10.1038/s41467-022-34740-5Nature Communications |         (2022) 13:6916 9https://doi.org/10.1038/s41467-022-34740-5http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Coherent momentum control of forbidden excitons Results Dark exciton brightening using on-Γ symmetry protected BIC at room temperature Directional and coherent emission of dark excitons by Friedrich-Wintgen BIC Coherence demonstration of directional emission from dark excitons Discussion Methods Sample fabrication Optical measurement Numerical simulation Data availability References Acknowledgements Author contributions Competing interests Additional information