Anomalous light cones and valley optical selection rules of interlayer excitons in. twisted heterobilayers

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1 Anomalous light cones and valley optical selection rules of interlayer excitons in twisted heterobilayers Hongyi Yu 1, Yong Wang 1, Qingjun Tong 1, Xiaodong Xu 2,3, Wang Yao 1* 1 Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China 2 Department of Physics, University of Washington, Seattle, Washington 98195, USA 3 Department of Material Science and Engineering, University of Washington, Seattle, Washington 98195, USA * Correspondence to: wangyao@hku.hk Abstract: Van der Waals heterostructures built with 2D materials are of wide scientific and technological interests, with potential to combine and extend the appealing properties of differing 2D building blocks. One example under intensive experimental investigation is heterobilayer of transition metal dichalcogenides where the observation of interlayer exciton with electron and hole constituents in different monolayers has implied rich device physics. Here we show that, because of the inevitable twisting and lattice mismatch in heterobilayers, interlayer excitons have six-fold degenerate light cones anomalously located at finite velocities on the parabolic energy dispersion. With the change of twisting angle, the polarization selection rules of these light cones evolve from circularly polarized to elliptically polarized, with helicity specified by the valley indices of the electron and hole. These finite-velocity light cones allow unprecedented possibilities to optically inject valley polarization and valley current, and the observation of both direct and inverse valley Hall effects, by exciting interlayer excitons. Our findings suggest potential excitonic circuits with valley functionalities, and unique opportunities to study exciton dynamics and condensation phenomena in semiconducting 2D heterostructures.

2 Introduction Monolayers of group-vib transition metal dichalcogenides (TMDs) have recently emerged as a new class of direct-gap semiconductors in the two-dimensional (2D) limit 1-5. These hexagonal 2D crystals have exotic properties associated with the valley degeneracy of the band edges, including the valley Hall effect 6,7, the valley magnetic moment 8-11, and the valley optical selection rules 6,12-16, leading to rich possibilities for valley-based device applications. The visible range bandgap further makes these 2D semiconductors ideal platforms for optoelectronics. Due to the strong Coulomb interaction, the optical response is dominated by exciton, the hydrogen-like bound state of an electron-hole pair. The demonstrated electrostatic tunability and optical controllability of valley configurations of excitons in monolayer TMDs have implied new optoelectronic device concepts not possible in other material systems Stacking different TMDs monolayers to form van der Waals heterostructures opens up a new realm to extend their already extraordinary properties 18. MoX 2 /WX 2 (X = Se, S) heterobilayers have been realized 19-25, which feature a type-ii band alignment with the conduction (valence) band edges located in MoX 2 (WX 2 ) layer. Exciton then has the lowest energy in an interlayer configuration (i.e. electron and hole in different layers), from which luminescence is observed 19,23,25. Due to the spatially indirect nature, interlayer excitons in MoSe 2 /WSe 2 heterobilayers have shown long lifetime exceeding nanosecond, repulsive interaction, and electrostatically tunable resonance 19, all of which are highly desirable for the realization of excitonic circuits and condensation An unprecedented aspect of this interlayer exciton system is that the heterobilayers in general have incommensurate structures due to lattice mismatch and twisting in the stacking which, together with the valley physics inherited from the monolayers, bring in radically new properties. Here we discover anomalous light coupling properties of interlayer excitons in twisted MoX 2 /WX 2 heterobilayers. We find these excitons have unique six-fold degenerate light cones, located at finite center-of-mass velocities on the parabolic energy dispersion. In these light cones, interlayer exciton can directly interconvert with photon (without phonon or impurity assistance), which has polarization selection rules that evolve from circularly polarized to elliptically polarized

3 with the change of the twisting angle, with helicity depending on the valley indices of the electron and hole. These light cones allow resonant optical injection of excitonic valley polarization and valley current with versatile controllability, as well as the observation of valley Hall and inverse valley Hall effects of interlayer excitons. Contrary to the known bright exciton systems, here the interlayer exciton lifetime is long at the energy minimum while optical injection and probe are allowed at finite velocities, suggesting unique opportunities to study exciton dynamics and condensation phenomena. For lattice matching heterobilayers of AA or AB like stacking, all light cones merge into a single one, and the quantum interference leads to dependence of the brightness and the polarization selection rules on the translation between the layers. Results Kinematical momentum and light cones. The heterobilayers can be characterized by the twisting angle 0 60 and the in-plane translation between the layers (Fig. 1a). We choose the origin of the in-plane coordinates to be a Mo atom, and the coordinate of its nearest W atom defines (see supplementary information). The optically active interlayer excitons are the ones with electrons (holes) from the ± valleys in MoX 2 (WX 2 ), where layer-hybridization is substantially quenched by the large conduction and valence band offsets between the layers (c.f. Fig. 1b) 3, We first give the description of interlayer valley excitons in twisted heterobilayers at vanishing interlayer hopping, and then discuss their light-coupling effects by introducing residue interlayer hopping as a perturbation. For various commensurate configurations, this perturbative approach is compared with ab initio calculation, which show excellent agreement (supplementary information). We use, ( h, ) to denote the electron (hole) Bloch state at a small momentum ( ) measured from the BZ corner ( ) in MoX 2 (WX 2 ) layer in the absence of interlayer hopping, where, =±1 are the valley indices of the electron and hole in their own BZs (Fig. 1c). The matrix element of the interlayer Coulomb interaction ( ) between electron and hole is h,,, h, =,, ( + ) ( ) ( ), where ( ) denote the reciprocal lattice vectors of MoX 2 (WX 2 ) layer. ( ),,,

4 ( ) h, ( ) h,, which are small factors ~ 0.1 when and are finite (supplementary note I). ( ) ( ), so ( + ) ( ) for finite. Therefore in the above Coulomb matrix element we only need to retain the term with = =0, h,,, h,, ( ). (1) Namely, the interlayer Coulomb interaction conserves +, the sum of the electron and hole momentums defined in the individual layers. When this interlayer Coulomb interaction binds an electron-hole pair, the exciton wavefunction is, = Φ ( ), h,, with the eigenenergy ( ) = ħ +Δ (see supplementary note I). + is the exciton mass, Δ the band-gap, the binding energy, and Φ ( ) describes the electron-hole relative motion. Besides the electron and hole valley indices and, an interlayer exciton state is labeled by, which is a good quantum number even in absence of translational invariance at general twisting angles, a direct consequence of Eq. (1). Because of the nearly parabolic band edges of MoX 2 and WX 2, is also the kinematical momentum associated with the center-of-mass group velocity + = ħ. With finite interlayer hopping, the light coupling of, is mediated by an intralayer valley exciton in MoX 2 layer through hole interlayer hopping, or in WX 2 through electron hopping. Its optical transition matrix element is: 0, =, ( ) +, ( ), (2) where is the electric dipole operator, 0 the vacuum state, and ( ) is the energy of the intralayer exciton in the MoX 2 (WX 2 ) layer. The light cones of interlayer excitons are defined as the -space regions where interconversion with photon can directly happen (without phonon or impurity assistance). This requires the intermediate intralayer exciton states in Eq. (2) to have vanishing momentum:

5 ± = respectively 36,37 : Φ( ) ±, h ±,. The electron and hole interlayer hopping matrix elements are,, =, ( ),, h, h, =, ( ),, where + ( + ). ( ) is the Fourier component of hopping integral in the twocenter approximation between a W and a Mo d-orbitals with magnetic quantum numbers and respectively (supplementary note II). The coupling between the inter- and intra-layer exciton in Eq. (2) is then,, = ( ), ( ), = ( ),,,, ( ), (3) where ( ) = Φ ( )Φ + and ( ) = Φ ( )Φ. The Kronecker delta in Eq. (3) dictates the light cones are at: =. (4) In Fig. 2a and 2d, the extended Dirac points and of the WX 2 and MoX 2 layers are shown in the extended zone scheme for a MoSe 2 /WSe 2 heterobilayer at =5 and a MoS 2 /WSe 2 heterobilayer at =0. When changes, ( ) move on concentric circles denoted as ( ). The nearest (, ) pairs on and correspond to main light cones. And the nearest (, ) pairs on and define the nth Umklapp light cones, where light coupling is assisted by Umklapp type process in the interlayer hopping (Fig. 2c). The brightness is determined by ( ), which decays fast with the increase of (see supplementary note III). So only the main, the 1 st and the 2 nd Umklapp light cones need to be considered, and others are too dark to be relevant. Fig. 2b and 2e shows the locations of these light cones as a function of twisting angle, in a

6 range 0.1 ( ) that corresponds to a kinetic energy ħ 60 mev. The six main light cones are located at = ( cos ) sin for near 0 ( = cos sin for near 60 ), and its 3 rotations. 1 st -Umklapp light cones also have six-fold degeneracy and are located at 2. The main and 1 st -Umklapp light cones are the lowest energy ones at near 0 or 60. There are twelve 2 nd -Umklapp light cones (Fig. 2c), and six of them become the lowest energy ones for lattice matching heterobilayers when is near the commensurate angle 21.8 or 38.2 (Fig. 2b). For 0.1, ( ) ~ ( ) ~ 1, the optical transition dipole moment (Eq. (2)) of the interlayer exciton is then ~ ( )/ ( ) ( ), where is that of the intralayer exciton. ( ) is a few hundred mev determined from optical and scanning tunneling measurements 19,23,25,30. With ( ) extracted from ab initio calculations (supplementary note III), we estimate that the transition dipole moment is ~ 0.05 at the main light cones, ~ at 1 st Umklapp, and ~ at 2 nd Umklapp light cones. Valley optical selection rules. The dipole matrix element of the intralayer exciton in each monolayer has the optical selection rule associated with its valley index: 0 ± = ( ± )/ 2, coupling to the ± circularly polarized photon only 6. Eq. (2) then becomes, 0, =, ( ) +, ( ), (5) which leads to the polarization selection rule of interlayer exciton, determined by the electron and hole valley indices and. Fig. 3a schematically illustrates interlayer excitons, with valley configuration =. These are the ones in the main and 1 st Umklapp light cones at near 0 and in 2 nd Umklapp light cones at near They have circularly polarized selection rule: the three, couple to + polarized photon only, and their time-reversal, couple to photon (Fig. 3a inset). Fig. 3b illustrates interlayer excitons, with =, which are the ones in the main and 1 st Umklapp

7 light cones at near 60 and in 2 nd Umklapp light cones at near As the two recombination pathways via the electron and hole interlayer hopping (c.f. Eq. (5)) have opposite polarization, their interference leads to elliptically polarized selection rules, with the ellipticity and major axis determined by the magnitude and the phase of ( ) which can be determined through ( ) comparison with ab initio calculations at commensurate stacking (see supplementary information). For MoSe 2 /WSe 2 heterobilayers, we find the ellipticity ~ 0.5 and the major axis as shown in Fig. 3b. Because of the 3-fold rotational symmetry of ( ), the three, always have their elliptical polarizations related by the -rotation (Fig. 3b inset), and, are their time-reversals. Their dipole strength and polarization have no dependence on the layer translation which appears in a phase factor only (c.f. Eq. (3)). Translation dependence at commensurate stacking. For MoSe 2 /WSe 2 or MoS 2 /WS 2 heterobilayers, neglecting the ~ 0.1% lattice mismatch, they have commensurate stacking at =0,60, 21.8 or At =0,60, all main and Umklapp light cones merge into a single cone at =0. At = 21.8, 38.2, the six 2 nd Umklapp light cones merge into a single one at =0 (c.f. Fig. 2b). For the light cone at =0, the optical transition dipole is the superposition of contributions from multiple (, ) pairs each of which is associated with a distinct phase factor (c.f. Eq. (3)), and their interference then gives rise to the dependence of the strength and polarization of the transition dipole moment. In reality, the interlayer separation also varies with which can affect the dipole strength 38,39, but not the selection rule. For =0, the light cone at =0 has two-fold valley degeneracy:, and,, which couple to + and photons, respectively. The transition dipole strength + +, and Fig. 3c shows the dependence. At =0 (i.e. AA stacking), the transition dipole is strongest, ~ For =60, the light cone at =0 also has two-fold valley degeneracy:, and,, with more interesting behavior. The two recombination pathways (c.f. Eq. (5)) have different

8 dependence:, ( ) + +,, ( ) + +. (6) Thus both the strength and the elliptical polarization of the transition dipole moment change with. At =( + )/3 (i.e. AB stacking), the transition moment is strongest, ~ 0.1. Note that elliptically polarized emission by, indicates the absence of the 3-fold rotational symmetry in the heterobilayer. The rotational symmetry is restored at =0 and ( + )/3, where the, emission becomes fully + and polarized respectively (c.f. Fig. 3d). At such stacking, either electron or hole interlayer hopping vanishes at the ± valleys 3, so only one of the recombination pathways is contributing (c.f. Eq. (3) and Fig. 3b). At =2( + )/3, electron and hole interlayer hopping both vanish at the ± valleys, so, becomes dark. Optical injection of exciton valley current. At the finite-velocity main light cones, the overall optical oscillator strength of interlayer excitons is about 2% of the intralayer one in a monolayer, which allows reasonably efficient resonant injection of interlayer exciton. The polarization selection rules further make possible valley selective injection and probe. Injections of exciton valley current are made possible by the elliptical polarization selection rules in heterobilayers with near 60. Under excitation by an elliptically or linearly polarized light, the absorption rates are all different at the six degenerate main light cones associated with different group velocities, giving rise to valley current. Using to denote the angle between the major axis of polarization of the light cone at = and that of a linearly polarized pumping light, a pure valley current is injected with rate ( ) ħ ( ) in the direction sin 2 +cos2, being the population injection rate. Optical valley Hall effect. Interlayer excitons in heterobilayers also provide an ideal platform to study the Berry phase effect in the Bloch band. TMD monolayers have valley dependent Berry curvatures in their conduction and valence band edges, which give rise to the valley Hall effect of the carriers 6,7. The electron and hole Berry curvatures Ω and Ω from the MoX 2 and WX 2

9 respectively are both ~ O(10) A, but can have a difference of ~ 50% 3. Inheriting the Berry curvature of its electron and hole constituents 39, the curvature of interlayer exciton is: Ω + Ω. All four valley configurations have finite and contrasted Berry curvatures. The resultant skewed motion is an exciton valley Hall effect, observable from the spatial pattern of emission polarization (Fig. 4c-d) 40,41. For heterobilayers ~ 60, the elliptically polarized emission on the two edges with opposite helicity is a phenomenon unique to this heterobilayer system. The optical injection of excitonic valley current through the finite-velocity light cones further makes possible the observation of inverse valley Hall effect, where the valley current induces a number current in perpendicular direction due to the skewed motion by the Berry curvature (Fig. 4e). Discussion For thermalized interlayer excitons, the population is peaked at the bottom of the energy dispersion, outside the light cones. Photon emission of these excitons is through a higher order process consisting of the phonon or impurity assisted scattering into the light cones, followed by radiative recombination. The inefficiency of such process can explain the long lifetime (exceeding nanosecond) observed in the time resolved photoluminescence (PL) 19. The energy of the emitted photon is ( ) plus (minus) the phonon energy if phonon absorption (emission) is involved. The distribution of exciton kinetic energy and the phonon energy can explain the large inhomogeneous broadening of PL. Our results also explain why strong interlayer exciton PL is observed only at twisting angle near 0 or 60 19, where the main light cones get sufficiently close to the bottom of energy dispersion. There are other twisting angles ( near 21.8 or 38.2 ) where the 2 nd Umklapp light cones are close to =0, however, the weak dipole of these light cones renders the PL inefficient. A different type of interlayer excitons (, or, ) can also form with the electron from the MoX 2 Λ valleys or hole from the WX 2 Γ valley, which have significant composition of chalcogen p-orbitals with larger interlayer hopping (Fig. 1b). Consequently, (, ) will hybridize with the dark intralayer one / ( / ) in the WX 2 (MoX 2 ) layer. However, the light-interaction of, (, ) is determined by its coupling to the bright ± valley exciton

10 ( ) from the WX 2 (MoX 2 ) layer, which is through the weak Umklapp type interlayer hopping. The analysis here can be straightforwardly applied to determine the light coupling properties of these interlayer excitons (supplementary note V). As the light cones of, and, are always the Umklapp type, their light coupling is expected to be weak. Acknowledgments: This work is mainly supported by the Croucher Foundation (Croucher Innovation Award), the Research Grant Council (HKU705513P) and University Grant Committee (AoE/P-04/08) of HKSAR, and University of Hong Kong (OYRA). X.X. is supported by NSF (DMR and EFRI ), and Cottrell Scholar Award. References: 1 Wang, Q. H., Kalantar-Zadeh, K., Kis, A., Coleman, J. N. & Strano, M. S. Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nature Nanotech. 7, , doi: /nnano (2012). 2 Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal dichalcogenides. Nature Phys. 10, , doi: /nphys2942 (2014). 3 Liu, G.-B., Xiao, D., Yao, Y., Xu, X. & Yao, W. Electronic structures and theoretical modelling of twodimensional group-vib transition metal dichalcogenides. Chem. Soc. Rev., doi: /c4cs00301b (2015). 4 Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically Thin MoS2: A New Direct-Gap Semiconductor. Phys. Rev. Lett. 105, , doi: /physrevlett (2010). 5 Splendiani, A. et al. Emerging Photoluminescence in Monolayer MoS2. Nano Lett. 10, , doi: /nl903868w (2010). 6 Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled Spin and Valley Physics in Monolayers of MoS 2 and Other Group-VI Dichalcogenides. Phys. Rev. Lett. 108, , doi: /physrevlett (2012). 7 Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, , doi: /science (2014). 8 Aivazian, G. et al. Magnetic Control of Valley Pseudospin in Monolayer WSe2. Nature Phys. 11, , doi: /nphys3201 (2015). 9 Srivastava, A. et al. Valley Zeeman Effect in Elementary Optical Excitations of a Monolayer WSe2. Nature Phys. 11, , doi: /nphys3203 (2015). 10 Li, Y. et al. Valley Splitting and Polarization by the Zeeman Effect in Monolayer MoSe2. Phys. Rev. Lett. 113, , doi: /physrevlett (2014). 11 MacNeill, D. et al. Breaking of valley degeneracy by magnetic field in monolayer MoSe2. Phys. Rev. Lett. 114, , doi: /physrevlett (2015). 12 Yao, W., Xiao, D. & Niu, Q. Valley-dependent optoelectronics from inversion symmetry breaking. Phys. Rev. B 77, , doi: /physrevb (2008). 13 Mak, K. F., He, K., Shan, J. & Heinz, T. F. Control of valley polarization in monolayer MoS2 by optical helicity.

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13 Fig. 1 Twisted heterobilayers and interlayer valley excitons. (a) A MoX 2 /WX 2 heterobilayer can be specified by the twisting angle and the interlayer translation. Red (blue) balls denote Mo (W) atoms. (b) ab initio band structure of lattice matching MoSe 2 /WSe 2 heterobilayer with = 0 and = 0 (AA stacking). The insets show that the conduction (valence) band edge wave functions at ± are predominantly localized in the MoSe 2 (WSe 2 ) layer. (c) Relevant electron and hole valleys for the interlayer exciton, shown in the first Brillouin zones of uncoupled MoX 2 and WX 2 layer at a finite twisting. (d) The interlayer electron-hole Coulomb interaction conserves the sum +, where ( ) is the electron (hole) momentum measured from the nearest ± (± ) point of MoX 2 (WX 2 ) layer.

14 Fig. 2 Anomalous light cones at finite velocities. (a) Solid (open) dots denote + ( + ) points of WSe 2 (MoSe 2 ) layer in the extended zone scheme, for a heterobilayer MoSe 2 /WSe 2 with = 5. The displacements between the nearest solid and open dots on the C 0 circle corresponds to the locations of main light cones in the -space, and those on circles C n corresponds to the nth Umklapp light cones (see text). (b) The low energy light cones (bright spots) in the -space (dark planes) at = 0, 5, 22.8, 37.2, 55, 60, respectively. The evolution of the locations of main (2 nd Umklapp) light cones as a function of are shown by the thick (thin) red lines, which cross at commensurate angles = 0, 60 ( = 21.8, 38.2 ). (c) Locations of the main (largest spots), 1 st Umklapp, and 2 nd Umklapp (smallest spots) light cones at near 0 or 60. (d),(e) Lattice mismatch MoS 2 /WSe 2 heterobilayer. All light cones are always at finite. Only the main light cones are in the range of Q-space shown in (e).

15 Fig. 3 Valley optical selection rules. (a), (b) Radiative recombination of interlayer excitons has two quantum pathways, each consisting of an interlayer hopping (dashed arrow) plus an intralayer recombination (solid arrow). (a) shows valley configuration at twisting angle near 0 ; and (b) shows at near 60. The insets show the emission polarization at the six main light cones. (c) The optical transition dipole strength of as a function of, in the lattice matching heterobilayers at = 0. (d) The optical transition dipole of in lattice matching heterobilayers at = 60. The three panels show respectively the dipole strength, the degree of polarization in circular basis, and the ellipcity and major axis (representd by short lines) of elliptical polarization.

16 Fig. 4 Optical injections and valley Hall effects. (a) Resonant injection of valley polarized interlayer excitons at the finite-velocity main light cones. The excitons become optically dark after relaxing to the energy minimum. (b) At near 60, linearly polarized light inject excitons with different rates at the six main light cones. The arrows illustrate the injected exciton currents at the corresponding light cones, with thickness denoting the magnitude. The net effect is a pure valley current. (c) Valley Hall effect of ±± in heterobilayers with near 0. The mechanical force on the interlayer exciton can be realized by biasing one layer only. (d) Valley Hall effect of ± in AB stacked heterobilayers. In heterobilayers with the elliptical polarized light cones at finite velocities, the spatial pattern of emission polarization depends on the exciton distribution in -space. (e) Exciton valley current injected by a linearly polarized light can induce number current through the inverse valley Hall effect, observable from the spatial map of the luminescence.