Anomalous Light Cones and Valley Optical Selection Rules of Interlayer Excitons in Twisted Heterobilayers

Hongyi Yu, Yong Wang, Qingjun Tong, Xiaodong Xu, and Wang Yao

Phys. Rev. Lett. 115, 187002 – Published 30 October 2015

Abstract

We show that, because of the inevitable twist and lattice mismatch in heterobilayers of transition metal dichalcogenides, interlayer excitons have sixfold degenerate light cones anomalously located at finite velocities on the parabolic energy dispersion. The photon emissions at each light cone are elliptically polarized, with the major axis locked to the direction of exciton velocity, and helicity specified by the valley indices of the electron and the hole. These finite-velocity light cones allow unprecedented possibilities for optically injecting 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.

Received 11 June 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.187002

Authors & Affiliations

Hongyi Yu¹, Yong Wang¹, Qingjun Tong¹, Xiaodong Xu^2,3, and Wang Yao^1,*

¹Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Hong Kong, China
²Department of Physics, University of Washington, Seattle, Washington 98195, USA
³Department of Materials Science and Engineering, University of Washington, Seattle, Washington 98195, USA

^*wangyao@hku.hk

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Vol. 115, Iss. 18 — 30 October 2015

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Images

Figure 1
(a) With a small twist and/or lattice mismatch, the Brillouin zone corners of the $Mo X_{2}$ layer ( $τ^{'} K^{'}$ , the open dots) shift away from those of the $W X_{2}$ layer ( $τ K$ , the solid dots). An electron and a hole can recombine if their momentum sum $k + k^{'}$ equals the displacement vector between a $τ K$ and a $τ^{'} K^{'}$ corner (the green arrows). (b) The interlayer Coulomb interaction (the wavy line) between the electron and the hole conserves their momentum sum [36]. An interlayer exciton formed with kinematic momentum $Q = \pm Q_{0}$ , $\pm {\hat{C}}_{3} Q_{0}$ , or $\pm {\hat{C}}_{3}^{2} Q_{0}$ can directly recombine to emit a photon. (c) These light cones at finite velocities are illustrated by the bright spots on the energy dispersion, while the exciton is optically dark at the energy minimum at $Q = 0$ .
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Figure 2
(a) Injection of pure valley current of excitons by linearly polarized light (the green and pink colors denoting the valleys). See the text. (b) Valley Hall effect in heterobilayers. With the light cones at finite velocities, the spatial pattern of emission polarization also depends on the exciton distribution in $Q$ space. (c) The inverse valley Hall effect.
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Figure 3
(a) A $Mo X_{2} / W X_{2}$ heterobilayer can be specified by the twist angle $θ$ and the interlayer translation $r_{0}$ . (b) The bright spots from large to small denote, respectively, the main, first-Umklapp, and second-Umklapp light cones in the $Q$ space, for ${WSe}_{2} / {MoSe}_{2}$ heterobilayers with $θ$ near 0° or 60°. (c) Solid (open) dots denote $τ κ$ ( $τ^{'} κ^{'}$ ) points of the ${WSe}_{2}$ ( ${MoSe}_{2}$ ) layer in the extended zone scheme for $θ = 5 °$ . The locations of main ( $n$ th Umklapp) light cones are given by the displacements between the nearest solid and open dots on the $C_{0}$ ( $C_{n}$ ) circles. (d) Low energy light cones at $θ = 0 °$ , 5°, 22.8°, 37.2°, 55°, 60°, respectively. Evolutions of the main (second-Umklapp) light cones with $θ$ are shown by the thick (thin) red lines. (e),(f) Lattice mismatched ${MoS}_{2} / {WSe}_{2}$ heterobilayer. All light cones are always at a finite $Q$ . (g) Recombination pathways of interlayer exciton $X_{- +}$ at $θ$ near 60°. Solid (dashed) arrows denote the dipole transition (interlayer hopping). The inset shows the elliptical polarizations of emissions at the main light cones. (h) $X_{+ +}$ at $θ$ near 0°. At the finite $Q$ light cones, the polarization and strength of the optical dipole are independent of $r_{0}$ .
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Figure 4
Dependence of the transition dipole strength and polarization as a function of layer translation $r_{0}$ in lattice matching ${MoSe}_{2} / {WSe}_{2}$ heterobilayers. (a) $X_{+ +}$ at $A A$ -like stacking ( $θ = 0 °$ ). The three panels show, respectively, the dipole strength, the degree of polarization in circular basis, and the ellipticity and major axis (represented by short lines) of elliptical polarization, from ab initio calculations [54]. (b) The same plots for $X_{- +}$ at $A B$ -like stacking ( $θ = 60 °$ ).
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