Introduction

In bilayer 2D semiconductors, the moiré pattern introduced by twist and/or lattice mismatch can endow excitons with a plethora of intriguing opportunities, from applications in optoelectronics to the exploration of quantum matters of fundamental interest1,2,3,4,5,6,7,8,9. In heterobilayers, the lowest energy excitons have an interlayer configuration carrying a permanent electric dipole, which underlies electric field tunability of exciton energy and pronounced exciton–exciton interaction10. In the moiré-defined periodic landscape11, these interlayer excitons are tightly confined in an ordered array of moiré traps, exhibiting behavior akin to quantum-dot-like single-photon emitters3,5,12,13. Such quantum emitters uniquely feature optical selection rules conditioned on the spin and valley indices as well as local atomic registries3,14,15, implying rich optoelectronic control possibilities. While the layer separation of electron and hole components has led to a weak coupling to light, the resultant long radiative lifetime is on the other hand favorable for the exploration of exciton many-body physics in Bose-Hubbard lattices with the strong dipolar interaction16,17,18.

When light-coupling is favored, the interlayer moiré excitons can be brightened through hybridization with a nearly resonant intralayer exciton. In heterobilayers, this is made possible under various compound choices that have nearly aligned conduction or valence bands8,19,20,21 (Fig. 1a). In moiré traps of different local stacking registries, the C3 rotational symmetry dictates different center-of-mass envelope forms of the intralayer component hybridized to a tightly trapped interlayer exciton wavepacket (Fig. 1c), whereas an s-wave envelope leads to brightening. The intralayer wavefunction overlap between electron and hole also underlies pronounced electron–hole Coulomb exchange that can non-locally transfer exciton either in momentum space (i.e. between valleys22,23,24), or in real space25,26,27,28. This is essentially the Förster non-radiative dipole-dipole coupling underlying the fluorescence resonance energy transfer29.

Fig. 1: Exciton hybridization in TMD moiré structures.
figure 1

a Schematics of hybrid excitons in TMD homobilayers (upper) and heterobilayers (lower). Black (white) dots denote the electron (hole). T denotes interlayer hopping. b Spatially varying interlayer hopping in R-stacking bilayers. Hopping vanishes at the high symmetry points B (MX) and C (XM), and has a p-wave form in their vicinity. The insets below show their local stacking. The phase of T is color-coded to show the counterclockwise and clockwise winding patterns around the B and C points, respectively. In the vicinity of A point of MM stacking, the hopping has an s-wave form. c Schematics of the center-of-mass envelope functions of the hybrid exciton at high symmetry points. The minor intralayer component exhibits an s-wave and a p-wave envelope at A point and B/C points, respectively, whereas the major interlayer component always has an s-wave envelope for the lowest energy state in the confinement. Black (white) dots represent the electron (hole), indicating their intralayer/interlayer components.

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Hybrid moiré excitons are also hosted by homobilayers of near 0° twisting, such as twisted MoTe2 (t-MoTe2) that has caught great attention for the fractional quantum anomalous Hall effects30,31,32,33. Because of the stacking-dependent interfacial electrical polarization34,35, interlayer excitons are trapped at the MX and XM stacking regions with opposite layer configuration (electric dipole), where trapping energy can largely compensate the binding energy difference from an intralayer exciton, leading to their hybridization36. The C3 rotational symmetry, however, dictates that the hybridized intralayer component has a p-wave envelope that leads to vanishing optical transition dipole, therefore protected from radiative recombination. Such homobilayers offer a platform to explore dipolar ordering of long-lived hybrid moiré excitons, where spontaneous C3 symmetry breaking manifests a unique optical signature36.

Here, we show that the moiré exciton is qualitatively altered by its intralayer component, in which sizable Förster-type multipole–multipole coupling enables the hybrid exciton to hop coherently and non-locally between moiré traps across layers and/or over 10 nm apart laterally, where kinetic propagation is completely suppressed. Valley-flip hopping channels are found as significant as the valley-conserving ones, and we identify their non-monotonic dependence on hopping distance, with phase pattern dependent on valley indices and envelope forms of the intralayer component. This leads to rich possibilities to tailor valley–orbit-coupling (VOC) forms, which can introduce non-trivial topology to the moiré exciton superlattice. We demonstrate this possibility for the long-lived excitons in twisted MoTe2 bilayers, where Förster-type quadrupole–quadrupole couplings link the two otherwise isolated exciton sublattices of opposite layer configurations and give rise to a rich topological phase diagram.

Results

Layer hybridization of moiré-trapped excitons

In a long-wavelength moiré pattern, the C3 rotational symmetry dictates that the potential energy minima and maxima of the exciton must correspond to the three high symmetry points in a supercell where local stacking registries are C3 invariant. We focus on moiré excitons trapped at these points with a major interlayer component of an s-wave envelope, and a minor intralayer component (throughout this paper, ‘envelope function’ refers specifically to the exciton’s center-of-mass wave function). Their hybridization is determined by the interlayer tunneling matrix element T between conduction or valence band edges. At the  ±K valleys, the large crystal momentum leads to a stacking selection rule that dictates T must vanish at two out of the three high symmetry stacking points37. In the first harmonic approximation38, T as a function of space coordinate R takes the form of \({\sum }_{j}\exp (i{C}_{3}^{j}\delta {{{\bf{K}}}}\cdot {{{\bf{R}}}})\)37,39, δK being the valley mismatch between layers. The example for the R-type bilayer is shown in Fig. 1b. In the vicinity of B (MX stacking) and C (XM) points, interlayer tunneling takes the p-wave forms T ∝ δRx + iδRy and δRx − iδRy, respectively, where δR denotes the displacement from the high symmetry points. In the vicinity of the A (MM) point, T is approximately a constant. Lattice reconstructions can quantitatively change the function T(R), but not these symmetry-dictated leading order forms near the three high symmetry points. These determine the envelope function of the minor intralayer component. For instance, hybrid moiré exciton has an intralayer component in p-wave envelope at B and C points in R-type bilayer (Fig. 1c).

Table 1 summarizes the symmetry-dictated envelope forms of intralayer component in moiré hybrid excitons at the high symmetry points in R- (near 0°) and H-type (near 60°) bilayers, which can also be deduced from optical selection rules (see Supplementary Note 1). In a hybrid moiré exciton, the allowed intralayer component must have an optical dipole with the same polarization as the interlayer component. However, the latter has a valley optical selection rule that also depends on the stacking registries (c.f. Table 1)3. At high symmetry points where the intra- and inter-layer valley selection rules differ, a p-type envelope is necessitated to reshape the optical dipole polarization of the intralayer component. Such hybrid excitons have a tiny optical dipole and long radiative lifetime (see Supplementary Note 2), just like their interlayer component. Nevertheless, having an intralayer component can activate Förster coupling, which enables a coherent non-local hopping.

Table 1 Symmetry-dictated properties of hybrid moiré excitons trapped at high symmetry points in R- and H-stacking bilayers
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Förster type multipole-multipole coupling of intralayer exciton wavepackets

Electron-hole Coulomb exchange can annihilate an intralayer exciton at valley τ of layer n and create another at \({\tau }^{{\prime} }\) valley of layer \({n}^{{\prime} }\) (Fig. 2a). In the basis of exciton momentum eigenstates \(\left\vert \tau ,n,{{{\bf{q}}}}\right\rangle\), such a Förster coupling takes the form: \({\hat{J}}_{{\tau }^{{\prime} },\tau }^{{n}^{{\prime} },n}({{{\bf{q}}}})={(-1)}^{\frac{{\tau }^{{\prime} }-\tau }{2}}{e}^{-i({\tau }^{{\prime} }{\theta }_{{n}^{{\prime} }}-\tau {\theta }_{n})}{e}^{-i({\tau }^{{\prime} }-\tau ){\varphi }_{q}}J{e}^{-qz}\frac{q}{K}\)25,28, where \(\tau ,{\tau }^{{\prime} }=\pm 1\), q is the exciton center-of-mass momentum with φq being the azimuthal angle, z is the distance between layer n and \({n}^{{\prime} }\), and θn denotes the twisting angle of layer n. Now we consider this coupling between localized exciton wavepacket states of C3 rotational symmetry (Fig. 2b), which are further classified by the azimuthal quantum number m: \(\vert {\psi }_{\tau ,n,m}({{{{\bf{r}}}}}_{c})\rangle ={\sum }_{{{{\bf{q}}}}}{e}^{-i{{{\bf{q}}}}\cdot {{{{\bf{r}}}}}_{c}}{f}_{m}(q){e}^{im{\varphi }_{q}}\left\vert \tau ,n,{{{\bf{q}}}}\right\rangle\), m = 0 or  ± 1, corresponding to s- or p-wave envelope. rc denotes the wavepacket center, and fm(q) is a real function accounting for the radial dependence. Förster coupling between a pair of exciton wavepacket states of in-plane displacement \({{{\bf{r}}}}=(r,{\varphi }_{r})={{{{\bf{r}}}}}_{c}^{{\prime} }-{{{{\bf{r}}}}}_{c}\) is of the form (see Supplementary Note 3 for derivation details):

$${J}_{\tau ,n,m}^{{\tau }^{{\prime} },{n}^{{\prime} },{m}^{{\prime} }}({{{\bf{r}}}}) = \left\langle {\psi }_{{\tau }^{{\prime} },{n}^{{\prime} },{m}^{{\prime} }}({{{{\bf{r}}}}}_{c}^{{\prime} })| \hat{J}| {\psi }_{\tau ,n,m}({{{{\bf{r}}}}}_{c})\right\rangle \\ ={e}^{-i({m}^{{\prime} }-m)\frac{\pi }{2}}{e}^{-i({\tau }^{{\prime} }{\theta }_{{n}^{{\prime} }}-\tau {\theta }_{n}) }{e}^{-i({\tau }^{{\prime} }-\tau +{m}^{{\prime} }-m){\varphi }_{r}}{{\mathfrak{J}}}_{\tau ,m}^{{\tau }^{{\prime} },{m}^{{\prime} }}(r,z,w)$$
(1)

The angular part is in the form of a phase factor, where \(({\tau }^{{\prime} }-\tau +{m}^{{\prime} }-m)\) are integers between  ±4. The radial part \({{\mathfrak{J}}}_{\tau ,m}^{{\tau }^{{\prime} },{m}^{{\prime} }}(r)\) is a real function, with dependence on the layer distance z and wavepacket width w. For calculations hereafter, we adopt the wavepacket envelopes to be eigenfunctions of 2D harmonic traps26: \({f}_{m}(q)\propto {w}^{| m| +1}{q}^{| m| }{e}^{-{w}^{2}{q}^{2}/2}\). Figure 2c plots the radial dependence of several representative intra- and inter-valley Förster coupling channels. Notably, compared to the intravalley ones, the intervalley channels peak at a larger hopping distance r ~5–10 nm, where the overlap between the initial and final state wavepackets is already negligible. We also identify scaling behaviors (see Supplementary Note 3): (1) \(w{{\mathfrak{J}}}_{\tau ,m}^{{\tau }^{{\prime} },{m}^{{\prime} }}\) is a function of the dimensionless distances r/w and z/w only; (2) for r > 10w, \({{\mathfrak{J}}}_{\tau ,m}^{{\tau }^{{\prime} },{m}^{{\prime} }}\) is asymptotic to \({r}^{-(3+| m| +| {m}^{{\prime} }| )}\), suggesting that p-envelope excitons are coupled via a quadrupole moment40. For hybrid excitons, their Förster coupling can be obtained by multiplying a factor \(\eta \equiv {\psi }_{i}{\psi }_{f}^{* }\), where ψi (ψf) is the normalized probability amplitude of intralayer component in the initial (final) state.

Fig. 2: Förster coupling of moiré exciton wavepackets.
figure 2

a Schematic of Coulomb exchange that annihilates an electron–hole pair in valley τ at layer n and creates one in valley \({\tau }^{{\prime} }\) of layer \({n}^{{\prime} }\). b The process in a effectively realizes non-local hopping—Förster coupling—of moiré exciton wavepackets, which is further determined by the envelope form (azimuthal quantum number m, wavepacket width w) of the intralayer component. c Förster coupling strength of various channels \({{\mathfrak{J}}}_{\tau ,m}^{{\tau }^{{\prime} },{m}^{{\prime} }}\), as function of in-plane distance (r) evaluated at out-of-plane distance z = 0. The subscripts (superscripts) denote the valley index τ (\({\tau }^{{\prime} }\)) and azimuthal index m (\({m}^{{\prime} }\)) of the initial (final) states. The inset shows the \({{\mathfrak{J}}}_{1,1}^{-1,-1}\) channel as function of the dimensionless in-plane and out-of-plane distances r/w, z/w. Wavepacket width w = 2 nm is used in calculation.

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Förster valley-orbit coupling in t-MoTe2

In the following, we focus on R-stacking t-MoTe2, where moiré excitons are trapped at the B (MX) and C (XM) points with opposite layer configurations (Fig. 3a), such that kinetic propagation has no nearest-neighbor terms, leaving the two sublattices uncoupled. Pronounced hybrdization of a p-wave intralayer component is expected36, where our analysis finds an azimuthal index locked to the valley m = τ (see Table 1). Through such intralayer component, Förster coupling not only links the two sublattices, but also introduces a pronounced VOC. The relevant intra- and inter-valley channels are \({{\mathfrak{J}}}_{1,1}^{1,1}\) and \({{\mathfrak{J}}}_{1,1}^{-1,-1}\), the latter having a phase factor dependent on hopping direction e4iφ (Fig. 3b, c.f. Eq. (1)). The Förster VOC in the honeycomb superlattice can be written as (details in Supplementary Note 5):

$${H}_{{i}^{{\prime} }i}^{voc}={b}_{{i}^{{\prime} }}^{{{\dagger}} }\left[{\alpha }_{{i}^{{\prime} }i}-{\beta }_{{i}^{{\prime} }i}\left({\tau }_{x}\cos 4{\varphi }_{{i}^{{\prime} }i}+{\tau }_{y}\sin 4{\varphi }_{{i}^{{\prime} }i}\right)\right]{b}_{i},$$
(2)

where \({b}_{i}\equiv {({b}_{i,K},{b}_{i,-K})}^{T}\) is the exciton annihilation operator at site i, τx,y,z are the Pauli matrices for the valley pseudospin, and \({\varphi }_{{i}^{{\prime} }i}\) denotes the hopping angle from site i to \({i}^{{\prime} }\). \({\alpha }_{{i}^{{\prime} }i}=\eta {{\mathfrak{J}}}_{1,1}^{1,1}\) and \({\beta }_{{i}^{{\prime} }i}=\eta {{\mathfrak{J}}}_{1,1}^{-1,-1}\), and \({\mathfrak{J}}\) are evaluated at the corresponding hopping distance, whose non-monotonic dependence can be used to tailor the relative strengths of α and β through tuning the twisting angle. η is the weight of the intralayer component in the alike initial and final states, and it depends on several key energy scales (c.f. ref. 36): (i) the amplitude of interlayer carrier hopping; (ii) the depth of moiré trapping potential; (iii) the binding energy difference between inter- and intra-layer excitons. The last quantity can be tunable through the dielectric environment, which underlies the possibility to tailor η value. In regime of small η, the effect of Förster coupling can be neglected as in earlier discussion of moire excitons in twisted MoTe236. Hereafter, we consider a regime of relatively large η where Förster coupling can qualitatively change the properties of hybrid exciton lattice.

Fig. 3: Hybrid moiré exciton lattice in the twisted MoTe2 homobilayer.
figure 3

a Schematics of hybrid moiré excitons trapped at the B and C sites in t-MoTe2. Black (white) dots denote the electron (hole), and they are distributed in top (orange) and bottom (green) layer. The symbols of inter- and intra-layer configurations correspond to wavepackets of the form \(\vert {\psi }_{\tau ,m}^{{{{\rm{inter}}}}/{{{\rm{intra}}}}}({{{{\bf{r}}}}}_{c})\rangle ={\sum }_{{{{\bf{q}}}}}{e}^{-i{{{\bf{q}}}}\cdot {{{{\bf{r}}}}}_{c}}f(q){e}^{im{\varphi }_{q}}\vert {X}_{\tau ,{{{\bf{q}}}}}^{{{{\rm{inter}}}}/{{{\rm{intra}}}}}\rangle\), rc denoting wavepacket center, and this gauge choice leads to the rc-dependent phase factor in the linear superposition (see Supplementary Note 2). b Nearest-neighbor intravalley and intervalley Förster coupling between moiré exciton states at B and C site, which have p+ (p−) intralayer envelope at K(−K) valley. c Kinetic propagation amplitude as a function of moiré period. The inset shows the kinetic propagation as a next-nearest-neighbor complex hopping, τ = ±1 is the valley index.

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Topological lattice of hybrid moiré excitons

With the p-wave intralayer component, moiré hybrid excitons can have a long radiative lifetime, favorable for the exploration of correlation and topological matters of these composite bosons41. Here we show how nontrivial topology can emerge from the Förster VOC in the exciton superlattice, which can be tunable by twisting angle, as well as modest interlayer bias. The latter introduces, through the opposite electric dipoles on the two sublattices, a staggered onsite energy εi = M and −M, respectively on B and C sites. The exciton tight-binding Hamiltonian is then,

$$H= {\sum}_{i}{\varepsilon }_{i}{b}_{i}^{{{\dagger}} }{b}_{i}-t{\sum}_{\left\langle \left\langle {i}^{{\prime} }i\right\rangle \right\rangle }{b}_{{i}^{{\prime} }}^{{{\dagger}} }\exp \left(i\frac{2\pi }{3}{\nu }_{{i}^{{\prime} }i}{\tau }_{z}\right){b}_{i}\\ +{\sum}_{\left\langle {i}^{{\prime} }i\right\rangle }{b}_{{i}^{{\prime} }}^{{{\dagger}} }\left[\alpha -\beta ({\tau }_{x}\cos 4{\varphi }_{{i}^{{\prime} }i}+{\tau }_{y}\sin 4{\varphi }_{{i}^{{\prime} }i})\right]{b}_{i}$$
(3)

The second term is the valley-conserving kinetic propagation in the moiré potential landscape, which is absent between sublattices, and we keep its the leading order effect within each sublattice, i.e., a NNN hopping. Its strength t is an exponential function of moiré period (Fig. 3c)3. The momentum space mismatch between the valleys of two layers leads to a constant hopping phase, where \({\nu }_{{i}^{{\prime} }i}=-{\nu }_{i{i}^{{\prime} }}=\pm 1\) following the convention shown in Fig. 3c3 (Supplementary Note 2). The last term is the Förster coupling which we retain only the NN term where \({\varphi }_{{i}^{{\prime} }i}=0,\pm \frac{2\pi }{3}\). The \(4{\varphi }_{{i}^{{\prime} }i}\) dependence is then equivalent to \({\varphi }_{{i}^{{\prime} }i}\), such that intervalley Förster coupling behaves like a Dresselhaus spin–orbit coupling.

At M = 0 and β = 0, this Hamiltonian has two decoupled valley subspaces described by Haldane model of opposite Chern numbers respectively42, so the system is in a quantum valley Hall phase (QVH), with a topological gap at the mini-Brillouin zone corners  ±KM. It is known that the staggered potential M tends to open a trivial gap at  ±KM42. To elucidate the general situation at finite M and β, we track the evolution of the eigenstates at KM point under these two control parameters (Fig. 4c). Specifically, the first group of states, Ψ1 and Ψ2 with eigenenergies E1,2 = −6t ∓ M, are polarized in a single sublattice with valley-sublattice locking. The other two states, Ψ3 and Ψ4 with energy \({E}_{3,4}=3t\mp \sqrt{9{\beta }^{2}+{M}^{2}}\), are spread over both sublattices. The crossover between the two groups defines phase-space boundaries (c.f. Fig. 4b, c):

$$\frac{M}{t}=\pm \left[\frac{9}{2}-\frac{1}{2}{\left(\frac{\beta }{t}\right)}^{2}\right].$$
(4)

Inside the area enclosed by the two parabolas, Ψ3 and Ψ4 have higher energy than the valley-sublattice polarized states Ψ1 and Ψ2, and we find the system in QVH phase with gapless edge states (Fig. 4d, see Supplementary Note 6 for details). Upon the inversion of Ψ3 with Ψ1 or Ψ2 across the boundary defined by Eq. (4), the edge states become gapped.

Fig. 4: Dispersion and phase diagram of hybrid moiré exciton lattice.
figure 4

a An exemplary dispersion of hybrid moiré exciton in t-MoTe2, color coded with the sublattice polarization \(\left\langle {\sigma }_{z}\right\rangle\). t = 0.1 meV, α = 1 meV, β = M = 0.1 meV. b Left: layer and valley configurations of three states at the mini-Brillouin zone corner KM enclosed by the dashed box in a. Red (blue) denote K (−K) valley excitons. Right: inversions between the three states, upon the increase of intervalley Förster coupling β. c Topological phase diagram as function of β and interlayer bias M. t = 0.1 meV and α = 1 meV. The two parabolas mark the points of inversion of Ψ3 with Ψ1 and Ψ2, respectively (c.f. Eq (4)). The two consecutive band dispersions in b are along the path marked by blue dots. The yellow region is a quantum valley Hall (QVH) phase. d Evolution of the topological edge states, along the path marked by red dots in (c). Color stands for the sublattice polarization \(\left\langle {\sigma }_{z}\right\rangle\). e Phase diagram as a function of twisting angle and interlayer bias M, for hybrid moiré excitons with a 25% intralayer component.

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Lastly, we can map this topological phase diagram to the parameter space spanned by the physical control parameters of twisting angle θ and interlayer bias M. θ determines the moiré period and therefore the NNN kinetic propagation amplitude t (Fig. 3c), as well as the intra- and inter-valley Förster coupling strength α and β between the NN (Fig. 2c). The details of their twisting angle dependence are given in Supplementary Note 7). The amplitudes α and β are also controlled by the hybridization weight η and envelope width w of the hybrid moiré exciton. Taking η = 0.25 and w = 2 nm leads to the topological phase diagram shown in Fig. 4e, where the QVH states emerge at twist angle θ ≥ 2.6°. We note several simplifications that may quantitatively affect the phase diagram. First of all, the hybridization weight η can be manipulated by the interlayer bias and the twist angle, and therefore implies its dependence on M and θ. This dependence may quantitatively affect how the phase diagram in the parameter space of M/t and β/t (Fig. 4c) is mapped to the phase diagram in the M and θ parameter space (Fig. 4e), due to the dependence of Förster coupling strength β on M and θ. Secondly, our estimation of the Förster coupling is based on wavepackets trapped at high symmetry MX and XM sites, which can serve as a good basis since Förster coupling is a non-local process not relying on wavefunction overlap. In general, the spread out of the wavefunction shall also depend on the twist angle and moiré potential that is affected by the interlayer bias M, which we did not take into account. On the other hand, as long as the dependence is not drastic, the qualitative feature of the phase diagram shall remain.