Interlayer hybridization and moiré superlattice minibands for electrons and excitons in heterobilayers of transition-metal dichalcogenides

Geometrical moiré patterns, generic for almost aligned bilayers of two-dimensional crystals with similar lattice structure but slightly different lattice constants, lead to zone folding and miniband formation for electronic states. Here, we show that moiré superlattice (mSL) effects in $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ and $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ heterobilayers that feature alignment of the band edges are enhanced by resonant interlayer hybridization, and anticipate similar features in twisted homobilayers of transition-metal dichalcogenides (TMDs), including examples of narrow minibands close to the actual band edges. Such hybridization determines the optical activity of interlayer excitons in TMD heterostructures, as well as energy shifts in the exciton spectrum. We show that the resonantly hybridized exciton energy should display a sharp modulation as a function of the interlayer twist angle, accompanied by additional spectral features caused by umklapp electron-photon interactions with the mSL. We analyze the appearance of resonantly enhanced mSL features in absorption and emission of light by the interlayer exciton hybridization with both intralayer $A$ and $B$ excitons in $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2, \mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2, \mathrm{MoSe}{}_2/\mathrm{MoS}{}_2, \mathrm{WS}{}_2/\mathrm{MoS}{}_2$, and $\mathrm{WSe}{}_2/\mathrm{MoSe}{}_2$.

Figure 1

Twisted bilayer of TMDs ${\mathrm{MoX}}_2$ and ${\mathrm{WX}}_2^{\prime }$. Top left: band alignment, where almost resonant conduction band states of same spin and valley quantum numbers are identified by the same color. Top right: atomic arrangement and bond orientation for almost parallel (P, $\theta \approx 0^{\circ }$) orientation of the two crystals. Basis Bravais vectors for each layer are indicated as ${\widehat{\mathbf{a}}}_n$ and ${\widehat{\mathbf{a}}}_n^{\prime }$. Transition-metal atoms are shown in gray and blue, and chalcogens in orange and red. Center: band alignment and atomic arrangement for the almost antiparallel (AP, $\theta \approx {60}^{\circ }$) heterobilayer. Bottom: hexagonal Brillouin zones of the two crystals ($\mathrm{BZ}$ and ${\mathrm{BZ}}^{\prime }$), where the band edges appear at the Brillouin zone corners $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$. Interlayer hybridization produces electron Bragg scattering from the moiré superlattice, illustrated in the left sketch. This leads to band folding, and the formation of the superlattice mini Brillouin zone (mBZ) on the right.

Figure 2

Comparison between the low-energy absorption spectra (in arbitrary units) of TMD heterobilayers with resonant ($\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, left) and nonresonant ($\mathrm{MoSe}{}_2/\mathrm{MoS}{}_2$, right) conduction band edges, as function of the interlayer twist angle. In $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, hybridized excitons form close to perfect alignment ($\theta =0^{\circ }$) and antialignment ($\theta ={60}^{\circ }$), leading to the avoided crossings marked with green arrows. The white arrows point to absorption lines enabled by moiré umklapp processes, corresponding to higher exciton momentum states that become visible as they are folded onto zero momentum by the moiré superlattice. For $\mathrm{MoSe}{}_2/\mathrm{MoS}{}_2$, the $\mathrm{MoSe}{}_2 A$ and $B$ excitons lie hundreds of $\mathrm{meV}$ above the lowest momentum bright IX states, such that their hybridization is negligible, making the IX dark in our approximation. For large twist angles, the zero-momentum IX energies are raised toward the intralayer exciton energies, becoming semibright and eventually producing hX states.

Figure 3

Moiré patterns formed by parallel (P) and antiparallel (AP) aligned TMD bilayers. Top- and bottom-layer metal atoms are shown in blue and gray, with corresponding chalcogens shown in red and orange. For each stacking type, the interlayer conduction and valence band alignment and spin ordering near the $\text{MX}{}_2$ layer $\tau =+1$ valley is shown for $\text{M}=\mathrm{Mo}$ and ${\text{M}}^{\prime }=\mathrm{W}$. The moiré unit cell is indicated by the black rhombus, and the encircled areas correspond to regions of the heterobilayer with local registry between the two lattices. For parallel alignment, these local registries correspond to $AA, BA$, and $AB$ stacking, analogous to the case of bilayer graphene. By contrast, for antiparallel alignment we find regions with local $2H, A{A}^{\prime }$, and $B{B}^{\prime }$ stacking, as shown in the insets.

Figure 4

(a) Virtual processes giving rise to the intralayer scattering terms (23). A state in band $\mathrm{c}$ with wave vector $\mathbf{k}$ (black dot) tunnels to the band ${\mathrm{c}}^{\prime }$ state ${\mathbf{k}}^{\prime }=-\mathrm{\Delta }\mathbf{K}+\mathbf{k}+{\mathbf{b}}_1$ (faint red circle), subsequently hopping back to band $\mathrm{c}$, with final wave vector $\mathbf{k}-{\mathbf{b}}_2$. (b) Energy diagram of the virtual processes, for $\theta \approx 0^{\circ }$ or $\theta \approx {60}^{\circ }$. The effective model (24) breaks down at the crossings between bands $\mathrm{c}$ and ${\mathrm{c}}^{\prime }$ (red dashed circle). A small detuning ${\delta }_{\mathrm{c}}$ allows a crossing at or near the band edges, such that the model cannot describe low-energy electrons. Because of larger values of ${\delta }_{\mathrm{v}}$, the model correctly describes low-energy holes. See also Figs. 11–14. (c) Extended mBZ scheme, representing the top valence minibands that couple through (23) at the $\kappa$ point. Same color lines represent degenerate levels at the $\kappa$ point, with the three blue ones being closest to the band edge, and mixing through the term (26).

Figure 5

(a) Perturbative parameters for all four $\tau =1$ monolayer band edges of $\mathrm{MoSe}{}_2/\mathrm{MoS}{}_2$, as functions of twist angle. The harmonic potential (24) is valid below the gray line, representing ${\mathcal{P}}_{\alpha s}^{+}=0.1$. (b), (c) Moiré miniband structure of P-stacked $\mathrm{MoSe}{}_2/\mathrm{MoS}{}_2$ at twist angle $\theta =1^{\circ }$ and AP-stacked $\mathrm{MoSe}{}_2/\mathrm{MoS}{}_2$ at $\theta ={59}^{\circ }$, respectively, obtained from direct diagonalization of both the full hybridization Hamiltonian and the effective harmonic-potential model. For the full Hamiltonian, spin-up (-down) bands are shown with triangles (circles). Spin-up and -down minibands of the harmonic-potential model are shown with dashed and solid cyan lines, respectively. In each case, the band diagrams on the right indicate the spin ordering of the hybridizing bands. We use [41] $t_{\mathrm{c}}=t_{\mathrm{v}}/2=26\mathrm{meV}$; all other parameters are listed in Table 3.

Figure 6

Perturbative parameters for the conduction and valence band edges of twisted bilayer $\mathrm{MoSe}{}_2$, as functions of the twist angle. For all bands, the perturbative approach described in Sec. 3 is valid only for strong misalignment angles ${10}^{\circ }\lesssim \theta \lesssim {50}^{\circ }$, and breaks down near close alignment and antialignment. Left and right panels show schematics of the homobilayer's atomic arrangement and band alignment for P and AP stacking, respectively.

Figure 7

(a) Moiré conduction and valence minibands of P-type twisted bilayer $\mathrm{MoSe}{}_2$, for twist angle $\theta =3.5^{\circ }$. Spin-up (spin-down) bands are shown with triangles (circles), and the symbol color represents the out-of-plane electric dipole moment of the state. (b) Corresponding density of states near the conduction band edge. We use [41] $t_{\mathrm{c}}=t_{\mathrm{v}}/2=26\mathrm{meV}$; all other parameters are listed in Table 3.

Figure 8

(a) Moiré conduction and valence minibands of AP-type twisted bilayer $\mathrm{MoSe}{}_2$, for twist angle $\theta =56.5^{\circ }$. Spin-up (spin-down) bands are shown with triangles (circles), and the symbol color represents the out-of-plane electric dipole moment of the state. (b) Corresponding density of states near the conduction band edge. We use [41] $t_{\mathrm{c}}=t_{\mathrm{v}}/2=26\mathrm{meV}$; all other parameters are listed in Table 3.

Figure 9

Perturbative parameters for the conduction and valence bands of twisted (a) $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and (b) $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ heterostructures. In both cases, the highest valence bands show weak interlayer mixing in close alignment and antialignment, and a perturbative treatment of interlayer hybridization is appropriate. By contrast, the lowest conduction bands mix resonantly in the moiré regime, and hybridization effects must be treated exactly.

Figure 10

Band alignment schematics for P- and AP-stacked twisted $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ heterostructures, and bilayer $\mathrm{MoSe}{}_2$. Respective atomic arrangements are also shown, for reference. (a) The spin ordering of the conduction bands in P-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and AP-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ corresponds to the case of P-type bilayer $\mathrm{MoSe}{}_2$. (b) A similar correspondence is found between AP-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$, P-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, and AP-type bilayer $\mathrm{MoSe}{}_2$.

Figure 11

(a) Moiré conduction and valence minibands of twisted P-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$, with twist angle $\theta =1^{\circ }$. Spin-up (-down) bands are shown with triangles (circles), with symbol color representing the state's out-of-plane electric dipole moment. The cyan curves in the lower panel show the harmonic-potential approximation to the highest valence bands, which hybridize weakly between layers. (b) Corresponding density of states near the conduction band edge.

Figure 12

(a) Moiré conduction and valence minibands of twisted AP-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$, with twist angle $\theta ={59}^{\circ }$. Spin-up (-down) bands are shown with triangles (circles), with symbol color representing the state's out-of-plane electric dipole moment. The cyan curves in the lower panel show the harmonic-potential approximation to the highest valence bands. (b) Corresponding density of states near the conduction band edge.

Figure 13

(a) Moiré conduction and valence minibands of twisted P-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, with twist angle $\theta =1^{\circ }$. Spin-up (-down) bands are shown with triangles (circles), with symbol color representing the state's out-of-plane electric dipole moment. The cyan curves in the lower panel show the harmonic-potential approximation to the highest valence bands. (b) Corresponding density of states near the conduction band edge.

Figure 14

(a) Moiré conduction and valence minibands of twisted AP-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, with twist angle $\theta ={59}^{\circ }$. Spin-up (-down) bands are shown with triangles (circles), with symbol color representing the state's out-of-plane electric polarization. The cyan curves in the lower panel show the harmonic-potential approximation to the highest valence bands. (b) Corresponding density of states near the conduction band edge.

Figure 15

Conduction minibands and DOS at critical bias voltage $V_{\mathrm{B}}$, for (a) P-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and (b) AP-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ heterostructures with different degrees of alignment. In both cases, $V_{\mathrm{B}}$ is weakly twist-angle dependent. Spin-up (-down) bands are shown with triangles (circles). The two branches of the band edge belong to a single spin-polarized band, analogously to the case of P-type TMD homobilayers (Fig. 7).

Figure 16

Conduction minibands and DOS at critical bias voltage $V_{\mathrm{B}}$, for (a) AP-$\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and (b) P-$\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ heterostructures with different degrees of alignment. Spin-up (-down) bands are shown with triangles (circles). The two branches of the band edge belong to different minibands of opposite spin polarization, similarly to the case of AP-type TMD homobilayers (Fig. 8).

Figure 17

Schematics of the different intralayer and interlayer excitons in Eq. (30) with valley quantum number $\tau =1$, for P-stacked $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$. Wavy lines indicate binding of the electron and hole by electrostatic interactions. The $\mathrm{MoSe}{}_2 A$ and $B$ excitons correspond to the states ${\text{X}}_{\downarrow \downarrow }^{++}$ and ${\text{X}}_{\uparrow \uparrow }^{++}$, whereas the $\mathrm{WS}{}_2 A$ and $B$ excitons are ${\text{X}}^{\prime }{}_{\downarrow \downarrow }^{++}$ and ${\text{X}}^{\prime }{}_{\uparrow \uparrow }^{++}$, respectively.

Figure 18

(a), (b) Main ($\tau =1$) bright exciton moiré bands for perfectly aligned and antialigned $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$. The dark exciton bands are also shown in gray. For reference, the atomic arrangements of P and AP stacking are presented in the panel (c) insets. (c), (d) Energy and activation temperature $T_{★}$ of the main optically active ($\mathbf{Q}=0$) bright exciton state, as functions of the twist angle. In all panels, the symbol color represents the state composition, with red (blue) corresponding to pure interlayer (${\mathrm{MX}}_2$-intralayer) excitons. Intermediate colors indicate strong mixing of IX and X species, corresponding to hybridized excitons hX.

Figure 19

Photoluminescence (PL) at room temperature (left) and absorption spectra (center) as functions of twist angle for $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ (top) and $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ (bottom), close to parallel and antiparallel alignment. PL curves for different twist angles are offset for clarity, and the decoupled ${\mathrm{MX}}_2-A$ exciton energy, obtained at $\theta ={30}^{\circ }$, is indicated by the vertical blue line. For $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$, a second PL feature is clearly visible at room temperature, for $\theta \approx 0^{\circ }$ and $|\theta -{60}^{\circ }|\approx 0^{\circ }$. A faint second PL peak appears also in $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, shown magnified (multiplied by a factor of 10) in the figure. These secondary spectral features are thermally activated, and disappear at low temperatures. The full low-energy exciton spectrum is overlaid on top of each absorption map (blue and yellow curves), showing multiple momentum-dark exciton states. Blue curves correspond to the 10 minibands obtained from the first ${\mathrm{MX}}_2-A$ exciton band (1 line), the lowest $\mathrm{IX}$ bands (3 lines), and the first folding of the ${\mathrm{MX}}_2-A$ band into the mBZ (6 lines), shown schematically in the top- and bottom-right panels. In the top-right panel, same-color lines indicate degenerate exciton states at the $\gamma$ point. Multiple hX states are identified by avoided crossings of the exciton minibands close to (anti)alignment, and sketched on the center-right panels. White arrows indicate absorption signatures from photon umklapp proceses associated to one of the latter six exciton bands, which is maximally hybridized with the first ${\mathrm{MX}}_2-A$ exciton, thus becoming bright. This absorption line is direct evidence of the moiré superlattice. All PL and absorption line shapes are assumed of Lorentzian form, with broadening of $5\mathrm{meV}$.

Figure 20

Absorption spectra of parallel (top) and antiparallel (bottom) $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$, for varying out-of-plane electric field. The energies of various intralayer and interlayer exciton states, in the limit of $t_{\mathrm{c}}=t_{\mathrm{v}}=0$, are shown for reference. Line shapes are assumed of Lorentzian form, with broadening of $5\mathrm{meV}$.

Figure 21

Absorption spectra of parallel (top) and antiparallel (bottom) $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$, for varying out-of-plane electric field. The energies of various intralayer and interlayer exciton states, in the limit of $t_{\mathrm{c}}=t_{\mathrm{v}}=0$, are shown for reference. Line shapes are assumed of Lorentzian form, with broadening of $5\mathrm{meV}$.

Figure 22

Absorption spectra of $\mathrm{WS}{}_2/\mathrm{MoS}{}_2$ (top) and $\mathrm{WSe}{}_2/\mathrm{MoSe}{}_2$ (bottom). For reference, blue and red curves show the energies of different intralayer and interlayer excitons, respectively, in the limit of $t_{\mathrm{c}}=t_{\mathrm{v}}=0$. Whereas for P-$\mathrm{WSe}{}_2/\mathrm{MoSe}{}_2$ a faint ${\mathrm{IX}}_{\downarrow \downarrow }^{++}$ line is visible at low energies, labeled IX, the corresponding line is absent in P-$\mathrm{WS}{}_2/\mathrm{MoS}{}_2$ due to its larger detuning with the $\mathrm{MoS}{}_2-A$ exciton. For both material pairs, the intralayer exciton lines display complex fine structures close to perfect alignment and antialignment, coming from moiré exciton minibands that mix strongly with the main $A$ or $B$ exciton.

Figure 23

Reflectivity spectra, typically measured in optics experiments, of $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ and $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ for various twist angles near perfect alignment and antialignment. The absorption peaks predicted in Fig. 19 translate into double features, where the exciton energy can be read as the point of maximum negative derivative between the peak and the trough.

Figure 24

Dependence of the low-energy three-peak structure in the absorption spectrum of perfectly aligned ($\theta =0^{\circ }$) $\mathrm{MoTe}{}_2/\mathrm{MoSe}{}_2$ on (a) the electron interlayer hopping energy $t_{\mathrm{c}}$, and (b) the $\mathrm{MoTe}{}_2$ and (c) $\mathrm{MoSe}{}_2$ conduction band masses. Reference mass values $m_{\mathrm{c}}^{{\mathrm{MoTe}}_2}$ and $m_{\mathrm{c}}^{{\mathrm{MoSe}}_2}$ are given in Table 3.

Figure 25

Dependence of the low-energy three-peak structure in the absorption spectrum of perfectly aligned ($\theta =0^{\circ }$) $\mathrm{MoSe}{}_2/\mathrm{WS}{}_2$ on (a) the electron interlayer hopping energy $t_{\mathrm{c}}$, and (b) the $\mathrm{MoSe}{}_2$ and (c) $\mathrm{WS}{}_2$ conduction band masses. Reference mass values $m_{\mathrm{c}}^{{\mathrm{MoSe}}_2}$ and $m_{\mathrm{c}}^{{\mathrm{WS}}_2}$ are given in Table 3.