Introduction

Monolayer group VIB transition metal dichalcogenides (TMDs) with broken inversion symmetry and strong spin-orbit coupling (SOC) show intriguing coupled spin and valley phenomena14. For example, TMDs on EuO (111) surface were predicted to show a large valley splitting over 300 meV8. However, the experimentally observed valley splittings of WSe2 and WS2 on top of bulk EuS are only 1615 and 2.5 meV T−1,16, respectively, probably due to substrate surface reconstruction and orientation-dependent exchange splitting17. Such issue could be easily avoided in van der Waals heterostructures with high-quality and atomically sharp interfaces. When two-dimensional (2D) ferromagnetic semiconductor CrI3 and CrBr3 were adopted as substrates, splittings about 4 meV for WSe211,18 and 2.9 meV for MoSe219 have been realized, which is very close to the calculated values ranging from 1.4 to 6.4 meV6,7,20.

Although several prototypical magnetic proximity systems have been studied, achieving a large valley splitting in 2D heterostructures is still in demand, which requires a general understanding on the strength and influencing factors of magnetic proximity effect, as well as its consequences on excitonic properties. To achieve this goal, model systems with commensurate lattices between group VIB TMDs and substrates are preferred, because of great challenges in theoretical simulations of coupled many-body systems. The recently explored 1H-VSe2 is an ideal candidate for the purpose. As a family member of group VB TMDs, it shares a similar lattice constant with those of group VIB TMDs21,22. Importantly, while it is paramagnetic in the bulk form, thin-film 1H-VSe2 exhibits a ferromagnetic ground state, which remains stable above room temperature21,22, rendering it a compelling platform for various high-temperature spintronic applications. For comparison, previous reported insulating magnetic substrates have a rather low Curie temperature, TC. For example, the TC’s of bulk EuO and EuS are 69K23,24 and 16.6 K25, while those of monolayer CrI3 and CrBr3 are around 45K26 and 34K27, respectively. Further, the band edges of VSe2 mainly contributed by V 3d orbitals are close in energy to band edges of MSe2 consisting of Mo/W 3d orbitals, suggesting the hop** and hybridization between these d orbitals with specific spin and thus the valley splitting could be tuned through external control knobs. Besides, VSe2 may be employed to achieve gate-controlled band topology manipulation through magnetic proximity-induced staggered exchange field28.

In this work, we present first-principles calculations on the magnetic proximity effect in MSe2 (M = Mo or W)/VSe2 heterostructures, demonstrating its manipulation by varying exchange interaction strength, strain, and electric field. The energy alignment and hybridization between the valleys from MSe2 and VSe2 can be well tuned, leading to a large valley splitting of 37.8 meV in the topmost valence band of monolayer MoSe2 and 55.8 meV in the second topmost valence band of monolayer WSe2, significantly larger than previous results. Moreover, large valley splittings of 203 meV (202 meV) for the intralayer A (B) exciton in MoSe2 and 156 meV for B exciton in WSe2 are realized when the excitonic effect is taken into consideration.

Results

Stacking registry and band alignment

It is noted monolayer VSe2 prefers in-plane ferromagnetic ordering21. Nevertheless, our calculations show the magnetic anisotropy energy (~0.6 meV) is small, and out-of-plane magnetization–magnetic field hysteresis loop at 300 K for monolayer VSe2 has been observed experimentally21. Thus, applying an external out-of-plane magnetic field can polarize the VSe2 magnetization into out-of-plane direction to induce valley splitting in MSe2. The fully relaxed lattice constants for MSe2 and VSe2 are 3.32 and 3.33 Å, respectively, which indicates the lattice mismatch between these two monolayers is negligibly small. Thus, 1 × 1 primitive cells of monolayer MSe2 and VSe2 are stacked to form vdW MSe2/VSe2 heterostructures, with the in-plane lattice parameter fixed to 3.32 Å. Six high-symmetry stacking configurations are considered, which can be divided into two types, i.e., the R-type stacking with two layers sharing the same orientation and the H-type with two layers having opposite orientations, as shown in Fig. 1a. To retain the C3 rotational symmetry, the metal (M) site of MSe2 layer can be vertically aligned with the V, Se, and hollow (h) sites of VSe2 layer. The corresponding structures are named as \(R(H)_V^M\), \(R(H)_{\rm{Se}}^M\), and \(R(H)_h^M\), respectively. The equilibrium interlayer spacing d0’s between two metal atoms in neighboring layers of the six stable stackings are listed in Supplementary Table 1. Clearly, the interlayer spacing d0’s of \(R_V^M\) and \(H_h^M\) are larger than those of the other four stakings with slight difference. The characteristic features of \(R_V^M\) and \(H_h^M\) are one Se atom in MSe2 is directly on top of another Se atom in the VSe2, resulting in the strong Coulomb repulsion between two layers.

Fig. 1: Stacking structures and band alignment of MSe2/VSe2 heterostructures.
figure 1

a Schematic top- and side-view of three R-type and three H-type heterobilayer registries. The Se atoms from MSe2 and VSe2 layers are shown in different colors. b Band edges relative to the vacuum energy level (set to zero) within different approximations for MoSe2, WSe2, and VSe2.

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Different stacking registries between the two constitutes have important influences on the band offset in these heterostructures. Figure 1b shows the band alignments using PBE, HSE, GW@PBE, and GW@HSE, which reveal strong dependence on the applied approximations. Specifically, both the MoSe2/VSe2 and WSe2/VSe2 show a type-III band alignment using the PBE method, even with the on-site Hubbard correction included for V d electrons. It is known that the alignment of valence band maximum (VBM) and conduction band minimum (CBM) with respect to the vacuum level are not accurate in the PBE level29. More reasonable electronic structure can be achieved using hybrid functionals, in which a certain portion of the exact Hartree exchange is mixed with local or semilocal exchange30. By varying the mixing coefficient α and range-separation Coulomb potential parameter μ, one can construct different HSE functionals incorporating different levels of exchange and correlation, denoted as HSE(α, μ). Our test results show that mixing coefficient α plays a more important role in determining the electronic structure (Supplementary Figs. 10–11). Previously, a simple method was proposed to obtain this mixing parameter by the inverse of effective static dielectric constant α = 1/ε∞31, which give an accurate description on the gap and structural properties. For VSe2, the effective static dielectric constant \(\varepsilon _\infty ^{{\rm{VSe}}_2}\) is calculated as 10.3, leading to an optimal α ≈ 0.1. Meanwhile, HSE functional with α = 0.1 could give good description of VSe2 bands, as suggested previously32. Hereafter, all HSE results are obtained using HSE (α = 0.1, μ = 0.2 Å−1) unless noted otherwise. Under HSE level, both the MoSe2/VSe2 and WSe2/VSe2 show a type-II band alignment. Further, many-body perturbation theory in GW approximation is a more practical routine to obtain band edges to compare with experimental data33,34,35. To take into account the influences of exchange-correlation interaction on the electronic structure of magnetic VSe2, two GW schemes, starting from either PBE or HSE functional results were chosen. The two GW schemes represent two different levels of exchange-correlation strength, which could be tuned by external fields, for example, strain36,37. The quasi-particle band gaps obtained from GW@HSE of MoSe2 and WSe2 monolayers are 2.40 and 2.31 eV, respectively, which are in good agreement with experimentally measured values38,39. In particular, for MoSe2/VSe2, GW@PBE yields a type-I band alignment, whereas GW@HSE gives type-II band alignment, suggesting significant influences of exchange and correlation. These results indicate the positions of band edges from the two constitutes of the heterostructures could feasibly be tuned, potentially leading to controllable hop** and hybridization strength.

Band structure and valley band splitting

The influences of interlayer orbital hybridization on the valley splitting are schematically presented in Fig. 2. With SOC, both the top valence band and bottom conduction band of MSe2 split into two subbands, which are oppositely spin polarized but energetically degenerate at K and –K valleys, denoted by blue dashed lines in Fig. 2. The two valence and conduction bands for MSe2 split by SOC are labeled as V1, V2, and C1, C2, while the nearest valence and conduction bands for VSe2 are denoted as V1′ and C1′. Assuming the ferromagnetic VSe2 is spin-up polarized, there are two scenarios for orbital hybridization depending on the alignment between spin-polarized bands. When the V1 band of MSe2 is closer to the V1′ band of VSe2, the effective interlayer hop** only happens between the MSe2 V1 band and the VSe2 V1′ band at K, because of the requirement of spin conservation. The hybridization results in two mixed bands with orbital contributions from both MSe2 and VSe2, as depicted by solid lines in Fig. 2a, leading to a considerable valley band splitting. Similarly, if the V2 band of MSe2 matches well with V1′ band of VSe2, a significant d-d orbital hybridization appears between the MSe2 V2 band and the VSe2 V1′ band only at –K, again resulting in notable valley splitting. Thus, the stacking-dependent interlayer coupling, the band alignment, and the spin conservation collaboratively determine the hybridization of electronic states, leading to varied valley splitting and rich excitonic structures.

Fig. 2: Schematic diagrams of two scenarios of interlayer hybridization.
figure 2

The energy levels for MSe2 and VSe2 before hybridization are shown by blue and red dashed lines, while hybridization between valence bands from MSe2 and VSe2 with aligned spin, lead to the hybridized bands indicated by colored solid lines. Two different scenarios are highlighted by the shadowed boxes. The corresponding band labels are shown to the right, with Vn/Cn (Vn′/Cn′) denoting valence/conduction bands for MSe2 (VSe2). ΔEV1 and ΔEV2 are the valley bands splitting of the topmost valence band (V1) and second topmost valence band (V2) for MSe2. The short arrows alongside the bands represent spin direction. The resulting circularly polarized valley-split A and B excitonic transitions are also shown.

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Figure 3a, b shows the calculated band structures for \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2 heterostructures using PBE and WSe2/VSe2 heterostructures under HSE level, respectively, while the results for the rest five stackings of MSe2/VSe2 are shown in the Supplementary Figs. 2, 3 and 6, 7. Considering the band edges for MoSe2 and WSe2 at K and –K valleys are predominantly composed of transition metal \(d_{z^2}\), dxy and \(d_{x^2 - y^2}\) orbitals3b, which is in accordance with Fig. 2b. In particular, ΔEV2 for \(H_{\rm{Se}}^W\) stacked WSe2/VSe2 reaches 42.3 meV. Surprisingly, when GW approximation is applied on top of HSE, the splitting values does not increase overall (Supplementary Figs. 8 and 9). This may result from the combined effect of exchange interaction and the quasi-particle correction, which needs further exploration in the future.

Interlayer hybridization tuned by exchange interaction

As discussed above, the strongest splitting appears in \(H_{\rm{Se}}^{\rm{Mo}}\) stacked MoSe2/VSe2 within PBE method and \(H_{\rm{Se}}^W\) stacked WSe2/VSe2 under HSE level (α = 0.1, μ = 0.2 Å−1), with strong interlayer orbital hybridization. Such hybridization depends critically on the alignment of spin-polarized energy levels from two constituents of heterostructures, which are greatly influenced by exchange interaction. To unravel such effects, one can tune the hybridization strength adopting the hybrid functionals methods with varying parameters (α, μ), taking \(H_{\rm{Se}}^M\) stacked heterostructures as examples in the following.

Figure 4 summarizes valley splitting values for \(H_{\rm{Se}}^M\) at different α values. Considering the lowest-energy bright A exciton of WSe2 (MoSe2) originates from transition between V1 and C2 (C1) bands, the splittings for WSe2 C2 and MoSe2 C1 are shown here. α coefficient dramatically changes the band alignment between the two layers (for more details, see Supplementary Figs. 10 and 11 as well as Tables 6 and 7). With increasing α, the CBMs at K and –K move down, and become even lower in energy than original CBM at M when α = 0.15, whereas the VBMs at K and –K rise up, and become closer to VBM at Γ. The strongest hybridization emerges between MoSe2 (WSe2) V2 band and VSe2 V1′ band at –K when α is around 0.06 (0.09). Meanwhile, the valley splittings of V2 band reach the maxima of 30.9 meV for MoSe2 and of 55.8 meV for WSe2, respectively. What’s more, the strong hybridization can even give rise to the sign change of valley polarization, as shown in Fig. 4 for ΔEV2, which leads to critical change in the transport and optical behaviors. Regarding μ parameter, the stronger the range-separated Coulomb potential (i.e., the smaller μ parameter) is, the larger the splitting is. Nevertheless, HSE calculations using different μ parameters give similar band structures (Supplementary Fig. 12), indicating that μ parameter does not affect much the interlayer hybridization.

Fig. 4: Valley splittings of \(H_{\rm{Se}}^M\) stacked MSe2/VSe2 under varied exchange strength.
figure 4

The valley splittings ΔEV1, ΔEV2, and ΔEC1(C2) for the topmost valence band, the second topmost valence band, and lowest conduction band (second lowest conduction band) of \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2 (a) and WSe2/VSe2 (b) using HSE with different α coefficients and fixed μ = 0.2 Å−1.

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The hybrid functionals involve a portion of exact exchange, and thus can also take the Hubbard U correction into account42. To shed light on this point, the representative orbital projected band structures for MoSe2/VSe2 at varied Ueff values are plotted in Supplementary Fig. 13 and the extracted valley band splittings are summarized in Supplementary Table 8. For Ueff between 1.4 and 1.5 eV, valence bands show strong hybridization at –K similar to the HSE cases with α around 0.06 (Supplementary Fig. 10). Because of its sensitivity to exchange and correlation effects, the actual ground-state band structures of the heterobilayers could depend on experimental conditions.

Interlayer hybridization tuned by strain and electric field

It should be pointed out that the tunability of the valley splitting through exchange interaction can be achieved experimentally. For example, one can feasibly apply an external pressure or in-plane strain onto the layered heterostructures to regulate the interlayer orbital hybridization. Applying strain/pressure can modify not only the interlayer distance or lattice constants but also the bandwidth, leading to the effective modulation of the electronic exchange and correlation interaction36,37,43,44. To verify this, Supplementary Fig. 14 shows the band structures of \(H_{\rm{Se}}^{\rm{Mo}}\) stacked MoSe2/VSe2 with a biaxial strain applied, which clearly display the change of band alignment and band hybridization. To demonstrate the relationship between HSE parameter and strain, Fig. 5a shows the variation of valley splitting as a function of biaxial strain. The notable hybridization appears when the strain is in the range of −2% to −3%, and the largest V2 band splitting increases to 46.5 meV. The compressive strain tends to enhance the kinetic energy of electrons and make the electrons more delocalized, which effectively suppresses electronic exchange. Similarly, the largest valley splitting is achieved using small exchange parameter with α = 0.06.

Fig. 5: Valley splittings of \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2 under external field.
figure 5

The valley splittings ΔEV1, ΔEV2, and ΔEC1 as a function of (a) in-plane biaxial strain and (b) electric field applied perpendicular to the heterostructure. All the calculations were carried out using HSE (α = 0.1, μ = 0.2 Å−1) approximation.

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Furthermore, in such heterostructures, their properties can be easily tuned by gate voltage45,46,47. Figure 5b shows electric field induced valley splitting in \(H_{\rm{Se}}^{\rm{Mo}}\) stacked MoSe2/VSe2. It can be seen that the valley splitting strongly depends on the electric field strength and direction. Under positive electric field (from VSe2 to MoSe2), VSe2 V1′ band moves toward the top valence bands of MoSe2, resulting in the orbital hybridization with V2 and V1 sequentially, as shown in Fig. S15. Accordingly, giant band splitting of 31.7 and 37.8 meV for V2 and V1 bands are realized at 0.12 and 0.42 V Å–1, respectively. Interestingly, valley splitting experiences sign change when strong hybridization occurs, as indicated by the gray area in Fig. 5b. Meanwhile for conduction bands, the bands extrema of the two layers almost coincide with each other at about 0.4–0.5 V Å–1, leading to band hybridization as indicated by the green box in Fig. S15, and thus the conduction band valley splitting is also slightly enhanced.

Valley exciton splitting

The magnetic proximity effects in these heterostructures can be probed by their optical response, especially the valley intralayer exciton splitting of TMDs. In the absence of external magnetic field, the K and –K valley excitons are degenerate in monolayer MSe2, and the spectra (including peak shape and peak position) for left and right circularly polarized light show no difference. In the hybridized heterostructures, the effective hybridization at one valley renders the interband optical transition from either of the two hybridized valence bands to conduction band possible, as indicated by black and grey arrows in Fig. 2. While the transition at the other valley without hybridization is only optically allowed for bands contributed by one constitute layer. Accordingly, strong splitting of the exciton states from K and −K valleys emerges. The valley exciton splitting in TMDs is defined as the energy difference between low-energy bright A or B excitons from K and −K valleys, i.e., \(\Delta E^{A/B} = E_K^{A/B} - E_{ - K}^{A/B}\). Figure 6 shows the optical absorbance of right and left circularly polarized light (σ+ and σ−) for the representative \(H_{\rm{Se}}^M\) stacked heterostructures, while the rest are shown in Supplementary Figs. 16 and 17. The vertical solid blue and red lines denote the low-energy bright excitons from K and –K valleys, respectively. While the first two excitons are mainly contributed by VSe2 monolayer, the four higher-energy excitons mostly originate from MSe2 monolayers. Note that A and B exciton absorption peaks from MSe2 monolayers already overlap with the continuum excitation of VSe2 monolayer rather than well-separated ones. All heterostructures exhibit clear exciton valley splitting in MSe2 monolayers except for the \(H_h^M\)stacked bilayers, which is consistent with its negligible electronic band valley splitting. The exciton valley splitting values are summarized in Supplementary Table 11. In addition, in the case of an in-plane magnetization of monolayer VSe2, the valley degeneracy in MSe2 is not lifted, and the dark excitons in MoSe2 gain some oscillator strength (Supplementary Fig. 18), as suggested in ref. 10. This is another interesting phenomenon deserving further investigation.

Fig. 6: The absorption spectra and excitonic oscillating strength of \(H_{\rm{Se}}^M\) stacked MSe2/VSe2.
figure 6

Optical absorbance spectra excited by left-handed σ+ and right-handed σ− light for \(H_{\rm{Se}}^{\rm{Mo}}\) stacked MoSe2/VSe2 heterostructures within GW-BSE@PBE (a) and GW-BSE@HSE (α = 0.06, μ = 0.2 Å−1) (b), for \(H_{\rm{Se}}^W\) stacked WSe2/VSe2 within GW-BSE@PBE (c) and GW-BSE@HSE (α = 0.09, μ = 0.2 Å−1) (d). The vertical grey vertical line indicates the oscillating strength of each exciton for in-plane polarized light absorption, while the blue and red vertical lines highlight the low-energy bright excitons from K and −K valleys.

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It has been discussed in previous section that \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2 heterostructures shows a strong hybridization between topmost valence bands at K valley using PBE method, leading to a significant valley V1 band splitting. After taking into account the excitonic effects, it exhibits giant A exciton valley splitting reaching 203 meV, while B exciton shows slight splitting (see Fig. 6a). In addition, the exciton splitting in VSe2 monolayer is also enhanced by two fold48. Under HSE (α = 0.06, μ = 0.2 Å−1) approximation, a large B exciton splitting of 202 meV and a tiny A exciton splitting (Fig. 6b) appear, because of the effective hybridization between MoSe2 V2 band and VSe2 V1′ band at –K. The hybridization pushes up the MoSe2 V2 band, and at the same time moves down VSe2 V1′ band at –K, as depicted in Fig. 2b. Accordingly, the MoSe2 B exciton at –K lies significantly below that at K valley, while the lowest-energy bright intralayer exciton in VSe2 at –K exceed the exciton at K in energy, leading to reversed exciton splitting in VSe2 (compare Fig. 6a and Fig. 6b). For \(H_{\rm{Se}}^W\) stacked WSe2/VSe2, A and B intralayer excitons in WSe2 show almost comparable splitting (Fig. 6c) due to the absence of interlayer hybridization under PBE level. On the other hand, GW-BSE on top of HSE (α = 0.09, μ = 0.2 Å−1), within which strong hybridization appears, gives rise to a large B exciton splitting of 156 meV similar to that of \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2.

To show the importance of the interlayer hybridization on excitonic transition, Supplementary Tables 10–13 provide the decomposition of the exciton oscillator strength with respect to single particle transition for \(H_{\rm{Se}}^M\) stacked heterobilayers. Under GW-BSE@PBE level, the lowest-energy bright exciton in MoSe2/VSe2 under σ− polarized light excitation denoted as \(A_K^{{\rm{VSe}}_2}\) exciton displays a considerable interlayer contribution from band V1 to C1′, besides the main intralayer transition from band V1′ to C1′. Meanwhile, for \(A_K^{{\rm{MoSe}}_2}\) exciton, the interlayer contribution from band V1′ to C1 is comparable to the intralayer component from band V1 to C1 transition. Accordingly, \(A_K^{{\rm{VSe}}_2}\) and \(A_K^{{\rm{MoSe}}_2}\) excitons show hybridization between the corresponding intra- and interlayer exciton transitions because of the interlayer-hybridized hole. In contrast, \(A_{ - K}^{{\rm{MoSe}}_2}\) exciton shows a negligible mixing with VSe2, due to the absence of effective band interlayer hybridization at –K. Eventually, there exists a large valley exciton splitting between \(A_K^{{\rm{MoSe}}_2}\) and \(A_{ - K}^{{\rm{MoSe}}_2}\). Under HSE (α = 0.06, μ = 0.2 Å−1) level, both \(A_{ - K}^{{\rm{VSe}}_2}\) and \(B_{ - K}^{{\rm{MoSe}}_2}\) exciton exhibits obvious mixed contributions from two monolayers, which significantly differs from \(A_K^{{\rm{VSe}}_2}\) and \(B_K^{{{\rm{MoSe}}}_2}\) excitons, leading to the significant B valley excitons splitting hitting 202 meV shown in Fig. 6b. \(H_{\rm{Se}}^W\) stacked WSe2/VSe2 shows the similar trend within GW-BSE@HSE.

The above discussion shows A or B excitons exhibit clear intralayer characteristics without hybridization, and after strong hybridization they can be viewed as hybridized excitons with holes distributing over two layers and electrons located in one layer, a special type of interlayer exciton. Although these hybridized excitons are higher in energy than ground-state dark excitons in VSe2 layer and some pure interlayer excitons (see Supplementary Fig. 19), they can emerge as new peaks below or above the A and B excitons in absorption and PL spectra, due to their strong oscillator strength inherited from the intralayer exciton component49. Similar behaviors were observed in WSe2/CrI3 heterostructure, where the probed high-energy circularly polarized PL peak (at ~1.69 eV)11,41 is significantly higher in energy than the lowest-energy PL peak of CrI3 (~1.1 eV, ref. 50). Furthermore, as the previous section suggests that one can selectively tune the hybridization of V1 or V2 band, and thus giant A or B exciton splitting under external electric fields. Therefore, the large exciton valley splitting can be observed in the MoSe2 and WSe2 monolayer via optical methods, which allows for the control and detection of magnetization using electrical or optical excitation.

Discussion

In conclusion, our study on the valley splitting in MSe2 on top of ferromagnetic monolayer VSe2 with a strong and tunable magnetic proximity effect using advanced many-body methods is interesting in the following three aspects. First, although magnetic proximity effects in 2D materials have been studied by various theoretical computations, they are mostly limited to a single-particle level, neglecting the dominating role of excitons. Such single-particle picture cannot give an accurate description of the excitonic behaviors, which are probed by optical response in experiments. Accordingly, the GW/BSE results obtained here provide a deeper understanding and some guidance for realizing large magnetic proximity effects for excitons in 2D heterostructures. Second, the proposed VSe2/MSe2 offers a more practical platform to realize the strong proximity and their applications, compared to previously studied systems. For example, early work suggests a valley splitting as large as 300 meV can be achieved in MoTe2/Eu(111)8. However, due to surface reconstruction and orientation-dependent exchange splitting17, the observed valley splitting is at least one order of magnitude smaller, on the order of several meV16. Similar issue could also exist in other polar surfaces51,52. Such challenge could be easily tackled in 2D materials with atomically flat surfaces. Meanwhile, compared to low transition temperature of Eu-based ferromagnets or other 2D magnets, monolayer VSe2 shows room-temperature ferromagnetism, advantageous for the implementation of room-temperature devices. Third, the VSe2/MSe2 exhibits high tunability in their excitonic behaviors. Not only the hybridization strength could be feasibly tuned by strain or electric field, the exchange splitting sign could also be modulated by electric field, offering an interesting knob for device design. Remarkably, the control of interlayer exchange coupling could give rise to a strong intra- and interlayer exciton hybridization and sizable valley band as well as exciton splittings. For instance, for \(H_{\rm{Se}}^M\) stacked MoSe2/VSe2 the largest valley splittings of 37.8 and 46.5 meV can be achieved for V1 and V2 bands, respectively, and they are 27.4 and 55.8 meV for V1 and V2 bands in the \(H_{\rm{Se}}^W\) stacked WSe2/VSe2. Particularly, considering excitonic effect, giant valley exciton splittings of 203 meV (202 meV) for the A (B) exciton in MoSe2 and 156 meV for B exciton in WSe2 are realized. Our study provides ideas for searching vdW heterostructures with giant valley splittings, and the proposed MSe2/VSe2 bilayers serve as excellent candidates for valleytronics research.

Methods

Computational details

Our first principles calculations were carried out within the projector augmented wave method53 as implemented in VASP package54, with the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof55,56 adopted to describe the electron exchange and correlation effects. Electronic wave function was expanded on a plane-wave basis set with a kinetic energy cutoff of 400 eV. The convergence thresholds for electronic and ionic relaxations were chosen to be 1.0 × 10−7 eV and 0.001 eV Å−1. The Brillouin zones for the heterostructures were sampled by 12 × 12 × 1 Г-centered k-point meshes. The SOC effect was fully included in all our calculations and the van der Waals interaction was taken into account in the DFT-D3 scheme57. The 3s23p64s23d3 states of V, the 4s24p4 of Se, the 5p66s25d4 of W and the 4s24p65s14d5 of Mo were treated as valence states, respectively.

For hybrid functional calculations, the convergence thresholds for electronic minimizations were chosen to be 1.0 × 10−5 eV. The screened Coulomb potential by an inclusion of exact Hartree exchange could be simplified by using a mixing coefficient α and a range-separation Coulomb potential described by μ58. Dudarev et al.’s method59 was also applied to treat localized V d electrons with the effective U parameter set as 1.1322,60.

Optical properties from the GW-BSE

Single-shot GW (G0W0)35 approach on top of both PBE (GW@PBE) and hybrid functional (GW@HSE) single-electron approximations were adopted for quasiparticle band structure calculations. For the monolayers and \(H_{\rm{Se}}^M\) stackings, which are the main focus of our paper, more than 1000 bands were employed, while for the rest stackings a total number of empty bands more than twice the occupied bands were used. The optical properties were calculated by solving the Bethe–Salpeter Equation (BSE)61,62 based on GW correction. Fifteen valence bands and fifteen conduction bands were included in the BSE optical transition calculations. The optical absorbance for circularly polarized light can be calculated from63

$$A\left( E \right) = \frac{{Ed}}{{\hbar c}}Im\left( {\varepsilon _{xx} \pm i\varepsilon _{xy}} \right)$$
(1)

where ℏ,c, E, d, εxx, and εxy are the reduced Planck constant, the speed of light in vacuum, the energy of photon, the thickness of 2D heterostructures, the rescaled diagonal and nondiagonal terms of dielectric tensor for heterostructures, respectively. Im denotes the imaginary part of the dielectric tensor components. Considering the calculations for 2D heterostructures were performed under periodic boundary conditions with a sufficiently large interlayer distance l = 30 Å, the rescaled dielectric functions were obtained by eliminating the vacuum contribution through64

$$\varepsilon _{xx} = 1 + \frac{l}{d}\left( {\tilde \varepsilon _{xx} - 1} \right)$$
(2)
$$\varepsilon _{xy} = \frac{l}{d}\tilde \varepsilon _{xy}$$
(3)

where \(\tilde \varepsilon _{xx}\) and \(\tilde \varepsilon _{xy}\) are calculated dielectric functions of simulation cell with vacuum space.