Magnetic proximity effect has been demonstrated to be an effective routine to introduce valley splitting in two-dimensional van der Waals heterostructures. However, the control of its strength and the induced valley splitting remains challenging. In this work, taking heterobilayers combining monolayer MSe2 (M = Mo or W) with room-temperature ferromagnetic VSe2 as examples, we demonstrate that the valley splitting for both band edges and excitons can be modulated by the tuning of the interlayer orbital hybridization, achieved by inclusion of different amounts of exact Hartree exchange potential via hybrid functionals. The calculations suggest that large valley band splitting about 30 meV and valley exciton splitting over 150 meV can be induced in monolayer MSe2. Besides, we show such tuning of orbital hybridization could be experimentally realized by external strain. Our work reveals a way to control proximity effects and provides some guidance for the design of spintronic and valleytronic devices.
Article
Tunable valley band and exciton splitting by interlayer orbital hybridization
https://doi.org/10.21203/rs.3.rs-1775142/v1
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Monolayer group VIB transition metal dichalcogenides (TMDs) with broken inversion symmetry and strong spin-orbit coupling (SOC) show intriguing coupled spin and valley phenomena1–3. To address spin or valley degree of freedom, TMDs are usually selectively excited by circularly polarized light, leading to carriers/excitons with specific valley pseudospin. However, it is challenging to integrate light into electronic nanodevices. Another way is directly applying an external magnetic field to split the degenerate band edges with different valley pseudospins. Yet, the magnetic-field-induced splitting is only at the order of ∼ 0.2 meV/T4,5, showing limited promise in real applications. An alternative and robust approach is to build TMDs/magnets heterostructures taking advantage of the magnetic proximity effect, where the magnetic orders of magnets can significantly affect the behaviors of non-magnetic TMDs6–12. Applying this strategy, it has been confirmed in experiments that the valley splittings of WSe2 and WS2 on top of bulk EuS reach 16 meV/T13 and 2.5 meV/T14, respectively. When two-dimensional (2D) ferromagnetic semiconductor CrI3 and CrBr3 were adopted as substrates, splittings about 4 meV for WSe215 and 2.9 meV for MoSe216 have been realized.
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 TMDs17,18. Importantly, while it is paramagnetic in the bulk form, thin-film 1H-VSe2 exhibits a ferromagnetic ground state, which remains stable above room temperature17,18, 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 69K19, 20 and 16.6 K21, while those of monolayer CrI3 and CrBr3 are around 45K22 and 34K23, 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 d orbitals, suggesting the hopping and hybridization between these d orbitals with specific spin and thus the valley splitting could be tuned through external control knobs.
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 or strain. The energy alignment and hybridization between the valleys from MSe2 and VSe2 can be well tuned, leading to a large valley splitting of 29.1 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, a large valley splitting of 203 meV (202 meV) for the intralayer A (B) exciton in MoSe2 and 156 meV for B exciton in WSe2 is realized when the excitonic effect is taken into consideration.
Stacking registry and band alignment
The fully relaxed lattice constants for MSe2 and VSe2 are 3.32 Å 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. 1(a). 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 \({R\left(H\right)}_{V}^{M}\), \({R\left(H\right)}_{Se}^{M}\), and \({R\left(H\right)}_{h}^{M}\), 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.
Different stacking registries between the two constitutes have important influences on the band offset in these heterostructures. Figure 1(b) shows the band alignments using PBE, HSE, [email protected] and [email protected], 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 level24. 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 exchange25. 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). 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 data26–28. 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, strain29,30. In particular, for MoSe2/VSe2, [email protected] yields a type-I band alignment, whereas [email protected] 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 hopping and hybridization strength.
Band structure and valley band splitting
The influences of interlayer orbital hybridization on the valley splitting is 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 hopping 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. 2(a), 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.
Figures 3(a) and 3(b) show the calculated band structures for \({H}_{Se}^{M}\) 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 Sec. II. Considering the band edges for MoSe2 and WSe2 at K and –K valleys are predominantly composed of transition metal \({d}_{{z}^{2}}\), dxy and \({d}_{{x}^{2}-{y}^{2}}\) orbitals31, the weight decomposition of the contribution from these three orbitals is also provided in Figs. 3(a) and 3(b). It is clear that characteristic MSe2 band edges are largely preserved in heterostructures. The calculated orbital projected band structures can also be used to estimate the band alignment between MSe2 and VSe2. For example, the metallic behavior obtained using PBE in Fig. 3(a) and Supplementary Figs. S2 and S3, suggests a type-III alignment, which is consistent with the previous band alignment calculations in Fig. 1(b).
Notably, in these heterostructures, different valleys are now imprinted by proximity exchange splittings. Focusing on the valence bands, the valley splitting energy is defined as the energy difference between the topmost valence bands (V1) or second topmost valence bands (V2) of MoSe2 or WSe2 at K and − K valleys, that is, \({\varDelta E}_{Vn}={E}_{K}^{Vn}-{E}_{-K}^{Vn}\) with n equal to 1 or 2. The valley splitting values for six stable stackings are summarized in the Figs. 3(c) and 3(d), as well as the Supplementary Tables S2-S5. It can be found that the valley splitting in the heterostructures are strongly stacking dependent. Especially, the largest valley splitting under PBE approximation appears for the V1 band in \({H}_{Se}^{Mo}\) stacked MoSe2/VSe2, reaching 29.1 meV, which is much larger than the reported values in WSe2/CrI3 and MoSe2/CrBr36,7,15,16,32,33. The large valley splitting is related to the strong valence bands hybridization between MoSe2 V1 band and VSe2 V1′ band at K point, which results in the significant contribution from Mo d orbitals to the original top valence band of VSe2 as shown in Fig. 3(a). Such scenario corresponds to the scheme in Fig. 2(a). It can be further confirmed by the projected partial charge density distribution of the top two valence bands, which shows clear charge distribution over both MoSe2 and VSe2 layers (top panel of Supplementary Fig. S1). However, the hybridization is not allowed at –K point, as a result of the opposite spin directions between VSe2 V1′ and MoSe2 V1 states at –K. For comparison, no hybridization emerges for \({H}_{Se}^{W}\) stacked WSe2/VSe2, due to the large energy mismatch in their valence band edges. In addition, \({H}_{h}^{M}\) stacked MoSe2/VSe2 and WSe2/VSe2 show the smallest valley splitting. This is because they have the largest interlayer spacing, resulting in a weaker M-V proximity magnetic coupling. When the quasi-particle GW correction is taken into account, almost all the splitting values for both the valence and conduction bands become larger compared with PBE single-particle results (see Supplementary Tables S2 and S4).
Regarding to HSE results, similar enhancement is observed for most of the splitting values compared with PBE results [see Figs. 3(c) and 3(d)], except for \({H}_{Se}^{M}\) stacked MoSe2/VSe2 owing to the vanished hybridization at K in HSE level (Supplementary Fig. S6). Furthermore, it is found that the valley splittings in WSe2/VSe2 are larger than those in MoSe2/VSe2. Such enhancement can be ascribed to the stronger hybridization between WSe2 V2 band and VSe2 V1′ band at –K valley, where hole hopping from WSe2 to VSe2 is allowed for the aligned spin, as displayed in Fig. 3(b), which is in accordance with Fig. 2(b). In particular, ΔEV2 for \({H}_{Se}^{W}\) 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. S8 and S9). 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 \({H}_{Se}^{Mo}\) stacked MoSe2/VSe2 within PBE method and \({H}_{Se}^{W}\) 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 \({H}_{Se}^{M}\) stacked heterostructures as examples in the following.
Figure 4 summarizes valley splitting values for \({H}_{Se}^{M}\) 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. S10 and S11 as well as Tables S6 and S7). 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. S12), indicating that µ parameter does not affect much the interlayer hybridization.
The hybrid functionals involve a portion of exact exchange, and thus can also take the Hubbard U correction into account34. To shed light on this point, the representative orbital projected band structures for MoSe2/VSe2 at varied Ueff values are plotted in Supplementary Fig. S13. 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. S10). Because of its sensitivity to exchange and correlation effects, the actual ground-state band structures of the heterobilayers could depend on experimental conditions.
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 interaction29,30,35,36. To verify this, Supplementary Fig. S14 shows the band structures of heterostructures with a biaxial strain applied, which clearly display the change of band alignment and band hybridization.
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 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., \({\varDelta E}^{A/B}={E}_{K}^{A/B}-{E}_{-K}^{A/B}\). Figure 5 shows the optical absorbance of right and left circularly polarized light (σ+ and σ−) for the representative \({H}_{Se}^{M}\) stacked heterostructures, while the rest are shown in Supplementary Figs. S15 and S16. 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 S7.
It has been discussed in Sec. B that \({H}_{Se}^{M}\) 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. 5(a)]. In addition, the exciton splitting in VSe2 monolayer is also enhanced by two fold37. 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. 5(b)] 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. 2(b). 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. 5(a) and Fig. 5(b)]. For \({H}_{Se}^{W}\) stacked WSe2/VSe2, A and B intralayer excitons in WSe2 show almost comparable splitting [Fig. 5(c)] 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 \({H}_{Se}^{M}\) stacked MoSe2/VSe2.
To show the importance of the interlayer hybridization on excitonic transition, Supplementary Tables S8-S11 provide the decomposition of the exciton oscillator strength with respect to single particle transition for \({H}_{Se}^{M}\) stacked heterobilayers. Under [email protected] level, the lowest-energy bright exciton in MoSe2/VSe2 under σ− polarized light excitation denoted as \({A}_{K}^{{VSe}_{2}}\) exciton displays a considerable interlayer contribution from band V1 to C1′, besides the main intralayer transition from band V1′ to C1′. Meanwhile, for \({A}_{K}^{{MoSe}_{2}}\) exciton, the interlayer contribution from band V1′ to C1 is comparable to the intralayer component from band V1 to C1 transition. Accordingly, \({A}_{K}^{{VSe}_{2}}\) and \({A}_{K}^{{MoSe}_{2}}\) excitons show hybridization between the corresponding intra- and interlayer exciton transitions because of the interlayer-hybridized hole. In contrast, \({A}_{-K}^{{MoSe}_{2}}\) 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 \({A}_{K}^{{MoSe}_{2}}\) and \({A}_{-K}^{{MoSe}_{2}}\). Under HSE (α = 0.06, µ = 0.2) level, both \({A}_{-K}^{{VSe}_{2}}\) and \({B}_{-K}^{{MoSe}_{2}}\) exciton exhibits obvious mixed contributions from two monolayers, which significantly differs from \({A}_{K}^{{VSe}_{2}}\) and \({B}_{K}^{{MoSe}_{2}}\) excitons, leading to the significant B valley excitons splitting hitting 202 meV shown in Fig. 5(b). \({H}_{Se}^{W}\) stacked WSe2/VSe2 shows the similar trend within [email protected]
These strongly hybridized intra- and interlayer excitons 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 component38. Meanwhile, because of their characteristic interlayer distribution, they could show a high tunability under external electric fields39. Thus, 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.
In conclusion, using first principles methods we systematically investigated the valley splitting in MSe2 on top of ferromagnetic monolayer VSe2 to demonstrate a strong and tunable magnetic proximity effect. The magnitude of valley splitting relies on the strength of interlayer orbital hybridization and can be controlled feasibly by applying varied exchange and correlation interaction or in-plane strain. In principle, one can further tune the band offset by a vertical electric field to modulate the degree of hybridization40. Remarkably, the control of interlayer exchange coupling could give rise to a strong intra- and interlayer exciton hybridization and sizable exciton valley splitting. 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.
Our first principles calculations were carried out within the projector augmented wave method41 as implemented in VASP package42, with the generalized gradient approximation (GGA) functional of Perdew, Burke, and Ernzerhof43,44 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. 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 scheme 45. 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 µ46. Dudarev et al.’s method47 was also applied to treat localized V d electrons with the effective U parameter set as 1.1318,48.
Single-shot GW (G0W0)28 approach on top of both PBE ([email protected]) and hybrid functional ([email protected]) single-electron approximations were adopted for quasiparticle band structure calculations. For the monolayers and \({H}_{Se}^{M}\) 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)49,50 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 from51
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 through52
where \({\tilde{\epsilon }}_{xx}\) and \({\tilde{\epsilon }}_{xy}\) are calculated dielectric functions of simulation cell with vacuum space.
DATA AVAILABILITY
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (11974197 and 51920105002), Guangdong Innovative and Entrepreneurial Research Team Program (No. 2017ZT07C341), and the Bureau of Industry and Information Technology of Shenzhen for the 2017 Graphene Manufacturing Innovation Center Project (No. 201901171523).
AUTHOR CONTRIBUTIONS
D.W. were responsible for the calculations and data analyses. X.Z. and D.W. contributed to the discussion and the writing of the manuscript.
COMPETING INTERESTS
The authors declare no competing interests.
ADDITIONAL INFORMATION
Supplementary information The online version contains supplementary material available at https://doi.org/***.
Correspondence and requests for materials should be addressed to Xiaolong Zou.
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