Band valley modification under strain in monolayer WSe₂

Transition-metal dichalcogenides (TMDs), MX₂ (M = Mo, W and X = S, Se), have rapidly received the attention of the physical community, thanks to their attractive electronic and optical material properties. Monolayer (ML) TMDs possess direct bandgap in the infrared and visible light spectra^1–7 with the presence of local band extrema or “valleys” in their band structures, the weak screening, and the electronic valley dependent optical selection rules for interband transitions.^7–12 Therefore, ML TMDs are potentially suitable for a variety of applications in photonics, optoelectronics,^13–15 spintronics, and valleytronics.^13,16–21

In the class of TMDs, ML WSe₂ exhibits fascinating electronic and optical properties such as higher surface area, a direct bandgap of about 1.6 eV, high electrical conductivity, strong excitonic effects,^21,22 and remarkably high absorption coefficient in the visible and near-infrared regions. The geometric structure of ML WSe₂ consists of nanosheets with three atomic Se–W–Se layers held together by weak van der Waals forces. Such highly anisotropic morphology of ML WSe₂ marks rich mechanical responses of these monolayers under deformation, leading to a strong modification of its electronic and optical properties. Consequently, the tunable electronic and optical properties are desirable for optoelectronic and spintronic applications. Recently, structural modification, which enhances the optoelectronic properties of WSe₂, was actively investigated experimentally.^12,23–27 However, the changes in these optoelectronic properties such as the redshift, intensity enhancement, localization, and symmetrization of the exciton resonances under strain are still poorly understood theoretically. Furthermore, the spin–orbit coupling (SOC) effect of ML WSe₂ induced by the manifestation of orbital hybridization under deformation has not been systematically studied. Although the band structure modification without SOC effect of ML WSe₂ upon biaxial strain has been revealed,²⁸ there is still a lack of systematic theoretical study on the changes in the band structure caused by isotropic strain including the SOC effect. A systematic study on the modification of the band structure under strain in ML WSe₂ is optical and electronic property insight, contributing to the development of optoelectronic and spintronic devices based on ML TMDs.

The band structure of ML TMDs was calculated by using different exchange-correlation functionals.^6,7,28–30 It was reported that the calculations using the generalized gradient approximation (GGA) based on the Perdew–Burke–Ernzerhof (PBE) formula can depict well the band structures of ML TMDs.⁷ Noticeably, the attractive force between constituent nanosheets of ML TMDs is dominated by the van der Waals (vdW) interaction. Neglecting the vdW effect may cause an inaccurate estimation of the interlayer distance and interaction,²² and consequently, may affect the accuracy of the electronic structure calculation. A simple way to improve the accuracy of the electronic structure and provide the modification of the band structures caused by strain using the density functional theory (DFT) calculation method is to take into account not only the SOC effect but also the non-local vdW interaction.

In this work, we employed the non-empirical vdW density functional theory to investigate the band valley modification, orbital hybridization, and appearance/disappearance of excitons in ML WSe₂ upon the isotropic strain. First, we calculated the geometrical and electronic structures of WSe₂ by the generalized gradient approximation (GGA) and the five non-empirical vdW functionals by taking into account the SOC. Then, we employed the optB88-vdW functional,^31,32 which generates the closest lattice parameters and bandgap of WSe₂ to the experimental observations, to calculate the electronic structures. We revealed the spin splitting, dominant contributions of orbitals to the local band extrema near the Fermi level, and exciton diversity in strain free WSe₂. Second, we investigated the electronic band structures of ML WSe₂ under isotropic strain. The band valley modification, reduction of the bandgap, and direct–indirect transition of the bandgap under compression strain were explained thoroughly. The change in orbital hybridization is also discussed. Based on the band valley modification induced by strain, we discussed the mechanism of the energetic shift, symmetrization, and localization of resonance shapes of exciton resonances. The enhancement of the magnitude of bright exciton resonances upon the tensile is revealed. Furthermore, we predicted the tendency of the energetic shifts, asymmetric broadening, and intensity reduction of the exciton resonances under compression strain.

All DFT calculations were done by using the projector augmented wave method³³ as implemented in the Vienna Ab initio Simulation Package (VASP).^34,35 To generate the accurate electronic band structures, we adopted five non-empirical vdW functionals—revPBE-vdW,³⁶ optPBE-vdW, optB88-vdW, optB86b-vdW,^31,32 and DF2-vdW³⁷—to compute the lattice constants and band structure of strain free ML WSe₂. The DFT calculation based on the generalized gradient approximation (GGA) according to the Perdew–Burke–Ernzerhof formula^38,39 was also performed for comparison. The energy cutoff is set to 500 eV in all calculations. For the Brillouin zone integration, a 20 × 20 × 1 Γ-centered Monkhorst–Pack k-point mesh was used. The lattice parameters and the internal coordinates were fully relaxed until the atomic force on each atom is less than $0.01eV/Ȧ$ and the change in the total energy is below 10⁻⁵ eV. The calculation scheme that gives the lattice parameters and bandgap closest to the experimental data is then adopted to investigate the modifications of the band structure of ML WSe₂ upon the strain.

The crystal structure of ML WSe₂ is illustrated in Fig. 1(a)—it has a hexagonal arrangement of W atoms sandwiched between two Se layers as reported previously.^1,7 The distance between the two nearest layers of Se is 3.369 $Ȧ$⁠, and the bond angle of Se–W–Se is 82.58° as calculated by the optB88-vdW functional. The optimized lattice constants and the band gaps of ML WSe₂ with and without the SOC effect for six different functionals are given in Table I. Figure 1(b) sketches the highly symmetrical Brillouin zone (BZ) points for a better presentation of the band structures. In our calculations, ML WSe₂ has a direct (D) bandgap at the K point as illustrated in Fig. 1(c) for most investigated exchange-correlation functionals, except for the one calculated by the optB86b-vdW functional which exhibits an indirect (ID) bandgap at K and Q points. In the presence of the SOC effect, the calculated bandgap is narrowed down by ∼0.1 eV with respect to the non-SOC case, implying that ML WSe₂ is strongly affected by the SOC effect. More importantly, the lattice constants and the bandgap calculated by the optB88-vdW functional show excellent agreement with the experimental results.⁴⁰ Therefore, we hereafter employ the optB88-vdW functional to calculate the band structures and explore the modification of band valleys, which demonstrate the exciton appearance/disappearance in ML WSe₂ upon the strain.

The lack of space inversion symmetry of ML WSe₂ leads to a noticeable spin splitting observation at the symmetry points in the first BZ when the SOC effect was employed.^6,7,41 To highlight the spin polarization at the energy levels close to the Fermi energy, we plot the spin projected band structures in the energy range of [−3, 3] eV relative to the Valence Band Maximum (VBM) in Fig. 2, where each orbital is projected onto the spin-up and spin-down states. The contribution of spin up and spin down electrons is indicated by the color scale by which the bright color represents the occupation of electrons with spin up and the dark color indicates that of electrons with spin down in each state.

We pay great attention to the spin splitting of the band extrema near the gap, whose energy levels are occupied by the flexible electrons or holes, two fundamental components of excitons Compared to the band structure without the SOC effect in Fig. 1, band structures calculated with the SOC effect in Fig. 2 show a significant spin splitting of the VBM at the K point into two subbands 1a-1b at energy levels with their separation of 0.458 eV. Subband 1a is preferably occupied by an electron with spin up, whereas the next lower singly occupied molecular orbital (SOMO) level 1b is by a spin down. Another spin splitting band maximum in the valence band (VB) is labeled 2a-2b at the Q point. Among these four subbands in the VB, subbands 1a, 1b, and 2a have a strong band dispersion. The conduction band edge has three SOC splitting band minima (band valleys), including conduction band maximum (CBM) 1a′-1b′ at K, 2a′-2b′ at Q, and 3a′-3b′ centered at the midpoint between M and Γ. CBM 1a′ can be occupied by an electron with spin down, whereas CBM 1b′ is preferably occupied by an electron with spin up. The spin up and spin down states in the conduction band maximum (CBM) at K lie at the energy levels very close to each other with a separation of about 30 meV—the size comparable to room-temperature thermal energy. The spin splitting of the conduction band edge 2a′-2b′ at the Q point is quite large, about 0.210 eV. Subband 2a′ is preferably occupied by an electron with spin up, whereas subband 2b′ can be occupied by an electron with spin down. Figure 2 also displays several spin splitting band valleys with a strong dispersion, which locates higher than the conduction band edge, such as 4a′-4b′ centered at K, 5a′-5b′ around Q, 6a′-6b′, and 7a′-7b′. The presence of a great number of the band valleys with strong dispersion in the conduction band (CB) as shown in Fig. 2 implies a diversity of exciton types in ML WSe₂.

It was reported that ML WSe₂ is a p-type semiconductor.⁴² The electronic properties of this material can be further understood from the analysis of the band structure. We can see that the VBM is occupied by W orbitals as shown in Figs. 1(c) and 1(d). The electronic properties of the material involve the outermost shell electrons of orbitals that have the contribution to the energy bands near the gap. Figure 3 demonstrates the occupation of each p, d, and s electron at the energy levels around the gap. Figure 3(a) shows that the occupation of s electrons is extremely small in this energy range, whereas d electrons dominantly distribute as shown in Fig. 3(c).

The contributions of d_xy, d_yz, d_xz, $dz2$⁠, and $dx2−y2$ orbitals of W atoms as well as p_x, p_y, and p_z orbitals of Se atoms to energy band extrema around the gap are clearly shown in Figs. 4 and 5. It is obvious that the energy difference between the spin up and spin down states in the partial d orbitals is the smallest for $dz2$ and the largest for d_yz and d_xz, which is in excellent agreement with the calculated results in Ref. 41. The CBM is mostly occupied by electrons of $dz2$ orbital. The VBM is dominated by d_xy and $dx2−y2$ orbitals of W atoms as well as p_x and p_y orbitals of Se atoms. The presence of these partial orbitals in the VBM at K emphasizes a hybridization between d orbitals of W and p orbitals of Se. Similar behaviors in the VBM can be found for the band valleys 2a′-2b′, 3a′-3b′, and 6a′-6b′.

The photon absorption of ML WSe₂ promotes the transition of electron from the VB to the CB, leaving a hole at the VB. The Coulomb attraction between an excited electron and an induced hole creates a bound exciton because of the weak screening. The presence of a large number of SOC splitting band extrema near the bandgap indicates that various types of exciton states can be formed. The exciton diversity in WSe₂ has been experimentally confirmed.^{11,24,43–49} It was reported that the binding energy of the excitons is relatively high, about 0.37 eV,¹¹ two orders of magnitude larger than that in ordinary ML materials. Therefore, the exciton states can exist at room temperature. In this section, we focus on the appearance of optical and momentum excitons, thanks to the analysis of the spin projected band structure shown in Fig. 2.

Because ML WSe₂ is a direct bandgap semiconductor, the photon absorption can lead to the vertical transitions of electrons from the VBM to the CBM, creating various optical excitons of $dz2$ electron of W. The binding between a spin up hole at the VBM 1a and a spin up excited electron at the CBM 1b′ [Fig. 6(a)] is referred to as the A exciton, whereas the coupling between a spin down hole at the VBM 1b and a spin down electron at the CBM 1a′ [Fig. 6(b)] is known as the B exciton. Both the A and B excitons are spin-allowed bright optical excitons. The recombination of the A and B excitons does not require the spin-direction flip, radiating visible photons. The energies of photons emitted from the recombination of the A and B excitons are estimated to be about 1.663 and 2.091 eV, respectively, which agree well with the experimental measurements.^11,23,24,27 The tight binding between the spin down electron at the CBM 1a′ and the spin up hole at the VBM 1a creates the first dark exciton [Fig. 6(c)], whereas the coupling of the excited electron with spin up at the CBM 1b′ and the hole with spin down at the VBM 1b generates the second dark exciton [Fig. 6(d)]. Because two types of dark excitons consist of a hole and an electron with opposite spin directions, the recombination of these excitons requires spin flipping, which consumes energy. As a consequence, the recombination of these two dark excitons does not emit photons in the visible spectrum. The appearance of the dark excitons was experimentally confirmed^9,10,45,49 and they were brightened by applying an in-plane magnetic field.¹⁰ 

Figure 2 shows that the energy state 4b′ at the K point, which is 2.633 eV higher than the VBM 1a, can be occupied by spin up electrons due to the optical transitions of electrons from the VBM 1a-1b. The tight bounding between the excited electron at the state 4b′ and the hole at the VBM can lead to the appearance of two different optical excitons, which were observed in experimental work.²⁴ The first type is the spin-forbidden exciton, named the C exciton, consisting of a spin up electron at the energy state 4b′ and a spin down hole at the energy state 1b. Accompanying the spin flipping of the electron, the recombination of a C exciton may emit a photon whose energy is estimated to be about 2.450 eV. The second type is the spin-allowed optical exciton called the D exciton. The D exciton appears due to the tight bounding between a spin up electron at the subband 4b′ and a spin up hole at the VBM 1a. The energy of the photon radiated from the recombination of the D exciton is estimated to be about 2.898 eV.

Noticeably, the local band maximum 2a and the SOC splitting valley 7a′-7b′ are located at the same point of the first BZ as shown in Fig. 2. The absorption of photons can promote the direct transition of electrons from the state 2a to the states 7a′ or 7b′, generating additionally two types of optical excitons. The first type is the spin-allowed exciton and the second one is the spin-forbidden exciton. In our calculation, the energy separation of the subband pairs 7a′-2a and 7b′-2a is significantly larger than the energy of violet photons. Therefore, it is impossible to visually observe the resonances due to the recombination of these excitons in spectral measurements.

The lowest unoccupied state 2a′ at Q stands above the CBM 1b′ at K only a few tens meV as illustrated in Fig. 2. Therefore, the indirect transitions, which require the momentum shift, of electrons from the VBM 1a-1b to the conduction band edge 2a′-2b′ at Q may happen. The excited electrons at the CBM 1b′, which arises from the vertical transitions from the VBM, can occupy the state 2a′ after absorbing or radiating phonon. The interaction between the spin up electron at the state 2a′ and the spin up hole at VBM 1a creates spin-allowed KQ momentum exciton. The appearance of the spin-allowed KQ momentum excitons 2a′-1a leads to the geometrical asymmetry of the A exciton resonance observed in experimental work.²⁷ Furthermore, the excited electrons at the CBM 1a′ can occupy the energy state 2a′ after absorbing the photon and exchanging phonon. The Coulomb interaction between the spin down electron at the state 2b′ and the spin up hole at VBM 1a binds them into spin-forbidden KQ momentum exciton 2b′-1a. Analogously, momentum excitons 1b-2a′ and 1b-2b′ may appear. In addition to the indirect transition from the VBM 1a-1b, the indirect transition of electrons from the state 2a to CB can also occur. Excited electrons due to indirect transitions may occupy the energy valleys near the gap such as 3a′-3b′, 5a′-5b′, and 6a′-6b′. The coupling of these electrons and the holes at the states 1a, 1b, and 2b additionally create several types of momentum excitons. Because of the diversity of momentum excitons in ML WSe₂, the four bright exciton resonances observed in experimental work are asymmetric.²⁴ 

Strain is known as an effective way to modify the band structure⁵⁰ of 2D materials. By comparing the band structure of strainless WSe₂ shown in Fig. 2 to the band structure of strained material, we can see the band structure modification, particularly the band valley modification under strain. Therefore, the influence of strain on the optoelectronic properties of ML WSe₂ can be understood and predicted.

Figure 7 illustrates the spin projected band structures of ML WSe₂ at the lattice constant a of 3.158, 3.191, and $3.291Ȧ$⁠, corresponding to the compression of −5%, −4%, and −1%, respectively. Overall, we found that there is a modification of the band valleys dominated by d_xy, $dz2$⁠, and $dx2−y2$ orbitals. The modification includes the spin splitting change and the vertical shift of the band extrema. The modification of these band extrema becomes larger when the compression intensity is stronger. The calculation shows that, under compression, ML WSe₂ exhibits the indirect bandgap at K − Q points in the first BZ, which is in good agreement with previous DFT calculations.⁵¹ The bandgap reduction is more pronounced when larger compression strain amplitude is applied (Table II). Moreover, the magnitude of spin splitting of the CBM 1a′-1b′ increases, whereas that of the VBM decreases upon compression. The dispersion of the VBM exhibits a slight change. The band valley 2a-2b, whose states are occupied by d_xy and $dx2−y2$ electrons, shifts toward the Fermi level and becomes less dispersed. However, the band extrema 2a-2b disappears under compression of −5%. The band valley 2a′-2b′ in the CB dominated by d_xy and $dx2−y2$ orbitals shifts toward the Fermi level, whereas the band valley 1a′-1b′ dominated by $dz2$ orbital strongly lifts. The subband 2a′ is even 2.013 eV higher than the VBM 1a upon compression of −5% as shown in Fig. 7(a). The dispersion of the band valleys 2a′-2b′, 3a′-3b′, and 6a′-6b′ increases, whereas the band valley 1a′-1b′ exhibits a flattening. The uplift of the band valley 1a′-1b′ and the downshift of the band valley 2a′-2b′ upon compression lead to the transition from direct to indirect bandgap at a critical compression strain of 0.01%. Overall, under compression strain, the band valleys dominated by $dz2$ orbital are less dispersed, even flattened, and shift far away from the Fermi level, whereas the band valleys dominated by d_xy and $dx2−y2$ orbitals are more dispersed and shift closer to the Fermi level. It implies that the electrons occupying the energy levels of the band valleys 2a′-2b′, 3a′-3b′, and 6a′-6b′ are more flexible, whereas the electrons occupying the states of band valleys 1a′-1b′ and 4b′ are less flexible upon compression strain. In addition, Fig. 7 shows a vertical downshift of the subband 4b′, which is dominated by d_xy and $dx2−y2$ orbitals. The energy of the subband 4b′ is higher than that of the VBM 1a, about 2.321 eV as shown in Fig. 7(a), demonstrating a vertical downshift of 0.312 eV. Clearly, the band valley modification under compression induced by orbital hybridization enhances the indirect transition of electrons from VB to CB.

The spin projected band structures of the material at lattice constant a of 3.336, 3.457, and $3.490Ȧ$ under the tensile strain of 1%, 4%, and 5%, respectively, are shown in Fig. 8. It can be seen that, under tensile strain, ML WSe₂ retains a direct bandgap at K in the first BZ, although there is a significant modification of the band valleys around the gap. Similar to the case of applying compression strain, the bandgap is reduced and its reduction is larger when the tension intensity is stronger (see Table II). Moreover, we find that the modification of the band structure caused by tensile strain dominantly involves the vertical shift and the spin splitting change of the band valleys dominated by d_xy, $dz2$⁠, and $dx2−y2$ orbitals. Table II indicates that the spin splitting of the VBM at K point increases, whereas the spin splitting of the CBM 1a′-1b′ decreases upon the tensile strain. As shown in Fig. 8, the subband pair 2a-2b dominated by d_xy and $dx2−y2$ orbitals is more dispersed, whereas the dispersion of the VBM at K has no considerable change. The band valley 1a′-1b′ dominated by $dz2$ orbital strongly shifts down in the vertical direction. The downshift of the band valley 1a′-1b′ results in the bandgap reduction and enhances the dispersion of this valley as shown in Fig. 8. On the contrary, the band valleys 2a′-2b′, 3a′-3b′, and 6a′-6b′ are less dispersed, even flattened upon the tension of 5%. It implies that, under tensile strain, the electrons occupying the energy states of the band valley 1a′-1b′ are more flexible, whereas the electrons occupying the states of the band valleys 2a′-2b′, 3a′-3b′, and 6a′-6b′ become less flexible. In short, under tensile strain, the band valleys dominated by $dz2$ orbital are more dispersed and shift toward the Fermi level, whereas the band valleys dominated by d_xy and $dx2−y2$ orbitals are less dispersed, even flattened. Such a change in the band structure indicates that the tensile strain may enhance the direct transition but counteract the indirect transition of electrons from the VB to the CB. In addition, Fig. 8 also shows a vertical uplift of the subband 4b′ dominated by d_xy and $dx2−y2$ orbitals upon the tensile strain. The magnitude of the uplifting is about 0.164 eV under 5% tensile strain.

The modification of the band structures under strain can be understood by analyzing the orbital hybridization. Upon isotropic strain, the space inversion asymmetry of ML WSe₂ changes. It leads to an increase/decrease of the spin splitting of the band energy extrema. In addition, strain causes the increase/decrease of the distance between two adjacent Se–Se atoms in the xy plane as indicated in Table II. The p_x and p_y orbitals may be dilated/concentrated. Therefore, the hybridization of p_x and p_y orbitals of two adjacent atoms reduces/increases. Consequently, the band valleys dominated by p_x and p_y orbitals would be modified. For orbitals in the z direction, they are compressed/relaxed because the bond length between W and Se atoms along this axis is reduced/increased (see Table II). Such a change may enhance/reduce the hybridization of the partial orbitals in the z direction between the atoms. Therefore, the band valleys dominated by $dz2$ orbital of W atoms would change. In short, the change in the orbital hybridization as well as the breaking of space inversion symmetry as applying isotropic strain may lead to the modification of the band valleys dominated by d_xy, $dx2−y2$⁠, and $dz2$ of W atoms mentioned above.

The modifications of the radiation spectrum of ML WSe₂ under tensile strain were observed.^{22–24,26,27} It was pointed out that the intensity of the A and B excitons’ resonances increases. Moreover, the two resonances become more symmetric, more localized, and have redshift. The blueshift of the C and D exciton resonances is inconsiderable.²⁴ Here, we focus on illumination as well as prediction of the exciton transformations caused by tensile strain, thanks to analyzing the modifications of the spin projected band structure.

As mentioned above, the strong downshift of the CBM at K and the upshift of the conduction band edge at Q upon strain indicate that the direct transition of electrons from the VBM 1a-1b to the CBM 1a′-1b′ are noticeably enhanced, whereas the indirect transitions of electrons from the VBM 1a-1b to the CB are counteracted. By accompanying the flattening of the band valleys 3a′-3b′, 6a′-6b′, and 2b′, the tensile strain can cause the disappearance of a great number of momentum excitons, especially KQ momentum excitons. However, the total number of optical excitons, including dark and bright excitons, in the CBM 1a′-1b′ significantly increases. Therefore, upon tensile, the A and B exciton resonances are more symmetric and more localized. Moreover, the intensity of the two resonances is enhanced. We find that a considerable redshift of the A and B exciton resonances involves a considerable reduction of the bandgap as applying the tensile strain. Because the reduction of the bandgap is proportional to the tensile strength, the redshift of the A and B exciton resonances is also proportional to the tensile strain amplitude. This effect was also observed experimentally in Refs. 12 and 13. In addition, the energy separation between the photon radiated from the recombination of exciton A and that of exciton B is predicted to be enlarged due to the enhancement in the SOC splitting of the VBM.

The increase in the DOS of the subband 4b′with an increase in the lattice constant a, as shown by color in Figs. 7 and 8, indicates that, upon tensile, the number of the C and D optical excitons may increase. Therefore, we predict that the tensile effect can lead to the intensity enhancement of the C and D exciton resonances. By accompanying the decrease in the density of momentum excitons, the C and D exciton resonances may be more localized and more symmetric. The upshift of the state 4a′ with respect to the VBM with an increase in the lattice constant a, as shown in Fig. 8, indicates the blueshift for C and D exciton resonances. Furthermore, the density of optical excitons in the states 2a-2b may increase due to the appearance of the double extrema 2a-2b with strong dispersion as shown in Figs. 8(b) and 8(c).

In Section III B 2 a, we have investigated and successfully compared our theoretical prediction of the exciton formation in ML WSe₂ under tensile strain to the experimental observations in Refs. 12 and 13. However, under compression, due to the complexity of orbital interactions when the atoms come closer to their nearest neighbors⁵⁰ and the technological limit of current experimental systems to maintain the flatness of these monolayers, the exciton transformation under compression has not been observed experimentally. Thus, in this section, we present a theoretical prediction of exciton formation under compression.

As mentioned above, the downshift of the band valley 2a′-2b′ at Q point and the flattening of the band valley 1a′-1b′ caused by compression enhance the indirect transitions of electrons from the VBM 1a-1b to the CB, especially to the CBM 2a′-2b′ at Q, in ML WSe₂ under compression and hinder the direct transition of electrons from the VBM 1a-1b to the conduction band edge 1a′-1b′ at K. As a consequence, there is an additional creation of a great number of momentum excitons and a disappearance of the dark and bright optical excitons in the band valley 1a′-1b′. Therefore, the A and B exciton resonances may be more asymmetric and broadened. Moreover, the intensity of these two resonances would decrease. The more pronounced decrease in the spin occupation of the subband 1a′ with respect to that of the subband 1b′ in Fig. 7 reveals that the density of the A excitons decreases more than that of the B excitons under compression. Noticeably, the flattening of the subband 1b′ and the extremely low spin occupation of the subband 1a′, as illustrated in Fig. 7(a), enable us to predict that both bright and dark optical excitons in the band valley 1a′-1b′ may be vanished under −5% compression strain. Therefore, A and B exciton resonances may be quenched. The uplift of the conduction band edge at K with respect to the VBM explains the blueshifts of A and B bright excitons. Furthermore, the decrement of the SOC splitting at the conduction band edge at K (as given in Table II) explains the reduction of the energy difference between the photon emitted from the recombination of the exciton A and that of the B exciton with an increase in the compression strain.

Under compression strain, the vertical downshift with a decrease in the spin occupation in the subband 4b′ (in Fig. 7) reveals that there is a redshift of both the C and D exciton resonances and the total number of C and D excitons may be greatly decreased. As a result, the intensity of the C and D exciton resonances would reduce as in the case of A and B exciton resonances. In combination with the gradual increase in the momentum exciton density as discussed above, the C and D exciton resonances may be less symmetric and broadened. In particular, the extremely low spin contribution in the subband 4b′ upon −5% compression strain [in Fig. 7(a)] diminishes the existence of the C and D excitons. In addition, the decrement of the optical exciton density in state 2a may occur due to the flattening of the local extreme 2a as shown in Fig. 7(a). In short, the compression strain can lead to the disappearance of the optical excitons and the additional creation of a great number of momentum excitons.

In this work, the band valley modification of ML WSe₂ under isotropic strain was explored by DFT calculations including the SOC effect and vdW exchange-correlation functionals. The spin splitting, dominated orbital interactions to the band extrema near the Fermi level, and exciton diversity of ML WSe₂ were explored. We found that the vertical downshift of the states dominated by the d and p partial orbitals of the W and Se atoms, respectively, at band edges under strain can lead to a reduction of the band gaps. ML WSe₂ under tensile strain exhibits a direct bandgap. Moreover, the modification of the band valleys around the gap induced by the tensile strain results in the disappearance of the momentum excitons and the additional appearance of a great number of optical excitons. Therefore, the experimental observations of the optical redshift or blueshift, intensity enhancement, localization, and symmetrization of the radiated exciton resonances induced by the tensile strain were thoroughly interpreted. For ML WSe₂ under compression strain, the vertical shifts of the band valleys at the conduction band edge lead to the direct-to-indirect transition of the bandgap. The band valley deformation caused by compression may enhance the indirect transition but counteract the direct transition of electrons from the VB to the CB. This compression leads to the energetic shift, asymmetric broadening, and intensity reduction of the exciton resonances. The band valley deformations are explained by the enhancement and the reduction of the orbital hybridization between W and Se atoms with the application of strain. These results emphasize the crucial contribution of spin–orbit coupling on the band structures as well as the optical properties of ML WSe₂ upon mechanical perturbation.

This work was financially supported by Hanoi University of Industry under Scientific Research and Technological Development Project No. 31-2021-RD/HD-DHCNHN. This research was also partially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.01-2018.315. V.A.D. would like to thank the Center for Computational Physics, Institute of Physics, Vietnam Academy of Science and Technology, for supporting a high-performance computing system.

The authors have no conflicts to disclose.

Thi Nhan Tran: Data curation (lead); Formal analysis (lead); Investigation (lead); Writing – original draft (lead). Minh Triet Dang: Supervision (equal); Validation (equal). Quang Huy Tran: Formal analysis (equal); Investigation (equal). Thi Theu Luong: Formal analysis (supporting); Validation (supporting). Van An Dinh: Conceptualization (lead); Supervision (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.