The interlayer exciton in TMDs heterostructures is formed in indirect two-step processes; that is, the intralayer exciton is firstly formed in one sublayer, and then the photogenerated electrons transfer into the other sublayer as shown in Fig. 1a ① and ② processes. The process has been confirmed by many experimental and theoretical studies.7, 14, 24, 26, 28 In fact, the formation mode can be clearly illustrated from the geometry and electronic structure of the TMDs. Taking MoS2 as an example (see Fig. 1b), monolayer MoS2 is a sandwich structure with the middle Mo atom layer surrounded by two outer S atomic layers. The intermediate Mo atom layer mainly contributes to valence and conduct band edges. Hence, after forming type II TMDs heterostructures, the direct interlayer photoexcitation at band edges (see Fig. 1a, the ③ process) will be difficult due to the large spatial distance between the two Mo atomic layers. Therefore, interlayer exciton in TMDs heterostructures can only be formed in intralayer excitation and interlayer transfer mechanism.

To improve the probability of interlayer photoexcitation at band edges, the band edge components should be mainly composed of interface atoms. This can reduce the photoexcitation spatial distance of carriers. InSe, possessing high electron mobility (103-104 cm2V− 1s− 1), on/off ratio (~ 108), quantum Hall effect, and anomalous optical response 29, 30, has attracted widespread attention in electronic and photoelectric applications. More fortunately, the *p*z orbital of surface Se atoms mainly contribute to its band edge states (see Fig. 1c). This can reduce the spatial distance of photoexcitation channel after forming a vertical heterojunction with the other 2D semiconductor.

2D Sb semiconductor, a group-VA material, has been successfully prepared by various methods, such as liquid or mechanical exfoliation,31–33 vdW epitaxy growth.34 As a potential 2D photoelectric material, it not only has good photoresponse property33, 35, but also possesses high environmental stability. 36 More interesting, Sb is suitable for constructing heterojunction with InSe to explore interlayer excitation (see Fig. 1d). First of all, the VBM of Sb is located at the Γ point (see Fig. 1g), which is in agreement with the CBM of InSe. Secondly, 2D Sb is a buckled honeycomb structure, and each Sb atom is bonded to three neighboring Sb atoms. Then, the nonbonding lone pair electrons are left at the surface, which can increase the interlayer coupling after forming a vertical heterojunction. And lastly, the lattice constants of Sb (4.12 Å) and InSe (4.07 Å) are very well matched with each other. Then, by adjusting the twist angles (see Fig. 1e), the interlayer interaction can be periodically regulated, and the superlattice pattern can appear simultaneously. Therefore, the InSe/Sb heterojunction will be a suitable system to explore the direct formation of interlayer excitons and the interlayer twist regulation of interlayer excitons.

We define a highly symmetrical structure as the 0° twist angle structure as shown in Figure S1. The Sb and Se atoms at the interface of heterostructure are staggered arrangement forming strong interlayer coupling. Since 2D InSe and Sb are hexagonal lattice, after twisting 60° for one sublayer, the heterostructure can form a new highly symmetrical structure. Therefore, we evenly take seven twist angles from 0° to 60° (that is, 0.0°, 10.9°, 19.1°, 30.0°, 40.9°, 49.1° and 60.0°, see Figure S1-S8) to investigate the formation and regulation of interlayer exciton in InSe/Sb heterostructure. According to the previously widely studied twist stacking preparation strategy for 2D materials, such as graphene, hexagonal boron nitride, TMDs, the InSe/Sb heterostructures with different twist angles have the opportunity to be prepared by using the stamping dry-transfer method.19, 37, 38

As expected, InSe/Sb heterostructure is a type II heterojunction (see Fig. 2a). The VBM and CBM are located at Γ point, which are contributed by Sb and InSe sublayers, respectively. The band gap of InSe/Sb heterostructure can be significantly adjusted from 0.63 to 0.91 eV by the different twist angles (see Fig. 2a-2g). As presented in Fig. 2h, the highly symmetrical structures possess stronger binding energy (around − 45 meV per atom) than asymmetric structures (around − 35 meV per atom). It's worth noting that even for asymmetric structures, the interlayer interactions are much stronger than typical 2D vdW interaction, such as bilayer MoS2 or graphene/MoS2 heterojunctions (around − 20 meV per atom). The strong interlayer interaction of InSe/Sb heterostructure will facilitate direct interlayer photoexcitation.

As exhibited in Fig. 2i, the photoexcitation threshold of the InSe/Sb heterostructures (around 0.6 -1.0 eV) is obviously smaller than that of isolated 2D InSe and Sb layers (larger than 1.5 eV). That is, the intralayer photoexcitation cannot contribute to the first photoabsorption peak of InSe/Sb heterostructures. Simultaneously, the electronic band gap of these heterostructures is also in the range from ~ 0.6 to ~ 1.0 eV. Therefore, the first photoabsorption peak should derive from the direct interlayer photoexcitation at band edges of heterostructures. Specifically, for the interlayer photoexcitation at around Γ point, the square of the transition dipole moments (*P*2: see Fig. 2j). reaches up to ~ 300–500 Debye2, which has similar transition strength with the intralayer photoexcitation of isolated 2D InSe and Sb layers. Therefore, the direct interlayer bright photoexcitation can occur in InSe/Sb heterostructures. Note that, for InSe/Sb heterostructure with 0.0°, 10.9°, 30.0°, 49.1° and 60.0° twist angles, the first photoexcitation is bright excitation from VBM to CBM. For InSe/Sb heterostructure with 19.1° and 40.9° twist angles, the first bright excitation arises from the transition channel from VBM-4 to CBM, and VBM-3 to CBM, respectively. Therefore, the bandgap value is smaller than the first peak position for InSe/Sb heterostructure with 19.1° and 40.9° twist angles. For the calculation of transition dipole moment, we use the sum of transition between the first four valence bands and one conduction band for InSe/Sb heterostructure with 19.1° and 40.9° twist angles,

The interlayer twist can shift the peak position up to ~ 400 meV. In previous researches, the shifted range of the first photoluminescence peak is ~ 60 meV in twisted MoSe2/WS2 heterostructure,)16, ~ 90 meV in MoS2/WSe2 heterostructure39 and ~ 100 meV in WS2 bilayer38. Interesting, for InSe/Sb heterostructure, MoSe2/WS2 heterostructure and WS2 bilayer, although their shifted range of the first peak varies widely, the trend is consistent with each other. That is, for the highly symmetrical interlayer stacking heterostructure with twist angle near 0° or 60°, the energy of peak is small, while for the low symmetrical interlayer stacking heterostructures, the energy of peak is gradually increasing. the highly symmetrical interlayer stacking mode increases interlayer interaction and promotes the wavefunction coupling of band edge electronic states. this can be further confirmed by the transition probability between band edges (seeing Fig. 2j). for highly symmetrical interlayer stacking heterostructures (0° and 60°), the Γ-Γ interlayer photoexcitation possesses a larger transition probability (*P*2: >500 Debye2) compared with that of other heterostructures (*P*2: 300–400 Debye2) .

As presented in Fig. 3a, after forming heterojunction, the InSe and Sb sublayer all contribute electrons to the interface central region, which is different common phenomenon of charge transfer from one sublayer to the other.40, 41 Then, the stacked charge wavefunction distribution in the interface region can provide a buffer to increase the wavefunction overlap of interlayer transition channel, thereby in turn enhancing the transition probability of direct interlayer excitation.

The interlayer twist obviously changes the distribution of charge density difference. As shown in Fig. 3 and S9-S15, for the highly symmetrical interlayer stacking heterostructures with 0° and 60° twist angles, there accumulates more charges at interface central region compared with that of the low symmetrical modes. This further indicates that the interlayer excitation of the highly symmetrical stacked structures has a greater transition probability, consistent with the result of the transition dipole moment in Fig. 2j. Since the lattice parameter of InSe and Sb sublayer is similar to each other, for the low symmetrical modes, these heterostructures form a superlattice pattern (see Fig. 3). According to the different superlattice patterns of sectional charge distribution, the various interlayer twist angles indeed bring about different regulatory effects for electronic structure and interlayer photoexcitation properties.

After interlayer photoexcitation, the photogenerated electron and hole are localized in InSe and Sb sublayer, respectively. The lifetime of photogenerated carriers plays an essential role in various applications, such as optoelectronics, photovoltaics, and sensors. We perform the NAMD simulation by initiating the photogenerated electron at the conduction band edge of heterojunction (see Fig. 4b-4h). After the photogenerated electron recombines with the hole located in the valance band edge, the time interval is defined as the lifetime of photogenerated carriers.

For the InSe/Sb heterojunctions with different twist angles, we find that only less than 6% of photogenerated carriers recombine in 5 ns simulating time (see Fig. 4a). Although the precise lifetime is hard to obtain based on the current simulation period, we can estimate the timescale by fitting an exponential function: P(t) = exp(-t/τ). As presented in Fig. 4b-4h, for different twist angle structures, the lifetime of photogenerated carriers is as high as 96.2 to 1181.9 ns. That is, the interlayer exciton can be adjusted over a wide range by interlayer twist.

Generally, the photogenerated carrier hopping probability between two electronic states inversely depends on the square of nonadiabatic coupling (NAC) based on Fermi's golden rule. As exhibited in Fig. 5a, the largest value of NAC is less than 2.5 meV for all InSe/Sb heterojunctions with different twist angles. Such a small NAC value ensures the long-lived photogenerated carrier lifetimes. Moreover, compared with the highly symmetrical structures, the low symmetrical structures generally have a smaller NAC coefficient. To facilitate comparison, we present the averaged absolute value of NAC between the adjacent 10 band edge states in Fig. 5b-5h. The InSe/Sb heterojunction with 60° (40°) twist angle possesses the largest (smallest) NAC coefficient between VBM and CBM electronic states, which is in agreement with the longest (shortest) lifetime of InSe/Sb heterojunctions. That is, the interlayer twist has a strong regulatory effect on the photogenerated carrier lifetime. Simultaneously, the average NAC between other neighboring states is larger than that of VBM and CBM. This also facilitates the other photogenerated carriers transferring into the VBM and CBM band edge states, forming the Γ-Γ interlayer exciton.

The NAC element is dependent on the energy gap difference of hopping channel (\({\epsilon }_{k}-{\epsilon }_{j}\)), the electron-phonon coupling (\(\langle {\phi _j}|{\nabla _R}H|{\phi _k}\rangle\)) and nuclear velocity (*Ṙ**I*) describing as follows:

\({d_{jk}}=\langle {\phi _j}|\frac{\partial }{{\partial t}}|{\phi _k}\rangle =\sum\limits_{I} {\frac{{\langle {\phi _j}|{\nabla _R}H|{\phi _k}\rangle }}{{{\varepsilon _k} - {\varepsilon _j}}}\mathop {{R_I}}\limits^{ \bullet } }\)

Where H is the Kohn-Sham Hamiltonian,\({\phi _k}\),\({\phi _j}\),\({\epsilon }_{j}\),\({\epsilon }_{k}\) are the wave-functions and corresponding eigenvalues for j and k electronic states, respectively, and *Ṙ**I* is velocity of the nuclei. For InSe/Sb heterojunctions with different twist angles, since the elements and the temperature are the same with each other, the difference in NAC should derive from the energy gap difference and electron-phonon coupling term.

As discussed above, the band gap of InSe/Sb heterojunctions can be gradually adjusted from 0.63 to 0.91 eV with the change of twist angle. That is, the small energy gap difference will increase the carrier hopping probability, which is consistent with the trend of photo-generated carrier lifetime with the change of twist angle.

For the electron-phonon coupling term, its strength can be reflected by the energy fluctuation of involved electronic states. As shown in Fig. 4b-4h, the fluctuation of VBM with the time evolution is stronger than that of CBM. Hence, the vibration of Sb sublayer will play a more critical role in lifetime. To visualize the phonon modes dominating the fluctuation, the Fourier transforms (FT) of the autocorrelation functions are calculated to the time-dependent evolution of the band edges energy difference (Fig. 5i). For all InSe/Sb heterojunctions, the vibrational peaks are mainly concentrated around 50 and 200 cm− 1, attributed to the interlayer breathing mode, *A*1g mode of InSe42 and Sb 34sublayer, respectively. Clearly, the phonon spectral density changes from high to low as the different twist angle structures from high to low symmetric stacking mode. This is also consistent with the trend of photogenerated carrier lifetime.

Moreover, the pure-dephasing time between initial and final electronic states, similar to Huang-Rhys factor and Frank-Condon factor, is also associated with the lifetime of nonradiative recombination. The fast loss of quantum coherence can suppress the hopping rate. As presented in Fig. 5j, for all InSe/Sb heterojunctions, the range of pure dephasing time is from 10 to 27 fs by Gaussian fitting, exp(-0.5(t/τ)). This is obviously smaller than that of MoS2/WS2 heterojunctions with different twist angles from 37 to 58 fs.18 Hence, the small pure dephasing time also contributes to the long lifetime of photogenerated carriers for InSe/Sb heterojunctions.

For 2D materials, the excitonic effect is of significance due to the enhanced Coulomb interactions. Unfortunately, for InSe/Sb heterojunctions with various twist angles, it is not yet feasible because of excessive computational expense associated with NAMD involving thousands of electronic structure calculations. Although the single-particle picture cannot accurately obtain the lifetime of photogenerated electron and hole recombination, the change trends of lifetime should be reliable as the twist angle changes. In previous researches, Zhu et al. have studied the effect of twist angle on the photogenerated electron and hole recombination in MoS2/WS2 heterostructure and WS2 bilayer17 based on single-particle picture. They found that the larger bandgaps or smaller nonadiabatic couplings make the slower electron and hole recombination in twisted than high-symmetry structures. The trend of their results is consistent with the experiments19, 38.