1 Introduction

With the rapid development in the field of spintronics and optoelectronics, the study of materials with spin–orbit coupling (SOC) has attracted more and more attention, and various novel physical phenomena caused by the SOC effect in related materials are increasingly reported [1]. The SOC effect is a physical property in two- or three-dimensional materials, which originates from the relativistic motion of electrons and is important as an effective internal magnetic field to promote the electron-spin operation [2,3,4,5]. As a kind of the SOC effect, the Rashba effect is induced by surface asymmetry or structural asymmetry. Due to the locking between spin and electron momentum, this effect has caused a great deal of interest in semiconductor spintronic and optoelectronics devices [6,7,8,9,10].

In recent years, van der Waals (vdW) heterostructures formed by vertical stacks of two-dimensional materials have facilitated extensive researches. VdW heterostructures not only retain the excellent optoelectronic properties of each layer due to weak interlayer coupling, but also possess other unique physical properties of the heterostructures themselves [11,12,13,14,15,16,17,18,19]. Two-dimensional transition metal dichalcogenides (TMDs) as one type of the most widespread materials in nano device applications are rich in electrical performance, adjustable band gap, good thermal stability and strength of spin-orbit coupling effect, and other unique advantages. TMDs show excellent electronic and optical properties among many two-dimensional materials, and are expected to be used in semiconductor devices and spintronic devices [20,21,22,23,24,25,26,27,28,29,30,31,32]. Heterostructures composed of TMDs with different layer materials and their modulation have become a frontier of research today, which can be performed by strain engineering [33,34,35,36,37,38], doping [39], and external electric field [40]. Among them, strain engineering is an effective method to change the electronic, spin and photoelectric properties.

In this paper, we consider one construction of the MoSSe/WSe\(_2\) vdW heterostructures, this structure has been shown to be thermodynamically stable [41, 42]. We study the electronic structure and optical properties of MoSSe/WSe\(_2\) heterostructures by first principles, and discuss the variation of Rashba spin splitting and optical absorption under biaxial strain. The calculation results show that the Rashba spin splitting is seen at the valence band \(\Gamma \) point; moreover, the heterostructure is a type II energy band alignment. In addition, strain affects the heterogeneous Rashba spin splitting, indicating that the intensity of its Rashba SOC is tunable. On the other hand, compared with MoSSe and WSe\(_2\) monolayer, the optical absorption of MoSSe/WSe\(_2\) vdW heterostructures is improved. Red-shift and blue-shift of the heterostructure in optical absorption under the biaxial strain can be used for designing strain-tunable spintronic and optoelectronic devices.

Fig. 1
figure 1

a–f Top and side views of the six stacked structures of MoSSe/WSe\(_2\) vdW heterostructures [41]. Their unit cells are represented by the rhombuses in the top panel and rectangles in the bottom panel

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2 Computational method

All calculations are performed based on density functional theory (DFT) by using the Linear Combination of Atomic Orbitals (LCAO) method as implemented in QuantumATK software package [43], the obtained results are consistent with the plane-wave method. The generalized gradient approximation (GGA) with Perdew–Burke–Ernzerhof parametrization (PBE) functional is used for the exchange-correlation functional calculations. The DFT-D2 dispersion correction method is applied throughout the calculations. The density grid truncation energy is set to 360 Ry, and a 15\(\times \)15\(\times \)1 \(\Gamma \)-centered Monkhorst–Pack K-mesh grid is used in the Brillouin zone for structural optimization and performance calculations. The convergence thresholds for energy and atomic forces are fixed at \(10^{-4}\) eV and \(10^{-5}\) eV/Å , respectively. 20 Å vacuum layer along z -axis is added to interrupt the artifacts of the periodic boundary conditions, SOC is taken into account in this work.

3 Results and discussion

3.1 Structural stability

Both MoSSe and WSe\(_2\) monolayers have the hexagonal honeycomb structure in the x-y plane, the geometrically optimized MoSSe has lattice constant of \(a=b\)=3.25 Å and a direct band gap of 1.56 eV, while the WSe\(_2\) has lattice constant of 3.32 Å  and a direct band gap of 1.56 eV. To construct the MoSSe/WSe\(_2\) heterostructure, the lattice mismatch degree between two monolayers can be defined as \(\Delta a=\frac{2(a_{WSe_2}-a_{MoSSe})}{a_{WSe_2}+a_{MoSSe}}\), where \(a_{WSe_2}\) and \(a_{MoSSe_2}\) denote the lattice constants of WSe\(_2\) and MoSSe\(_2\) monolayers, respectively. The lattice mismatch rate is about 2.13% , which is in a reasonable range and indicates that MoSSe/WSe\(_2\) heterostructures can be formed well. There are six possible stacking manners of MoSSe/WSe\(_2\) heterostructures considering different alignments between layers, as shown in Fig. 1 [41]. If the MoSSe structures are rotated 180 degrees in Fig. 1, six new stacking orders can be obtained, Se atomic contact between two layers. However, we only discuss the six stacking structures in Fig. 1, for them the interaction between the two monolayers is stronger.

Table 1 The lattice constants a, interlayer distances d, and binding energy \(E\) \(_b\) of MoSSe/WSe\(_2\) vdW heterostructure for six constructed patterns
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Fig. 2
figure 2

a Phonon spectrum of MoSSe/WSe\(_2\) heterostructure. b Project band structure considering spin-orbit coupling heterostructure, MoSSe monolayer in red, WSe\(_2\) monolayer in blue, inset is the Brillouin zone. c Schematic of spin texture and Rashba spin splitting. d Partial state density for heterostructures

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The most stable stacking structures can be determined by the binding energy calculated from

$$\begin{aligned} \begin{aligned} E_b=E_{MoSSe/WSe_2}-E_{MoSSe}-E_{WSe_2}, \end{aligned} \end{aligned}$$
(1)

where \(E_{MoSSe/WSe_2}\) is the total energy of MoSSe/WSe\(_2\) vdW heterostructures, \(E_{MoSSe}\) (\(E_{WSe_2}\)) is the total energy of Janus MoSSe (WSe\(_2\)) monolayer. The lattice constants a, interlayer distances d, and binding energy \(E\) \(_b\) of the heterostructures are calculated as shown in Table 1. All six stacking structures show negative binding energies, indicating that the MoSSe/WSe\(_2\) vdW heterostructures are thermodynamically stable. The six stacking structures in Fig. 1b with lattice constant 3.289 Å  has the lowest binding energy, and is the most stable. In the following calculations, we all use the MoSSe/WSe\(_2\) vdW heterostructure in Fig. 1b.

3.2 Rashba spin splitting, band alignment and work function without strain

To assess on the structural stability of the MoSSe/WSe\(_2\) vdW heterostructure in another approach, we calculate the phonon spectra of the heterostructure, as shown in Fig. 2a. There is no imaginary mode in the entire Brillouin zone, indicating that the MoSSe/WSe\(_2\) vdW heterostructure is thermodynamically stable. The energy band structure of the MoSSe/WSe\(_2\) vdW heterostructure at the high symmetry points in the Brillouin zone considering the spin-orbit coupling is shown in Fig. 2b, different colors are used to mark the contributions from different monolayers, that is, red for the MoSSe monolayer and blue for the WSe\(_2\) monolayer. Direct band gaps are observed, with the conduction band minimum (CBM) and valence band maximum (VBM) located at the K point of the Brillouin zone, the band gap is 0.47 eV. Schematic of spin texture and Rashba spin splitting are shown in Fig. 2c. The two bands have an intersection, at the so-called Dirac point, that is energy degeneracy and no momentum shift, so the energy difference between energy maximum and the Dirac point is defined as the Rashba splitting energy \(E\) \(_R\), the momentum offset between the two positions is \(K\) \(_R\). The Rashba parameter can be expressed as \(\alpha _R=2E_R/K_R\), it can characterize the strength of the Rashba SOC, \(\alpha _R\) of the heterostructure is 0.47eV/Å . The partial density of states of the MoSSe/WSe\(_2\) vdW heterostructure is shown in Fig. 2d, it can be seen that the conduction band bottom is mainly contributed by the Mo-d orbitals of the MoSSe monolayer, and the valence band top is mainly contributed by the W-d orbitals of the WSe\(_2\) monolayer.

Fig. 3
figure 3

a Energy band alignment and work function of the MoSSe/WSe\(_2\) heterostructure, MoSSe monolayer in red, and WSe\(_2\) monolayer in blue. b The electrostatic potential in the vertical direction of the MoSSe/WSe\(_2\) heterostructure

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The alignment of the bands near the Fermi level is shown in Fig. 3a. By comparing the energy bands of the two monolayers, one can see that the heterostructure perfectly retains the characteristics of the two monolayers. The conduction band of the heterostructure mainly originates from the WSe\(_2\) monolayer, while the valence band mainly originates from the MoSSe monolayer. It is found that the heterostructure is a typical type II energy band (staggered band gap) alignment. The type II energy band alignment can promote the separation and migration of electron and hole pairs, effectively reducing the recombination rate of carriers. The band gap of the heterostructure is significantly smaller than that of the two monolayers, when irradiated by light, the electron excitation in the heterostructure is easier than that of two monolayers.

The work function is an important parameter to determine the optoelectronic properties of a material, from it the phenomenon of electron transfer can be well explained, smaller work function can improve the optoelectronic properties of materials. The work function is defined as

$$\begin{aligned} \begin{aligned} W=E_{\mathrm{{VAC}}}-E_\mathrm{{F}}, \end{aligned} \end{aligned}$$
(2)

where the \(E_{\mathrm{{VAC}}}\) and the \(E_\mathrm{{F}}\) are the vacuum level and the Fermi level a system, respectively. The calculation results show that the work functions of MoSSe monolayer and WSe\(_2\) monolayer are 5.22 eV and 4.33 eV, respectively. The work function of MoSSe/WSe\(_2\) heterostructure is 4.76 eV, which is between those of the MoSSe monolayer and WSe\(_2\) monolayer. This shows intuitively that the difference between the work functions of the two monolayers. After the contact between the MoSSe monolayer and the WSe\(_2\) monolayer, electrons will diffuse from the WSe\(_2\) monolayer to the MoSSe monolayer, and holes from the MoSSe monolayer to the WSe\(_2\) monolayer. Thus, it leads to a downward shift of the Fermi energy level of the WSe\(_2\) monolayer and an upward shift of the Fermi energy level of the MoSSe monolayer. The final equilibrium state is reached at the 4.76 eV energy level, which is attributable to the spontaneous diffusion behavior of electrons and holes between layers.

The MoSSe monolayer has a work function about 0.745 eV due to the broken mirror symmetry, i.e., the S and Se atoms get electrons from the Mo atoms, but the potential energy is smaller and the work function is larger on the S side than that on the Se side. The electrostatic potential along the vertical direction of the heterostructure is shown in Fig. 3b. The heterostructure leads to the destruction of mirror symmetry in the vertical direction, making the potential energies of the two sides unequal. The potential difference \(\Delta \) \(P\) between the heterostructure interfaces is 5.3 eV, and the electrostatic potential of the WSe\(_2\) monolayer is higher than that of the MoSSe monolayer. Generation of a built-in electric field directed from the WSe\(_2\) monolayer to the MoSSe monolayer implies the electron transfer from the WSe\(_2\) monolayer to the MoSSe monolayer.

Fig. 4
figure 4

a Strain energy and band gap of MoSSe/WSe\(_2\) vdW heterostructure at different strains. b Magnitude of the heterostructure work function at different strains. c Enlarged view of spin-orbit splitting in the valence band near the \(\Gamma \) point at different strains. d The Rashba parameter (\(\alpha _R\)) under different strains

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3.3 Rashba spin splitting, energy gap, and optical absorption with strain

Applying strain is an effective approach to modulate the electronic properties of 2D materials by artificially changing the lattice constants. We impose an in-plane biaxial strain on the MoSSe/WSe\(_2\) heterostructure, and investigate the changes of its electronic and optical properties with strain. The strain is defined as \(\varepsilon =[(a-a_0)/a_0]100\%\), where a is the lattice constant under a strain, \(a_0\) is the lattice constant for non-strain, \(\varepsilon \) is negative for compressive strain, and is positive for tensile strain. The strain energy and band gap of the heterostructure at different strains are shown in Fig. 4a. The strain energy can indicate whether the strain exceeds the elastic limit of the structure and is calculated as

$$\begin{aligned} \begin{aligned} E_s=E_\varepsilon -E_0, \end{aligned} \end{aligned}$$
(3)

where \(E_s\) is the strain energy, \(E_\varepsilon \) is the energy of the heterostructure at a certain strain, and \(E_0\) is the energy of the heterostructure at equilibrium. It can be seen that the functional relationship between strain energy and strain is a smooth parabola, and we apply the strain to 8%, the strain energy and strain as a function of the smooth parabola remains the same. This means that the process of applying a certain degree of biaxial strain is a reversible elastic deformation, and will not destroy the structure of the heterostructure. It can be observed that the band gap gradually increases with increasing strain from −6 to −3%, and then gradually decreases, even disappears, and becomes metallic at the compressive strain of 5%, arrives at a semiconductor–metal transition. These results suggest that biaxial strain can change the size and type of a band gap.

Fig. 5
figure 5

Optical properties of the WSe\(_2\) monolayer, MoSSe monolayer, and MoSSe/WSe\(_2\) vdW heterostructure. a The real parts of the complex dielectric function. b The imaginary parts of the complex dielectric function. c Optical absorption of the MoSSe/WSe\(_2\) vdW heterostructure with biaxial strain. d The real parts of the complex dielectric function. e The imaginary parts of the complex dielectric function. f Optical absorption

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In addition, the strain also changes the spintronic properties of the MoSSe/WSe\(_2\) vdW heterostructure. The work function of the MoSSe/WSe\(_2\) heterostructure under various biaxial strains with considering SOC is shown in Fig. 4b, and it is found that the work function gradually increases under tensile strain and decreases under compressive strain, the change in the magnitude of the work function leads to a change in electron transfer. The magnification of the spin–orbit splitting in the valence band near the \(\Gamma \) point under different strains is shown in Fig. 4c, it is obvious that the spin–orbit splitting decreases significantly in the tensile strain range, and increases gradually in the compressive range. The magnitude of the Rahba spin-orbit splitting is closely related to the built-in electric field in the heterostructure. It can be understood as the tensile strain condition, the distance decrease, the magnitude of the built-in electric field increases as the interlayer distance decrease, which leads to the decrease of the spin-orbit splitting. This indicates that the magnitude of Rashba spin-orbit splitting can be effectively regulated by applying a biaxial strain, which has great application in spintronic devices. The Rashba parameters at different strains are shown in Fig. 4d, and it can be found that the parameter decreases with increasing tensile strain and increases with increasing compressive strain.

The strain and spin–orbit coupling lead to changes in the electronic structure of the heterostructure, which in turn has a great impact on its optical properties. The frequency-dependent dielectric function reflects the response of the medium to light, and the formula is

$$\begin{aligned} \begin{aligned} \varepsilon (\omega )=\varepsilon _1(\omega )+\varepsilon _2(\omega ), \end{aligned} \end{aligned}$$
(4)

where \(\omega \) is the frequency of the incident light, \(\varepsilon _1(\omega )\) and \(\varepsilon _2(\omega )\) are the real and imaginary parts of the dielectric function, respectively. \(\varepsilon _1(\omega )\) mainly responds to the phase modulation that is the dispersion effect, \(\varepsilon _2(\omega )\) mainly reflects the amplitude modulation that is absorption loss or gain. The related optical absorption equation can be expressed as

$$\begin{aligned} \begin{aligned} \alpha (\omega )=\frac{\sqrt{2}\omega }{c}\left\{ {\left[ \varepsilon _1^2(\omega )+\varepsilon _2^2(\omega )\right] ^\frac{1}{2}-\varepsilon _1(\omega )}\right\} ^\frac{1}{2}. \end{aligned} \end{aligned}$$
(5)

The real and imaginary parts of the dielectric function and the optical absorption are shown in Fig. 5a–c. Blue represents the WSe\(_2\) monolayer, red represents the MoSSe monolayer, and black represents the MoSSe/WSe\(_2\) vdW heterostructure, the solid line indicates that the spin–orbit coupling is considered, and the dashed line does not consider the spin-orbit coupling. The real part of the dielectric function is shown in Fig. 5a, it can be observed that the real part of the heterostructure is negative in the range of 2.9–6 eV, this shows that the absorption is dominant, and this is reflected in the optical absorption. The static dielectric function of the heterostructure is 3.89, which is greater than that of two the monolayers, indicating that the heterostructure has a stronger binding ability to charges and a stronger polarization ability. The imaginary part of the dielectric function is shown in Fig. 5b, when the light is perpendicular to the surface of the heterostructure, most of the peaks of the imaginary part are distributed in the range of 1.65–5 eV, and in the visible light range. The imaginary part of the heterostructure is greatly improved compared to the two monolayers, consistent with the absorption coefficient.

The optical absorption is shown in Fig. 5c, the strongest peak appears at 3.05 eV, up to 6.11\(\times \) \(10^5\) cm\(^{-1}\), located in the visible light region, and the absorption range is from the visible light to the ultraviolet region. Optical absorption is significantly improved compared to the two monolayers, since the van der Waals interaction, the orbital hybridization, and interlayer coupling between the monolayers lead to new interband jumps, so the MoSSe/WSe\(_2\) vdW heterostructure has a higher optical absorption coefficient. The value of the intersection between the optical absorption and the x-axis is the optical band gap, the optical band gap of the WSe\(_2\) monolayer is 1.56 eV, that of the MoSSe monolayer is 1.56 eV, and that of MoSSe/WSe\(_2\) vdW is 0.5 eV. It is well known that when the optical band gap is small, photoelectrons can be excited with less energy. Compared to the two monolayers, the optical band gap of the heterostructure is significantly reduced, suggesting that we can use lower energies to excite photoelectrons in the heterostructure. It is found that the optical absorption peak of the heterostructure decreases under the action of the SOC, which affects the electronic structure.

The real and imaginary parts of the dielectric function and optical absorption of the heterostructure under a biaxial strain are shown in Figs. 5d–f. The real part of the dielectric function is shown in Fig. 5d, the static dielectric function increases under tensile strain and decreases under compressive strain. The imaginary part of the dielectric function is shown in Fig. 5e, the peak in the visible light range increases continuously with increasing tensile strain, and decreases continuously with increasing compressive strain, consistent with optical absorption. As shown in Fig. 5f and its partially enlarged insertion, the optical absorption of the MoSSe/WSe\(_2\) vdW heterostructure is presented at different biaxial strains. We find that the first absorption peak of the heterostructure increases with increasing tensile strain and the absorption edge is red shifted; with the increase of compressive strain, the first absorption peak of the heterostructure decreases and the absorption edge is blue shifted. It is shown that the MoSSe/WSe\(_2\) vdW heterostructure is a material with good optoelectronic properties for strain-tunable optoelectronic devices, broadening the application range of heterostructures.

4 Conclusions

In this paper, we have inspected the six configurations of MoSSe/WSe\(_2\) vdW heterostructures by the first-principles calculations based on the density functional theory, and found the most stable stacking. We have studied the Rashba spin splitting and optical absorption of this heterostructure, and obtained a direct band gap of 0.47 eV with the type II energy band alignment under the consideration of spin-orbit coupling. The Rashba splitting is found at the \(\Gamma \) point and it can effectively promote the separation and migration of electron and hole pairs. We have observed the change of the Rashba spin splitting after the application of biaxial strain, then verified that the tension suppresses the Rashba spin splitting and compression promotes the Rashba spin splitting. Compared with the MoSSe and WSe\(_2\) monolayers, the optical absorption of the MoSSe/WSe\(_2\) vdW heterostructure is improved in the visible light range, the optical absorption peak of the heterostructure decreases under the action of SOC. Under biaxial strain, stretching red-shifts the heterostructure optical absorption, and compression blue-shifts the heterostructure optical absorption. The best results are at the tensile strain 4%, where the Rashba spin splitting is maximum and there is maximum optical absorption at around 6 eV. Its tunable electronic and photonic properties provide many application opportunities for strain-tunable electron spin and optoelectronics.