Effect of the strain on spin-valley transport properties in MoS2 superlattice

The effect of the strain on the spin and valley dependent transport properties, including the conductance and polarization, through a monolayer MoS2 superlattice under Rashba spin–orbit coupling is theoretically investigated. It is found that the conductance strongly depends on the spin and valley degrees of freedom, and spin-inversion can be achieved by MoS2 superlattice. Also, the spin and valley dependent conductance in a monolayer MoS2 superlattice can be efficiently adjusted via strain and the number of the superlattice barriers. Moreover, it is demonstrated that both the magnitude and sign of the spin and valley polarization depend on the strain strength, the number of barriers, and electrostatic barrier height. Both full spin and valley polarized current (with 100% or − 100% efficiency) can be realized in a MoS2 superlattice under strain.

In recent years, two-dimensional (2D) materials have been attracted extensive interests due to their potential applications in various research fields. Graphene 1,2 is currently the most important member of the 2D materials family. Since monolayer graphene is a gapless semiconductor, it has no useful applications in the semiconductor industry, logic, and spintronic devices. Within the 2D materials, monolayer molybdenum disulfide (MoS 2 ) was successfully synthesized via several experimental techniques [3][4][5][6][7] . Unlike graphene, the monolayer MoS 2 is a direct bandgap semiconductor with a tunable bandgap 8 . Due to the heavy transition-metal atoms, the monolayer MoS 2 has a strong spin-orbit coupling (SOC) 9 . Furthermore, in the monolayer MoS 2 , in the Brillouin zone two inequivalent valleys (K and K′) are separated by a large momentum 10 . Additionally, due to the symmetry breaking and large SOC in the monolayer MoS 2 , it is possible to control and tune the spin and valley polarization properties in the monolayer MoS 2 -based systems [11][12][13] . Moreover, the monolayer MoS 2 , in the presence of the Rashba spin-orbit coupling (RSOC), is a fascinating material for spintronics applications. In the monolayer MoS 2 the RSOC can easily be induced and tuned via an external electric field 14 or a ferromagnetic exchange field 15 . On the other hand, the electronic, optical, and transport properties of the monolayer MoS 2 can be modulated by applying an external strain [16][17][18][19][20] . The strain can be induced in the MoS 2 sheet by substrate 21 or during the CVD growth 22 . In recent years, electron, spin and valley-dependent transport properties were reported extensively in monolayer MoS 2 structures, both experimentally and theoretically [23][24][25][26][27][28][29][30][31][32][33][34][35][36] . Fontana et al. 23 experimentally investigated the transport properties of the electron and hole in a gated MoS 2 Schottky barrier and found that in this structure, the source and drain electrodes' materials are essential keys in controlling the transport through the conduction or valence band. Rotjanapittayakul et al. 27 theoretically studied the magnetoresistance and spin injection in a MoS 2 junction, and demonstrated that a magnetoresistance and spin injection efficiency of the order of 300% and 80%, respectively, can be observed in a MoS 2 -based tunnel junction. The effect of the Rashba spin-orbit interaction on the thermoelectric properties of monolayer MoS 2 nanoribbon is described in Ref. 31 , in which the authors presented that the magnitude and sign of Seebeck thermopower can be tuned by adjusting the structure's parameters. Besides, the superlattices structures 37 provide a new way for controlling the transport properties. Recently a great deal of attention has been addressed to the transport properties in superlattice-based 2D materials [38][39][40][41][42][43][44][45][46][47][48][49][50] . Yu and Liu 41 studied the spin transport properties through monolayer and bilayer graphene superlattice. They showed that the monolayer and bilayer graphene superlattice with zigzag boundaries could be used for perfect spin-filtering. Zhang et al. 42 demonstrated that a controllable spin and valley polarized current could be obtained in a silicene superlattice in the presence of the electric and magnetic field. The effect of the strain on the electronic properties in the MoS 2 -WSe 2 moiré superlattice was investigated by Waters et al. 49 . They found that in-plane strain and out-of-plane deformations significantly impact the MoS 2 -WSe 2 moiré superlattice's

Model and methods
In this work, we are interested in the spin-and valley-dependent transport properties in a MoS 2 superlattice with RSOC in the presence of a uniaxial strain. A series of metallic gate voltages on the top of monolayer MoS 2 with a suitable substrate can be used to get a monolayer MoS 2 superlattice in the presence of RSOC and strain. In our model, the RSOC region with a gate voltage and strain (barrier region) is separated by a normal monolayer MoS 2 (N MoS 2 ), in which there is no RSOC and strain (well region). The schematic of our proposed device structure is shown in Fig. 1. The growth direction of the superlattice is along the x-axis. We assume that the strain is applied in the armchair direction. The strain tensor, є, can be written as follows 20 where α is the angle between the x-axis and the direction of the strain. For the armchair direction strain α = 0 . ε is the strain strength and µ = 0.25 is the Poisson's ratio for the MoS 2 51 . In the considered structure, the lowenergy effective Hamiltonian of the carriers near the K and K′ valleys can be written as: with Here, σ = (σ x , σ y , σ z ) and s = (s x , s y , s z ) are the Pauli matrices for the sublattice and the spin spaces, respectively. v F ≈ 5.3 × 10 5 m/s denotes the Fermi velocity in a monolayer MoS 2 , η = +1/ − 1 is the valley index ( η = 1 for K and η = −1 for K′ valley) and s z = +1(−1) denotes the electron with the spin-up (down). = 833 meV is the energy gap 12 in the monolayer MoS 2 , R is RSOC strength and = 37.5 meV is the spin-splitting energy of the valence band caused by the spin-orbit coupling 12,15 . U 0 displays the electrostatic potential barrier's height, Î is a unitary matrix, U(α) = diag(1, e iα ) denotes the unitary matrix, which performs a rotation in the sublattice space 52 and for the monolayer MoS 2 x = 2.2 and y = − 0.5 53 . Also, the longitudinal k x ( k s ′ (s) ) and the transverse k y ( q y ) components of the wave vectors in the N MoS 2 regions (in the ERSOC regions) with k ( k ′ s ′ (s) ), respectively. k and k ′ s ′ (s) are given by: Let us now consider electrons with the angle of incidence of ϕ , spin s and energy E will go towards the monolayer MoS 2 superlattice from the left side. The spin and valley dependent wave function in the RSOC ( ψ ± s,η ) and the normal ( ψ ± Ns,η ) regions can be given by: The spin and valley dependent transmission probability, T s ′ sη , (with the spin s =↑, ↓ to be transmitted to the spin s ′ =↑, ↓ ) through the monolayer MoS 2 superlattice with N electrostatic barriers can obtained by applying the boundary conditions and using the transfer matrix approach 54,55 . Then, the spin and valley dependent conductance of a monolayer MoS 2 superlattice under strain and RSOC is defined as 56 :

Results and discussion
In the following, we consider the effect of both the strain and the RSOC on the spin and valley dependent conductance and polarization through the monolayer MoS 2 superlattice, as shown in Fig. 1. Here, we fix the parameters as; E = 1.5 , U 0 = 3.5 , R = 50 meV, the normalized barrier and the normal region width as k F0 b = 5 (k F0 = E/ v F ) and k F0 w = 3 , respectively. First, we investigate the valley and spin-dependent conductance as a function of the strain strength ( ε ) with a different number of electrostatic potential barriers. As shown in Fig. 2, the conductance for the valley K 1 depends on the spin degree of freedom and the number of the electrostatic barriers. Also for N > 2 , the conductance without and with the spin-flip, shows an oscillatory behavior with respect to ε . Due to the strain and the normal regions' interface, more resonant peaks appear in the conductance by increasing the number of barriers in the superlattice. It is evident from Fig. 2 that the G ↑↑k 1 /G 0 and G ↓↑k 1 /G 0 decrease by increasing the strain strength. For a large strain and when number of superlattice barriers is big enough ( N ≥ 4 ), spin-dependent conductance tends to zero. This is due to the evanescent states in the strain region. According to Fig. 2a, it is clear that for ε ≥ 0.08 , the value of G ↑↑K 1 /G 0 = 0 and G ↓↑K 1 /G 0 � = 0 . In other words, electrons could transmit through the monolayer MoS 2 superlattice only with spin-flip, In this case the spin state of outgoing electrons were inverted by using monolayer MoS 2 superlattice. So, the monolayer MoS 2 superlattice acts as a spin inverter. The conductance for the valley K 2 is presented in Fig. 3. It is observed that for ε > 0.04 the spin-dependent conductance, without and with spin-flip, in a monolayer MoS 2 superlattice has a zero value due to the evanescent waves. This leads to a gap in the spin-dependent conductance with respect to the

Conclusion
In summary, we have investigated the spin and valley-dependent transport properties in a monolayer MoS 2 superlattice under uniaxial strain and RSOC. We found that the strain has a significant effect on the spin and valley dependent conductance, without and with the spin-flip. Furthermore, we showed that the valley and spin dependent conductance have a gap regarding the strain, which allows the valley and spin conductance to www.nature.com/scientificreports/ Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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