A monolayer of molybdenum disulfide (MoS₂), which has a honeycomb crystal structure, is a prototypical transition-metal dichalcogenide (TMD). Its phonon-limited room temperature mobility is dominated by the optical deformation potential and polar optical scattering versus Frohlich interaction.^[1] The dominating deformation potentials originate from couplings to the intervalley longitudinal optic (LO) and homopolar phonons.^[1] It should be noted that due to the valley degeneracy in the conduction band, both intervalley and intravalley scatterings of carriers should be considered for calculating the electron–phonon coupling constant.^[1] Roldan et al. have studied the origin of superconductivity in heavily doped MoS₂ by considering the electron–phonon and electron–electron interactions.^[2] They have shown that the intravalley (intervalley) electron–phonon coupling constant is equal to −0.36 (−0.13).^[2] It has been shown that heavily gated thin films of MoS₂ become superconductive.^[3,4] The Rashba spin–orbit coupling (RSOC) arouse in the experiment due to the presence of the strong gating field of the order of 10 MeV/cm.^[5–7] The RSOC induced two superconducting phases which can be topologically nontrivial.^[8] Lu et al. have shown that the Zeeman field, which is originated from the intrinsic SOC induced by breaking in-plane inversion symmetry, pins the spin orientation of the electrons to the out-of-plane direction and protects the Ising superconductivity in gated MoS₂.^[9] Also, it has been shown how the spin-triplet p-wave pairing symmetry affects the superconducting excitations.^[1]

From civil practical point of view, increasing the critical temperature of superconductors (T_c) is a necessary condition. For increasing T_c, both the density of states (DOS) and the vibration frequency of a superconducting material should be high.^[11] For non-metallic materials or materials which have low DOS, applying strain and doping can be an effective method to induce superconductivity.^[12] The elastic bending modulus of monolayer MoS₂ has been studied and it has been shown that the binding modulus is equal to 9.61 eV.^[13] Woo et al. have used a first-principle approach and studied the elastic properties of layered two-dimensional materials.^[14] They have found that the Poisson’s ratios of graphene, h-BN, and 2H-MoS₂ along the out-of-plane direction are negative, near zero, and positive, respectively, whereas their in-plane Poisson’s ratios are all positive.^[14] Two important and main crystal deformations are mechanical deformation and curvature of crystal lattice. The bond lengths of TMDs are changed under mechanical deformations and in consequence their electronic structures are changed due to the corrections in the electronic Hamiltonian. However, the curvature of the crystal lattice mixes the orbital structure of the electronic Bloch bands. Pearce et al. have presented an effective low energy Hamiltonian for describing the effects of mechanical deformations and curvature on the electronic properties of monolayer TMDs.^[15] In sodium intercalated bilayer MoS₂, the electron–phonon interaction strength changes and the superconductivity is significantly enhanced due to the strain effect.^[16] Similar work has been reported about calcium doped MoS₂ bilayer recently.^[14] Finally, Kang et al. have reported the discovery of Holstein polarons (a small composite particle of an electron that carriers a cloud of phonons) in a surface-doped layered semiconductor MoS₂^[18] and He et al. have shown that in a wide range of experimentally accessible regime where the in-plane magnetic field is higher than the Pauli limit field but lower than second critical magnetic field, a 2H-structure monolayer NbSe₂ or similarly TaS₂ becomes a nodal topological superconductor.^[19]

In this paper, we study the spin-singlet p + ip topological superconductivity in monolayer MoS₂ using a simple tight-binding Hamiltonian when the crystal is (not) under uniaxial strain.

Xiao et al. have considered the molecular orbitals d_z² and as basis functions of conduction and valence bands, respectively (τ = ± 1 is the valley index). They have shown that, to the first order in k, the C_3h symmetry dictates a two-band k · p Hamiltonian.^[2] Cappelluti et al. have shown that the 4d_3z² −r² (82%), 3p_x, 3p_y (12%), and other orbitals (6%) contribute in constructing the minimum of the conduction band (MCB).^[21] Roldan et al. have studied the superconductivity effect in heavily doped MoS₂ by considering the d_3z² − r² orbital as the main orbital component.^[2] The possible topological superconductivity phase^[8] and strongly enhanced superconductivity^[12] in MoS₂ have been studied. In both studies, the d₃z² − r² orbital is considered as the main orbital component.^[8,12] Therefore, we consider a single band Hamiltonian model including the intrinsic SOC (ISOC) and RSOC near K and K′ points and show that there are two band inversion (nodal) points which separate unoccupied states from occupied ones in the phase diagram. Using the threefold rotation symmetry of the crystal and the properties of rotation fixed points Γ, K, and K′, we calculate the Chern number.^[22] It is shown that when the chemical energy is greater (smaller) than the ISOC strength, the Chern number is equal to four (two) and otherwise it is equal to zero. Since, the 4d_z² orbital is the dominant component of the states we only consider the Mo-layer with triangular lattice structure for doing the numerical calculations. The Mo-layer has both flat and zigzag edges but we consider a zigzag nanoribbon of Mo-layer and introduce a simple tight-binding Hamiltonian for studying the topological superconductivity in MoS₂. It is shown that when the superconductivity energy gap is smaller than the ISOC and the chemical potential is greater than the ISOC, the zero energy Majorana states exist. Finally, we show that the topological superconducting phase is preserved under uniaxial strain. Using a simple tight binding method and crystal symmetry for studying the topological properties and justifying the effect of uniaxial strain on the topological properties are the novelties of the article in comparison with the other published ones.^[8] The structure of this article is as follows. The analytical calculations are presented in Section 2. In Section 3, the tight-binding Hamiltonian is introduced. The results are explained and discussed in Section 4. In Section 5, the summary is presented.

As it has been shown, the 4d_z² orbital is the dominant component of the states near the CBM and the VBM located at K and K′ points.^[2,8,12] Therefore, we consider a zigzag nanoribbon of Mo-layer with triangular crystal structure (Fig. 1(a)) and use a simple tight-binding Hamiltonian for studying the topological superconductivity in MoS₂. Rostami et al. have shown that if the basis ψ_Mo = (d_z², d_x² − y², d_xy) is used, the Hamiltonian of Mo–Mo hopping can be written in k-space as^[31]

Another simple model can be used to study the effect of the chemical potential, respect to the ISOC strength, on the topological properties of MoS₂. In this model, only the d_z² orbital is considered, and the term −μ + ε β_so is placed on the diagonal of . For hopping between two Mo-nearest neighbor atoms, the hopping energy t = (3V_dd δ + V_ddσ)/2 is used in all directions^[29] and the above explained method is followed again. It should be noted that in calculations, Δ = Δ₀ e^{i β_ij},^[8] V_ddδ = 0.228 eV, and V_ddσ = −0.895 eV.^[29]

The eigenvalues of Eq. (4) at K and K′ points were found in Section 2. Figure 2 shows them schematically. As Fig. 2 shows, the curves of negative energies touch the curves of positive energies at two points A and B which are called the band-inversion (nodal) points. However, the points are placed on the line E/β_so = 0. Therefore, band closing happens at K and K′ points. Since the K and K′ points are rotation fixed points, it is possible to label the states at the fixed points by their rotation eigenvalues which are e^{iπ(2p −1)/n} with p = 1,2,…, n (fold).^[22] If R stands for the rotation matrix for both electron and hole, a BdG Hamiltonian is rotation-invariant if it satisfies^[28]

Fig. 2.

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	Fig. 2. Energy dispersion curve (phase diagram) of superconducting MoS2 at K (K′)-point. The dashed and filled lines show the negative and positive energies, espectively. The points A and B stand for band-inversion (nodal) points. The general equation E/βso = ± (1 ± μ/βso) is used here.

Let us consider a nanoribbon of Mo-atoms with zigzag edge as shown in Fig. 1(a) and use Eqs. (10) (in real-space) and (11) to find the energy dispersion curve E(k). It can be shown that the edge states can also be found for the flat edge of the triangular lattice of Mo layer.^[8] Therefore, the similar behavior is expected. Figure 3(a) shows the energy dispersion curve of the nanoribbon. As it shows, the valence and conduction bands intersect with each other and since the DOS at zero energy has significant value (Fig. 3(b)), there are surface states and the ribbon behaves as a metal. It should be noted that the RSOC is not considered here. Since we assume β_so = 60 meV,^[29] the difference between spin up and down electrons is negligible, and it is not shown.

Fig. 3.

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	Fig. 3. (a) Energy dispersion curve of a zigzag nanoribon of Mo atoms and (b) density of states.

By using the model of zigzag nanoribbon described in Eqs. (12) and (13), the energy dispersion curve of the BdG Hamiltonian is found without RSOC. Figures 4(a) and 4(b) show the E(k) curves for β_so > Δ₀ and β_so < Δ₀. As Fig. 4(a) shows, there are four zero energy states when β_so > Δ₀ while it decreases to two for β_so < Δ₀. In addition, figure 4(c) shows that there are zero energy states when β_so = Δ₀. Therefore, it can be concluded that the model can predict the same topological properties which were found by using the Eqs. (16)–(21) above. The zero energy states are referred to the Majorana states.

Fig. 4.

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	Fig. 4. Energy dispersion curve of BdG Hamiltonian with (a) βso = 60 meV, Δ0 = 6 meV, (b) βso = 6 meV, Δ0 = 60 meV, and (c) βso = 60 meV, Δ0 = 60 meV. The RSOC = 0 here. Each supercell includes 9 Mo atoms.

Here, the simpler model which was introduced at the end paragraph of Section 3 is used to study the effect of chemical energy in respect to the ISOC strength. Figure 5(a) shows the energy dispersion curve when β_so = 60 meV, Δ₀ = 6 meV, and μ = 87 meV.^[30] As it shows, since μ > β_so, there are four zero energy states while for μ = 57 meV, there are two zero energy states (Fig. 5(b)). The model shows that when μ ≫ β_so, a gap is opened in the curve (it is not shown in the figures). Finally, as Fig. 6 shows, the DOS has significant values at zero energy for both cases, i.e., μ > β_so and μ < β_so. Therefore, the simple model can explain the analytical results qualitatively (at least). Through fourth-order Gingzburg–Landau analysis, Yuan et al. have shown that the time reversal symmetry is spontaneously broken in the E irreducible representation of C_3v and this phase is characterized by Chern numbers and supports Majorana edge states.^[8] In Fig. 4, the edge states can be recognized for E = 0 eV which are related to the different Chern numbers. These edge states are attributed to the Majorana state.^[8]

Fig. 5.

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	Fig. 5. Energy dispersion curve of BdG Hamiltonian with (a) μ = 87 meV and (b) μ = 57 meV. Here, βso = 60 meV, Δ0 = 6 meV, and RSOC strength is equal to zero. Each supercell includes 9 Mo atoms.

Fig. 6.

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	Fig. 6. Density of states versus energy for μ > βso in filled blue color and for μ < βso in dashed red color. Here, RSOC strength is equal to zero. Each supercell includes 9 Mo atoms.

It should be noted that, the intrinsic unconventional superconductivity has been observed in twisted bilayer Graphene.^[34] Po et al. have invoked SU(4) ferromagnetism interaction^[35] and Xu et al. have invoked an effective anti-ferromagnetic interaction^[36] for explaining the effect, while both have considered an effective triangular superlattice.^[34,35] We considered a triangular lattice too. But, for doing the numerical calculations we considered the (d_z², d_{x² − y²}, d_xy) orbitals and assumed that the pairing happens only between the 4d_z² orbitals while Xu et al. have considered a two-orbital Hubbard model.^[36] Also, we considered the spin-singlet p-wavelike phase while they have expected the d-wave pairing to be favored.

Finally, the effect of mechanical transformation on the topological properties of monolayer MoS₂ is investigated. An uniaxial tension field (T) is applied to the Mo-plane. The vector T has the angle θ with the x-axis which is in parallel with the zigzag edge of the nanoribbon. It can be shown that the strain tensor (ε) can be written as^[33]

It should be noted that one can calculate the effect of biaxial strain similarly. For example, under trigonal deformation such as , by using Eq. (8), one can easily show that F^ε = u₀(y − x). Also, for arc-shape deformation , where R is the arc radius, . But, the deformed new Hamiltonian is now given by Eq. (50) of Ref. [15] and the effect of curvature and gauge fields should be considered. Finally, the effect of deformation on the topological properties can be studied by calculating the amount of each element of the new Hamiltonian, if the deformation does not change the symmetry of the lattice.

As the electronic properties of two-dimensional materials are studied near K and K′ points in this research, the behavior of energy dispersion curve near these points is considered. But, for studying the effect of other orbitals, complete Hamiltonian, i.e., Eq. (2) of Ref. [31], should be considered. Then, equations (34)–(37) of Ref. [15] can be used to calculate the effect of deformation. Obviously, the calculation method is more complicated than the provided method in this article and needs more justification. The effect of doping can be only studied by using the ab-inito methods such as density functional theory, which is out of the scope of this article.

It has been shown that the 4d_z² orbital is the dominant component of the states near the CBM and the VBM located at K and K′ points in the monolayer MoS₂. We have considered the triangular lattice of Mo-plane and studied the topological superconductivity in monolayer MoS₂. By considering the spin-singlet pairing at K and K′ points, we have shown that the BdG-Hamiltonian matrix is diagonal and has four distinct eigenvalues which are functions of chemical energy and ISOC strength. By using the rotation symmetry of the fixed points K and K′, it has been shown that the Chern number is equal to four (two) when the chemical potential μ is greater (smaller) than ISOC strength β_so and otherwise it is equal to zero. Also, we have introduced two simple tight-binding BdG-Hamiltonian models for finding the zero energy states (i.e., Majorana states) and confirming the analytical results. In the first model, the ISOC was considered by choosing the d_{x² − y²} and d_xy orbitals of Mo-atoms. We have shown that when β_so is greater than pairing potential Δ₀, there are four zero energy states. Under the same condition and using the second single-band tight-binding Hamiltonian, we have shown that for both μ > β_so and μ < β_so, there are zero energy states and for μ ≫ β_so, a gap is opened in the energy dispersion curve. Finally, we have shown that under small uniaxial strain which can be parallel or perpendicular to the zigzag edge of Mo-nanoribbon, the topological properties are preserved.