Topological superconductivity in monolayer T_d−MoTe₂

Abstract

Topological superconductivity has attracted significant attention due to its potential applications in quantum computation, but its experimental realization remains challenging. Recently, monolayer T_d−MoTe₂ was observed to exhibit gate tunable superconductivity, and its in-plane upper critical field exceeds the Pauli limit. Here, we show that an in-plane magnetic field beyond the Pauli limit can drive the superconducting monolayer T_d−MoTe₂ into a topological superconductor. The topological superconductivity arises from the interplay between the in-plane Zeeman coupling and the unique Ising plus in-plane spin-orbit coupling (SOC) in the monolayer T_d−MoTe₂. The Ising plus in-plane SOC plays the essential role to enable the effective p_x + ip_y pairing. As the essential Ising plus in-plane SOC in the monolayer T_d−MoTe₂ is generated by an in-plane polar field, our proposal demonstrates that applying an in-plane magnetic field to a gate tunable 2D superconductor with an in-plane polar axis is a feasible way to realize topological superconductivity.

Introduction

Topological superconductors hosting unpaired Majorana zero modes are known as a potential platform for topological quantum computation^{1,2,3,4,5,6,7,8,9,10,11}, which have stimulated the intense pursuit of topological superconductivity (TSC) in real materials^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}. The key element to realize TSC lies in the p-wave pairing in the Fermi surfaces^1,3,12, but the intrinsic p-wave superconductors are extremely rare and yet to confirm. In the past decades, intensive efforts have been made attempting to induce the effective p-wave pairing via the proximity effect with an s-wave superconductor^{12,13,14,15,16,17,18,19,20,26,27}. Among a wide variety of proposals, one ingenious setup is the hybrid superconductor-semiconductor system with strong spin-orbit coupling (SOC) subjected to a Zeeman field^{14,15,16,17,18,19}, where the Zeeman coupling results in odd number of Fermi surfaces and makes the proximity pairing in one single Fermi surface become the effective p-wave pairing. The complexity of the hybrid superconductor-semiconductor structure and its reliance on the pairing proximity effect, however, largely limits the experimental realization of TSC. Recently, several two dimensional (2D) materials are found to exhibit the gate tunable superconductivity ^{29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52}. Therefore, seeking an intrinsic 2D superconductor that is highly tunable to support the TSC phase becomes an alternative approach that bypasses the unfavorable conditions of the hybrid superconductor-semiconductor device.

MoTe₂ is a layered transition metal dichaocogenide that exhibits rich topological and superconductivity properties^{47,48,49,50,51,52,53,54,55}. The MoTe₂ of T_d structure is an intrinsic superconductor in the bulk^54,55, and the superconductivity gets greatly enhanced as the T_d−MoTe₂ is exfoliated down to the monolayer^{48,49,50,51,52}. In the dual-gate device of monolayer T_d−MoTe₂, superconductivity has been achieved in both the electron and hole dominated region^48,50. Owing to the gate tunability, the electron and hole superconducting states can be converted to each other^48,50. According to the first principle band structure calculations, the Fermi surfaces of monolayer T_d−MoTe₂ are composed of one hole pocket centered at Γ along with two flanking electron pockets^47,48,56,57. Given the condition that the chemical potential is gate-tuned to be around the energy of Γ, an in-plane magnetic field induced Zeeman gap at Γ would leave one single hole Fermi surface, which creates the prerequisite to realize the effective p-wave pairing in the monolayer T_d−MoTe₂.

The monolayer T_d−MoTe₂ involves a special type of SOC that is a mixture of the Ising SOC and in-plane SOC^47,48. In the gated monolayer T_d−MoTe₂, the gate-induced out-of-plane electric field⁵⁸, the intrinsic distortion of the crystal structure⁵⁹, and the external substrate effects⁴⁸ all contribute to the breaking of inversion symmetry, leading to Ising plus in-plane SOC. For the hole superconducting states in the monolayer T_d−MoTe₂, the Ising plus in-plane SOC resembles the out of plane Dresselhaus SOC and in-plane Rashba SOC in the hybrid superconductor-$\left(110\right)$-grown-semiconductor system¹⁸, which can be driven into a topological superconducting state by applying an in-plane magnetic field. In the superconducting T_d−MoTe₂ of atomic thickness, it has been demonstrated that the Ising plus in-plane SOC gives rise to an enhanced in-plane upper critical field with an emergent two fold symmetry⁴⁷. Since the Ising SOC component protects the pairing from the in-plane magnetic field^30,31,32,33, it becomes promising that the gate tunable superconducting monolayer T_d−MoTe₂ in an in-plane magnetic field can enter the TSC phase.

Here, we propose that the gate tunable superconducting monolayer T_d−MoTe₂ can be driven into a topological superconductor by applying an in-plane magnetic field. We show that the unique Ising plus in-plane SOC in the monolayer T_d−MoTe₂ is essential to the in-plane magnetic field induced TSC phase. For the monolayer T_d−MoTe₂ with a polar axis tilted to its atomic layer^47,48,49,58, we point out that for the TSC phase transition, the preferred direction of the in-plane magnetic field is along the in-plane projection of the polar axis. By performing the mean field calculation, we demonstrate that the in-plane magnetic field induced TSC phase occupies a considerable region in the phase diagram of the superconducting monolayer T_d−MoTe₂. As entering the TSC phase is always accompanied by the in-plane magnetic field driven gap closing, the tunneling spectroscopy to measure the density of states (DOS) is suggested to experimentally detect the phase transition to the TSC. It is worth emphasizing the key advantages held in our scheme of TSC realization in the monolayer T_d−MoTe₂: i) the TSC phase is derived from an intrinsic 2D superconductor; ii) the monolayer nature circumvents the undesired orbital depairing effect of the in-plane magnetic field; iii) the monolayer T_d−MoTe₂ is highly tunable through the dual-gate and in-plane magnetic field.

Results

Realization of TSC in systems with SOC

To illustrate the essential role of the Ising plus in-plane SOC in our TSC realization in the monolayer T_d−MoTe₂, we first start with reviewing the seminal scheme of effective p_x + ip_y pairing in the Rashba superconductor^14,15,16,17. For a crystalline system with SOC, its general Hamiltonian in the Bloch basis $[{c}_{{{{\bf{k}}}},\uparrow },{c}_{{{{\bf{k}}}},\downarrow }]$ can be written as

$$H\left({{{\bf{k}}}}\right)={\epsilon }_{{{{\bf{k}}}}}-\mu +{{{{\bf{g}}}}}_{{{{\bf{k}}}}}\cdot {\bf{\sigma}} +\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}{{{\bf{B}}}}\cdot {\bf{\sigma}} ,$$

(1)

with ϵ_k being the band energy in terms of the Bloch wave vector k, μ being the chemical potential, g_k denoting the SOC, and ${\bf{\sigma}} = ({\sigma }_{x},{\sigma }_{y},{\sigma }_{z})$ being the Pauli matrices in the spin space. The last term in Eq. (1) represents the magnetic field induced Zeeman coupling, where g = 2 is the Landé g factor and μ_B is the Bohr magneton. For a 2D electron gas with a z-directional polar axis, an electric field along the polar axis generates the Rashba SOC ${{{{\bf{g}}}}}_{{{{\bf{k}}}}}=\alpha ({k}_{y},-{k}_{x},0)$. The Rashba SOC pins the electrons’ spins to the in-plane direction and gives a helical spin texture as shown in Fig. 1a. Given an isotropic band dispersion ${\epsilon }_{{{{\bf{k}}}}}=\frac{{\hslash }^{2}{{{{\bf{k}}}}}^{2}}{2m}$, the Zeeman effect of a magnetic field parallel to the z-directional polar axis lifts the Kramers degeneracy at k = 0 and opens the Zeeman gap. When the chemical potential is tuned into the Zeeman gap, there is only one single Fermi surface present in the system. Crucially, due to the Rashba SOC, the z-directional magnetic field cannot fully polarize the electrons’ spins to the z-direction, so the electrons with canted spins in the single Fermi surface are allowed to have finite singlet pairing^14,15,16,17. Under the condition of weak pairing $\sqrt{{\Delta }^{2}+{\mu }^{2}} < \frac{1}{2}g{\mu }_{{{{\rm{B}}}}}| {{{\bf{B}}}}|$, the pairing in the single Fermi surface becomes the effective p_x + ip_y pairing, making the Rashba superconductor become a topological superconductor^14,15,16,17.

Fig. 1: The schematic diagram of realizing topological superconductivity in systems with spin-orbit coupling.

a The schematic band dispersions with Rashba SOC (upper panel) and the Rashba SOC induced spin texture in the Fermi surface (lower panel). The system with Rashba SOC has a z-directional polar axis. Applying a magnetic field parallel to the z-directional polar axis introduces a Zeeman gap at k = 0. b The schematic band dispersions and the corresponding spin textures derived from a 2D system with an in-plane mirror symmetry (the C_1v symmetry). The mirror plane contains an in-plane polar axis that lies in the atomic plane. The resulting in-plane polar field generates the unique Ising plus in-plane SOC, where its z-directional and in-plane components of the spin texture are plotted in the upper and lower panel respectively. Similar to the case of Rashba SOC, an in-plane magnetic field parallel to the in-plane polar axis leads to a Zeeman gap at k = 0. c In the superconducting state, the effective p_x + ip_y pairing is induced by the magnetic field applied parallel to the polar axis. The upper and lower panel corresponds to the effective p_x + ip_y pairing in the 2D system with the Rashba SOC and the Ising plus in-plane SOC respectively.

One long standing challenge for the TSC realization in the Rashba superconductor is known to be the orbital depairing effects introduced by the z-directional magnetic field¹⁸. A modification to the Rashba superconductor scheme is to introduce an extra out of plane Dresselhaus SOC so that the transition to the TSC phase can be induced by an in-plane magnetic field¹⁸. However, an intrinsic 2D superconductor with such in-plane and out of plane SOC that enables the in-plane magnetic field driven topological phase transition remains yet to know. Microscopically, the SOC arises from the relativistic interaction between the electron’s spin and the crystal electric field ${{{\bf{E}}}}\left({{{\bf{r}}}}\right)$. The approximate form of the SOC field takes ${{{{\bf{g}}}}}_{{{{\bf{k}}}}}\propto {{{\bf{E}}}}\left({{{\bf{r}}}}\right)\times \left(\hslash {{{\bf{k}}}}+\hat{{{{\bf{p}}}}}\right)$ with $\hat{{{{\bf{p}}}}}$ being the momentum operator⁶⁰. For the 2D Rashba superconductor, it is clear that its Rashba SOC is produced by the z-directional polar field. In a 2D system, the z-directional SOC that resembles the out of plane Dresselhaus SOC can only be generated by an in-plane polar field. As a result, in a 2D system, a polar axis with in-plane projection is required for the z-directional SOC. Such in-plane projection of the polar axis is referred as the in-plane polar axis.

In the crystalline system, the SOC field g_k is known to respect the symmetry constraint^61,62

$${{{{\bf{g}}}}}_{{{{\bf{k}}}}}=\det \left(\hat{R}\right)\hat{R}{{{{\bf{g}}}}}_{{\hat{R}}^{-1}{{{\bf{k}}}}},$$

(2)

with $\hat{R}$ being the orthogonal matrix that represents the point group transformation. In the quasi-2D case of atomic thickness, it has been pointed out that only C₁, C_1v, C₂ and C_2v symmetries can have the in-plane polar axis and allow the nonzero z-component of g_k up to the linear order of k⁶³. The explicit forms of g_k expanded from Γ are listed in Table 1. It is clear from Table 1 that g_k derived from the C₁, C_1v or C₂ symmetry is the Ising plus in-plane SOC that involves both the z-directional and in-plane components, while that from the C_2v symmetry is the pure Ising SOC⁶⁴ with only the z component of g_k nonzero. Superconductors with the pure Ising SOC can have the nodal TSC phase driven by an in-plane magnetic field⁶⁵. Since the mechanism of the effective p_x + ip_y pairing in the Rashba superconductor tells that the realization of the fully gapped TSC phase requires g_k to have at least two nonzero components, the g_k in the quasi-2D system with the C₁, C_1v or C₂ symmetry meets the criteria. It is noteworthy that gated monolayer 1H TMDs with C_3v symmetry have both the Ising and Rashba SOC, but applying an in-plane magnetic field can not give the fully gapped TSC phase⁶⁵. As a result, the seek for the in-plane magnetic field driven TSC phase focuses on the intrinsic 2D superconductivity with the C₁, C_1v or C₂ symmetry.

Table 1 The explicit forms of the SOC g_k ⋅ σ in a quasi-2D system

Additionally, the Ising plus in-plane SOC in the quasi-2D system with the C_1v symmetry, as can be seen from the spin texture of ${{{{\bf{g}}}}}_{{{{\bf{k}}}}} = ({\lambda }_{x}{k}_{y},{\lambda }_{y}{k}_{x},{\lambda }_{z}{k}_{y})$ in Fig. 1b, is very similar to the combination of the out of plane Dresselhaus SOC and in-plane Rashba SOC in the hybrid superconductor-(110)-grown-semiconductor system¹⁸. The C_1v group only has one in-plane mirror symmetry, so the in-plane polar axis is defined along the intersection of the mirror plane and the layer plane. For an intrinsic 2D superconductor with C_1v symmetry that can have its chemical potential gate-tuned to be around the energy of Γ, it is expected that applying an in-plane magnetic field parallel to the in-plane polar axis can lead to the TSC phase in the similar mechanism as the hybrid superconductor-(110)-grown-semiconductor system¹⁸. As is schematically shown in Fig. 1c, the in-plane magnetic field parallel to the in-plane polar axis induces the effective p_x + ip_y pairing, similar to the effect of the z-directional magnetic field parallel to the z-directional polar axis in the Rashba superconductor. Since it is promising to engineer a 2D superconductor with the C_1v symmetry into a topological superconductor, the superconducting monolayer T_d−MoTe₂ that belongs to the C_1v point group turns out to be a suitable candidate for TSC.

Electronic structure of the monolayer T_d−MoTe₂

The crystal structure of the monolayer T_d−MoTe₂ hosts only one in-plane mirror symmetry along x as is shown in Fig. 2a. Isolated from the inversion symmetry breaking bulk T_d−MoTe₂, the inversion symmetry of the monolayer T_d−MoTe₂ gets slightly broken^48,49,50,59. The inversion symmetry breaking can be ascribed to the gate-induced z-directional electric field⁵⁸, the intrinsic crystal distortion⁵⁹, and the external substrate effects⁴⁸. Here for simplicity of computation, the inversion symmetry breaking is considered to mainly come from the gate induced out of plane electric field. In the dual-gate device, the applied E_z field induces a tilted dipole moment⁵⁸ as illustrated in Fig. 2b. The in-plane component of the E_z field induced dipole moment gives rise to the z-directional Ising SOC, while its out of plane component generates the in-plane SOC. Under the assumption of the applied E_z = 0.1 V ⋅ Å⁻¹, we perform the first principle density functional theory (DFT) calculation to obtain the band structure of the monolayer T_d−MoTe₂ (see the methods section). The results are depicted in Fig. 2c. Consistent with the previous calculations^47,48,56,57, the Fermi surfaces are composed of one hole pocket at Γ and two electron pockets at either side of Γ. Such metallic band structure of monolayer T_d−MoTe₂ is supported by the recent transport measurements^48,66. Deviating from Γ, spin splittings due to the SOC are observed to appear. On the Fermi surfaces, the Ising plus in-plane SOC is clearly demonstrated by the spin textures plotted in Fig. 2d.

Fig. 2: The crystal structure and electronic band structure of monolayer T_d−MoTe₂.

a The top view of the crystal structure for monolayer T_d−MoTe₂. The red dashed line labeled by M_y denotes the in-plane mirror symmetry. b Due to the nonalignment of the Te atoms in the vertical direction, an out-of-plane electric field E_z induces a net in-plane polar field. c The electronic band dispersions along Y-Γ-X-S for the monolayer T_d−MoTe₂. A z-directional electric field E_z = 0.1 V ⋅ Å⁻¹ is considered in the DFT calculations. d The spin textures of the Γ centered hole pocket and the two flanking electronic pockets.

Since the TSC phase arises from the effective p_x + ip_y pairing in the Fermi surface that encloses the time reversal invariant momenta^{14,15,16,17,18}, our primary focus is on the hole pocket centered at Γ. In the Bloch basis of $[{c}_{{{{\bf{k}}}},\uparrow },{c}_{{{{\bf{k}}}},\downarrow }]$ near Γ, an effective Hamiltonian respecting the C_1v symmetry can be constructed to describe the hole bands as

$${H}_{0}\left({{{\bf{k}}}}\right) = -\frac{{\hslash }^{2}{k}_{x}^{2}}{2{m}_{x}}-\frac{{\hslash }^{2}{k}_{y}^{2}}{2{m}_{y}}-\mu \hfill \\ + {\lambda }_{x}{k}_{y}{\sigma }_{x}+{\lambda }_{y}{k}_{x}{\sigma }_{y}+{\lambda }_{z}{k}_{y}{\sigma }_{z},$$

(3)

with m_x = 0.75m_e, m_y = 1.07m_e being the effective mass (here m_e is the electron mass), and λ_x = 1.72 × 10⁻²eV ⋅ Å, λ_y = 4.13 × 10⁻²eV ⋅ Å, λ_z = 7.87 × 10⁻²eV ⋅ Å denoting the strength of SOC. In experiment, the SOC is tunable by the gate induced E_z field. In the experimental accessible range^43,44,49 of ${E}_{z}\in \left[-0.1,0.1\right]{{{\rm{V}}}}\cdot$ Å⁻¹, our first principle DFT calculations show that the Ising plus in-plane SOC in the effective Hamiltonian in Eq. (3) is sufficient to capture the spin texture in the hole pocket (see the Supplementary Note 1 and Supplementary Note 2).

TSC realization in the monolayer T_d−MoTe₂

The monolayer T_d−MoTe₂ is found to enter the superconductivity phase with the transition temperature being 5K ~ 8K^48,49,50. Distinctly different from its isostructural monolayer T_d-WTe₂ where the superconductivity occurs only in the electron type carriers^34,35, the monolayer T_d−MoTe₂ hosts intrinsic superconductivity in both the electron and hole dominated region^48,49,50, indicating that there exists finite pairing in the hole pocket at Γ. On the other hand, due to the protection of the Ising SOC component, the in-plane upper critical field of the superconducting monolayer T_d−MoTe₂ gets strongly enhanced: for the hole superconducting states with T_c ≈ 7K, the measured in-plane upper critical field B_c2 is extrapolated to reach B_c2/B_P ≈ 2 ~ 3^48,50, where B_P ≈ 1.84T_c is the Pauli paramagnetic limit^67,68. Since the monolayer T_d−MoTe₂ has an atomic thickness around 0.68 nm that is far smaller than the measured coherence length of 8 ~ 16 nm^48,50, the orbital magnetic field effect gets suppressed and the Zeeman coupling with the superconducting states plays the dominate role. It is worth noting that the pairing mechanisms in the monolayer T_d−MoTe₂ and T_d-WTe₂ have been extensively discussed in literatures^{47,48,50,69,70,71}, but no consensus has been reached. Here, we adopt the most common assumption that the dominant pairing in the monolayer T_d−MoTe₂ is of the conventional s-wave type. In the Nambu basis of $[{c}_{{{{\bf{k}}}},\uparrow },{c}_{{{{\bf{k}}}},\downarrow },{c}_{-{{{\bf{k}}}},\uparrow }^{{{\dagger}} },{c}_{-{{{\bf{k}}}},\downarrow }^{{{\dagger}} }]$, the Bogliubov-de Gennes (BdG) Hamiltonian for the hole pocket pairing in the presence of an x-directional Zeeman coupling takes the form:

$${H}_{{{{\rm{BdG}}}}}\left({{{\bf{k}}}}\right) = \left(-\frac{{\hslash }^{2}{k}_{x}^{2}}{2{m}_{x}}-\frac{{\hslash }^{2}{k}_{y}^{2}}{2{m}_{y}}-\mu \right){\tau }_{z}+\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}B{\tau }_{z}{\sigma }_{x}\\ +{\lambda }_{x}{k}_{y}{\sigma }_{x}+{\lambda }_{y}{k}_{x}{\tau }_{z}{\sigma }_{y}+{\lambda }_{z}{k}_{y}{\sigma }_{z}+\Delta {\tau }_{y}{\sigma }_{y},$$

(4)

where τ_z is the Pauli matrix in the particle-hole space and Δ denotes the pairing order parameter. Similar to the Rashba superconductor, the superconducting state in Eq. (4) undergoes the topological phase transition and enters the TSC phase for $\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}| B| > \sqrt{{\Delta }^{2}+{\mu }^{2}}$. Specifically, at μ = 0, the topological phase transition is estimated to occur around $B=\sqrt{2}{B}_{{{{\rm{P}}}}}$⁶⁵, which is smaller than the in-plane B_c2 measured in the superconducting monolayer T_d−MoTe₂^48,50. It indicates that the hole pocket pairing in the superconducting monolayer T_d−MoTe₂ can survive to become the effective p_x + ip_y pairing after the in-plane magnetic field driven topological phase transition.

The above analysis of the in-plane magnetic field driven TSC phase based on the effective Hamiltonian of the Γ pocket gets further supported by the mean field phase diagram calculation. At a given temperature T, the evolution of the superconducting pairing order parameter Δ with an applied in-plane magnetic field can be determined by minimizing the free energy density (see the methods section)

$$F=\frac{| \Delta {| }^{2}}{U}-\frac{1}{2\beta \Omega }{\sum }_{{{{\bf{k}}}},\nu }\log \left(1+{e}^{-\beta {\xi }_{\nu ,{{{\bf{k}}}}}}\right),$$

(5)

with U, $\beta =1/\left({k}_{{{{\rm{B}}}}}T\right)$ and Ω denoting the intra-band attractive interaction, the thermodynamic beta and the volume of the system respectively. Here ξ_ν,k are the eigenvalues of the BdG Hamiltonian that is derived from a twelve-band tight binding model of the monolayer T_d−MoTe₂ (see the methods section). In the calculation, the intra-band attractive interaction U is determined by fitting the superconducting critical temperature at zero magnetic field to be 7.6K. The chemical potential μ is fixed to be at the energy of Γ that locates at the top of the hole bands. The resulting B−T phase diagram of the superconducting state is shown in Fig. 3a. It clearly shows that the in-plane B_c2 is enhanced to reach 3B_P. By solving $| \Delta \left(T,B\right)| =\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}| B|$, we obtain the topological phase transition line. Between the in-plane B_c2 and the topological phase transition line, it can be seen that there exists a considerable superconducting region supporting the TSC phase. At T = 0K, given an in-plane magnetic field in the TSC region, the BdG quasiparticle energy spectrum is plotted in Fig. 3b, where a chiral Majorana edge state appears as expected for the TSC phase.

Fig. 3: The phase diagram of monolayer T_d−MoTe₂ in an x-directional magnetic field.

In the region between the in-plane upper critical field ${B}_{{{{\rm{c}}}}2}\left(T\right)$ (red dashed line) and the topological phase transition line (black solid line) in a, the superconducting monolayer T_d−MoTe₂ becomes a topological superconductor. b The BdG quasiparticle energy spectrum of a monolayer T_d−MoTe₂ ribbon along x. The applied x-directional magnetic field is $B=2.1\Delta /\left(g{\mu }_{{{{\rm{B}}}}}\right)$.

In reality, the gate tunability in the superconducting monolayer T_d−MoTe₂ facilitates our proposal of the in-plane magnetic field driven TSC. First, the gate tunability enables the conversion of the superconducting state between the electron type and the hole type^48,50, indicating that the gate can tune the chemical potential close to the energy of Γ at the top of the hole bands. In the presence of an in-plane magnetic field, opening a Zeeman gap at Γ reduces the DOS, so the magneto-conductance exhibits a dip once the chemical potential is tuned to lie within the Zeeman gap (see the Supplementary Note 2 and Supplementary Fig. 2). Second, the gate-induced E_z field can tune the strength of the SOC. For the Ising plus in-plane SOC ${{{{\bf{g}}}}}_{{{{\bf{k}}}}} = ({\lambda }_{x}{k}_{y},{\lambda }_{y}{k}_{x},{\lambda }_{z}{k}_{y})$, its y and z components that are perpendicular to the x-directional magnetic field play the major role to enhance the in-plane B_c2 and promote the effective p_x + ip_y pairing, while its x component suppresses the pairing in the presence of an x-directional magnetic field. By varying E_z in our first principle DFT calculation, we show that E_z ∈ ± [0.05, 0.07]V ⋅ Å⁻¹ can reduce the x component of g_k so that the in-plane B_c2 can be further enhanced and the TSC region gets enlarged (see the Supplementary Note 2 and Supplementary Fig. 3). For the superconducting monolayer T_d−MoTe₂ with dual-gate, the chemical potential, the Ising plus in-plane SOC, and the in-plane magnetic field are all individually tunable. Such high tunability makes the superconducting monolayer T_d−MoTe₂ a promising platform to realize the long sought topological superconductivity.

Given a 2D TSC phase in the monolayer T_d−MoTe₂, one can further cut it into a quasi-1D wire with its width smaller than the decay length of the Majorana edge state. The decay length of the Majorana edge state is estimated to be about 300 nm for the pristine topological superconducting monolayer T_d−MoTe₂ at $B=2.1\Delta /\left(g{\mu }_{{{{\rm{B}}}}}\right)$. In such a quasi-1D superconducting wire, a pair of robust Majorana zero modes live in the ends of the wire⁷², indicating that the quasi-1D wire of topological superconducting monolayer T_d−MoTe₂ can serve as the building block for the Majorana based topological quantum computation^5,6,7.

Detection

For our proposed in-plane magnetic field driven TSC phase in the superconducting monolayer T_d−MoTe₂, we show that one can measure the evolution of DOS with B to identify the topological phase transition. It is known that the topological phase transition must undergo the gap close and reopen in the BdG quasiparticle energy spectrum. Given a small B, the gap in the BdG energy spectrum is not closed yet, so the electronic DOS exhibits a U-shape with zero DOS in the gap. At the critical value of B that enables the topological phase transition, the BdG quasiparticle has the Dirac cone like dispersion with the gap closing at k = 0, so the resulting DOS becomes the V-shape. After the topological phase transition, the gap reopens so the electronic DOS turns back to the U-shape. In Fig. 4a, we present the local electronic DOS in the bulk sample of superconducting monolayer T_d−MoTe₂. Here the electronic DOS is calculated by $N\left(E\right)=-\frac{1}{\pi }{{{\rm{Tr}}}}\left[{{{\rm{Im}}}}{G}^{{{{\rm{R}}}}}\left(E\right)\right]$ with ${G}^{{{{\rm{R}}}}}\left(E\right)$ being the retarded Green’s function (see the Supplementary Note 3). Clearly, the electronic DOS evolution in Fig. 4a shows the gap close and reopen, consistent with our expected signature of the in-plane magnetic field induced topological phase transition.

Fig. 4: The signatures of the in-plane magnetic field driven superconducting topological phase transition.

a The electronic DOS in the bulk at different in-plane magnetic fields. In the magnetic field driven topological phase transition, the electronic DOS shows the gap closing and reopening. At the topological phase transition with the gap closing at k = 0, the DOS exhibits the V-shape (red colored) as shown in the inset. During the topological phase transition, the electronic DOS change from the U-shape to the V-shape and then back to the U-shape. b The surface electronic DOS calculated at the edge along x direction. After the topological phase transition, a plateau centered at the zero energy emerges, which corresponds to the chiral Majorana edge states with the linear energy dispersion in the topological superconductivity. The surface electronic DOS after the topological phase transition are colored in red.

After the topological phase transition, the key feature of the TSC phase is known to be the chiral Majorana edge state that appears within the gap. At the edges of the sample, the local electronic DOS in the gap is expected to get strongly enhanced by the chiral Majorana edge states. By calculating the local electronic DOS at the edge of the supeconducting monolayer T_d−MoTe₂ (see the Supplementary Note 3), we find that after the topological phase transition, a plateau in the DOS emerges within the gap of the BdG quasiparticle energy spectrum as shown in Fig. 4b. The plateau appearing within the gap signifies the chiral Majorana edge states with the linear energy dispersion in the TSC phase. Since both the local electronic DOS in the bulk and at the edge can be detected by the tunneling spectroscopy^73,74,75, we suggest to carry out tunneling experiments to probe the in-plane magnetic field driven TSC phase.

Discussion

In the above sections, we have proposed that applying an in-plane magnetic field to the superconducting monolayer T_d−MoTe₂ is promising to achieve the TSC phase. In the proposal, the Ising plus in-plane SOC plays the crucial role in the TSC realization. The Ising SOC component is responsible for the enhancement of the in-plane B_c2 to guarantee the finite superconducting pairing after the magnetic field driven topological phase transition. It is evident that a larger Ising SOC component is more favorable for the TSC realization. For the moderate Ising SOC component in the monolayer T_d−MoTe₂, one can consider to put it in contact with another 2D material with strong Ising SOC so that the Ising SOC component in the monolayer T_d−MoTe₂ can get proximity-enhanced^43,44,76. Since the Ising SOC arises from the in-plane polar field, it is also possible to add uniaxial strain to increase the Ising SOC strength^77,78. With the Ising SOC component enhanced, the TSC region is expected to get further enlarged in the phase diagram of the monolayer T_d−MoTe₂.

In our proposal, for the Ising plus in-plane SOC ${{{{\bf{g}}}}}_{{{{\bf{k}}}}}=({\lambda }_{x}{k}_{y},{\lambda }_{y}{k}_{x},{\lambda }_{z}{k}_{y})$ in the monolayer T_d−MoTe₂, we suggest to apply the in-plane magnetic field along the in-plane mirror axis, namely the x direction, to realize the TSC phase. In the case of applying an in-plane magnetic field along the y direction, the whole system respects the in-plane mirror symmetry, so the y-directional magnetic field can only lead to the mirror symmetry protected nodal point degeneracies at k_y = 0 plane. It indicates that the fully gapped TSC phase requires the applied in-plane magnetic field to have a nonzero x-directional component. For the applied in-plane magnetic field slightly deviating from the x direction, the magnetic field can still induce the topological phase transition, but the resulting topological BdG energy spectrum is a bit tilted in the k_x direction (see the Supplementary Note 3 and Supplementary Fig. 5). The tilted BdG energy spectrum indicates that the magnetic field makes the center of the Fermi surface get shifted away from k = 0, so the pairing in the Fermi surface is possible to gain finite momentum under appropriate conditions^79,80,81.

The crucial element in our proposal is the Ising plus in-plane SOC in a gate tunable 2D superconductor. For the monolayer T_d−MoTe₂, its Ising plus in-plane SOC arises from the C_1v symmetry. Apart from the monolayer T_d−MoTe₂, the bilayer T_d−MoTe₂ also belongs to the C_1v point group and exhibits gate tunable superconductivity⁴⁷. The Fermi surfaces of the bilayer T_d−MoTe₂ include two hole pockets at Γ ^47,49, so it is possible that applying an in-plane magnetic field to the superconducting bilayer T_d−MoTe₂ gives rise to a topological superconductor with the Chern number C = 2. The seek for the intrinsic 2D superconductors with Ising plus in-plane SOC can extend beyond the 4 point groups listed in Table 1. For example, the monolayer 2M-WS₂ is a superconductor whose crystal structure respects the C_2h symmetry²⁵. In the presence of gate-induced out-of-plane electric field, the C_2h symmetry is reduced to the C_1v symmetry so that a gate tunable Ising plus in-plane SOC can be generated in monolayer 2M-WS₂. Analogous to our proposal of TSC realization in monolayer T_d−MoTe₂, we expect that an in-plane magnetic field can induce the TSC in monolayer 2M-WS₂ as well. Similar gate-induced Ising plus in-plane SOC has also been proposed in the monolayer 1T’-WTe₂⁸², where proximity induced pairing in the hole pockets can be turned into an effective p-wave pairing by an in-plane magnetic field. It is worth noting that, apart from the C_2h group, applying an out-of-plane electric field can reduce the point groups D₂ and D_2h to the 4 polar groups listed in Table 1 (both D₂ and D_2h reduce to C₂).

In a brief summary, we propose that for the gate tunable superconducting monolayer T_d−MoTe₂ with the C_1v symmetry, the TSC phase can be realized by applying an in-plane magnetic field. Based on the symmetry analysis, we point out that 2D superconductors with an in-plane polar axis are good platforms to host the TSC driven by an in-plane magnetic field.

Methods

The First principle calculation and effective tight binding model

The band structure of monolayer T_d−MoTe₂ is calculated by first-principle DFT as implemented in the Vienna Ab initio Simulation Package (VASP) using the projected-augmented wave method (PAW)⁸³. We adopt the Perdew-Burke-Ernzerhof’s (PBE) form of exchange-correlation functional within the generalized-gradient approximation (GGA). The Brillouin zone is sampled in a Monkhorst-Pack k-point of 9 × 15 × 1 and a large vacuum slab of 30 Å along the z direction is used to reduce image interactions under periodic boundary condition in the calculation. The monolayer crystal structure is fully relaxed with a maximum residual force of less than 0.01 eV ⋅ Å⁻¹. The energy cutoff of the plane wave basis is set to be 520 eV. To take into account the dual-gate induced z-directional electric field, we have considered an out-of-plane electric field in the range of [−0.1, 0.1]V ⋅ Å⁻¹ in the calculation.

After the first principle DFT calculation is done, a twelve-band tight binding model of monolayer T_d−MoTe₂ is built through Wannier function construction by using the Wannier 90 package⁸⁴. At the Fermi level around the Γ pocket, the dominant orbitals are the ${d}_{{x}^{2}-{y}^{2}}$ type from the Mo atom and p_y type from the Te atom. We project twelve bands near the Fermi level into these atomic orbitals. By a Wannier interpolation⁸⁴ of these twelve DFT bands, we construct a basis of localized Wannier functions, comprising four ${d}_{{x}^{2}-{y}^{2}}$-type orbitals that are derived from the Mo atoms and eight p_y-type orbitals derived from Te atoms. The spin degrees are already involved in the twelve orbitals. The hopping matrix of the twelve-band tight binding Hamiltonian is obtained in this way. After the Fourier transformation, we get the twelve-band tight binding Hamiltonian of the monolayer T_d−MoTe₂ in the k space.

The mean field free energy calculation for the superconducting pairing state

Given the twelve-band tight binding model of monolayer T_d−MoTe₂ in the Wannier basis, the Hamiltonian in the presence of an effective pairing attractive interaction can be written as

$${{{\mathcal{H}}}} = {\sum }_{{{{\rm{k}}}},i,o,{o}^{{\prime} },s,{s}^{{\prime} }}{c}_{{{{\bf{k}}}},o,s}^{{{\dagger}} }[{h}_{os,{o}^{{\prime} }{s}^{{\prime} }}({{{\bf{k}}}})-\mu +\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}{B}_{i}{\sigma }_{i,s{s}^{{\prime} }}]{c}_{{{{\bf{k}}}},{o}^{{\prime} },{s}^{{\prime} }}\\ -\frac{U}{\Omega }{\sum }_{{{{\bf{k}}}},{{{{\bf{k}}}}}^{{\prime} }}{\sum }_{o}{c}_{{{{\bf{k}}}},o,\uparrow }^{{{\dagger}} }{c}_{-{{{\bf{k}}}},o,\downarrow }^{{{\dagger}} }{c}_{-{{{{\bf{k}}}}}^{{\prime} },o,\downarrow }{c}_{{{{{\bf{k}}}}}^{{\prime} },o,\uparrow },$$

(6)

where ${c}_{{{{\bf{k}}}},o,\sigma }^{{{\dagger}} }$ is the creation operator and ${h}_{os,{o}^{{\prime} }{s}^{{\prime} }}\left({{{\bf{k}}}}\right)$ denotes the hopping matrix. Here o is the orbital index, s = ↑/↓ represents the spin index, and i = x, y, z denotes the spatial components of magnetic field. Please note that we have assumed a constant intraband s-wave pairing, which naturally arises from the intra-orbital attraction U here. In the mean field approximation, the partition function for the superconducting pairing state can be obtained through the path integral formalism⁸⁵

$${{{\mathcal{Z}}}} \approx \int{{{\mathcal{D}}}}\left[{\psi }_{{{{\bf{k}}}}}^{{{\dagger}} },{\psi }_{{{{\bf{k}}}}}\right]{e}^{-\frac{\beta \Omega | \Delta {| }^{2}}{U}+\frac{1}{2}{\sum }_{{{{\bf{k}}}},n}{\psi }_{{{{\bf{k}}}},n}^{{{\dagger}} }\left[i{\omega }_{n}-{{{{\mathcal{H}}}}}_{{{{\rm{BdG}}}}}({{{\bf{k}}}})\right]{\psi }_{{{{\bf{k}}}},n}}\\ = {e}^{-\frac{\beta \Omega | \Delta {| }^{2}}{U}}+\frac{1}{2}{\sum }_{{{{\bf{k}}}},n}{{{\rm{Tr}}}}\log \left[-\beta {G}^{-1}({{{\bf{k}}}},i{\omega }_{n})\right],$$

(7)

where ${\psi }_{{{{\bf{k}}}}}^{{{\dagger}} }={[{c}_{{{{\bf{k}}}},o,\uparrow }^{{{\dagger}} },{c}_{{{{\bf{k}}}},o,\downarrow }^{{{\dagger}} },{c}_{-{{{\bf{k}}}},o,\uparrow },{c}_{-{{{\bf{k}}}},o,\downarrow }]}^{{{{\rm{T}}}}}$ is the Nambu spinor, ${G}^{-1}(i{\omega }_{n},{{{\bf{k}}}})=i{\omega }_{n}-{h}_{{{{\rm{BdG}}}}}({{{\bf{k}}}})$ is the Matsubara Green’s function with the BdG Hamitonian taking the form

$${h}_{{{{\rm{BdG}}}}}\left({{{\bf{k}}}}\right)=\left(\begin{array}{cc}\tilde{h}\left({{{\bf{k}}}}\right)-\mu &-i{\sigma }_{y}\Delta \hfill\\ i{\sigma }_{y}\Delta \hfill &-{\tilde{h}}^{* }\left(-{{{\bf{k}}}}\right)+\mu \hfill \end{array}\right).$$

(8)

Here $\tilde{h}\left({{{\bf{k}}}}\right)=h\left({{{\bf{k}}}}\right)+\frac{1}{2}g{\mu }_{{{{\rm{B}}}}}{{{\bf{B}}}}\cdot {\bf{\sigma}}$ is the twelve-band tight binding Hamiltonian matrix that takes into account the Zeeman coupling. The free energy density $F=-\frac{1}{\beta \Omega }\log {{{\mathcal{Z}}}}$ gives Eq. (5) with ξ_ν,k being the eigenvalues of ${h}_{{{{\rm{BdG}}}}}\left({{{\bf{k}}}}\right)$ in Eq. (8). The self-consistent mean field equation of the pairing order parameter is given by $\frac{\partial F}{\partial \Delta }=0$. In the calculation, at B = 0T, we set T_c = 7.6K and then the effective attraction U can be determined by solving $\frac{\partial F}{\partial \Delta }=0$. By minimizing the free energy density at given T and in-plane magnetic field, the phase diagram of the superconducting pairing state can be obtained.