A numerical method for designing topological superconductivity induced by s-wave pairing

Abstract

Topological superconductors have garnered significant attention due to their potential for realizing topological quantum computation. However, a universal computational tool based on first-principles calculations for predicting topological superconductivity has not yet been fully developed, posing substantial challenges in identifying topological superconducting materials. In this paper, we present a numerical method to characterize the superconducting spectrum and topological invariants of two-dimensional (2D) slab systems using first-principles band structure, implemented in the open-source software WannierTools. To more accurately model the superconducting proximity effect, we integrate a phenomenological theory of SC pairing decay module into the program. Our approach can be applied to classical superconductor-topological insulator (SC-TI) heterostructures, SC-semiconductor heterostructures, and intrinsic topological superconductors. The program’s validity is demonstrated using the topological crystal insulator SnTe, the Rashba semiconductor InSb, and the superconductor NbSe₂ as examples. We anticipate that this tool will accelerate the discovery of topological superconductor candidates.

Introduction

The concept of topological superconductors was first introduced in 2000 through theoretical models by Read and Green in two dimensions (2D)¹ and by Kitaev in one dimension (1D)². These models describe spinless, time-reversal-breaking p-wave superconducting states, which support non-Abelian Majorana zero modes. In the 2D case, these modes are localized in the vortex cores, while in the 1D case, they appear at the edges of the system. The existence of Majorana zero modes in these topologically protected states is considered a key feature for potential applications in topological quantum computation.

Interest in topological superconductors (TSCs) has grown significantly in recent years, driven largely by their connection to Majorana fermions-exotic particles that are their own antiparticles^3,4,5,6. These Majorana zero modes are particularly attractive for topological quantum computation due to their potential as stable qubits, leveraging their unique non-Abelian statistics to enable fault-tolerant quantum information processing^7,8. More recently, chiral Majorana states (MSs) have also been explored for similar applications, further expanding the possibilities for quantum computation⁹. As a result, TSCs have garnered significant attention from the academic community^{10,11,12,13,14,15,16,17,18,19,20}. Traditionally, TSCs have been associated with p-wave superconductors, which are predicted to support Majorana zero modes^{21,22,23,24,25}. However, the intrinsic occurrence of p-wave superconductivity is rare, and the direct observation of Majorana states in such systems has proven challenging. Over the past decade, numerous strategies have been proposed to induce TSCs at the surfaces or interfaces of topological materials, including semiconductors with Rashba spin-orbit coupling (Rashba-SOC)^{26,27,28,29,30,31}, magnetic atom chains^{32,33,34,35,36}, and obstructed atomic insulators^37,38, all of which utilize the proximity effect with s-wave superconductors. Additionally, an alternative approach focuses on the realization of TSCs within superconducting topological metals, which combine topologically nontrivial electronic structures at the Fermi level with superconductivity in a single material. Examples of such systems include iron-based superconductors^{39,40,41,42,43,44,45}. This growing body of research highlights the importance of understanding the mechanisms and conditions under which TSCs can be realized, offering promising avenues for their application in next-generation quantum technologies.

Despite significant theoretical progress, experimental confirmation of topological superconductors (TSCs) remains relatively limited. A primary challenge lies in the fact that theoretical predictions are often based on simplified, effective models, which fail to fully capture the complexity of real materials. These models typically overlook intricate electronic structures and interactions that exist in actual systems, making experimental validation difficult.

In this paper, we introduce a numerical method to characterize the superconducting spectrum and topological invariants of two-dimensional (2D) slab systems based on first-principles band structure and phenomenological theory of superconducting proximity effects. This method is implemented within the open-source software WannierTools⁴⁶, which allows for a more accurate description of the electronic structure of materials through Wannier functions. By bridging the gap between theoretical models and real material properties, this approach is expected to significantly enhance the alignment between theoretical predictions and experimental results in the study of TSCs.

The flow of the program is depicted in Fig. 1. The process begins with the construction of Wannier functions from first-principles calculations, providing a precise representation of the electronic structure of the candidate material. Subsequently, s-wave superconducting pairing and spin splitting are introduced at the interface of the 2D slab, simulating superconducting and magnetic proximity effects, respectively. This enables the construction of the corresponding 2D Bogoliubov-de Gennes (BdG) Hamiltonian, which is used to investigate the system’s superconducting properties.

Fig. 1: Flow chart illustrating the numerical method.

The chart outlines the sequential steps involved, starting from first-principles calculations to the construction of Wannier functions, the introduction of superconducting pairing and spin splitting, and culminating in the analysis of the superconducting spectrum and topological invariants.

Using this method, we explore a physical platform for realizing 2D chiral topological superconducting phases. Specifically, we focus on a heterostructure composed of a superconductor, candidate material, and ferromagnetic insulator. Our approach provides a robust framework for predicting and characterizing TSCs in experimentally accessible systems, facilitating further experimental investigation and potential technological applications.

Results

BdG Hamiltonian in slab system

A widely used and efficient method for constructing tight-binding (TB) Hamiltonians for real materials is the maximally localized Wannier functions (MLWFs)⁴⁷, as implemented in the Wannier90 software package⁴⁸. This package provides interfaces to various first-principles calculation codes, including Vienna ab initio simulation package (VASP), Wien2k, and Quantum ESPRESSO (QE), among others. As a result, MLWF Hamiltonians can be automatically generated from first-principles calculations⁴⁷, which give the real-space hopping parameters of electrons in real materials as:

$${H}_{mn}({\boldsymbol{R}})=\langle {{\mathcal{W}}}_{m{\boldsymbol{0}}}| {\hat{H}}_{KS}| {{\mathcal{W}}}_{n{\boldsymbol{R}}}\rangle$$

(1)

where ${\hat{H}}_{KS}$ represents the Kohn–Sham (KS) Hamiltonian derived from first-principles calculations, H_mn(R) denotes the projection of the KS Hamiltonian onto the MLWF basis, and R indicates the lattice vector in the crystal structure. In the lattice gauge, the Fourier transform gives the bulk Hamiltonian:

$${H}_{mn}({\boldsymbol{k}})=\sum _{{\boldsymbol{R}}}{H}_{mn}({\boldsymbol{R}}){e}^{-i{\boldsymbol{kR}}}$$

(2)

To investigate the superconducting proximity effects and magnetic proximity effects in superconductor/candidate material/ferromagnetic insulator heterojunctions, it is essential to employ open boundary conditions that effectively simulate the surface or interface of the material. For convenience, the three-dimensional vector R is divided into the longitudinal vector R_⊥ and the transverse vector R_∥, where R_⊥ is perpendicular to the material surface and R_∥ is parallel to it. By applying open boundary conditions in the R_⊥ direction, a two-dimensional slab system is created, in which periodicity is maintained solely in the in-plane R_∥ direction. However, it is important to note that the experimentally relevant dissociation plane may not align with the basis vectors defined by first-principles calculations. Consequently, the basis vectors ${{\boldsymbol{R}}}_{1,2,3}^{{\prime} }$ must be redefined to accommodate this discrepancy:

$$\begin{array}{rcl}{{\boldsymbol{R}}}_{1}^{{\prime} }&=&{U}_{11}{{\boldsymbol{R}}}_{1}+{U}_{12}{{\boldsymbol{R}}}_{2}+{U}_{13}{{\boldsymbol{R}}}_{3}\\ {{\boldsymbol{R}}}_{2}^{{\prime} }&=&{U}_{21}{{\boldsymbol{R}}}_{1}+{U}_{22}{{\boldsymbol{R}}}_{2}+{U}_{23}{{\boldsymbol{R}}}_{3}\\ {{\boldsymbol{R}}}_{3}^{{\prime} }&=&{U}_{31}{{\boldsymbol{R}}}_{1}+{U}_{32}{{\boldsymbol{R}}}_{2}+{U}_{33}{{\boldsymbol{R}}}_{3}\end{array}$$

(3)

where ${{\boldsymbol{R}}}_{1}^{{\prime} }$ and ${{\boldsymbol{R}}}_{2}^{{\prime} }$ lattice vectors form R_∥ and ${{\boldsymbol{R}}}_{3}^{{\prime} }$ corresponds to the R_⊥ direction. With the basis-vector transformation matrix U, it is possible to generalize the physical properties of any dissociation surface.

The Hamiltonian for the two-dimensional (2D) thin film system can be derived by applying open boundary conditions in the ${{\boldsymbol{R}}}_{3}^{{\prime} }$ direction and subsequently performing a Fourier transform:

$${H}_{mn}^{slab}({{\boldsymbol{k}}}_{\parallel })={\sum }_{{{\boldsymbol{R}}}_{1}^{{\prime} },{{\boldsymbol{R}}}_{2}^{{\prime} }}{H}_{mn}({{\boldsymbol{R}}}_{\parallel }^{{\prime} }){e}^{-i{{\boldsymbol{k}}}_{\parallel }{{\boldsymbol{R}}}_{\parallel }^{{\prime} }}$$

(4)

In this expression, the notation ${{\boldsymbol{R}}}_{\parallel }^{{\prime} }$ represents either ${{\boldsymbol{R}}}_{1}^{{\prime} }$ or ${{\boldsymbol{R}}}_{2}^{{\prime} }$. Different values of ${{\boldsymbol{R}}}_{3}^{{\prime} }$ correspond to distinct film layers, which constitute the longitudinal dimension of the 2D slab Hamiltonian. We denote the layer indices along the ${{\boldsymbol{R}}}_{3}^{{\prime} }$ as i and j.

For a system comprising N_s layers, the matrix representation of the Hamiltonian for the 2D slab can be expressed as:

$${H}_{mn}^{slab}({{\boldsymbol{k}}}_{\parallel })=\left[\begin{array}{cccc}{H}_{mn}^{11}({{\boldsymbol{k}}}_{\parallel })&{H}_{mn}^{12}({{\boldsymbol{k}}}_{\parallel })&\cdots \,&{H}_{mn}^{1{N}_{s}}({{\boldsymbol{k}}}_{\parallel })\\ {H}_{mn}^{21}({{\boldsymbol{k}}}_{\parallel })&{H}_{mn}^{22}({{\boldsymbol{k}}}_{\parallel })&\cdots \,&{H}_{mn}^{2{N}_{s}}({{\boldsymbol{k}}}_{\parallel })\\ \vdots &\vdots &\ddots &\vdots \\ {H}_{mn}^{{N}_{s}1}({{\boldsymbol{k}}}_{\parallel })&{H}_{mn}^{{N}_{s}2}({{\boldsymbol{k}}}_{\parallel })&\cdots \,&{H}_{mn}^{{N}_{s}{N}_{s}}({{\boldsymbol{k}}}_{\parallel })\end{array}\right]$$

(5)

In this context, the diagonal matrix elements correspond to the intra-layer Hamiltonians, while the off-diagonal matrix elements represent the inter-layer Hamiltonians. The specific formulation of these matrix elements can be expressed as follows:

$${H}_{mn}^{ij}({{\boldsymbol{k}}}_{\parallel })=\sum _{{{\boldsymbol{R}}}^{{\prime} }=\{{{\boldsymbol{R}}}_{1}^{{\prime} },{{\boldsymbol{R}}}_{2}^{{\prime} },(i-j){{\boldsymbol{R}}}_{3}^{{\prime} }\}}{H}_{mn}({{\boldsymbol{R}}}^{{\prime} }){e}^{-i{{\boldsymbol{k}}}_{\parallel }{{\boldsymbol{R}}}^{{\prime} }}$$

(6)

In the presence of atomic-scale exchange effect resulting from magnetic proximity effects, the Hamiltonian for the surface layer, defined by i = j = N_s, is expressed as follows:

$${H}_{mn}^{{N}_{s}{N}_{s}}({{\boldsymbol{k}}}_{\parallel },{\boldsymbol{M}})={H}_{mn}^{{N}_{s}{N}_{s}}({{\boldsymbol{k}}}_{\parallel })+{\boldsymbol{M}}\cdot {\boldsymbol{\sigma }}{I}_{orbital}$$

(7)

where M represents the spin splitting, σ denotes the Pauli matrices in spin space, and I is the identity matrix in the Wannier orbital space.

To obtain the BdG Hamiltonian for the superconductor/candidate material/ferromagnetic insulator heterojunction, we induce superconducting pairing across the entire system, incorporating the interaction of exchange effect. In the Nambu representation, the BdG Hamiltonian can be expressed as follows:

$${H}_{BdG}^{slab}=\left[\begin{array}{cc}{H}_{mn}^{slab}({{\boldsymbol{k}}}_{\parallel },{\boldsymbol{M}})-\mu &\Delta \\ {\Delta }^{\dagger }&-{H}_{mn}^{slab}{(-{{\boldsymbol{k}}}_{\parallel },{\boldsymbol{M}})}^{* }+\mu \end{array}\right]$$

(8)

where μ denotes the chemical potential of the system. Since the superconducting substrates used to realize topological superconductors are mostly s-wave superconductors, intra-orbital s-wave pairing, denoted by Δ, is used throughout this paper.

Superconducting proximity effect

The superconducting gap induced by the proximity effect decays gradually with distance from the superconductor interface, a behavior that can be expressed as⁴⁹:

$$\begin{array}{rlr}{\Delta }_{{\rm{decay}}}&={\Delta }_{{\rm{SC}}}\frac{\cosh [\alpha (z-d)]}{\cosh [\alpha d]}\,,\,0\, < \,z\, < \,d&\\ {\alpha }^{2}&=\frac{t}{{{{\Xi }}}_{0}}(\frac{t}{{{{\Xi }}}_{0}}+\frac{3}{l}),\end{array}$$

(9)

Let z denote the distance from the surface of the SC, while d represents the thickness of the candidate material. The parameter t is defined as the ratio T/T_c, where T is the temperature and T_c is the critical temperature. Additionally, Ξ₀ denotes the superconducting correlation length, and l indicates the mean free path in the candidate material. We utilize these parameters to analyze the superconducting proximity effect in the NbSe₂/Bi₂Se₃ heterostructure. In Fig. 2, we demonstrate that this layer-dependent superconducting gap is consistent with the experimental findings of NbSe₂/Bi₂Se₃^50,51, using the values Ξ₀ = 7.7 nm⁵² and l = 16 nm⁵¹. Consequently, when accounting for the decay of superconducting pairing in heterostructures, Δ in Eq. (8) should be replaced with Δ_decay.

Fig. 2: decay of superconducting proximity effect.

a, c The experimental superconducting energy gap of different layers in Refs. ⁵¹ and ⁵⁰. b, d Our fitting results.

Superconducting topological invariant

As summarized in Table 1, superconductors can be categorized into class D and class D and class DIII based on the presence or absence of time-reversal symmetry⁵³. For two-dimensional (2D) systems, when time-reversal symmetry is broken, the system falls under the ${\mathbb{Z}}$ classification, with the corresponding topological invariant being the Chern number⁵⁴. Conversely, when time-reversal symmetry is preserved, the system is classified as ${{\mathbb{Z}}}_{2}$, where the topological invariant is the ${{\mathbb{Z}}}_{2}$ topological number defined by Kane and Mele⁵⁵. Furthermore, Qi et al. demonstrated that in a superconducting system possessing both time-reversal and inversion symmetries, s-wave superconducting pairing results in a topologically trivial phase⁵⁶.

Table 1 Time-reversal-breaking (TRB) and time-reversal-invariant (TRI) topological periodic table of superconductors

There are many methods for calculating topological invariants^57,58,59,60, among which the Wannier charge centers (Wilson loop) method^57,58 can calculate the Z₂ topological number of a system as well as the Chern number. In the following, the topological invariants of the BdG Hamiltonian are calculated using the Wannier charge centers (Wilson loop) method in the 2D slab system. In WannierTools, we take the algorithm presented in Refs. ^57,61. The hybrid Wannier functions⁶² are defined as:

$$\left\vert n{k}_{x}{l}_{y}\right\rangle =\frac{1}{2\pi }\mathop{\int}\nolimits_{0}^{2\pi }d{k}_{y}{e}^{-i{k}_{y}{l}_{y}}\left\vert {\psi }_{n{{\boldsymbol{k}}}_{\parallel }}^{BdG}\right\rangle$$

(10)

where $\vert {\psi }_{n{{\boldsymbol{k}}}_{\parallel }}^{BdG}\rangle$ is the Bloch wave function of BdG Hamiltonian Eq. (8), k_x and k_y represent the reciprocal lattice vectors in the x and y directions of the 2D plane formed by k_∥. The hybrid Wannier centers are defined as:

$$\begin{array}{l}{\bar{y}}_{n}({k}_{x})=\langle n{k}_{x}0| y| n{k}_{x}0\rangle \\\qquad\quad=\frac{i}{2\pi }\mathop{\int}\nolimits_{-\pi }^{\pi }d{k}_{y}\langle {u}_{n,{k}_{x},{k}_{y}}^{BdG}| {\partial }_{{k}_{y}}| {u}_{n,{k}_{x},{k}_{y}}^{BdG}\rangle \end{array}$$

(11)

where $\vert {u}_{n,{k}_{x},{k}_{y}}^{BdG}\rangle$ is the periodic part of Bloch function $\vert {\psi }_{n{{\boldsymbol{k}}}_{\parallel }}^{BdG}\rangle$. In practice, the integration over k_y is transformed by a summation over the discretized k_y. Eq. (11) can be reformulated using the discretized Berry phase formula⁶³:

$${\bar{y}}_{n}({k}_{x})=\frac{1}{2\pi }{\rm{Im}}\ln {\prod }_{j}{M}_{nn}^{(j)}$$

(12)

where the gauge-dependent overlap matrix ${M}_{mn}^{(j)}=\langle {u}_{m,{k}_{x},{k}_{{y}_{i}}}^{BdG}| {u}_{n,{k}_{x},{k}_{{y}_{i+1}}}^{BdG}\rangle$ is introduced. The method for calculating ${\bar{y}}_{n}({k}_{x})$ is well-established^{46,57,63,64,65}, we can get the topological properties of a 2D system from the evolution of ${\bar{y}}_{n}({k}_{x})$ along a k_x string. The details of such classification of Wannier charge centers or Wilson loop are discussed in Refs. ^57,58,65. More information can be found in Ref. ⁶¹.

In our program, the atomic positions in the slab system are given in Cartesian coordinates, and the positions of the particles and holes are assumed to correspond in a one-to-one manner. Additionally, the number of occupied states in the BdG spectrum is always half of the total number of BdG spectrum, meaning that the Wilson loop is computed for half of the total BdG spectrum.

Fu-Kane’s proposal to induce TSC by s-wave SC

Due to the superconducting proximity effect, the Bogoliubov-de Gennes (BdG) Hamiltonian at the interface of a 3D strong topological insulator (TI) and an s-wave superconductor can be written as⁶⁶:

$${H}_{interface}(k)={k}_{x}{\tau }_{0}{\sigma }_{x}+{k}_{y}{\tau }_{z}{\sigma }_{y}-\mu {\tau }_{z}{\sigma }_{0}+\Delta {\tau }_{y}{\sigma }_{y}$$

(13)

This Hamiltonian is written in the Nambu basis $({c}_{\uparrow ,k}^{\dagger },{c}_{\downarrow ,k}^{\dagger },{c}_{\uparrow ,-k},{c}_{\downarrow ,-k})$, where τ and σ represent the Pauli matrices of particle-hole space and spin space respectively, μ represents the chemical potential and Δ represents the superconducting pairing strength. For μ ≫ Δ, take the gauge transformation ${a}_{k}={c}_{\uparrow ,k}+{e}^{i{\theta }_{k}}{c}_{\downarrow ,k}$ for this Hamiltonian (Eq. (13)), where θ_k is defined by ∣k∣(cosθ_k, sinθ_k) = (k_x, k_y), an effective p + ip-wave pairing can be obtained:

$${H}_{eff}^{{p}_{x}+i{p}_{y}}(k)=\left[\begin{array}{cccc}-| k| -\mu &0&0&-\Delta {e}^{i{\theta }_{k}}\\ 0&| k| -\mu &\Delta {e}^{i{\theta }_{k}}&0\\ 0&\Delta {e}^{-i{\theta }_{k}}&-| k| +\mu &0\\ -\Delta {e}^{-i{\theta }_{k}}&0&0&| k| +\mu \end{array}\right]$$

(14)

The physical interpretation of this process involves treating Δ as a perturbation and projecting the Hamiltonian onto the eigenstates of the unperturbed system. However, a key distinction between this Hamiltonian and that of p + ip-wave superconductors is that the former retains time-reversal symmetry. As a result, the chiral boundary states characteristic of p + ip-wave superconductivity are prohibited in this system. When the exchange effect M_zτ_zσ_z is introduced to the system, it breaks time-reversal symmetry. In this context, it is evident that topological phases can emerge in systems belonging to class D, and these topological phases can be characterized by the Chern number⁶⁷. The condition for the phase transition is that $| {M}_{z}| > \sqrt{{\mu }^{2}+{\Delta }^{2}}$. Through the analysis above, it is evident that two-dimensional systems with Rashba spin-orbit coupling (SOC) exhibit a situation very similar to that described here. By introducing an exchange effect to break time-reversal symmetry, topological superconductivity can be achieved⁶⁸.

Nagaosa’s proposal to induce TSC by s-wave SC

Consider a superconductor-TI film-ferromagnet heterostructure as illustrated in Fig. 3⁶⁹. The low-energy physics of this system is governed by the two Dirac surface states of the topological insulator. Accordingly, the BdG Hamiltonian of the system in the Nambu basis $({c}_{t\uparrow ,k}^{\dagger },{c}_{t\downarrow ,k}^{\dagger },{c}_{b\uparrow ,k}^{\dagger },{c}_{b\downarrow ,k}^{\dagger },{c}_{t\uparrow ,-k},{c}_{t\downarrow ,-k},{c}_{b\uparrow ,-k},{c}_{b\downarrow ,-k})$ is given by:

$$\begin{array}{rc}{H}_{twosurfaces}(k)=&v({k}_{x}{\sigma }_{y}{\tau }_{z}-{k}_{y}{\sigma }_{x}{\tau }_{0}){s}_{z}+{t}_{c}{\sigma }_{0}{s}_{x}{\tau }_{z}\\ &+{M}_{z}{\sigma }_{z}{\tau }_{z}\frac{{s}_{z}-{s}_{0}}{2}+\Delta {\sigma }_{y}{\tau }_{y}\frac{{s}_{z}+{s}_{0}}{2}\\ &-{\sigma }_{0}(\mu {s}_{0}+\delta E{s}_{z}){\tau }_{z}\end{array}$$

(15)

τ, σ, and s represent the Pauli matrices of particle-hole space, spin space, top and bottom space, respectively. M_z represents the spin splitting on the bottom surface, Δ represents the s-wave superconducting pairing strength on the top surface, t_c represents the coupling between the top and bottom surfaces, μ represents the chemical potential, and δE represents the energy difference between the top and bottom surfaces. This Hamiltonian describes the interface between two Dirac cones characterized by distinct mass terms: Δ on the top and M_z on the bottom. According to the criteria established in Ref. ⁷⁰, this configuration corresponds to a chiral topological superconducting state.

Fig. 3: A sketch plot of Nagaosa’s proposal to realize a topological superconductor.

The top surface of TI is superconducting due to the proximity effect, and the bottom one feels an out-of-plane exchange field from the ferromagnet. The image is sourced from Ref. ⁶⁹.

Let’s briefly analyze the topological properties of this system. In the limit where both t_c and δE approach zero, the Hamiltonian H_{two surfaces}(k) for the top surface can be expressed as shown in Eq. (13). As discussed in the previous section, we conclude that the top surface is topologically trivial. The Hamiltonian for the bottom surface can be written as:

$${H}_{\rm{bottom}}(k)={k}_{x}{\tau }_{0}{\sigma }_{x}+{k}_{y}{\tau }_{z}{\sigma }_{y}-\mu {\tau }_{z}{\sigma }_{0}+{M}_{z}{\tau }_{z}{\sigma }_{z}$$

(16)

This Hamiltonian is also block diagonal, corresponding to two Dirac Hamiltonians with opposite helicity and opposite masses. Specifically, if M_z is not zero, the system must exhibit topological characteristics, which are determined by the mass term ($C=\frac{1}{2}sign({M}_{z})$ for each Dirac Hamiltonian). Moreover, if the superconducting proximity effect can extend to the bottom surface, then, analogous to the scenario discussed in the previous section, the system enters a topological superconducting state when the condition $| {M}_{z}| > \sqrt{{\mu }^{2}+{\Delta }^{2}}$ is satisfied.

Our program is based on the heterostructure depicted in Fig. 3, where both the superconducting proximity effect and the magnetic proximity effect are introduced in the candidate materials situated in the middle layer. Ultimately, the topological superconducting properties of the system are calculated using the Hamiltonian in the form presented in Eq. (8). In the following sections, we will validate the reliability of our approach through model calculations as well as computations on real materials, such as Bi₂Se₃ and SnTe.

Example: TI model

Before conducting calculations on real materials, we first perform numerical simulations and analyze the BdG Hamiltonian of TB model for the topological insulator. These numerical results are instrumental in establishing a clear physical understanding of the system and serve to validate the accuracy of our computational implementation. The model is written on a simple cubic lattice with two orbitals per primitive cell:

$${H}_{{\rm{TI}}}(k)=m(k){\tau }_{z}{\sigma }_{0}+{\tau }_{x}{\boldsymbol{d(k)}}\cdot {\boldsymbol{\sigma }}$$

(17)

where τ (σ) are Pauli matrices in the orbital (spin) space, $m(k)=M+{m}_{0}{\sum }_{i}\cos ({k}_{i}),d{(k)}^{i}=2t\sin {k}_{i},i=x,y,z$, and t, m₀, and M are parameters of the model. The model is in the strong TI phase if $-3\, < \,\frac{M}{{m}_{0}}\, < \,-1$. In our calculations, we employed the parameters t = 0.5, M = 2.5, m₀ = −1. The resulting bulk band spectrum is presented in Fig. 4a. To investigate the surface states, we constructed a two-dimensional slab system consisting of 20 layers, with the boundary opened in the (001) direction. The calculated band spectrum for this slab system is shown in Fig. 4b.

Fig. 4: Numerical results of the TI model.

a Band structure corresponding to the TI model. b Band structure of the 20-layer slab in the (001) direction. c, d BdG spectrum, and the Wilson loop with M_z = 0, the corresponding Chern number is 0. e BdG spectrum for M_z = 0.132, the system is at the critical point of a topological phase transition. f Energy of the BdG Hamiltonian of the system at Γ point as a function of the surface spin splitting M_z, This can be viewed as a topological phase diagram. g, h BdG spectrum, and the Wilson loop with M_z = 0.2, the corresponding Chern number is 1. (In panels c, e, and g, the red and blue colors represent the top and bottom surface states, respectively).

This section examines the topological superconducting properties of the slab Hamiltonian by substituting it into Eq. (8), while taking into account the magnetic proximity effect and superconducting pairing. In our model calculations, we temporarily neglect the decay of the superconducting gap by assuming a temperature of zero. We introduce a uniform superconducting pairing Δ = 0.1 to the slab system. With the chemical potential set to μ = 0 and surface spin splitting M_z = 0, the BdG spectrum of the system exhibits a superconducting gap and remains in a topologically trivial phase, as illustrated in Fig. 4c, d.

Subsequently, we utilize the phase diagram plotting function to calculate the BdG spectrum at the Γ as a function of the spin splitting. The results are presented in Fig. 4f. It is observed that the BdG spectrum reaches a critical point of phase transition when M_z equals 0.132, as shown in Fig. 4e. Upon exceeding this value, the system transitions into a topological superconducting state characterized by a Chern number of 1. The corresponding BdG spectrum and Wilson loop for this state are shown in Fig. 4g, h.

From the preceding analysis, the physical mechanism underlying the emergence of topological superconducting states is elucidated. The system transitions into a topological superconducting phase whenever the spin splitting M_z induces a closing and reopening of the BdG spectrum. This principle remains consistent across the examples discussed.

Example: Bi₂Se₃

In this section, we demonstrate the program’s capability to analyze topological superconductivity in real material systems, using Bi₂Se₃ as a representative example. For the calculations involving real materials, we aim to emulate the actual heterogeneous structure depicted in Fig. 3. To achieve this, we incorporate the spatially decaying superconducting pairing from Eq. (9) and the short-range exchange effect at the ferromagnet interface. Our goal is to induce a two-dimensional chiral topological superconducting state at the interface between the candidate material and the ferromagnet. Bi₂Se₃, a topological insulator, serves as our testbed. Utilizing first-principles calculations, we derived the TB model for the p-orbitals of Bi and Se atoms using Wannier90. The bulk band spectrum and the 10-layer slab band spectrum in the (001) direction, obtained from this TB model, are presented in Fig. 5a, b, respectively, with the Dirac cone emerging in the gap⁷¹.

Fig. 5: Numerical results of Bi₂Se₃.

a Band structure corresponding to Bi₂Se₃. b Band structure of the 10-layer slab in the (001) direction. c The decay of superconducting pairing with the number of layers. d Energy of the BdG Hamiltonian of the system at Γ point as a function of the surface spin splitting M_z, This can be viewed as a topological phase diagram. The blue lines represent the phase diagrams for superconducting pairing that is uniformly distributed without decay, while the red lines correspond to the phase diagrams when decay is taken into account. e, f BdG spectrum, and the Wilson loop with M_z = 0.3 eV when decay is not considered. It is a topologically trivial phase. g, h BdG spectrum, and the Wilson loop with M_z = 0.3 eV, when decay is considered, the corresponding Chern number is −1.

By substituting the TB Hamiltonian of Bi₂Se₃ into Eq. (8) and considering Ns = 10, we derive the BdG Hamiltonian for a 10-layer Bi₂Se₃ system, taking into account the effects of superconducting and magnetic proximity. In this analysis, we employ the decaying form described in Eq. (9), with the following parameters: Δ_SC = 0.1 eV, t = 0.6, Ξ₀ = 7.7 nm⁵², l = 16 nm⁵¹. The dependence of the superconducting pairing on the number of layers is illustrated in Fig. 5c.

Building upon the analysis presented above, we can construct the topological phase diagram of the system, as depicted in Fig. 5d. In this diagram, the blue lines represent the phase boundaries obtained without considering decay effects, while the red lines indicate the phase boundaries when decay is taken into account. It clearly demonstrates the correction brought by the decay parameter, which reduces the necessary spin splitting for the TSC phase. This is consistent with our analysis in Section D of Chapter II, where the topological properties of the system are primarily determined by the characteristics at the interface between the topological insulator and the ferromagnetic insulator. When $| {M}_{z}| > \sqrt{{\mu }^{2}+{\Delta }^{2}}$, the system enters the topological superconducting phase. The decay of the superconducting proximity effect will reduce the Δ at this interface, thereby reducing the necessary spin splitting for the TSC phase. For a fixed parameter of M_z = 0.3 eV, the BdG spectrum and Wilson loop corresponding to the case without decay are illustrated in Fig. 5e, f, which reveal a topologically trivial phase. Conversely, when decay effects are included, the BdG spectrum and Wilson loop are presented in Fig. 5g, h, indicating the emergence of a topological superconducting state characterized by a Chern number of −1. A comparison between Fig. 5e and g reveals that the introduction of the decay of superconducting proximity effect induces modifications to the low-energy BdG spectrum of the system at the Γ point, thereby leading to distinct topological phases. These results suggest that incorporating the decay of Δ facilitates the realization of the topological superconducting phase at smaller spin splitting, aligning more closely with the behavior observed in actual materials.

Example: SnTe

Based on the calculations presented in the previous two subsections, we have established that Dirac cone surface states provide a means to construct 2D chiral topological superconductors. In the following, we investigate whether systems with multiple surface Dirac cones can support 2D topological superconductors with high Chern numbers, using SnTe as a representative example.

SnTe is a topological crystal insulator protected by mirror symmetry, with four Dirac cone surface states on its (001) surface⁷². Using first-principles calculations, we constructed a TB model for the p-orbitals of Sn and Te atoms, implemented through the Wannier90 software. The resulting bulk band structure and the band spectrum of a 40-layer slab along the (001) direction, derived from this TB model, are presented in Fig. 6a, b, respectively.

Fig. 6: Numerical results of SnTe.

a Band structure corresponding to SnTe. b Band structure of the 40-layer slab in the (001) direction. c The decay of superconducting pairing with the number of layers. d Energy of the BdG Hamiltonian of the system at surface Dirac cone as a function of the surface spin splitting M_z, This can be viewed as a topological phase diagram. e, f BdG spectrum, and the Wilson loop with M_z = 0.08 eV, the corresponding Chern number is −4. g partial enlargement of f to see that the Chern number of the system at this point is −4.

Here, we adopt the decaying form of Eq. (9), with a superconducting gap Δ_SC = 0.01 eV, while keeping the other parameter values consistent with those used for Bi₂Se₃. The decay of the superconducting pairing as a function of the number of layers is shown in Fig. 6c.

Next, we derived the topological phase diagram of the system, as shown in Fig. 6d. The BdG spectrum and Wilson loop for M_z = 0.08 eV are shown in Fig. 6e, f. Due to the requirement of incorporating both Dirac cones within the same integration path for the Wilson loop calculations, the Wilson loop spectrum of SnTe exhibits a pronounced gradient in the region where the Dirac cones overlap. By focusing on the Wilson loop spectrum near the Γ point, we can clearly identify that the Chern number of the system is −4, as shown in Fig. 6g.

This result is expected, as the system lacks time-reversal symmetry, and the four surface Dirac cones are related solely by a fourfold rotational symmetry. Consequently, these four Dirac cones should each contribute an equal Berry curvature, resulting in the Chern number being a multiple of four.

Example: NbSe₂ model

In this section, we extend the realization scheme of topological superconductivity from topological insulators to quasi-two-dimensional materials with a Rashba SOC. We begin with the TB model of monolayer NbSe₂ and replicate the findings presented in Ref. ⁶⁸ using our approach. The electronic bands of NbSe₂ near the Fermi energy are primarily derived from Nb d-orbitals, and the relevant low-energy physics can be qualitatively captured by a minimal model that incorporates one effective d orbital per site on a triangular lattice.

$${H}_{{{\rm{NbSe}}}_{{\rm{2}}}}(k)={E}_{t}(k){\sigma }_{0}+{E}_{{R}_{x}}(k){\sigma }_{x}+{E}_{{R}_{y}}(k){\sigma }_{y}$$

(18)

We consider a simple third-neighbor hopping form for the normal band structure:

$$\begin{array}{l}{E}_{t}(k)=-\,2{t}_{1}[\cos {k}_{x}a+2\cos \frac{{k}_{x}a}{2}\cos \frac{{k}_{y}a\sqrt{3}}{2}]\\ \qquad\qquad-\,2{t}_{2}[\cos {k}_{y}\sqrt{3}a+2\cos \frac{{k}_{x}3a}{2}\cos \frac{{k}_{y}\sqrt{3}a}{2}]\\ \qquad\qquad-\,2{t}_{3}[\cos 2{k}_{x}a+2\cos {k}_{x}a\cos {k}_{y}\sqrt{3}a]\end{array}$$

(19)

The SOC terms are given by:

$$\begin{array}{rc}&{E}_{{R}_{x}}(k)=2{\alpha }_{R}\sqrt{3}\sin \frac{{k}_{y}a\sqrt{3}}{2}\cos \frac{{k}_{x}a}{2}\\ &{E}_{{R}_{y}}(k)=-2{\alpha }_{R}(\sin {k}_{x}a+\sin \frac{{k}_{x}a}{2}\cos \frac{{k}_{y}a\sqrt{3}}{2})\end{array}$$

(20)

For the subsequent calculations, we employed the parameters from⁶⁸: t₁ = −0.04 eV, t₂ = −0.132 eV, t₃ = − 0.012 eV, α_R = 0.06 eV. The band structure of the model is shown in Fig. 7a.

Fig. 7: Numerical results of the NbSe₂ model.

a Band structure corresponding to the NbSe₂ model. b The phase diagram of the whole system with respect to B_z and μ. c, d BdG band spectrum, and the Wilson loop with μ = 1.104 eV, the corresponding Chern number is 1. e, f BdG band spectrum, and the Wilson loop with μ = 0.636 eV, the corresponding Chern number is 2. g, h BdG band spectrum, and the Wilson loop with μ = −0.272 eV, the corresponding Chern number is −3.

Next, we set Δ = 0.001 eV, adjust the chemical potential, and utilize our phase diagram function to plot the phase transition points at various chemical potentials. By connecting these points, we obtain the phase diagram of the system as a function of B_z and μ, as shown in Fig. 7b. It is evident that this phase diagram is in perfect agreement with the one presented in Ref. ⁶⁸.

Finally, we computed the Chern numbers for different regions of the phase diagram by setting B_z = 0.01 eV. We identified three topologically non-trivial regions: the region with a Chern number of C = 1, as shown in Fig. 7c, d; the region with C = 2, as shown in Fig. 7e, f; and the region with C = −3, as shown in Fig. 7g, h.

As discussed in Ref. ⁶⁸, the absolute values of the different Chern numbers can be explained qualitatively. When the chemical potential intersects the band degeneracies at the three high-symmetry points shown in Fig. 7a, the Berry fluxes at these points contribute to the generation of distinct Chern numbers. In the first Brillouin zone, there is one Γ point, three M points, and two equivalent K and ${{\rm{K}}}^{{\prime} }$ points, so three different topological phases are produced.

Example: InSb

Building on the previous example, we have established that topological superconducting states can be realized in 2D systems with Rashba SOC. In this section, we examine indium antimonide (InSb), a material commonly used for the fabrication of Majorana nanowires, to demonstrate the feasibility of our proposed scheme in real materials.

The crystal structure of InSb adopts the zincblende form, which is equivalent to a face-centered cubic (FCC) lattice with one formula unit per primitive unit cell. Following the approach outlined in Ref. ⁷³, we employed HSE06 hybrid functionals for first-principles calculations and constructed a TB model for InSb using the s and p orbitals of In and the p orbitals of Sb. The band structure derived from this TB model is shown in Fig. 8a and is in good agreement with the results presented in Ref. ⁷³.

Fig. 8: Numerical results of InSb.

a Band structure corresponding to the TB model of InSb. b Band structure of the 10-layer slab in the (001) direction. c, d BdG band spectrum, and the Wilson loop with μ = 0.5475 eV, the corresponding Chern number is 1. e, f BdG band spectrum, and the Wilson loop with μ = 0.7256 eV, the corresponding Chern number is −1. g, h BdG band spectrum, and the Wilson loop with μ = 0.7961 eV, the corresponding Chern number is 1.

Next, we calculated the band structure of a 10-layer slab InSb along the (001) direction. To mitigate the effects of dangling bonds, we adopted the method described in⁷³. Specifically, we modified the on-site energies of the s and p_x orbitals of In atoms on the bottom surface by +5 eV, and the energy of p_y and p_z orbitals of Sb atoms on the top surface by −5 eV. The resulting modified band structure is shown in Fig. 8b.

Due to the extensive range of parameter regimes in which topological superconducting phases can be realized in real materials, we do not present a complete topological phase diagram here. Instead, we focus on the topological superconducting states derived from the lowest conduction bands at the Γ, X, and Y points, when the chemical potential is positive. For these calculations, we set Δ_SC = 0.01 eV and M_z = 0.02 eV, related parameters of superconducting pairing attenuation consistent with those used for Bi₂Se₃. The corresponding BdG spectra and Wilson loops for these three topological superconducting states are shown in Fig. 8c–h, respectively. It is noteworthy that the absolute value of the Chern number for all three states is 1, as there is only one Γ, X, and Y point in the first Brillouin zone.

Discussion

In conclusion, we have developed a program that is integrated into WannierTools⁷⁴ to compute the superconducting BdG spectrum and superconducting topological invariants of 2D systems. We demonstrate its application in analyzing the topological superconducting properties of superconductor/candidate material/ferromagnetic insulator heterostructures. As a specific example, we show that SnTe can be utilized to construct chiral topological superconductors with a high Chern number. These results validate the program’s functionality and rationale, positioning it as a valuable tool in the design and prediction of topological superconductors. Furthermore, this program provides theoretical support for the exploration of candidate materials for topological superconductors. Additionally, the computational framework can be extended to investigate other topological superconducting systems, such as magnetic topological insulator/superconductor heterostructures.

Methods

First-principle calculations

The first-principle calculations are based on density functional theory (DFT)⁷⁵ as implemented in the VASP⁷⁶. Generalized gradient approximation in the form of Perdew–Burke–Ernzerhof⁷⁷ is used to treat the exchange-correlation interactions except InSb. Core and valence electrons are treated by projector augmented wave (PAW) method⁷⁸ and plane-wave basis functions, respectively. Then a TB Hamiltonian is built from DFT results using the Wannier90 package⁴⁸.