Prediction of intrinsic topological superconductivity in Mn-doped GeTe monolayer from first-principles

Topological superconductors (TSCs) host gapless or zero-energy boundary quasiparticles that behave as Majorana fermions with non-Abelian statistics, promising for topological quantum computation. While theoretical understanding is advanced, experimental realization remains difficult because multiple microscopic parameters (Fermi level, magnetic field, temperature, etc.) must be precisely tuned. Conventional first-principles predictions for superconductors estimate Tc via McMillan or Migdal–Eliashberg formalisms, which are not straightforwardly applicable when strong spin-orbit coupling and magnetism are present. Moreover, most prior first-principles works focused on normal-state prerequisites (e.g., Rashba splitting) rather than the topology of superconducting quasiparticles, with effective TSC models relying on empirical parameters. The authors aim to bridge this gap by developing a versatile, material-specific first-principles approach to construct and solve a Bogoliubov–de Gennes (BdG) Hamiltonian including Rashba SOC, Zeeman splitting, and electron-phonon coupling, enabling direct prediction of both superconductivity and topology. They target intrinsic Class D TSCs (broken time-reversal symmetry) that do not require external fields or heterostructures. Given that a 2D Rashba system requires inversion symmetry breaking, a Zeeman gap, and superconductivity, Mn-doped GeTe (a ferroelectric, non-centrosymmetric IV–VI compound with known p-type superconductivity and ferromagnetism upon Mn doping) is identified as a promising platform. The study focuses on an experimentally exfoliated GeTe monolayer, shows it inherits key bulk characteristics, and predicts that with hole doping and dilute Mn, it becomes a Class D TSC with a finite Chern number and chiral Majorana edge modes, providing a quantitative phase diagram in temperature–Mn concentration space to guide experiments.

- Intrinsic versus extrinsic TSCs: Intrinsic TSCs can host nontrivial superconducting gaps without external fields or engineered heterostructures; candidates include spin-triplet p-wave systems and non-centrosymmetric superconductors. Extrinsic realizations often use SOC materials proximitized by s-wave superconductors and/or external magnetic fields (e.g., semiconductor nanowires with SOC, ferromagnetic atomic chains, nanoscale magnetic islands, proximitized topological surface/edge states). Observations under magnetic fields in FeTe0.55Se0.45, GeTe films, and β-Bi2Pd films suggest possible intrinsic Class D behavior without external fields. - GeTe background: GeTe is a ferroelectric, non-centrosymmetric (rhombohedral) IV–VI semiconductor with large Rashba splitting; superconductivity under p-type doping (Ge vacancies) has been known since the 1960s. Mn-doped GeTe exhibits ferromagnetism with Curie temperatures up to ~200 K in epitaxial layers (BaF2(111)) and Zeeman gap opening in Rashba bands due to coupled ferromagnetic and ferroelectric order. - Prior theory limitations: First-principles efforts typically analyze normal-state topological features or Rashba splitting but do not compute superconducting quasiparticle topology. Effective low-energy models use empirical parameters partially fitted to ab initio results. Conventional Tc estimation (McMillan, Migdal–Eliashberg) is limited when SOC and magnetism are essential. - Need: A material-specific, first-principles BdG framework that incorporates SOC, magnetism, and EPC to predict Tc, critical fields/dopings, and topological invariants.

- Overall framework: Construct a material-specific BdG Hamiltonian in momentum space using Wannier-function-based tight-binding Hamiltonians obtained from first-principles electronic structure, and include Rashba SOC, Zeeman splitting, and phonon-mediated s-wave pairing. Solve the superconducting gap self-consistently to obtain temperature-dependent gap and Tc, and compute topological invariants (Chern number) from the BdG bands. - Electronic structure: DFT calculations performed with VASP (PBE-GGA), plane-wave cutoff 400 eV; k-point meshes: monolayer 30×30×1 (bulk 18×18×18). A vacuum region >15 Å (monolayer/thin films) and dipole correction for films. Band structures and spin textures evaluated; Rashba bands identified near Γ. - Phonons and EPC: DFPT with Quantum ESPRESSO and optimized norm-conserving Vanderbilt pseudopotentials; kinetic energy cutoff 100 Ry. q-mesh 18×18×1 with k-mesh 18×18×1 for dynamical matrices; EPC on finer 36×36×1 k-grid and 60×60×1 q-grid via Fourier interpolation. Hole doping simulated by removing electrons with compensating jellium background. Computed phonon DOS F(ω), Eliashberg function α²F(ω), and cumulative EPC λ(ω); identified Kohn anomalies and dominant acoustic-mode coupling. Total EPC constant λ determined (convergence checked). - Wannierization and BdG construction: Use Wannier90 to fit DFT bands (without localization minimization) using Ge/Te p orbital bases to obtain real-space Hamiltonian H_WF; Fourier transform to H_WF(k). Build the BdG Hamiltonian H_BdG(k) in the Nambu basis with intra-orbital spin-singlet s-wave pairing Δ_ii only, enforcing particle-hole symmetry. For magnetism (Mn doping), add an effective Zeeman term B·σ (out-of-plane Bz) to H_WF(k) and reconstruct H_BdG(k) without TRS. - Self-consistent gap equation: Solve Δ self-consistently using a multiband gap equation derived from the BdG spectrum, summing quasiparticle states within a Debye window θ_D around zero energy. The pairing strength g is related to EPC via g = (λ − μ*)/N_F (μ* effective Coulomb pseudopotential), assumed similar for bands with similar orbital character. The formulation reduces to the standard BCS gap equation in the single-band limit (validated) and reproduces known superconductors (Pb, bulk GeTe, gated monolayer MoS2) as benchmarks. - Parameter choices for GeTe monolayer: Debye temperature θ_D ~200 K (as bulk GeTe). Total EPC λ from DFPT is 1.39; to account for correlations, λ is heuristically reduced by ~45.5% to ~0.76 (benchmarking MoS2), yielding g ~0.4 (bulk GeTe g ~0.49). Only intra-orbital s-wave pairing considered. Hole doping of 0.1 per primitive cell places EF at the Rashba Dirac point. - Modeling Mn doping: Determine Mn high-spin state S=5/2 via reproducing bulk Ge1−xMnxTe using virtual crystal approximation (VCA). Fit Zeeman gaps δ_R and δ_I at Γ versus Mn content x from first-principles (δ_R = 250×x meV, δ_I = 1550×x meV; x in percent). Calibrate relation between effective Bz and x from BdG fits: δ_R = 0.122 Bz and δ_I = 2.0 Bz (meV), giving Bz ≈ 2049 x meV (Rashba) and 775 x meV (Ising). Compute Δ(T) and Tc versus Bz to obtain critical x for pair breaking. - Topology: Diagonalize H_BdG(k) and compute Berry curvature and Chern number Nc by integrating over the Brillouin zone for bands below the superconducting gap. Use ribbon geometry to demonstrate chiral Majorana edge modes consistent with bulk-boundary correspondence. - Codes: VASP, Quantum ESPRESSO, Wannier90; convergence tests and supplementary details provided by the authors.

- GeTe monolayer structure and Rashba physics: Optimized lattice constant a ≈ 3.955 Å and buckling height h ≈ 1.565 Å. Strong Rashba splitting in valence bands with α_R ≈ 0.66–0.76 eV·Å. Doping 0.1 holes per primitive cell (≈ 7.4×10^13 cm−2) moves EF to the Rashba Dirac point and yields N_F ≈ 1.4 states/eV per cell, with helical spin textures on the Fermi surface. - Zeeman gap from Mn doping: For out-of-plane Mn high-spin S = 5/2, the Zeeman gaps at Γ scale linearly with Mn content x (in %): δ_R = 250 × x meV (Rashba bands), δ_I = 1550 × x meV (Ising-like bands), reflecting different out-of-plane spin components. - Electron-phonon coupling (EPC): Total EPC constant λ = 1.39 from DFPT; λ(ω) reaches ~1.13 by ω ≈ 10 meV (~81% of λ), indicating dominant acoustic-mode coupling with Kohn anomalies near Γ that enhance EPC. After accounting heuristically for correlation effects, λ_eff ≈ 0.76 used for superconducting estimates. - Superconductivity (time-reversal symmetric case): With θ_D ≈ 200 K and μ* as in bulk GeTe, the self-consistent solution gives zero-temperature gap Δ ≈ 18.6 μeV for both Rashba and Ising bands and Tc ≈ 120 mK for 0.1-hole-doped monolayer GeTe. - Effect of Zeeman field (Mn doping) on superconductivity: Superconductivity is gradually suppressed with increasing Bz due to pair breaking. Critical fields: Rashba bands fully suppressed for Bz > 0.35 meV; Ising bands for Bz > 0.23 meV. Using Bz–x calibration, critical Mn contents are x_c ≈ 0.017% (Rashba) and ≈ 0.03% (Ising). - Topological superconductivity (Class D): When δ/2 > Δ is satisfied at the Rashba Dirac point, the BdG quasiparticle bands are topologically nontrivial. Using illustrative parameters Δ = 0.2 meV and Bz = 7.5 meV (δ ≈ 0.9 meV), the Chern number Nc = −1 and two chiral Majorana edge modes are obtained in a wide ribbon, confirming intrinsic Class D TSC. - Phase diagram and operating window: At T → 0, the Rashba-band SC phase persists for x < x_c ≈ 0.017%, and the TSC phase emerges for x > x_min ≈ 0.014% (where δ/2 > Δ). Considering the ferromagnetic Curie temperature T_FM(x) ≈ 333 x K and Tc(x), their crossover yields x′_min ≈ 0.014% as well. The TSC is predicted at a lower-limit transition temperature of ~40 mK for x ≈ 0.015%. - Experimental guidance: Ge1−xMnxTe monolayers can be grown on BaF2(111) by MBE; chiral Majorana edge modes could be probed via Josephson effects, charge transport, and magnetic-flux control. The approach also suggests other Class D TSC candidates, such as MnBi2Te4/Bi2Te3/NbSe2 heterostructures and magnetized Rashba-split surface states (e.g., HoRh2Si2 surfaces) under proximity-induced pairing.

The work addresses the challenge of making material-specific predictions of topological superconductivity by developing and validating a first-principles BdG framework that integrates SOC, Zeeman exchange, and EPC. Applying it to GeTe monolayers, the study quantifies the interplay between Rashba splitting, phonon-mediated s-wave pairing, and Mn-induced Zeeman gaps necessary for Class D topology (δ/2 > Δ). It establishes realistic operating conditions—extremely dilute Mn concentrations (≈0.014–0.017%) and millikelvin temperatures—under which intrinsic TSC with Nc = −1 and chiral Majorana edge modes can occur. The phase diagram in (T, x) space ties together superconductivity and ferromagnetism (T_FM ∝ x), delineating a practical window for experiments. The method’s ability to compute topological invariants directly from ab initio-derived BdG Hamiltonians makes it broadly relevant for searching intrinsic TSCs in systems with Rashba or topological surface states, and for estimating critical parameters (Tc, critical fields/dopings) beyond conventional Tc-only approaches. The indication that TSC could be robust against parity mixing in non-centrosymmetric systems further supports feasibility in GeTe-based platforms.

The authors introduce a versatile, material-specific first-principles framework to construct and solve BdG Hamiltonians including SOC, Zeeman exchange, and EPC, enabling predictions of both superconductivity and topological invariants. Using this approach, they predict that hole-doped GeTe monolayers with dilute Mn become intrinsic Class D TSCs characterized by Nc = −1 and chiral Majorana edge modes. They quantify key parameters: Δ ≈ 18.6 μeV, Tc ≈ 120 mK (TRS case), Mn-induced Zeeman gaps scaling with x, suppression of SC at Bz ≳ 0.35 meV (Rashba), and a TSC window for x ≈ 0.014–0.017% with a lower-limit transition temperature ~40 mK. A temperature–Mn concentration phase diagram provides concrete guidance for experimental realization. The methodology is broadly applicable to systems with phonon-mediated pairing and SOC/magnetism, and the study highlights additional candidate platforms (e.g., MnBi2Te4/Bi2Te3/NbSe2 heterostructures, magnetized Rashba surfaces) for future exploration. Future work could refine correlation treatments, include anisotropic/multiband pairing, assess magnetic anisotropy and thickness effects, and extend to fully ab initio Eliashberg–BdG workflows.

- Approximations in EPC and correlations: The total EPC λ obtained from DFPT is heuristically reduced by ~45.5% (benchmarked to MoS2) to account for correlation effects; this may not precisely capture material-specific renormalizations in GeTe. - Debye window and μ*: Debye temperature and Coulomb pseudopotential values are taken from bulk GeTe for the monolayer, which may introduce quantitative deviations. - Pairing model: Only intra-orbital spin-singlet s-wave pairing is considered; potential anisotropies, inter-orbital components, or parity-mixed pairing in non-centrosymmetric systems are not explicitly treated. - Zeeman modeling and disorder: Mn doping is modeled via an effective out-of-plane Zeeman term (and VCA) without explicit treatment of impurity disorder, local exchange fluctuations, or possible inhomogeneity. - SOC and Wannierization: Wannier functions are used without localization minimization, approximating separation into pseudo-spin orbitals in the presence of SOC; this can introduce small errors. - Operating regime: The predicted TSC window requires ultralow temperatures (~40 mK) and extremely dilute Mn concentrations (~0.015%), which may pose experimental challenges and sensitivity to sample quality. - Illustrative topology parameters: Larger Δ and Bz values are used in some calculations to more clearly demonstrate topology; actual experimental values may be smaller, affecting the topological gap size.