Majorana modes with side features in magnet-superconductor hybrid systems

The study investigates whether realistic, material-specific modeling of magnet–superconductor hybrid (MSH) systems can predict and explain the spatial structure of Majorana zero modes (MZMs), particularly the emergence of side features rather than point-like end modes. Topological superconductors host non-Abelian MZMs of interest for topological quantum computing, and MSH systems—magnetic atom chains on s-wave superconductors with surface spin-orbit coupling—provide an experimentally accessible platform. Prior work relied largely on simplified, often strictly 1D models that couple magnetism and superconductivity to the same orbitals and cannot capture complex spatial features (e.g., double-eye patterns). This work aims to develop a first-principles-based approach combining density functional theory (DFT), Wannier projections, and Bogoliubov–de Gennes modeling to assess topology, gap sizes, and real-space MZM structure for Mn chains on Nb(110), and to validate predictions via scanning tunneling spectroscopy (STS). The importance lies in understanding how real-material complexity and small superconducting gaps affect MZM localization, challenging the notion that point-like end-localized zero modes are a necessary hallmark of 1D topological superconductors.

The paper situates itself within theoretical proposals and experimental observations of MZMs in atomic chains on superconductors. Simplified models (helical Shiba chains, RKKY systems, self-organized topological superconductivity) predict topological phases with end-localized MZMs, and experiments on Fe/Pb(110) reported signatures consistent with such predictions. However, later high-resolution studies revealed nontrivial spatial structures such as the 'double-eye' feature, attributed to local order parameter suppression and tunneling geometry effects—features not captured by minimal models. Ab initio-based modeling of MSH structures is scarce, with a few first-principles efforts exploring topological phases in transition-metal chains and properties of Mn/Fe on Nb(110). Experiments on Mn/Nb(110) showed topological Shiba bands and length-dependent oscillations of low-energy states. The present work builds on these by deriving a multi-orbital, material-specific model to capture orbital-dependent magnetism, spin-orbit coupling, and proximity effects, enabling explanation of side-feature MZMs and their dependence on superconducting gap and chain length.

• Ab initio electronic structure: Fully relativistic all-electron DFT (FPLO) with GGA exchange-correlation; slab supercells for Mn chains along [001] on Nb(110) with d_Mn–Mn = 3.30 Å. The top half of the slab is relaxed; Mn occupancy of the hollow site is 1.4 eV/Mn more favorable than on-top. Ferromagnetic order is 29.4 meV/Mn lower than antiferromagnetic; out-of-plane moments are ~0.1 meV/Mn lower than in-plane.
• Wannier projection and tight-binding: Projective Wannier functions used to derive a multi-orbital tight-binding model including Mn 3d and select Nb 4d orbitals that bond with Mn. This yields a 40-band tight-binding model with direction-dependent overlaps, Mn spin splitting, and Nb spin-orbit coupling, reproducing Mn 3d spectral weight from DFT.
• Superconducting 80-band BdG model: The 40-band model is converted to an 80-band Bogoliubov–de Gennes Hamiltonian by adding onsite s-wave pairing only on Nb orbitals (Mn receives proximity via coupling). The bare pairing amplitude Δ (denoted A in figures) is a phenomenological parameter not equal to the physical gap; it effectively represents the 3D Nb substrate’s proximity effect. The model includes hoppings up to fifth neighbors from DFT.
• Topology and spectra: For periodic boundary conditions, compute spectral gaps and the class-D topological invariant (M = ±1) in momentum space; for finite chains (open boundaries), diagonalize real-space Hamiltonians (dimension 80N for chain length N), compute real-space topological invariant, and map zero-energy local density of states (LDOS) patterns.
• Simplified effective model: Guided by ab initio results, construct a reduced one-orbital-per-site model with a four-site unit cell: one magnetic site (Zeeman term J) and three 'substrate' sites hosting s-wave pairing Δ, with nontrivial hopping structure (including Rashba SOC α and chemical potential μ) beyond nearest neighbors. Analyze the topological phase diagram versus parameters and quantify 'side feature weight' (fraction of zero-energy LDOS on side substrate atoms). Explore dependence on chain length N and substrate size/boundary conditions.
• Experiments (STM/STS): Assemble linear Mn_N and Fe_N chains on Nb(110) via lateral atom manipulation at T ≤ 320 mK. Use superconducting Nb tips for enhanced energy resolution. Acquire constant-current topography and deconvoluted dI/dV maps and spectra (V_stab = −6 mV, I_stab = 1 nA, V_mod = 20 μV). Track evolution of low-energy side features with chain length by incrementally adding atoms to one end and measuring at the opposite end to minimize perturbations.
• Comparisons: Compare theoretical zero-energy LDOS maps and oscillatory behavior versus N with experimental dI/dV spatial maps and spectra to assess consistency and interpret low-energy side features as Majorana-related.

• Ab initio magnetism and structure: Mn chains on Nb(110) favor ferromagnetic ground state by 29.4 meV/Mn relative to antiferromagnetic; out-of-plane alignment slightly preferred by ~0.1 meV/Mn. The relaxed Nb surface is distorted relative to bulk, and Mn occupies hollow sites (1.4 eV/Mn more favorable).
• Topological phase diagram (80-band model): As a function of bare pairing amplitude Δ (A), the system is topological (M = −1) for 0 < Δ ≲ 120 meV and for Δ > 130 meV, with only a narrow trivial window 120–130 meV. For periodic chains, spectral gaps are tiny for Δ < 130 meV (7–188 μeV; mean ~69 μeV) and larger for Δ > 130 meV (33 μeV–1.83 meV; mean ~1.19 meV).
• Real-space LDOS and side features: For sufficiently long chains (≥ ~10 sites), real-space and momentum-space topology agree. In many topological cases (especially with small to moderate effective gaps), zero-energy spectral weight is strongly suppressed on magnetic atoms and accumulates on Nb atoms along both sides of the chain, producing pronounced 'side features'. Examples with open boundaries: Mn34 (Δ = 320 meV) shows three side maxima; Mn51 (Δ = 300 meV) five maxima; Mn52 (Δ = 270 meV) six maxima; corresponding topological gaps are ~3.10, 1.37, and 0.88 meV, respectively (before self-consistent suppression).
• Gap renormalization: A self-consistent suppression of Δ near the magnetic chain is expected to reduce effective gaps by factors of ~3–4, bringing theoretical gaps into the range of experimental values (e.g., ~180 μeV multi-orbital YSR band near Mn[001] chains on Nb(110)), much smaller than the bulk Nb gap (1.51 meV).
• Simplified model statistics: Approximately 40% of the topological parameter region exhibits notable side-feature weight, and ~10% of the explored parameter space has 70–99% of zero-energy spectral weight on side substrate atoms. Side features persist irrespective of substrate size or boundary conditions, ruling out simple confinement effects. As Δ decreases, hybridization increases and zero-energy LDOS displays periodic oscillatory maxima along the chain; increasing Δ suppresses bulk maxima, ultimately confining MZMs to ends while maintaining side localization ('double-eye'-like).
• Coherence length: In the small-gap regime, large coherence lengths (∼v/Δ_gap) exceeding chain lengths lead to strong MZM hybridization and oscillatory side-feature patterns; larger gaps reduce coherence length, enhancing end localization.
• Experimental validation: STM/STS on Mn51 chains along [001] shows low-energy side features in deconvoluted dI/dV maps at E ≈ 128 μeV, closely matching theoretical LDOS side patterns. The lowest-energy side-feature state oscillates around zero and does not converge to E=0 up to N=36, consistent with theory for small gaps and chains shorter than the coherence length. Fe16 chains of identical structure also show enhanced zero-energy dI/dV at sides of both ends, mirroring theoretical predictions, and indicating a common origin across Mn and Fe chains on Nb(110).

The results demonstrate that realistic, multi-orbital modeling of Mn/Nb(110) chains yields a topological superconducting phase over a wide range of pairing strengths and that the spatial form of the low-energy Majorana states is highly non-universal. In the experimentally relevant small-gap regime, MZMs tend to avoid the magnetic chain and localize on adjacent Nb atoms, producing prominent side features. This behavior arises from the competition between proximity-induced superconductivity and magnetism on the magnetic sites, and from the multi-orbital composition of low-energy states. The observation of similar side features in both Mn and Fe chains on Nb(110), together with the theoretical phase diagrams and LDOS maps, supports interpreting these features as hybridized Majorana modes rather than simple superpositions of Yu–Shiba–Rusinov states. The framework also reconciles prior 'double-eye' observations on Fe/Pb(110) as the large-gap limit of the same side-localization mechanism. Crucially, the presence or absence of point-like end-localized zero modes is not a definitive diagnostic of topology; instead, the topological invariant must be evaluated. The dependence of side features on Δ and orbital weights explains counter-examples lacking side features and underscores that material-specific details govern the spatial form of MZMs while preserving their topological character.

This work establishes a first-principles-based route to identify and analyze topological superconductivity in magnet–superconductor hybrid chains. By deriving an 80-band superconducting model from DFT for Mn chains on Nb(110) and validating with STM/STS, the study shows that topological phases are prevalent and that MZMs often manifest as side-localized features rather than point-like end modes in small-gap regimes. A simplified effective model, guided by ab initio inputs, explains the transition from oscillatory, delocalized side features to end-confined 'double-eye'-like localization as the gap increases and coherence length decreases. The findings challenge conventional identification criteria for 1D topological superconductors and suggest that side-feature MZMs may be widespread in realistic MSH systems. Future work should include self-consistent superconducting order modeling, comprehensive 3D tunneling simulations, exploration of different materials and chain directions, and assessments of whether side-feature MZMs can be braided and used for fault-tolerant operations.

• The superconducting pairing is introduced phenomenologically on Nb orbitals without self-consistent determination; effective gaps near magnetic chains are inferred to be reduced but not calculated self-consistently within the main text.
• The DFT-derived tight-binding basis is truncated to selected Mn 3d and bonding Nb 4d orbitals (40-band), omitting a full 666-band model that would include all relevant Mn and Nb orbitals; this may affect quantitative details of orbital weights and couplings.
• The mapping between the bare pairing parameter Δ and the physical superconducting gap is nontrivial; Δ is not directly comparable to the experimental gap without considering renormalization.
• Finite-size chains exhibit strong hybridization of end modes for small gaps, complicating unambiguous experimental resolution of a topological gap; long chains are required to clearly separate MZMs from bulk states.
• The ab initio model targets Mn/Nb(110) and does not explicitly describe other substrates (e.g., Pb), so cross-system comparisons (e.g., double-eye in Fe/Pb) are qualitative.
• Side features depend sensitively on Δ and orbital composition; not all chain geometries or materials will exhibit them, consistent with reported counter-examples.