In 2014, scanning tunneling microscopy (STM) on self-assembled ferromagnetic iron (Fe) atomic chains on superconducting Pb(110) revealed zero-bias conductance peaks (ZBPs) localized at the chain ends, interpreted as signatures of Majorana quasi-particles (MQPs) in a proximity-induced one-dimensional topological superconductor. The authors themselves enumerated what remained unresolved: energy resolution (about 100 μeV at 1.4 K) too coarse to resolve possible end-mode splitting; no momentum-resolved measurement of the chain band structure to test the odd number of Fermi-level band crossings required for topology; only preliminary chain-length dependence data; unclarified multi-channel end modes and their symmetry protection; unrefined ab initio modeling of the real chain–substrate structure; and untested extensions (two-dimensional magnetic islands, even- versus odd-crossing systems, FFLO-like modulated pairing, and ultimately manipulation and braiding). A decade of follow-up work resolved parts of this agenda — milli-kelvin STM found no detectable splitting, spin-resolved STM found enhanced spin polarization, and momentum-resolved Shiba-band spectroscopy and controlled length-dependent splitting were achieved on atom-by-atom constructed chains on Nb(110) — while independent theory and experiments showed that trivial Yu–Shiba–Rusinov physics can mimic several of the proposed diagnostics. The problem is to conclusively establish or refute the topological-Majorana interpretation of the end modes of ferromagnetic atomic chains on superconductors, including a momentum-resolved band-structure determination for the original Fe/Pb platform, discrimination from trivial zero modes, and, ultimately, demonstration of manipulation and non-Abelian (braiding) statistics in atomic-scale magnetic structures on superconductors.

Background

A Majorana zero mode is a superconducting quasiparticle that is its own antiparticle; in one dimension, a chain of sites with spinless p-wave pairing (the Kitaev chain) binds a Majorana zero mode at each end, and these modes are predicted to obey non-Abelian exchange statistics, meaning that exchanging two of them transforms the joint many-body quantum state in a way that depends on the order of the exchanges — the basis of proposals for fault-tolerant topological qubits. A practical route to effective spinless p-wave pairing is a magnetic atomic chain placed on a conventional s-wave superconductor with strong spin–orbit coupling: each magnetic atom induces a spin-split bound state inside the superconducting gap (a Yu–Shiba–Rusinov (YSR), or Shiba, state), and when the hybridized YSR bands form an effectively spinless band crossing the Fermi level an odd number of times, the proximitized chain enters a one-dimensional topological superconductor. Ferromagnetic chains (rather than the helical spin order treated in most earlier proposals) can work when the substrate supplies Rashba-type spin–orbit coupling, in which spin and momentum are coupled by a surface electric-field gradient.

Nadj-Perge, Drozdov, Li, Chen, Jeon, Seo, MacDonald, Bernevig, and Yazdani (Science 346, 602 (2014)) realized this platform by self-assembling ferromagnetic Fe chains, tens of nanometers long, on Pb(110) and imaging them with scanning tunneling microscopy (STM), a technique that maps the local electronic density of states of a surface with atomic resolution. They established four conditions: ferromagnetism of the chain, spin–orbit coupling from the Pb surface, a superconducting gap opening in the bulk of the chain when Pb goes superconducting, and a zero-bias conductance peak (ZBP) localized at the chain ends. Tight-binding and density-functional-theory (DFT) modeling of the measured double-peak structure of the chain's normal-state density of states suggested that the number of minority-spin band crossings at the Fermi level EFE_F is odd, as required for topology. Control experiments argued against a Kondo origin of the ZBP (a Kondo state is a screening cloud of conduction electrons around a magnetic atom that can also produce a zero-bias resonance; the peak disappears when superconductivity is suppressed), against suppressed-gap trivial ends, and against disorder. Very short chains (about 30–40 Angstrom) showed suppressed ZBPs, interpreted as end-state hybridization. Their model predicted the Majorana wavefunction decays from the end with both a power-law component (Fermi-length scale) and an exponential component (p-wave coherence length).

The paper itself, however, flagged the limits of this evidence. The energy resolution (about 100 microelectronvolts with a superconducting tip at 1.4 K, limited by thermal broadening) could not resolve any small splitting of the end modes that would come from the two ends hybridizing, from multiple channels per end, or from symmetry-breaking perturbations; the background zero-bias conductance also blurs the signal (so-called quasi-particle poisoning of the Majorana state). The number of Fermi-level crossings was inferred from tight-binding fits to real-space spectra, not measured; the authors state that momentum-resolved density-of-states measurements with the STM would be required to probe the band structure directly. The chain-length dependence was preliminary. The ab initio modeling neglected surface relaxation, surface dipoles, lattice mismatch, and modifications of the s-wave pairing near the Pb surface. The outlook further proposed two-dimensional ferromagnetic islands on Pb whose edges could host propagating Majorana modes, contrast experiments on systems with even versus odd numbers of crossings (even-crossing systems should be topologically trivial but candidates for a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state, a superconducting state with spatially modulated pairing), and ultimately manipulation and braiding of the Majorana quasi-particles (MQPs) in ring-like magnetic atomic structures on thin superconducting films, using rotation of a parallel magnetic field to move boundaries between trivial and topological regions.

The reason this agenda is not closed a decade later is that zero-bias peaks at chain ends are not a unique diagnostic. Later theoretical and experimental work established that trivial YSR-band physics, spin–orbit-coupled single impurities, and disorder can produce zero-energy, end-localized, even spin-polarized states. Milli-kelvin STM and spin-polarized STM on the same Fe/Pb system produced results consistent with Majoranas (no detectable splitting; spin polarization exceeding the chain's ferromagnetic polarization), and momentum-resolved quasiparticle-interference spectroscopy and controlled length-dependent splitting measurements were achieved on atom-by-atom constructed Mn chains on Nb(110), but the conclusive topological verification for the original platform — and any demonstration of non-Abelian statistics — is still missing.

Problem Statement

Determine, by direct experiment and consistent microscopic modeling, whether the zero-bias end states of ferromagnetic atomic chains proximitized by a superconductor — the self-assembled Fe/Pb(110) chains of the source experiment and magnetic atomic chains on superconductors generally — are topological Majorana zero modes, and whether the platform can be advanced to controllable, non-Abelian anyons. The scientific components are: (1) milli-kelvin, high-energy-resolution STM spectroscopy that resolves or tightly bounds any splitting of the end modes and quantifies quasi-particle poisoning, testing the predicted hybridization and decay behavior of Majorana modes; (2) a momentum-resolved measurement (STM-based or otherwise) of the chain's band structure that directly determines the number of Fermi-level band crossings and hence tests the odd-crossing condition for one-dimensional topological superconductivity in the Fe/Pb chains; (3) a systematic, quantitative chain-length-dependence study of the end-state amplitude and decay, extracting the localization lengths (power-law and exponential components) and comparing them with model predictions; (4) experimental clarification of multi-channel and near-degenerate end modes in realistic (e.g. zigzag) geometries, including which symmetries protect them and which perturbations split them; (5) refined ab initio modeling of the measured chain–substrate structure, accounting for surface relaxation, surface dipoles, lattice mismatch, and modification of the s-wave pair potential near the Pb surface; (6) the proposed extensions — two-dimensional ferromagnetic islands on Pb searched for edge Majorana modes, systems with even versus odd numbers of Fermi crossings to contrast topological and trivial proximitized phases, and FFLO-like modulated pairing in even-crossing systems; and (7), as the ultimate objective, manipulation and braiding of the Majorana quasi-particles, demonstrating their non-Abelian statistics. These results are jointly aimed at one scientific objective: an unambiguous verdict on the Majorana interpretation of the atomic-chain platform and its usability for topological quantum information.

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