Majorana modes with side features in magnet-superconductor hybrid systems

Topological superconductors, hosting Majorana zero modes (MZMs), are highly sought-after for their potential in topological quantum computing. MZMs exhibit non-Abelian braiding statistics, making them promising candidates for building fault-tolerant qubits. Magnet-superconductor hybrid (MSH) systems offer a particularly attractive platform due to their well-understood components (magnetic atoms like Mn, Fe, Co on conventional superconductors like Nb, Re, Pb), amenability to atomic-level manipulation, and suitability for scanning tunneling microscopy (STM) measurements. Despite significant theoretical and experimental advancements, one-dimensional MSH structures still face challenges in sample quality and spectral gap size. Most theoretical simulations rely on simplified models that neglect the complexity and multi-dimensionality of real materials. Often, these models unrealistically couple Zeeman and superconducting order parameters to the same orbitals. While simplified models demonstrate topological phases with MZMs at chain ends, they cannot capture observed experimental features, such as the 'double eye' feature. This work presents experimental dI/dV scanning tunneling spectroscopy data for Mn/Nb(110) and Fe/Nb(110) systems, revealing characteristic low-energy states on the magnetic chain's sides. Ab initio modeling of Mn chains on Nb(110) is performed, leading to an effective 80-band Bogoliubov-de Gennes model. This model, along with a simplified derived model, demonstrates that the spatial structure of MZMs sensitively depends on the spectral gap size, with low-energy features ranging from side localization to end localization, the latter resembling the 'double eye' feature. The strong agreement between the modeling and experimental findings for Mn chains on Nb(110), along with similarities to other MSH structures, questions the assumption of point-like MZMs as the hallmark of one-dimensional topological superconductors. The authors propose that side features might be a common characteristic of topologically non-trivial MSH systems in the small-gap regime, arising from the complex interplay between MZM orbital composition and repulsion from the magnetic chain.

The literature review extensively covers existing theoretical and experimental work on Majorana zero modes (MZMs) and magnet-superconductor hybrid (MSH) systems. It highlights the use of simplified models in theoretical studies, often neglecting the intricacies of real materials and the multi-dimensional nature of the structures. The limitations of these simplified models in accurately capturing experimentally observed features, such as the ‘double eye’ observed in Fe chains on Pb(110), are discussed. The review also explores previous experimental efforts to identify MZMs in various MSH systems, noting both successes and challenges encountered in achieving sufficient sample quality and the necessary spectral gap size to isolate MZMs from other low-lying excited states. Key papers referenced include those by Kitaev, Nadj-Perge et al., Pientka et al., and Feldman et al., among others, showcasing the existing theoretical frameworks and experimental observations that form the backdrop for the current research. The review underscores the need for more sophisticated modeling techniques that incorporate the complexities of real materials to accurately predict and understand the behavior of MZMs in MSH systems.

The research employed a combination of density functional theory (DFT) calculations, tight-binding model construction, and scanning tunneling microscopy (STM) experiments. DFT calculations, using the full-potential local orbital (FPLO) method with a generalized gradient approximation (GGA) to the exchange-correlation functional, were employed to investigate the electronic structure of Mn chains on Nb(110). Projective Wannier functions were used to map the DFT results onto a tight-binding model. Due to computational limitations, the model was simplified to include only the Mn 3d orbitals and the relevant Nb 4d orbitals, resulting in a 40-band tight-binding model. This 40-band model was then expanded into an 80-band Bogoliubov-de Gennes model, incorporating a superconducting onsite pairing term applied only to Nb orbitals to simulate the proximity effect. The topological phase diagram was determined by calculating the gap size and the topological invariant (M) characteristic of topological class D. Real-space electronic structure and the spatial distribution of MZMs were analyzed. A simplified, one-orbital model was also developed, inspired by the DFT-based model, to gain a deeper understanding of the side features. This simplified model allowed for systematic tuning of parameters to study the interplay of various factors such as Rashba spin-orbit coupling, magnetic moment, chemical potential, and superconducting pairing amplitude. The experimental setup involved a home-built ultra-high vacuum (UHV) STM facility operated at 320 mK. Linear chains of Mn and Fe atoms were assembled on a clean Nb(110) surface using lateral atom manipulation techniques. STM topography images and differential tunneling conductance (dI/dV) spectra and maps were acquired to probe the electronic structure of the chains. Superconducting Nb tips were used to enhance energy resolution. The measured dI/dV data were deconvolved numerically to approximate the sample's local density of states (LDOS).

The DFT-based 80-band model revealed that the Mn chain on Nb(110) system is a topological superconductor for a wide range of superconducting order parameter values. The MZMs exhibit non-universal spatial patterns, with a striking accumulation of spectral weight on both sides of the magnetic chain—these 'side features'. Analysis of the 80-band model and a simplified, derived model showed that the spatial structure of MZMs is highly sensitive to the spectral gap size. Small gap sizes lead to MZMs localizing along the sides of the chain, while larger gaps result in end localization. STM experiments on Mn51 and Fe16 chains on Nb(110) confirmed the existence of side features in the low-energy LDOS, showing remarkable agreement with theoretical predictions. The evolution of side feature energies with chain length, experimentally observed and theoretically reproduced, exhibits an oscillatory behavior consistent with MZMs with small superconducting gaps and coherence lengths exceeding the chain length. A simplified one-orbital model provided further insight into the formation of side features, suggesting that the small superconducting gap (proximity effect) competes with magnetism, pushing the MZMs to localize on the sides of the chain. The model's phase diagram shows a significant portion of the topological region exhibiting noticeable side features, suggesting this may be a common phenomenon in MSH systems with small gap sizes. The observed similarity between the side features and previously reported 'double-eye' features in Fe/Pb(110) suggests a common underlying topological origin for these features.

The findings significantly challenge the conventional understanding of MZMs in one-dimensional topological superconductors, where point-like zero-energy modes at chain ends are considered a necessary hallmark. The observed side features in both Mn/Nb(110) and Fe/Nb(110) systems, supported by robust theoretical modeling, demonstrate that MZMs can exist in various forms, including delocalized, hybridized states along the chain and localized states at the chain ends or even localized on the sides of the chains. The topological nature of the system is determined not by the spatial form of the lowest energy state but by the calculated topological invariant. The strong agreement between the ab initio calculations and the experimental STM data provides compelling evidence for the existence of MZMs with side features, indicating this is not an exceptional phenomenon but rather a potentially common characteristic of MSH systems under specific conditions (small spectral gap). This opens new avenues for exploring and designing MSH systems for topological quantum computing, where the manipulation of MZMs would be crucial. The potential utility of MZMs with side features for braiding and quantum computing warrants further theoretical and experimental investigation.

This research demonstrates, through a combination of ab initio calculations and experimental STM measurements, that Majorana zero modes (MZMs) in magnet-superconductor hybrid (MSH) systems can exhibit side features, challenging the conventional view of point-like MZMs. The strong agreement between theoretical predictions and experimental observations underscores the importance of considering the complex interplay between orbital composition and system parameters in understanding MZM behavior. The discovery of side features as a potentially common characteristic of MSH systems with small spectral gaps raises crucial questions about the usability of such MZMs for topological quantum computing and necessitates further research into their properties and potential applications.

The computational cost limited the size of the DFT calculations and the inclusion of all relevant orbitals. The 80-band model is a simplification of the real material, potentially neglecting subtle effects. The experimental STM measurements may be influenced by tip effects and tunneling geometry, although the authors attempted to mitigate this through numerical deconvolution. The simplified model provides insights but is not a complete representation of the real MSH system. The study focuses on specific MSH systems, and generalization to other materials requires further investigation.