Majorana Zero Mode Detected with Spin Selective Andreev Reflection in the Vortex of a Topological Superconductor

Observation of In-Gap States in a Two-Dimensional CrI2/NbSe2 Heterostructure

Low-dimensional magnetic structures coupled with superconductors are promising platforms for realizing Majorana zero modes, which have potential applications in topological quantum computing. Here, we report a two-dimensional (2D) magnetic-superconducting heterostructure consisting of single-layer chromium diiodide (CrI2) on a niobium diselenide (NbSe2) superconductor. Single-layer CrI2 nanosheets, which hold antiferromagnetic (AFM) ground states by our first-principles calculations, were epitaxially grown on the layered NbSe2 substrate. Using scanning tunneling microscopy/spectroscopy, we observed robust in-gap states spatially located at the edge of the nanosheets and defect-induced zero-energy peaks inside the CrI2 nanosheets. Magnetic-flux vortices induced by an external field exhibit broken 3-fold rotational symmetry of the pristine NbSe2 superconductor, implying the efficient modulation of the interfacial superconducting states by the epitaxial CrI2 layer. A phenomenological model suggests the existence of chiral edge states in a 2D AFM-superconducting hybrid system with an even Chern number, providing a qualitatively plausible understanding for our experimental observation.

Resilient Growth of a Highly Crystalline Topological Insulator-Superconductor Heterostructure Enabled by an Ex Situ Nitride Film

Highly crystalline and easily feasible topological insulator-superconductor (TI-SC) heterostructures are crucial for the development of practical topological qubit devices. The optimal superconducting layer for TI-SC heterostructures should be highly resilient against external contamination and structurally compatible with TIs. In this study, we provide a solution to this challenge by showcasing the growth of a highly crystalline TI-SC heterostructure using refractory TiN (111) as the superconducting layer. This approach can eliminate the need for in situ cleavage or growth. More importantly, the TiN surface shows high resilience against contaminations during air exposure, as demonstrated by the successful recyclable growth of Bi2Se3. Our findings indicate that TI-SC heterostructures based on nitride films are compatible with device fabrication techniques, paving the way to the realization of practical topological qubit devices in the future.

Nonlocal correlations transmitted between quantum dots via short topological superconductor

We study the quasiparticle states and nonlocal correlations of a hybrid structure, comprising two quantum dots interconnected through a short-length topological superconducting nanowire hosting overlaping Majorana modes. We show that the hybridization between different components of this setup gives rise to the emergence of molecular states, which are responsible for nonlocal correlations. We inspect the resulting energy structure, focusing on the inter-dependence between the quasiparticles of individual quantum dots. We predict the existence of nonlocal effects, which could be accessed and probed by crossed Andreev reflection spectroscopy. Our study would be relevant to a recent experimental realization of the minimal Kitaev model [T. Dvir et al., Nature 614, 445 (2023) ], by considering its hybrid structure with side-attached quantum dots.

Vortex entropy and superconducting fluctuations in ultrathin underdoped Bi2Sr2CaCu2O8+x superconductor

Vortices in superconductors can help identify emergent phenomena but certain fundamental aspects of vortices, such as their entropy, remain poorly understood. Here, we study the vortex entropy in underdoped Bi2Sr2CaCu2O8+x by measuring both magneto-resistivity and Nernst effect on ultrathin flakes (≤2 unit-cell). We extract the London penetration depth from the magneto-transport measurements on samples with different doping levels. It reveals that the superfluid phase stiffness ρs scales linearly with the superconducting transition temperature Tc, down to the extremely underdoped case. On the same batch of ultrathin flakes, we measure the Nernst effect via on-chip thermometry. Together, we obtain the vortex entropy and find that it decays exponentially with Tc or ρs. We further analyze the Nernst signal above Tc in the framework of Gaussian superconducting fluctuations. The combination of electrical and thermoelectric measurements in the two-dimensional limit provides fresh insight into high temperature superconductivity.

Topological band inversion and chiral Majorana mode in hcp thallium

The chiral Majorana fermion is an exotic particle that is its own antiparticle. It can arise in a one-dimensional edge of topological materials, and especially that in a topological superconductor can be exploited in non-Abelian quantum computation. While the chiral Majorana mode (CMM) remains elusive, a promising situation is realized when superconductivity coexists with a topologically non-trivial surface state. Here, we perform fully non-empirical calculation for the CMM considering superconductivity and surface relaxation, and show that hexagonal close-packed thallium (Tl) has an ideal electronic state that harbors the CMM. Thekz=0plane of Tl is a mirror plane, realizing a full-gap band inversion corresponding to a topological crystalline insulating phase. Its surface and hinge are stable and easy to make various structures. Another notable feature is that the surface Dirac point is very close to the Fermi level, so that a small Zeeman field can induce a topological transition. Our calculation indicates that Tl will provide a new platform of the Majorana fermion.

Tomasch Oscillations as Above-Gap Signature of Topological Superconductivity

The identification of topological superconductors usually involves searching for in-gap modes that are protected by topology. However, in current experimental settings, the smoking-gun evidence of these in-gap modes is still lacking. In this Letter, we propose to support the distinction between two-dimensional conventional s-wave and topological p-wave superconductors by above-gap transport signatures. Our method utilizes the emergence of Tomasch oscillations of quasiparticles in a junction consisting of a superconductor sandwiched between two metallic leads. We demonstrate that the behavior of the oscillations in conductance as a function of the interface barriers provides a distinctive signature for s-wave and p-wave superconductors. Specifically, the oscillations become weaker as the barrier strength increases in s-wave superconductors, while they become more pronounced in p-wave superconductors, which we prove to be a direct manifestation of the pairing symmetries. Our method can serve as a complimentary probe for identifying some classes of topological superconductors through the above-gap transport.

Majorana Zero Modes Induced by the Meissner Effect at Small Magnetic Field

One key difficulty in realizing Majorana zero modes (MZMs) is the required high magnetic field, which causes serious issues, e.g., shrinks the superconducting gap, reduces topological region, and weakens their robustness against disorders. In this Letter, we propose that the Meissner effect can bring the topological superconducting phase to a superconductor/topological-insulator/superconductor (SC/TI/SC) hybrid system. Remarkably, the required magnetic field strength (<10  mT) to support MZMs has been reduced by several orders of magnitude compared to that (>0.5  T) in the previous schemes. Tuning the phase difference between the top and bottom superconductors can control the number and position of the MZMs. In addition, we account for the electrostatic potential in the superconductor/topological-insulator (SC/TI) interface through the self-consistent Schrödinger-Poisson calculation, which shows the experimental accessibility of our proposal. Our proposal only needs a small magnetic field of less than 10 mT and is robust against the chemical potential fluctuation, which makes the SC/TI/SC hybrid an ideal Majorana platform.

Toward large-scale, ordered and tunable Majorana-zero-modes lattice on iron-based superconductors

Majorana excitations are the quasiparticle analog of Majorana fermions in solid materials. Typical examples are the Majorana zero modes (MZMs) and the dispersing Majorana modes. When probed by scanning tunneling spectroscopy, the former manifest as a pronounced conductance peak locating precisely at zero-energy, while the latter behaves as constant or slowly varying density of states. The MZMs obey non-abelian statistics and are believed to be building blocks for topological quantum computing, which is highly immune to the environmental noise. Existing MZM platforms include hybrid structures such as topological insulator, semiconducting nanowire or 1D atomic chains on top of a conventional superconductor, and single materials such as the iron-based superconductors (IBSs) and 4Hb-TaS2. Very recently, ordered and tunable MZM lattice has also been realized in IBS LiFeAs, providing a scalable and applicable platform for future topological quantum computation. In this review, we present an overview of the recent local probe studies on MZMs. Classified by the material platforms, we start with the MZMs in the iron-chalcogenide superconductors where FeTe0.55Se0.45and (Li0.84Fe0.16)OHFeSe will be discussed. We then review the Majorana research in the iron-pnictide superconductors as well as other platforms beyond the IBSs. We further review recent works on ordered and tunable MZM lattice, showing that strain is a feasible tool to tune the topological superconductivity. Finally, we give our summary and perspective on future Majorana research.

Intrinsic surface p-wave superconductivity in layered AuSn4

The search for topological superconductivity (TSC) is currently an exciting pursuit, since non-trivial topological superconducting phases could host exotic Majorana modes. However, the difficulty in fabricating proximity-induced TSC heterostructures, the sensitivity to disorder and stringent topological restrictions of intrinsic TSC place serious limitations and formidable challenges on the materials and related applications. Here, we report a new type of intrinsic TSC, namely intrinsic surface topological superconductivity (IS-TSC) and demonstrate it in layered AuSn4 with Tc of 2.4 K. Different in-plane and out-of-plane upper critical fields reflect a two-dimensional (2D) character of superconductivity. The two-fold symmetric angular dependences of both magneto-transport and the zero-bias conductance peak (ZBCP) in point-contact spectroscopy (PCS) in the superconducting regime indicate an unconventional pairing symmetry of AuSn4. The superconducting gap and surface multi-bands with Rashba splitting at the Fermi level (EF), in conjunction with first-principle calculations, strongly suggest that 2D unconventional SC in AuSn4 originates from the mixture of p-wave surface and s-wave bulk contributions, which leads to a two-fold symmetric superconductivity. Our results provide an exciting paradigm to realize TSC via Rashba effect on surface superconducting bands in layered materials.

Chemical Modulation of Metal-Insulator Transition toward Multifunctional Applications in Vanadium Dioxide Nanostructures

The metal-insulator transition (MIT) of vanadium dioxide (VO2 ) has been of great interest in materials science for both fundamental understanding of strongly correlated physics and a wide range of applications in optics, thermotics, spintronics, and electronics. Due to the merits of chemical interaction with accessibility, versatility, and tunability, chemical modification provides a new perspective to regulate the MIT of VO2 , endowing VO2 with exciting properties and improved functionalities. In the past few years, plenty of efforts have been devoted to exploring innovative chemical approaches for the synthesis and MIT modulation of VO2 nanostructures, greatly contributing to the understanding of electronic correlations and development of MIT-driven functionalities. Here, this comprehensive review summarizes the recent achievements in chemical synthesis of VO2 and its MIT modulation involving hydrogen incorporation, composition engineering, surface modification, and electrochemical gating. The newly appearing phenomena, mechanism of electronic correlation, and structural instability are discussed. Furthermore, progresses related to MIT-driven applications are presented, such as the smart window, optoelectronic detector, thermal microactuator, thermal radiation coating, spintronic device, memristive, and neuromorphic device. Finally, the challenges and prospects in future research of chemical modulation and functional applications of VO2 MIT are also provided.

Topological Superconductors from a Materials Perspective

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

Pressure-induced physical properties in topological semi-metal TaM2 (M = As, Sb)

In this study, DFT based first principles calculations are used for measuring the structural, elastic, mechanical, electronic, optical and thermodynamic features of topological semimetal TaM₂ (M = As, Sb) under various pressures. We conducted the first investigation into the physical properties of the topological semimetal TaM₂ (M = As, Sb) under pressure. Formation energy and Born stability criteria justify the compound's thermodynamic and mechanical stability. We used elastic constants, elastic moduli, Kleinman parameter, machinability index, and Vickers hardness to investigate the mechanical properties of topological semimetal TaM₂. Poisson's and Pugh's ratios reveal that both compounds change from brittle to ductile in response to pressure. The increasing nature of elastic moduli suggests that TaM₂ becomes stiffer under stress. The pressure has a significant effect on the anisotropy factor for both materials. Band structure analysis shows that both compounds are Weyl semi-metals and the d orbital contributes significantly to the formation of the Fermi level, as shown by the density of states (DOS) analysis. Investigation of electronic characteristics provides important support for dissecting optical performance. Both the reflectivity and absorption spectra shift upwards in energy when pressure is increased. The refractive index value decreases and becomes flat in the higher energy region. Based on their refractive indices, both of these materials demonstrate as a high-density optical data storage medium when exposed to the right light source. The thermodynamic properties including sound velocity, and Debye temperature all exhibit an increasing nature with applied pressure. Due to their high Debye temperatures, the components under study have a rather high melting point.

Controllable dimensionality conversion between 1D and 2D CrCl3 magnetic nanostructures

The fabrication of one-dimensional (1D) magnetic systems on solid surfaces, although of high fundamental interest, has yet to be achieved for a crossover between two-dimensional (2D) magnetic layers and their associated 1D spin chain systems. In this study, we report the fabrication of 1D single-unit-cell-width CrCl₃ atomic wires and their stacked few-wire arrays on the surface of a van der Waals (vdW) superconductor NbSe₂. Scanning tunneling microscopy/spectroscopy and first-principles calculations jointly revealed that the single wire shows an antiferromagnetic large-bandgap semiconducting state in an unexplored structure different from the well-known 2D CrCl₃ phase. Competition among the total energies and nanostructure-substrate interfacial interactions of these two phases result in the appearance of the 1D phase. This phase was transformable to the 2D phase either prior to or after the growth for in situ or ex situ manipulations, in which the electronic interactions at the vdW interface play a nontrivial role that could regulate the dimensionality conversion and structural transformation between the 1D-2D CrCl₃ phases.

Evidences of Topological Surface States in the Nodal-Line Semimetal SnTaS2 Nanoflakes

Exploring the topological surface state of a topological semimetal by the transport technique has always been a big challenge because of the overwhelming contribution of the bulk state. In this work, we perform systematic angular-dependent magnetotransport measurements and electronic band calculations on SnTaS₂ crystals, a layered topological nodal-line semimetal. Distinct Shubnikov-de Haas quantum oscillations were observed only in SnTaS₂ nanoflakes when the thickness was below about 110 nm, and the oscillation amplitudes increased significantly with decreasing thickness. By analysis of the oscillation spectra, together with the theoretical calculation, a two-dimensional and topological nontrivial nature of the surface band is unambiguously identified, providing direct transport evidence of drumhead surface state for SnTaS₂. Our comprehensive understanding of the Fermi surface topology of the centrosymmetric superconductor SnTaS₂ is crucial for further research on the interplay of superconductivity and nontrivial topology.

Statistical Majorana Bound State Spectroscopy

Tunnel spectroscopy data for the detection of Majorana bound states (MBS) is often criticized for its proneness to misinterpretation of genuine MBS with low-lying Andreev bound states. Here, we suggest a protocol removing this ambiguity by extending single shot measurements to sequences performed at varying system parameters. We demonstrate how such sampling, which we argue requires only moderate effort for current experimental platforms, resolves the statistics of Andreev side lobes, thus providing compelling evidence for the presence or absence of a Majorana center peak.

Topological superconducting vortex from trivial electronic bands

Superconducting vortices are promising traps to confine non-Abelian Majorana quasi-particles. It has been widely believed that bulk-state topology, of either normal-state or superconducting ground-state wavefunctions, is crucial for enabling Majorana zero modes in solid-state systems. This common belief has shaped two major search directions for Majorana modes, in either intrinsic topological superconductors or trivially superconducting topological materials. Here we show that Majorana-carrying superconducting vortex is not exclusive to bulk-state topology, but can arise from topologically trivial quantum materials as well. We predict that the trivial bands in superconducting HgTe-class materials are responsible for inducing anomalous vortex topological physics that goes beyond any existing theoretical paradigms. A feasible scheme of strain-controlled Majorana engineering and experimental signatures for vortex Majorana modes are also discussed. Our work provides new guidelines for vortex-based Majorana search in general superconductors.

Tunable Photoresponse in a Two-Dimensional Superconducting Heterostructure

The photo-induced superconducting phase transition is widely used in probing the physical properties of correlated electronic systems and to realize broadband photodetection with extremely high responsivity. However, such photoresponse is usually insensitive to electrostatic doping due to the high carrier density of the superconductor, restricting its applications in tunable optoelectronic devices. In this work, we demonstrate the gate voltage modulation to the photoresponsivity in a two-dimensional NbSe₂-graphene heterojunction. The superconducting critical current of the NbSe₂ relies on the gate-dependent hot carrier generation in graphene via the Joule heating effect, leading to the observed shift of both the magnitude and peak position of the photoresponsivity spectra as the gate voltage changes. This heating effect is further confirmed by the temperature and laser-power-dependent characterization of the photoresponse. In addition, we investigate the spatially-resolved photocurrent, finding that the superconductivity is inhomogeneous across the junction area. Our results provide a new platform for designing tunable superconducting photodetector and indicate that the photoresponse could be a powerful tool in studying the local electronic properties and phase transitions in low-dimensional superconducting systems.

Competing Vortex Topologies in Iron-Based Superconductors

In this Letter, we establish a new theoretical paradigm for vortex Majorana physics in the recently discovered topological iron-based superconductors (TFeSCs). While TFeSCs are widely accepted as an exemplar of topological insulators (TIs) with intrinsic s-wave superconductivity, our theory implies that such a common belief could be oversimplified. Our main finding is that the normal-state bulk Dirac nodes, usually ignored in TI-based vortex Majorana theories for TFeSCs, will play a key role of determining the vortex state topology. In particular, the interplay between TI and Dirac nodal bands will lead to multiple competing topological phases for a superconducting vortex line in TFeSCs, including an unprecedented hybrid topological vortex state that carries both Majorana bound states and a gapless dispersion. Remarkably, this exotic hybrid vortex phase generally exists in the vortex phase diagram for our minimal model for TFeSCs and is directly relevant to TFeSC candidates such as LiFeAs. When the fourfold rotation symmetry is broken by vortex-line tilting or curving, the hybrid vortex gets topologically trivialized and becomes Majorana free, which could explain the puzzle of ubiquitous trivial vortices observed in LiFeAs. The origin of the Majorana signal in other TFeSC candidates such as FeTe_{x}Se_{1-x} and CaKFe_{4}As_{4} is also interpreted within our theory framework. Our theory sheds new light on theoretically understanding and experimentally engineering Majorana physics in high-temperature iron-based systems.

Topological superconductivity and large spin Hall effect in the kagome family Ti6X4 (X = Bi, Sb, Pb, Tl, and In)

Topological superconductors (TSC) become a focus of research due to the accompanying Majorana fermions. However, the reported TSC are extremely rare. Recent experiments reported kagome TSC AV₃Sb₅ (A = K, Rb, and Cs) exhibit unique superconductivity, topological surface states (TSS), and Majorana bound states. More recently, the first titanium-based kagome superconductor CsTi₃Bi₅ with nontrivial topology was successfully synthesized as a perspective TSC. Given that Cs contributes little to electronic structures of CsTi₃Bi₅ and binary compounds may be easier to be synthesized, here, by first-principle calculations, we predict five stable nonmagnetic kagome compounds Ti₆X₄ (X = Bi, Sb, Pb, Tl, and In) which exhibit superconductivity with critical temperature Tc = 3.8 K - 5.1 K, nontrivial Z 2 band topology, and TSS close to the Fermi level. Additionally, large intrinsic spin Hall effect is obtained in Ti₆X₄, which is caused by gapped Dirac nodal lines due to a strong spin-orbit coupling. This work offers new platforms for TSC and spintronic devices.

Crossover from Ising- to Rashba-type superconductivity in epitaxial Bi2Se3/monolayer NbSe2 heterostructures

A topological insulator (TI) interfaced with an s-wave superconductor has been predicted to host topological superconductivity. Although the growth of epitaxial TI films on s-wave superconductors has been achieved by molecular-beam epitaxy, it remains an outstanding challenge for synthesizing atomically thin TI/superconductor heterostructures, which are critical for engineering the topological superconducting phase. Here we used molecular-beam epitaxy to grow Bi₂Se₃ films with a controlled thickness on monolayer NbSe₂ and performed in situ angle-resolved photoemission spectroscopy and ex situ magnetotransport measurements on these heterostructures. We found that the emergence of Rashba-type bulk quantum-well bands and spin-non-degenerate surface states coincides with a marked suppression of the in-plane upper critical magnetic field of the superconductivity in Bi₂Se₃/monolayer NbSe₂ heterostructures. This is a signature of a crossover from Ising- to Rashba-type superconducting pairings, induced by altering the Bi₂Se₃ film thickness. Our work opens a route for exploring a robust topological superconducting phase in TI/Ising superconductor heterostructures.

Anomalous universal conductance as a hallmark of non-locality in a Majorana-hosted superconducting island

The non-local feature of topological states of matter is the key for the topological protection of quantum information and enables robust non-local manipulation in quantum information. Here we propose to manifest the non-local feature of a Majorana-hosted superconducting island by measuring the temperature dependence of Coulomb blockade peak conductance in different regimes. In the low-temperature regime, we discover a coherent double Majorana-assisted teleportation (MT) process, where any independent tunneling process always involves two coherent non-local MTs; and we also find an anomalous universal scaling behavior, i.e., a crossover from a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\max (T,eV)]}^{6}$$\end{document}[max(T,eV)]6 power-law to a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\max (T,eV)]}^{3}$$\end{document}[max(T,eV)]3 power-law conductance behavior when energy scale increases — in stark contrast to the usual exponential suppression due to certain local transport. In the high-temperature regime, the conductance is instead proportional to the temperature inverse, indicating a non-monotonic temperature-dependence of the conductance. Both the anomalous power law and non-monotonic temperature-dependence of the conductance can be distinguished from the conductance peak in the traditional Coulomb block, and therefore, together serve as a hallmark for the non-local feature in the Majorana-hosted superconducting island.

The ability to detect the non-local nature of topological states in electron transport is highly desirable for topological quantum computation. Hao et al. propose a two-terminal transport scheme to detect the non-locality of a topological superconducting island via anomalous scaling of the tunnelling conductance.

Current through a hybrid four-terminal Josephson junction with Majorana nanowires

We investigate the current through a hybrid four-terminal Josephson junction with semiconductor nanowires, in which the junction is connected with two superconducting electrodes and two normal electrodes. The semiconductor nanowire, which is subject to an external magnetic field with Rashba spin-orbit coupling and proximity-induced superconductivity, can host Majorana bound states. When all the nanowires lie in topological nontrivial region, a 4π-periodic current can be observed through the normal terminal and a 2π-periodic current through the superconducting terminal. When a rotating magnetic field is applied to the junction, the supercurrent through different terminals varies with the variation of the magnetic field direction. Only when the magnetic field is applied at certain angles, we find that the 4π-periodic current will appear through the normal terminal.

Emergent Multifunctional Magnetic Proximity in van der Waals Layered Heterostructures

Proximity effect, which is the coupling between distinct order parameters across interfaces of heterostructures, has attracted immense interest owing to the customizable multifunctionalities of diverse 3D materials. This facilitates various physical phenomena, such as spin order, charge transfer, spin torque, spin density wave, spin current, skyrmions, and Majorana fermions. These exotic physics play important roles for future spintronic applications. Nevertheless, several fundamental challenges remain for effective applications: unavoidable disorder and lattice mismatch limits in the growth process, short characteristic length of proximity, magnetic fluctuation in ultrathin films, and relatively weak spin-orbit coupling (SOC). Meanwhile, the extensive library of atomically thin, 2D van der Waals (vdW) layered materials, with unique characteristics such as strong SOC, magnetic anisotropy, and ultraclean surfaces, offers many opportunities to tailor versatile and more effective functionalities through proximity effects. Here, this paper focuses on magnetic proximity, i.e., proximitized magnetism and reviews the engineering of magnetism-related functionalities in 2D vdW layered heterostructures for next-generation electronic and spintronic devices. The essential factors of magnetism and interfacial engineering induced by magnetic layers are studied. The current limitations and future challenges associated with magnetic proximity-related physics phenomena in 2D heterostructures are further discussed.

Coexistence of Robust Edge States and Superconductivity in Few-Layer Stanene

High-quality stanene films have been actively pursued for realizing not only quantum spin Hall edge states without backscattering, but also intrinsic superconductivity, two central ingredients that may further endow the systems to host topological superconductivity. Yet to date, convincing evidence of topological edge states in stanene remains to be seen, let alone the coexistence of these two ingredients, owing to the bottleneck of growing high-quality stanene films. Here we fabricate one- to five-layer stanene films on the Bi(111) substrate and observe the robust edge states using scanning tunneling microscopy/spectroscopy. We also measure distinct superconducting gaps on different-layered stanene films. Our first-principles calculations further show that hydrogen passivation plays a decisive role as a surfactant in improving the quality of the stanene films, while the Bi substrate endows the films with nontrivial topology. The coexistence of nontrivial topology and intrinsic superconductivity renders the system a promising candidate to become the simplest topological superconductor based on a single-element system.

Controllable majorana vortex states in iron-based superconducting nanowires

To reveal the non-Abelian braiding statistics of Majorana zero modes (MZMs), it is crucial to design a Majorana platform, in which MZMs can be easily manipulated in a broad topological nontrivial parameter space. This is also an essential step to confirm their existence. In this study, we propose an iron-based superconducting nanowire system with Majorana vortex states to satisfy desirable conditions. This system has a radius-induced topological phase transition, giving a lower bound for the nanowire radius. In the topological phase, the iron-based superconducting nanowires have only one pair of MZMs over a wide range of radii, chemical potential and external magnetic fields. The wave function of MZMs has a sizable distribution at the side edge of the nanowires. This property enables the control of the interaction of MZMs in neighboring vortex nanowires and paves the way for Majorana fusion and braiding.

Integration of Topological Insulator Josephson Junctions in Superconducting Qubit Circuits

The integration of semiconductor Josephson junctions (JJs) in superconducting quantum circuits provides a versatile platform for hybrid qubits and offers a powerful way to probe exotic quasiparticle excitations. Recent proposals for using circuit quantum electrodynamics (cQED) to detect topological superconductivity motivate the integration of novel topological materials in such circuits. Here, we report on the realization of superconducting transmon qubits implemented with (Bi_0.06Sb_0.94)₂Te₃ topological insulator (TI) JJs using ultrahigh vacuum fabrication techniques. Microwave losses on our substrates, which host monolithically integrated hardmasks used for the selective area growth of TI nanostructures, imply microsecond limits to relaxation times and, thus, their compatibility with strong-coupling cQED. We use the cavity-qubit interaction to show that the Josephson energy of TI-based transmons scales with their JJ dimensions and demonstrate qubit control as well as temporal quantum coherence. Our results pave the way for advanced investigations of topological materials in both novel Josephson and topological qubits.

Non-Abelian Braiding in Spin Superconductors Utilizing the Aharonov-Casher Effect

Spin superconductor (SSC) is an exciton condensate state where the spin-triplet exciton superfluidity is charge neutral while spin 2(ℏ/2). In analogy to the Majorana zero mode (MZM) in topological superconductors, the interplay between SSC and band topology will also give rise to a specific kind of topological bound state obeying non-Abelian braiding statistics. Remarkably, the non-Abelian geometric phase here originates from the Aharonov-Casher effect of the "half-charge" other than the Aharonov-Bohm effect. Such topological bound state of SSC is bound with the vortex of electric flux gradient and can be experimentally more distinct than the MZM for being electrically charged. This theoretical proposal provides a new avenue investigating the non-Abelian braiding physics without the assistance of MZM and charge superconductor.

High-T c superconductor Fe(Se,Te) monolayer: an intrinsic, scalable and electrically tunable Majorana platform

Iron-based superconductors have been identified as a novel platform for realizing Majorana zero modes (MZMs) without heterostructures, due to their intrinsic topological properties and high-T c superconductivity. In the two-dimensional limit, the FeTe1-x Se x monolayer, a topological band inversion has recently been experimentally observed. Here, we propose to create MZMs by applying an in-plane magnetic field to the FeTe1-x Se x monolayer and tuning the local chemical potential via electric gating. Owing to the anisotropic magnetic couplings on edges, an in-plane magnetic field drives the system into an intrinsic high-order topological superconductor phase with Majorana corner modes. Furthermore, MZMs can occur at the domain wall of chemical potentials at either one edge or certain type of tri-junction in the two-dimensional bulk. Our study not only reveals the FeTe1-x Se x monolayer as a promising Majorana platform with scalability and electrical tunability and within reach of contemporary experimental capability, but also provides a general principle to search for realistic realization of high-order topological superconductivity.

Universal Conductance Scaling of Andreev Reflections Using a Dissipative Probe

The Majorana search is caught up in an extensive debate about the false-positive signals from nontopological Andreev bound states. We introduce a remedy using the dissipative probe to generate electron-boson interaction. We theoretically show that the interaction-induced renormalization leads to significantly distinct universal zero-bias conductance behaviors, i.e., distinct characteristic power law in temperature, for different types of Andreev reflections, that show a sharp contrast to that of a Majorana zero mode. Various specific cases have been studied, including the cases in which two charges involved in an Andreev reflection process maintain or lose coherence, and the cases for multiple Andreev bound states with or without a Majorana. A transparent list of conductance features in each case is provided to help distinguish the observed subgap states in experiments, which also promotes the identification of Majorana zero modes.

Topological Superconducting Transition Characterized by a Modified Real-Space-Pfaffian Method and Mobility Edges in a One-Dimensional Quasiperiodic Lattice

A modified real-space-Pfaffian method is applied to characterize the topological superconducting transition of a one-dimensional p-wave superconductor with quasiperiodic potentials. We found that the Majorana zero-energy mode exists in the topological non-trivial phase, and its spatial distribution is localized at ends of the system, whereas in the topological trivial phase, there is no Majorana zero mode. Furthermore, we numerically found that due to the competition between the localized quasi-disorder and the extended p-wave pairing, there are mobility edges in the energy spectra. Our theoretical work enriches the research on the quasiperiodic p-wave superconducting models.

Chiral Majorana edge modes and vortex Majorana zero modes in superconducting antiferromagnetic topological insulator

The antiferromagnetic topological insulator (AFTI) is topologically protected by the combined time-reversal and translational symmetryTc. In this paper we investigate the effects of thes-wave superconducting pairings on the multilayers of AFTI, which breaksTcsymmetry and can realize quantum anomalous Hall insulator with unit Chern number. For the weakly coupled pairings, the system corresponds to the topological superconductor (TSC) with the Chern numberC= ±2. We answer the following questions whether the local Chern numbers and chiral Majorana edge modes of such a TSC distribute around the surface layers. By the numerical calculations based on a theoretic model of AFTI, we find that when the local Chern numbers are always dominated by the surface layers, the wavefunctions of chiral Majorana edge modes must not localize on the surface layers and show a smooth crossover from spatially occupying all layers to only distributing near the surface layers, similar to the hinge states in a three dimensional second-order topological phases. The latter phase, denoted by the hinged TSC, can be distinguished from the former phase by the measurements of the local density of state. In addition we also study the superconducting vortex phase transition in this system and find that the exchange field in the AFTI not only enlarges the phase space of topological vortex phase but also enhances its topological stability. These conclusions will stimulate the investigations on superconducting effects of AFTI and drive the studies on chiral Majorana edge modes and vortex Majorana zero modes into a new era.

Topographic signatures and manipulations of Fe atoms, CO molecules and NaCl islands on superconducting Pb(111)

Topological superconductivity emerging in one- or two-dimensional hybrid materials is predicted as a key ingredient for quantum computing. However, not only the design of complex heterostructures is primordial for future applications but also the characterization of their electronic and structural properties at the atomic scale using the most advanced scanning probe microscopy techniques with functionalized tips. We report on the topographic signatures observed by scanning tunneling microscopy (STM) of carbon monoxide (CO) molecules, iron (Fe) atoms and sodium chloride (NaCl) islands deposited on superconducting Pb(111). For the CO adsorption a comparison with the Pb(110) substrate is demonstrated. We show a general propensity of these adsorbates to diffuse at low temperature under gentle scanning conditions. Our findings provide new insights into high-resolution probe microscopy imaging with terminated tips, decoupling atoms and molecules by NaCl islands or tip-induced lateral manipulation of iron atoms on top of the prototypical Pb(111) superconducting surface.

Spin-triplet superconductivity in K2Cr3As3

A spin-triplet superconductor can harbor Majorana bound states that can be used in topological quantum computing. Recently, K₂Cr₃As₃ and its variants with critical temperature T_c as high as 8 kelvin have emerged as a new class of superconductors with ferromagnetic spin fluctuations. Here, we report a discovery in K₂Cr₃As₃ single crystal that the spin susceptibility measured by ⁷⁵As Knight shift below T_c is unchanged with the magnetic field H₀ applied in the ab plane but vanishes toward zero temperature when H₀ is along the c axis, which unambiguously establishes this compound as a spin-triplet superconductor described by a vector order parameter d→ parallel to the c axis. Combining with point nodal gap, we show that K₂Cr₃As₃ is a new platform for the study of topological superconductivity and its possible technical application.

Nanoresonator Enhancement of Majorana-Fermion-Induced Slow Light in Superconducting Iron Chains

We theoretically investigate Fano resonance in the absorption spectrum of a quantum dot (QD) based on a hybrid QD-nanomechanical resonator (QD-NR) system mediated by Majorana fermions (MFs) in superconducting iron (Fe) chains. The absorption spectra exhibit a series of asymmetric Fano line shapes, which are accompanied by the rapid normal phase dispersion and induce the optical propagation properties such as the slow light effect under suitable parametric regimes. The results indicated that the slow light induced by MFs can be obtained under different coupling regimes and different detuning regimes. Moreover, we also investigated the role of the NR, and the NR behaving as a phonon cavity enhances the slow light effect.

Majorana Zero Modes in Ferromagnetic Wires without Spin-Orbit Coupling

We present a novel controllable platform for engineering Majorana zero modes. The platform consists of a ferromagnetic metallic wire placed among conventional superconductors, which are in proximity to ferromagnetic insulators. We demonstrate that Majorana zero modes emerge localised at the edges of the ferromagnetic wire, due to the interplay of the applied supercurrents and the induced by proximity exchange fields with conventional superconductivity. Our mechanism does not rely on the pairing of helical fermions by combining conventional superconductivity with spin-orbit coupling, but rather exploits the misalignment between the magnetization of the ferromagnetic insulators and that of the ferromagnetic wire.

Coexistence of Superconductivity and Nontrivial Band Topology in Monolayered Cobalt Pnictides on SrTiO3

As an intrinsically layered material, FeSe has been extensively explored for potentially revealing the underlying mechanisms of high transition temperature (high-T_c) superconductivity and realizing topological superconductivity and Majorana zero modes. Here we use first-principles approaches to identify that the cobalt pnictides of CoX (X = As, Sb, Bi), none of which is a layered material in bulk form. Nevertheless, all can be stabilized as monolayered systems either in freestanding form or supported on the SrTiO₃(001) substrate. We further show that each of the cobalt pnictides may potentially harbor high-T_c superconductivity beyond the Cu- and Fe-based superconducting families, and the underlying mechanism is inherently tied to their isovalency nature with the FeSe monolayer. Most strikingly, each of the monolayered CoX's on SrTiO₃ is shown to be topologically nontrivial, and our findings provide promising new platforms for realizing topological superconductors in the two-dimensional limit.

Phase interference for probing topological fractional charge in a BiSbTeSe2-based Josephson junction array

Fractional charges can be induced by magnetic fluxes at the interface between a topological insulator (TI) and a type-II superconductor due to axion electrodynamics. In a Josephson junction array with a hole in the middle, these electronic states can have phase interference in an applied magnetic field with4×2πperiod, in addition to the 2πinterference of the Cooper pairs. Here, we test an experimental configuration for probing the fractional charge and report the observation of phase interference effect in superconducting arrays with a hole in the middle in both Au- and TI-based devices. Our numerical simulations based on resistive shunted capacitive junction model are in good agreement with the experimental results. However, no clear sign of an axion charge-related interference effect was observed. We will discuss possible reasons and perspectives for future experiments.

Majorana zero modes in impurity-assisted vortex of LiFeAs superconductor

The iron-based superconductor is emerging as a promising platform for Majorana zero mode, which can be used to implement topological quantum computation. One of the most significant advances of this platform is the appearance of large vortex level spacing that strongly protects Majorana zero mode from other low-lying quasiparticles. Despite the advantages in the context of physics research, the inhomogeneity of various aspects hampers the practical construction of topological qubits in the compounds studied so far. Here we show that the stoichiometric superconductor LiFeAs is a good candidate to overcome this obstacle. By using scanning tunneling microscopy, we discover that the Majorana zero modes, which are absent on the natural clean surface, can appear in vortices influenced by native impurities. Our detailed analysis reveals a new mechanism for the emergence of those Majorana zero modes, i.e. native tuning of bulk Dirac fermions. The discovery of Majorana zero modes in this homogeneous material, with a promise of tunability, offers an ideal material platform for manipulating and braiding Majorana zero modes, pushing one step forward towards topological quantum computation.

Despite the discovery of Majorana zero modes (MZM) in iron-based superconductors, sample inhomogeneity may destroy MZMs during braiding. Here, authors observe MZM in impurity-assisted vortices due to tuning of the bulk Dirac fermions in a homogeneous superconductor LiFeAs.

Optical Responses of Chiral Majorana Edge States in Two-Dimensional Topological Superconductors

Majorana fermions exist on the boundaries of two-dimensional topological superconductors (TSCs) as charge-neutral quasiparticles. The neutrality makes the detection of such states challenging from both experimental and theoretical points of view. Current methods largely rely on transport measurements in which Majorana fermions manifest themselves by inducing electron-pair tunneling at the lead-contacting point. Here we show that chiral Majorana fermions in TSCs generate enhanced local optical response. The features of local optical conductivity distinguish them not only from trivial superconductors or insulators but also from normal fermion edge states such as those in quantum Hall systems. Our results provide a new applicable method to detect dispersive Majorana fermions and may lead to a novel direction of this research field.

Low-energy in-gap states of vortices in superconductor-semiconductor heterostructures

The recent interest in the low-energy states in vortices of semiconductor-superconductor heterostructures are mainly fuelled by the prospects of using Majorana zero modes for quantum computation. The knowledge of low-lying states in the vortex core is essential as they pose a limitation on the topological computation with these states. Recently, the low-energy spectra of clean heterostructures, for superconducting-pairing profiles that vary slowly on the scale of the Fermi wavelength of the semiconductor, have been analytically calculated. In this work, we formulate an alternative method based on perturbation theory to obtain concise analytical formulas to predict the low-energy states including explicit magnetic-field and gap profiles. We provide results for both a topological insulator (with a linear spectrum) as well as for a conventional electron gas (with a quadratic spectrum). We discuss the spectra for a wide range of parameters, including both the size of the vortex and the chemical potential of the semiconductor, and thereby provide a tool to guide future experimental efforts. We compare these findings to numerical results.

Atomically Thin Quantum Spin Hall Insulators

Atomically thin topological materials are attracting growing attention for their potential to radically transform classical and quantum electronic device concepts. Among them is the quantum spin Hall (QSH) insulator-a 2D state of matter that arises from interplay of topological band inversion and strong spin-orbit coupling, with large tunable bulk bandgaps up to 800 meV and gapless, 1D edge states. Reviewing recent advances in materials science and engineering alongside theoretical description, the QSH materials library is surveyed with focus on the prospects for QSH-based device applications. In particular, theoretical predictions of nontrivial superconducting pairing in the QSH state toward Majorana-based topological quantum computing are discussed, which are the next frontier in QSH materials research.

Electronic and Magnetic Characterization of Epitaxial CrBr3 Monolayers on a Superconducting Substrate

The ability to imprint a given material property to another through a proximity effect in layered 2D materials has opened the way to the creation of designer materials. Here, molecular-beam epitaxy is used for direct synthesis of a superconductor-ferromagnet heterostructure by combining superconducting niobium diselenide (NbSe₂ ) with the monolayer ferromagnetic chromium tribromide (CrBr₃ ). Using different characterization techniques and density-functional theory calculations, it is confirmed that the CrBr₃ monolayer retains its ferromagnetic ordering with a magnetocrystalline anisotropy favoring an out-of-plane spin orientation. Low-temperature scanning tunneling microscopy measurements show a slight reduction of the superconducting gap of NbSe₂ and the formation of a vortex lattice on the CrBr₃ layer in experiments under an external magnetic field. The results contribute to the broader framework of exploiting proximity effects to realize novel phenomena in 2D heterostructures.

Interference effect on the Andreev reflections induced by Majorana bound states

We investigate the effect of quantum interference on the Andreev reflections (ARs) induced by Majorana bound states (MBSs), by considering their additional coupling via a quantum-dot molecule. It is found that due to the direct and indirect couplings of MBSs, a quantum ring is constructed in this system. Consequently, the interference effect makes important contribution to the ARs, especially in the presence of the local magnetic flux. All the results are manifested as the tight dependence of the differential conductance and Fano factors on the magnetic flux phase factor, dot-MBS couplings, and the dot level, respectively. Moreover, at the zero-bias limit, the magnitudes of the Fano factors and their relation can be efficiently altered by the interference properties. We believe that quantum interference is important for manipulating the Andreev reflection behaviors of the MBSs.

Recent Advances in 2D Superconductors

The emergence of superconductivity in 2D materials has attracted much attention and there has been rapid development in recent years because of their fruitful physical properties, such as high transition temperature (T_c ), continuous phase transition, and enhanced parallel critical magnetic field (B_c ). Tremendous efforts have been devoted to exploring different physical parameters to figure out the mechanisms behind the unexpected superconductivity phenomena, including adjusting the thickness of samples, fabricating various heterostructures, tuning the carrier density by electric field and chemical doping, and so on. Here, different types of 2D superconductivity with their unique characteristics are introduced, including the conventional Bardeen-Cooper-Schrieffer superconductivity in ultrathin films, high-T_c superconductivity in Fe-based and Cu-based 2D superconductors, unconventional superconductivity in newly discovered twist-angle bilayer graphene, superconductivity with enhanced B_c , and topological superconductivity. A perspective toward this field is then proposed based on academic knowledge from the recently reported literature. The aim is to provide researchers with a clear and comprehensive understanding about the newly developed 2D superconductivity and promote the development of this field much further.

Quantum spin Hall and quantum anomalous Hall states in magnetic Ti2Te2O single layer

Magnetic topological insulators, such as MnBi₂Te₄have attracted great attention recently due to their application to the quantum anomalous Hall (QAH) effect. However, the magnetic quantum spin Hall (QSH) effect in two-dimensional (2D) materials has not yet been reported. Here based on first-principle calculations we find that Ti₂Te₂O, a van der Waals layered compound, can cherish both the QAH and QSH states, depending on the magnetic order in its single layer. If the single layer was in a chessboard antiferromagnetic (FM) state, it is a QSH insulator which carries two counterpropagating helical edge states. The spin-orbit-couplings induced bulk band gap can approach as large as 0.31 eV. On the other hand, if the monolayer becomes FM, exchange interactions would push one pair of bands away from the Fermi energy and leave only one chiral edge state remaining, which turns the compound into a Chern insulator (precisely, it is semimetallic with a topologically direct band gap). Both magnetic orders explicitly break the time reversal symmetry and split the energy bands of different spin orientations. To our knowledge, Ti₂Te₂O is the first compound that predicted to possess both intrinsic QSH and QAH effects. Our works provide new possibilities to reach a controllable phase transition between two topological nontrivial phases through magnetism tailoring.

Spin-orbit coupling induced splitting of Yu-Shiba-Rusinov states in antiferromagnetic dimers

Magnetic atoms coupled to the Cooper pairs of a superconductor induce Yu-Shiba-Rusinov states (in short Shiba states). In the presence of sufficiently strong spin-orbit coupling, the bands formed by hybridization of the Shiba states in ensembles of such atoms can support low-dimensional topological superconductivity with Majorana bound states localized on the ensembles’ edges. Yet, the role of spin-orbit coupling for the hybridization of Shiba states in dimers of magnetic atoms, the building blocks for such systems, is largely unexplored. Here, we reveal the evolution of hybridized multi-orbital Shiba states from a single Mn adatom to artificially constructed ferromagnetically and antiferromagnetically coupled Mn dimers placed on a Nb(110) surface. Upon dimer formation, the atomic Shiba orbitals split for both types of magnetic alignment. Our theoretical calculations attribute the unexpected splitting in antiferromagnetic dimers to spin-orbit coupling and broken inversion symmetry at the surface. Our observations point out the relevance of previously unconsidered factors on the formation of Shiba bands and their topological classification.

The influence of spin-orbit coupling on the hybridization of Shiba states in dimers of magnetic atoms on superconducting surfaces remains unexplored. Here, the authors reveal a splitting of atomic Shiba orbitals due to spin-orbit coupling and broken inversion symmetry in antiferromagnetically coupled Mn dimers placed on a Nb(110) surface.

Observation of Distinct Spatial Distributions of the Zero and Nonzero Energy Vortex Modes in (Li_{0.84}Fe_{0.16})OHFeSe

The energy and spatial distributions of vortex bound state in superconductors carry important information about superconducting pairing and the electronic structure. Although discrete vortex states, and sometimes a zero energy mode, had been observed in several iron-based superconductors, their spatial properties are rarely explored. In this study, we used low-temperature scanning tunneling microscopy to measure the vortex state of (Li,Fe)OHFeSe with high spatial resolution. We found that the nonzero energy states display clear spatial oscillations with a period corresponding to bulk Fermi wavelength; while in contrast, the zero energy mode does not show such oscillation, which suggests its distinct electronic origin. Furthermore, the oscillations of positive and negative energy states near E_{F} are found to be clearly out of phase. Based on a two-band model calculation, we show that our observation is more consistent with an s_{++} wave pairing in the bulk of (Li, Fe)OHFeSe, and superconducting topological states on the surface.

Scanning Tunneling Microscopy as a Single Majorana Detector of Kitaev's Chiral Spin Liquid

We propose a local detection scheme for the Majorana zero mode (MZM) carried by a vison in Kitaev's chiral spin liquid (CSL) using scanning tunneling microscopy (STM). The STM introduces a single Majorana into the system through hole-charge injection and the Majorana interacts with the MZM to form a stable composite object. We derive the exact analytical expression of single-hole Green's function in the Mott insulating limit of Kitaev's model, and show that the differential conductance has split peaks, as a consequence of resonant tunneling through the vison-hole composite. The peak splitting turns out comparable to the Majorana gap in CSL, well within the reach of experimental observation.