Creating Majorana modes from segmented Fermi surface

Majorana bound states provide a fertile ground for both investigation of fundamental phenomena as well as for applications in quantum computation. However, despite enormous experimental and theoretical efforts, the currently available Majorana platforms suffer from a multitude of issues that prevent full realization of their potential. Therefore, improved Majorana systems are still highly sought after. Here we present a platform for creating Majorana bound states from 2D gapless superconducting state in spin-helical systems under the in-plane magnetic or Zeeman field. Topological 1D channels are formed by quantum confinement of quasiparticles via Andreev reflection from the surrounding fully gapped superconducting region. Our proposal can be realized using narrow strips of magnetic insulators on top of proximitized 3D topological insulators. This setup has key advantages that include: small required fields, no necessity of fine-tuning of chemical potential, removal of the low-energy detrimental states, and large attainable topological gap.

Phase Signature of Topological Transition in Josephson Junctions

Topological superconductivity holds promise for fault-tolerant quantum computing. While planar Josephson junctions are attractive candidates to realize this exotic state, direct phase measurements as the fingerprint of the topological transition are missing. By embedding two gate-tunable Al/InAs Josephson junctions in a loop geometry, we measure a π jump in the junction phase with an increasing in-plane magnetic field B_{∥}. This jump is accompanied by a minimum of the critical current, indicating a closing and reopening of the superconducting gap, strongly anisotropic in B_{∥}. Our theory confirms that these signatures of a topological transition are compatible with the emergence of Majorana bound states.

Topological Transition of Superconductivity in Dirac Semimetal Nanowire Josephson Junctions

We report the topological transition by gate control in a Cd_{3}As_{2} Dirac semimetal nanowire Josephson junction with diameter of about 64 nm. In the electron branch, the quantum confinement effect enforces the surface band into a series of gapped subbands and thus nontopological states. In the hole branch, however, because the hole mean free path is smaller than the nanowire perimeter, the quantum confinement effect is inoperative and the topological property maintained. The superconductivity is enhanced by gate tuning from electron to hole conduction, manifested by a larger critical supercurrent and a larger critical magnetic field, which is attributed to the topological transition from gapped surface subbands to a gapless surface band. The gate-controlled topological transition of superconductivity should be valuable for manipulation of Majorana zero modes, providing a platform for future compatible and scalable design of topological qubits.

Theoretical investigation of the scanning tunneling microscopy of Majorana bound states in topological superconductor vortices

Scanning tunneling microscopy (STM) is an indispensable tool in detecting Majorana bound states (MBSs) in vortices of topological superconductors. By reducing the computational complexity via non-uniform grids, we systematically study the tunnel coupling as well as the temperature dependence of the differential conductance of MBSs in two dimensional devices. Numerical results show that the conductance peak approaches the quantized value 2e ²/h in strong coupling limit at low temperatures which are characteristic features of MBSs. More interestingly, a conductance local minimum in the spatially scanning is observed when the STM tip is placed at the vortex center. The dip structure can be enhanced with increased temperature or enlarged vortex size. We ascribe this observation to the sensitivity of the Andreev reflection processes of carriers at the vortex center where the thermal energy could be comparable to the vanishing pair potential. We also investigate the STM of two-vortex systems where the hybridization of the vortices can lead to oscillatory behavior of the state energy. With small inter-vortex distances, the original MBSs in vortices can merge into topologically trivial states and the conductance peak can be significantly suppressed.

Spatially dispersing Yu-Shiba-Rusinov states in the unconventional superconductor FeTe0.55Se0.45

By using scanning tunneling microscopy (STM) we find and characterize dispersive, energy-symmetric in-gap states in the iron-based superconductor FeTe_0.55Se_0.45, a material that exhibits signatures of topological superconductivity, and Majorana bound states at vortex cores or at impurity locations. We use a superconducting STM tip for enhanced energy resolution, which enables us to show that impurity states can be tuned through the Fermi level with varying tip-sample distance. We find that the impurity state is of the Yu-Shiba-Rusinov (YSR) type, and argue that the energy shift is caused by the low superfluid density in FeTe_0.55Se_0.45, which allows the electric field of the tip to slightly penetrate the sample. We model the newly introduced tip-gating scenario within the single-impurity Anderson model and find good agreement to the experimental data.

Atomic-scale spin-polarization maps using functionalized superconducting probes

A scanning tunneling microscope (STM) with a magnetic tip that has a sufficiently strong spin polarization can be used to map the sample's spin structure down to the atomic scale but usually lacks the possibility to absolutely determine the value of the sample's spin polarization. Magnetic impurities in superconducting materials give rise to pairs of perfectly, i.e., 100%, spin-polarized subgap resonances. In this work, we functionalize the apex of a superconducting Nb STM tip with such impurity states by attaching Fe atoms to probe the spin polarization of atom-manipulated Mn nanomagnets on a Nb(110) surface. By comparison with spin-polarized STM measurements of the same nanomagnets using Cr bulk tips, we demonstrate an extraordinary spin sensitivity and the possibility to measure the sample's spin-polarization values close to the Fermi level quantitatively with our new functionalized probes.

Controlling in-gap end states by linking nonmagnetic atoms and artificially-constructed spin chains on superconductors

Chains of magnetic atoms with either strong spin-orbit coupling or spiral magnetic order which are proximity-coupled to superconducting substrates can host topologically non-trivial Majorana bound states. The experimental signature of these states consists of spectral weight at the Fermi energy which is spatially localized near the ends of the chain. However, topologically trivial Yu-Shiba-Rusinov in-gap states localized near the ends of the chain can lead to similar spectra. Here, we explore a protocol to disentangle these contributions by artificially augmenting a candidate Majorana spin chain with orbitally-compatible nonmagnetic atoms. Combining scanning tunneling spectroscopy with ab-initio and tight-binding calculations, we realize a sharp spatial transition between the proximity-coupled spiral magnetic order and the non-magnetic superconducting wire termination, with persistent zero-energy spectral weight localized at either end of the magnetic spiral. Our findings open a new path towards the control of the spatial position of in-gap end states, trivial or Majorana, via different chain terminations, and the realization of designer Majorana chain networks for demonstrating topological quantum computation.

Long-range focusing of magnetic bound states in superconducting lanthanum

Quantum mechanical systems with long-range interactions between quasiparticles provide a promising platform for coherent quantum information technology. Superconductors are a natural choice for solid-state based quantum devices, while magnetic impurities inside superconductors give rise to quasiparticle excitations of broken Cooper pairs that provide characteristic information about the host superconductor. Here, we reveal that magnetic impurities embedded below a superconducting La(0001) surface interact via quasiparticles extending to very large distances, up to several tens of nanometers. Using low-temperature scanning probe techniques, we observe the corresponding anisotropic and giant oscillations in the LDOS. Theoretical calculations indicate that the quasi-two-dimensional surface states with their strongly anisotropic Fermi surface play a crucial role for the focusing and long-range extension of the magnetic bound states. The quasiparticle focusing mechanism should facilitate the design of versatile magnetic structures with tunable and directed magnetic interactions over large distances, thereby paving the way toward the design of low-dimensional magnet-superconductor hybrid systems exhibiting topologically non-trivial quantum states as possible elements of quantum computation schemes based on Majorana quasiparticles.

Unveiling Odd-Frequency Pairing around a Magnetic Impurity in a Superconductor

We study the unconventional superconducting correlations caused by a single isolated magnetic impurity in a conventional s-wave superconductor. Because of the local breaking of time-reversal symmetry, the impurity induces unconventional superconductivity, which is even in both space and spin variables but odd under time inversion. We derive an exact proportionality relation between the even-frequency component of the local electron density of states and the imaginary part of the odd-frequency local pairing function. By applying this relation to scanning tunneling microscopy spectra taken on top of magnetic impurities immersed in a Pb/Si(111) monolayer, we show experimental evidence of the occurrence of the odd-frequency pairing in these systems and explicitly extract its superconducting function from the data.

Universal signatures of Majorana-like quasiparticles in strongly correlated Landau-Fermi liquids

Motivated by recent experiments in the Kitaev honeycomb lattice, Kondo insulators, and the 'Luttinger's theorem-violating' Fermi liquid phase of the underdoped cuprates, we extend the theoretical machinery of Landau-Fermi liquid theory to a system of itinerant, interacting Majorana-like particles. Building upon a previously introduced model of 'nearly self-conjugate' fermionic polarons, a Landau-Majorana kinetic equation is introduced to describe the collective modes and Fermi surface instabilities in a fluid of particles whose fermionic degrees of freedom obey the Majorana reality condition. At large screening, we show that the Landau-Majorana liquid harbors a Lifshitz transition for specific values of the driving frequency. Moreover, we find the dispersion of the zero sound collective mode in such a system, showing that there exists a specific limit where the Landau-Majorana liquid harbors a stability against Pomeranchuk deformations unseen in the conventional Landau-Fermi liquid. With these results, our work paves the way for possible extensions of the Landau quasiparticle paradigm to nontrivial metallic phases of matter.

Topologization of β-antimonene on Bi2Se3 via proximity effects

Topological surface states usually emerge at the boundary between a topological and a conventional insulator. Their precise physical character and spatial localization depend on the complex interplay between the chemical, structural and electronic properties of the two insulators in contact. Using a lattice-matched heterointerface of single and double bilayers of β-antimonene and bismuth selenide, we perform a comprehensive experimental and theoretical study of the chiral surface states by means of microscopy and spectroscopic measurements complemented by first-principles calculations. We demonstrate that, although β-antimonene is a trivial insulator in its free-standing form, it inherits the unique symmetry-protected spin texture from the substrate via a proximity effect that induces outward migration of the topological state. This "topologization" of β-antimonene is found to be driven by the hybridization of the bands from either side of the interface.

Superhigh Uniform Magnetic Cr Substitution in a 2D Mo2 C Superconductor for a Macroscopic-Scale Kondo Effect

Substitutional doping provides an effective strategy to tailor the properties of 2D materials, but it remains an open challenge to achieve tunable uniform doping, especially at high doping level. Here, uniform lattice substitution of a 2D Mo₂ C superconductor by magnetic Cr atoms with controlled concentration up to ≈46.9 at% by chemical vapor deposition and a specifically designed Cu/Cr/Mo trilayer growth substrate is reported. The concentration of Cr atoms can be easily tuned by simply changing the thickness of the Cr layer, and the samples retain the original structure of 2D Mo₂ C even at a very high Cr concentration. The controlled uniform Cr doping enables the tuning of the competition of the 2D superconductor and the Kondo effect across the whole sample. Transport measurements show that with increasing Cr concentration, the superconductivity of the 2D Cr-doped Mo₂ C crystals disappears along with the emergence of the Kondo effect, and the Kondo temperature increases monotonously. Using scanning tunneling microscopy/spectroscopy, the mechanism of the doping level effect on the interplay and evolution between superconductivity and the Kondo effect is revealed. This work paves a new way for the synthesis of 2D materials with widely tunable doping levels, and provides new understandings on the interplay between superconductivity and magnetism in the 2D limit.

On the Remarkable Superconductivity of FeSe and Its Close Cousins

Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origins and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years, focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe1−xSex, intercalated FeSe materials, and monolayer FeSe and FeTe1−xSex on SrTiO3. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

Exact eigenvectors and eigenvalues of the finite Kitaev chain and its topological properties

We present a comprehensive, analytical treatment of the finite Kitaev chain for arbitrary chemical potential and chain length. By means of an exact analytical diagonalization in the real space, we derive the momentum quantization conditions and present exact analytical formulas for the resulting energy spectrum and eigenstate wave functions, encompassing boundary and bulk states. In accordance with an analysis based on the winding number topological invariant, and as expected from the bulk-edge correspondence, the boundary states are topological in nature. They can have zero, exponentially small or even finite energy. Further, for a fixed value of the chemical potential, their properties are ruled by the ratio of the decay length to the chain length. A numerical analysis confirms the robustness of the topological states against disorder.

Magnetic field effect on trivial and topological bound states of superconducting quantum dot

We investigate the properties of a quantum dot embedded between the normal and superconducting leads which is additionally side-attached to the topological superconducting nanowire, hosting the Majorana modes. This setup enables formation of the trivial (finite-energy) bound states induced in the quantum dot through the superconducting proximity effect, coexisting/competing with the topological (zero-energy) mode transmitted from the topological superconductor. We analyze their interplay, focusing on a role played by the external magnetic field. To distinguish between these bound states we analyze the qualitative and quantitative features manifested in the subgap charge tunneling originating under nonequilibrium conditions from the Andreev (particle to hole) scattering processes.

Leakage of Majorana bound states in an inhomogeneous topological nanowire

We present an exact solution of the continuum Bogolyubov-de-Gennes Hamiltonian for Majorana bound states (MBSs) generated in a superconductor-semiconductor hybrid topological nanowire. The full energy spectra that include the band states and in-gap states could be obtained. We show that for relatively short wire length, the zero energy mode could be induced even in the topological trivial regime, which also indicates oscillatory dependence on the chemical potential. With the increase of the Zeeman field, the MBSs are almost fully spin-polarized and do not localize at the wire ends gradually. We also extend our discussion to the property of Majorana modes in an inhomogeneous nanowire, in which a local gate voltage is applied to one end of the nanowire. It is found that the local potential barrier or well could modulate the Majorana energy splitting periodically. The leakage of MBSs to the potential region is exponentially suppressed for the barrier case. A potential well could induce near-zero-energy bound states and these states merge with MBSs, leading to the delocalization of MBSs. In the potential well region, both the spin-up and spin-down components of the trivial states account for a significant proportion, which can be detected experimentally.

Direct Observation of One-Dimensional Peierls-type Charge Density Wave in Twin Boundaries of Monolayer MoTe2

One-dimensional (1D) metallic mirror-twin boundaries (MTBs) in monolayer transition-metal dichalcogenides exhibit a periodic charge modulation and provide an ideal platform for exploring collective electron behavior in the confined system. The underlying mechanism of the charge modulation and how the electrons travel in 1D structures remain controversial. Here, for the first time, we observed atomic-scale structures of the charge distribution within one period in MTB of monolayer MoTe₂ by using scanning tunneling microscopy/spectroscopy. The coexisting apparent periodic lattice distortions and U-shaped energy gap clearly demonstrate a Peierls-type charge density wave (CDW). Equidistant quantized energy levels with varied periodicity are further discovered outside the CDW gap along the metallic MTB. Density functional theory calculations are in good agreement with the gapped electronic structures and reveal that they originate mainly from a Mo 4d orbital. Our work presents hallmark evidence of the 1D Peierls-type CDW on the metallic MTBs and offers opportunities to study the underlying physics of 1D charge modulation.

Coupling of superconductivity and Coulomb blockade in Sn nanoparticles

Low dimensional superconductors have many unusual properties. When 0-dimensional superconductors reach the nanometer scale, the superconducting energy gap can be enhanced due to the shell effect. At the same time, the single electron Coulomb blockade effect can also be observed on metal nanoparticles if they are weakly coupled to the environment. So, if a superconducting nanoparticle is isolated well from the environment, the superconducting gap and the Coulomb gap would couple together, making the tunneling spectrum more complicated and interesting. Here Sn nanoparticles were deposited on the surface of STO (111). The charging energy of a nanoparticle mainly depends on its size and is comparable to the superconducting gap when the isolated particle is large enough. The superconducting energy gap can be deduced from the coupling tunneling spectrum and the shell effect is observed. The method to deduce the superconducting gap here is simpler than when fit using the Dynes density of states. Owing to the increased superconducting gap and critical field, the studied nanoparticles may find applications in studies of the properties of Majorana fermions.

Block-spiral magnetism: An exotic type of frustrated order

Competing interactions in quantum materials induce exotic states of matter such as frustrated magnets, an extensive field of research from both the theoretical and experimental perspectives. Here, we show that competing energy scales present in the low-dimensional orbital-selective Mott phase (OSMP) induce an exotic magnetic order, never reported before. Earlier neutron-scattering experiments on iron-based 123 ladder materials, where OSMP is relevant, already confirmed our previous theoretical prediction of block magnetism (magnetic order of the form [Formula: see text]). Now we argue that another phase can be stabilized in multiorbital Hubbard models, the block-spiral state. In this state, the magnetic islands form a spiral propagating through the chain but with the blocks maintaining their identity, namely rigidly rotating. The block-spiral state is stabilized without any apparent frustration, the common avenue to generate spiral arrangements in multiferroics. By examining the behavior of the electronic degrees of freedom, parity-breaking quasiparticles are revealed. Finally, a simple phenomenological model that accurately captures the macroscopic spin spiral arrangement is also introduced, and fingerprints for the neutron-scattering experimental detection are provided.

Handedness-dependent quasiparticle interference in the two enantiomers of the topological chiral semimetal PdGa

It has recently been proposed that combining chirality with topological band theory results in a totally new class of fermions. Understanding how these unconventional quasiparticles propagate and interact remains largely unexplored so far. Here, we use scanning tunneling microscopy to visualize the electronic properties of the prototypical chiral topological semimetal PdGa. We reveal chiral quantum interference patterns of opposite spiraling directions for the two PdGa enantiomers, a direct manifestation of the change of sign of their Chern number. Additionally, we demonstrate that PdGa remains topologically non-trivial over a large energy range, experimentally detecting Fermi arcs in an energy window of more than 1.6 eV that is symmetrically centered around the Fermi level. These results are a consequence of the deep connection between chirality in real and reciprocal space in this class of materials, and, thereby, establish PdGa as an ideal topological chiral semimetal.

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling

The combination of spin-orbit coupling with interactions results in many exotic phases of matter. In this Letter, we investigate the superconducting pairing instability of the two-dimensional extended Hubbard model with both Rashba and Dresselhaus spin-orbit coupling within the mean-field level at both zero and finite temperature. We find that both first- and second-order time-reversal symmetry breaking topological gapped phases can be achieved under appropriate parameters and temperature regimes due to the presence of a favored even-parity s+id-wave pairing even in the absence of an external magnetic field or intrinsic magnetism. This results in two branches of chiral Majorana edge states on each edge or a single zero-energy Majorana corner state at each corner of the sample. Interestingly, we also find that not only does tuning the doping level lead to a direct topological phase transition between these two distinct topological gapped phases, but also using the temperature as a highly controllable and reversible tuning knob leads to different direct temperature-driven topological phase transitions between gapped and gapless topological superconducting phases. Our findings suggest new possibilities in interacting spin-orbit coupled systems by unifying both first- and higher-order topological superconductors in a simple but realistic microscopic model.

Rigorous Results on Topological Superconductivity with Particle Number Conservation

Most theoretical studies of topological superconductors and Majorana-based quantum computation rely on a mean-field approach to describe superconductivity. A potential problem with this approach is that real superconductors are described by number-conserving Hamiltonians with long-range interactions, so their topological properties may not be correctly captured by mean-field models that violate number conservation and have short-range interactions. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the mean-field models, including a finite energy gap in a sector of fixed total particle number, the existence of long-range Majorana-like correlations between the ends of an open wire, and a change in the ground state fermion parity for periodic vs antiperiodic boundary conditions. These results show that many of the remarkable properties of mean-field models of topological superconductivity persist in more realistic models with number-conserving dynamics.

Two-Dimensional Noble-Metal Chalcogenides and Phosphochalcogenides

Noble-metal chalcogenides, dichalcogenides, and phosphochalcogenides are an emerging class of two-dimensional materials. Quantum confinement (number of layers) and defect engineering enables their properties to be tuned over a broad range, including metal-to-semiconductor transitions, magnetic ordering, and topological surface states. They possess various polytypes, often of similar formation energy, which can be accessed by selective synthesis approaches. They excel in mechanical, optical, and chemical sensing applications, and feature long-term air and moisture stability. In this Minireview, we summarize the recent progress in the field of noble-metal chalcogenides and phosphochalcogenides and highlight the structural complexity and its impact on applications.

In-Plane Zeeman-Field-Induced Majorana Corner and Hinge Modes in an s-Wave Superconductor Heterostructure

Second-order topological superconductors host Majorana corner and hinge modes in contrast to conventional edge and surface modes in two and three dimensions. However, the realization of such second-order corner modes usually demands unconventional superconducting pairing or complicated junctions or layered structures. Here we show that Majorana corner modes could be realized using a 2D quantum spin Hall insulator in proximity contact with an s-wave superconductor and subject to an in-plane Zeeman field. Beyond a critical value, the in-plane Zeeman field induces opposite effective Dirac masses between adjacent boundaries, leading to one Majorana mode at each corner. A similar paradigm also applies to 3D topological insulators with the emergence of Majorana hinge states. Avoiding complex superconductor pairing and material structure, our scheme provides an experimentally realistic platform for implementing Majorana corner and hinge states.

Robust Majorana edge modes with low frequency multiple time periodic driving

Floquet Majorana edge modes capture the topological features of periodically driven p-wave superconductors. We present a Kitaev chain with multiple time periodic driving terms. Our results demonstrate how multiple driving will affect Floquet bands in frequency space, leading to more robust Floquet Majorana edge modes against driving frequencyωin comparison with the single driving scenario. Meanwhile, we have proposed how to predict Majorana edge modes via the Zak phase of Floquet bands. Besides, in contrast to the cases with single driving term, where the constant phase can be gauged out by properly choosing the initial time, we have shown the relative phase between multiple driving can not be gauged out and will play a dominant role in deciding topological phase transitions. For the sake of completeness, we also investigate the high frequency limit. Analytical results on effective Hamiltonian can be obtained via Magnus expansion and relative phase induced topological transitions can be shown explicitly.

Imprinting Ferromagnetism and Superconductivity in Single Atomic Layers of Molecular Superlattices

Ferromagnetism and superconductivity are two antagonistic phenomena since ferromagnetic exchange fields tend to destroy singlet Cooper pairs. Reconciliation of these two competing phases has been achieved in vertically stacked heterostructures where these two orders are confined in different layers. However, controllable integration of these two phases in one atomic layer is a longstanding challenge. Here, an interlayer-space-confined chemical design (ICCD) is reported for the synthesis of dilute single-atom-doped TaS₂ molecular superlattice, whereby ferromagnetism is observed in the superconducting TaS₂ layers. The intercalation of 2H-TaS₂ crystal with bulky organic ammonium molecule expands its van der Waals gap for single-atom doping via co-intercalated cobalt ions, resulting in the formation of quasi-monolayer Co-doped TaS₂ superlattices. Isolated Co atoms are decorated in the basal plane of the TaS₂ via substituting the Ta atom or anchoring at a hollow site, wherein the orbital-selected p-d hybridization between Co and neighboring Ta and S atoms induces local magnetic moments with strong ferromagnetic coupling. This ICCD approach can be applied to various metal ions, enabling the synthesis of a series of crystal-size TaS₂ molecular superlattices.

Robust Zero Energy Modes on Superconducting Bismuth Islands Deposited on Fe(Te,Se)

Topological superconductivity is one of the frontier research directions in condensed matter physics. One of the unique elementary excitations in topological superconducting state is the Majorana Fermion (mode) which is its own antiparticle and obeys the non-Abelian statistics and is thus useful for constructing the fault-tolerant quantum computation. The evidence for Majorana Fermions (mode) in condensed matter is now quickly accumulated. We deposit Bi islands on the iron-based superconductors Fe(Te,Se) and find the easily achievable zero energy modes on the tunneling spectra on some Bi islands. The zero energy mode is robust and appears everywhere on the island. Temperature and magnetic field dependence of the zero energy mode are also investigated. We attribute these zero energy modes to the Majorana modes due to the proximity effect-induced topological superconductivity on the Bi islands with strong spin-orbit coupling effect.

Spin Transport in Ferromagnet-InSb Nanowire Quantum Devices

Signatures of Majorana zero modes (MZMs) have been observed in semiconductor nanowires (NWs) with a strong spin-orbital interaction (SOI) with proximity-induced superconductivity. Realizing topological superconductivity and MZMs in this platform requires eliminating spin degeneracy by applying a magnetic field. However, the field can adversely impact the induced superconductivity and places geometric restrictions on the device. These challenges could be circumvented by integrating magnetic elements with the NWs. Here, we report the first experimental investigation of spin transport across InSb NWs with ferromagnetic (FM) contacts. We observe signatures of spin polarization and spin-dependent transport in the quasi-one-dimensional ballistic regime. Moreover, we show that electrostatic gating tunes the observed magnetic signal and reveals a regime where the device acts as a spin filter. These results open an avenue toward developing MZM devices with spin degeneracy lifted locally without external fields. They could also enable spin-based devices that leverage spin-orbital states in quantum wires.

Superconducting proximity effect in flat band systems

We study theoretically proximity-induced superconductivity and its inverse effect in dice lattice flat band model by considering Josephson junction with an s-wave pairing in the superconducting leads. Using self-consistent tight-binding Bogoliubov-de Gennes method, we show that there is a critical value for chemical potential of the superconductors depending on paring interaction strength over which for undoped normal region the proximity effect is enhanced. Whereas if the supserconductor chemical potential is less than the critical one the proximity effect decreases regardless of normal region doping and in the meanwhile, the pairing amplitude of superconducting region increases significantly. Furthermore, we unveil that the supercurrent passing through the junction is large (vanishingly small) when the superconductor chemical potential is smaller (larger) than the critical value which increases as a function of normal region chemical potential.

Yu-Shiba-Rusinov bands in ferromagnetic superconducting diamond

The combination of different exotic properties in materials paves the way for the emergence of their new potential applications. An example is the recently found coexistence of the mutually antagonistic ferromagnetism and superconductivity in hydrogenated boron-doped diamond, which promises to be an attractive system with which to explore unconventional physics. Here, we show the emergence of Yu-Shiba-Rusinov (YSR) bands with a spatial extent of tens of nanometers in ferromagnetic superconducting diamond using scanning tunneling spectroscopy. We demonstrate theoretically how a two-dimensional (2D) spin lattice at the surface of a three-dimensional (3D) superconductor gives rise to the YSR bands and how their density-of-states profile correlates with the spin lattice structure. The established strategy to realize new forms of the coexistence of ferromagnetism and superconductivity opens a way to engineer the unusual electronic states and also to design better-performing superconducting devices.

Topological Semimetal Nanostructures: From Properties to Topotronics

Characterized by bulk Dirac or Weyl cones and surface Fermi-arc states, topological semimetals have sparked enormous research interest in recent years. The nanostructures, with large surface-to-volume ratio and easy field-effect gating, provide ideal platforms to detect and manipulate the topological quantum states. Exotic physical properties originating from these topological states endow topological semimetals attractive for future topological electronics (topotronics). For example, the linear energy dispersion relation is promising for broadband infrared photodetectors, the spin-momentum locking nature of topological surface states is valuable for spintronics, and the topological superconductivity is highly desirable for fault-tolerant qubits. For real-life applications, topological semimetals in the form of nanostructures are necessary in terms of convenient fabrication and integration. Here, we review the recent progresses in topological semimetal nanostructures and start with the quantum transport properties. Then topological semimetal-based electronic devices are introduced. Finally, we discuss several important aspects that should receive great effort in the future, including controllable synthesis, manipulation of quantum states, topological field effect transistors, spintronic applications, and topological quantum computation.

Signature of a pair of Majorana zero modes in superconducting gold surface states

Under certain conditions, a fermion in a superconductor can separate in space into two parts known as Majorana zero modes, which are immune to decoherence from local noise sources and are attractive building blocks for quantum computers. Promising experimental progress has been made to demonstrate Majorana zero modes in materials with strong spin-orbit coupling proximity coupled to superconductors. Here we report signatures of Majorana zero modes in a material platform utilizing the surface states of gold. Using scanning tunneling microscope to probe EuS islands grown on top of gold nanowires, we observe two well-separated zero-bias tunneling conductance peaks aligned along the direction of the applied magnetic field, as expected for a pair of Majorana zero modes. This platform has the advantage of having a robust energy scale and the possibility of realizing complex designs using lithographic methods.

Observation of short-range Yu-Shiba-Rusinov states with threefold symmetry in layered superconductor 2H-NbSe2

Yu-Shiba-Rusinov (YSR) states arise when magnetic impurities interact with superconductivity. The intricacy of coupling and the nature of the superconductivity determine the behavior of the YSR state, whose detailed correlations are not yet fully understood. Here, we study the YSR state of a single Fe adatom on the surface of 2H-NbSe₂ with combined low temperature scanning tunneling microscopy/spectroscopy, density functional theory calculations and tight-binding modeling. It is found that the Fe adatom occupies the hollow site of the Se surface layer. A prominent YSR state close to the Fermi level is observed. The YSR state exhibits a threefold symmetry along the diagonal direction of the Se lattice. The spatial decay of the YSR state follows a behavior in three-dimensional superconductivity. This behavior contrasts with a previous study of imbedded Fe impurities, whose YSR state shows a six-fold symmetry and a two-dimensional long-range decay. According to our theoretical modeling, the coupling configurations affect the adatom-substrate hopping and the interlayer coupling of the substrate. Both factors are crucial for the consequent behavior of the YSR state.

Induced Topological Superconductivity in a BiSbTeSe2-Based Josephson Junction

A 4 π -periodic supercurrent through a Josephson junction can be a consequence of the presence of Majorana bound states. A systematic study of the radio frequency response for several temperatures and frequencies yields a concrete protocol for examining the 4 π -periodic contribution to the supercurrent. This work also reports the observation of a 4 π -periodic contribution to the supercurrent in BiSbTeSe 2 -based Josephson junctions. As a response to irradiation by radio frequency waves, the junctions showed an absence of the first Shapiro step. At high irradiation power, a qualitative correspondence to a model including a 4 π -periodic component to the supercurrent is found.

Thermoelectric Effect in a Correlated Quantum Dot Side-Coupled to Majorana Bound States

We theoretically study the thermoelectric effect in a hybrid device composed by a topological semiconducting nanowire hosting Majorana bound states (MBSs) and a quantum dot (QD) connected to the left and right non-magnetic electrodes held at different temperatures. The electron-electron Coulomb interactions in the QD are taken into account by the non-equilibrium Green's function technique. We find that the sign change of the thermopower, which is useful for detecting the MBSs, will occur by changing the QD-MBS hybridization strength, the direct overlap between the MBSs at the opposite ends of the nanowire, and the system temperature. Large value of 100% spin-polarized or pure spin thermopower emerges even in the absence of Zeeman splitting in the QD or magnetic electrodes because the MBSs are coupled to electrons of only one certain spin direction in the QD due to the chiral nature of the Majorana fermions. Moreover, the magnitude of the thermopower will be obviously enhanced by the existence of MBSs.

Large spatial extension of the zero-energy Yu-Shiba-Rusinov state in a magnetic field

Various promising qubit concepts have been put forward recently based on engineered superconductor subgap states like Andreev bound states, Majorana zero modes or the Yu-Shiba-Rusinov (Shiba) states. The coupling of these subgap states via a superconductor strongly depends on their spatial extension and is an essential next step for future quantum technologies. Here we investigate the spatial extension of a Shiba state in a semiconductor quantum dot coupled to a superconductor. With detailed transport measurements and numerical renormalization group calculations we find a remarkable more than 50 nm extension of the zero energy Shiba state, much larger than the one observed in very recent scanning tunneling microscopy measurements. Moreover, we demonstrate that its spatial extension increases substantially in a magnetic field.

Local magnetic moments coupled to superconductors can form subgap Yu-Shiba-Rusinov states. Here the authors show that Shiba states made with an InAs nanowire quantum dot have large spatial extent, which is beneficial for making Shiba chains that are predicted to host Majorana zero modes.

Magnitude and Spatial Distribution Control of the Supercurrent in Bi2O2Se-Based Josephson Junction

Many proposals for exploring topological quantum computation are based on superconducting quantum devices constructed on materials with strong spin-orbit coupling (SOC). For these devices, full control of both the magnitude and the spatial distribution of the supercurrent is highly demanded, but has been elusive up to now. We constructed a proximity-type Josephson junction on nanoplates of Bi₂O₂Se, a new emerging semiconductor with strong SOC. Through electrical gating, we show that the supercurrent can be fully turned ON and OFF, and its real-space pathways can be configured either through the bulk or along the edges. Our work demonstrates Bi₂O₂Se as a promising platform for constructing multifunctional hybrid superconducting devices as well as for searching for topological superconductivity.

Weak Measurement Protocols for Majorana Bound State Identification

We propose a continuous weak measurement protocol testing the nonlocality of Majorana bound states through current shot noise correlations. The experimental setup contains a topological superconductor island with three normal-conducting leads weakly coupled to different Majorana states. Putting one lead at finite voltage and measuring the shot noise correlations between the other two (grounded) leads, devices with true Majorana states are distinguished from those without by strong current correlations. The presence of true Majorana states manifests itself in unusually high noise levels or the near absence of noise, depending on the chosen device configuration. Monitoring the noise statistics amounts to a weak continuous measurement of the Majorana qubit and yields information similar to that of a full braiding protocol, but at much lower experimental effort. Our theory can be adapted to different platforms and should allow for the clear identification of Majorana states.

Zero-energy bound states in the high-temperature superconductors at the two-dimensional limit

Majorana zero modes (MZMs) that obey the non-Abelian statistics have been intensively investigated for potential applications in topological quantum computing. The prevailing signals in tunneling experiments "fingerprinting" the existence of MZMs are the zero-energy bound states (ZEBSs). However, nearly all of the previously reported ZEBSs showing signatures of the MZMs are observed in difficult-to-fabricate heterostructures at very low temperatures and additionally require applied magnetic field. Here, by using in situ scanning tunneling spectroscopy, we detect the ZEBSs upon the interstitial Fe adatoms deposited on two different high-temperature superconducting one-unit-cell iron chalcogenides on SrTiO3(001). The spectroscopic results resemble the phenomenological characteristics of the MZMs inside the vortex cores of topological superconductors. Our experimental findings may extend the MZM explorations in connate topological superconductors toward an applicable temperature regime and down to the two-dimensional (2D) limit.

A Study of Interaction Effects and Quantum Berezinskii- Kosterlitz-Thouless Transition in the Kitaev Chain

The physics of the topological state of matter is the second revolution in quantum mechanics. We study the effect of interactions on the topological quantum phase transition and the quantum Berezinskii-Kosterlitz-Thouless (QBKT) transition in topological state of a quantum many-body condensed matter system. We predict a topological quantum phase transition from topological superconducting phase to an insulating phase for the interacting Kitaev chain. We observe interesting behaviour from the results of renormalization group study on the topological superconducting phase. We derive the renormalization group (RG) equation for QBKT through different routes with a few exact solutions along with the physical explanations, wherein we find the existence of two new important emergent phases apart from the two conventional phases of this model Hamiltonian. We also present results of a length-scale dependent study to predict asymptotic freedom like behaviour of the system. We do rigorous quantum field theoretical renormalization group calculations to solve this problem.

Artificial relativistic molecules

We fabricate artificial molecules composed of heavy atom lead on a van der Waals crystal. Pb atoms templated on a honeycomb charge-order superstructure of IrTe₂ form clusters ranging from dimers to heptamers including benzene-shaped ring hexamers. Tunneling spectroscopy and electronic structure calculations reveal the formation of unusual relativistic molecular orbitals within the clusters. The spin–orbit coupling is essential both in forming such Dirac electronic states and stabilizing the artificial molecules by reducing the adatom–substrate interaction. Lead atoms are found to be ideally suited for a maximized relativistic effect. This work initiates the use of novel two-dimensional orderings to guide the fabrication of artificial molecules of unprecedented properties.

Artificial molecules supported on templated surfaces attract enormous interest due to their tunable electronic properties. Here the authors use STM experiments and DFT calculations to show the formation of Pb artificial clusters on a IrTe₂ honeycomb template that are maximally stabilized by relativistic effects.

Vortex Dynamics of Charge Carriers in the Quasi-Relativistic Graphene Model: High-Energy k → · p → Approximation

Within the earlier developed high-energy- k → · p → -Hamiltonian approach to describe graphene-like materials, the simulations of non-Abelian Zak phases and band structure of the quasi-relativistic graphene model with a number of flavors N = 3 have been performed in approximations with and without gauge fields (flavors). It has been shown that a Zak-phases set for non-Abelian Majorana-like excitations (modes) in Dirac valleys of the quasi-relativistic graphene model is the cyclic group Z 12 . This group is deformed into Z 8 at sufficiently high momenta due to deconfinement of the modes. Since the deconfinement removes the degeneracy of the eightfolding valleys, Weyl nodes and antinodes emerge. We offer that a Majorana-like mass term of the quasi-relativistic model affects the graphene band structure in the following way. Firstly, the inverse symmetry emerges in the graphene model with Majorana-like mass term, and secondly the mass term shifts the location of Weyl nodes and antinodes into the region of higher energies.

Direction and symmetry transition of the vector order parameter in topological superconductors CuxBi2Se3

Topological superconductors have attracted wide-spreading interests for the bright application perspectives to quantum computing. Cu0.3Bi2Se3 is a rare bulk topological superconductor with an odd-parity wave function, but the details of the vector order parameter d and its pinning mechanism are still unclear. Here, we succeed in growing CuxBi2Se3 single crystals with unprecedented high doping levels. For samples with x  = 0.28, 0.36 and 0.37 with similar carrier density as evidenced by the Knight shift, the in-plane upper critical field Hc2 shows a two-fold symmetry. However, the angle at which the Hc2 becomes minimal is different by 90° among them, which indicates that the d-vector direction is different for each crystal likely due to a different local environment. The carrier density for x  = 0.46 and 0.54 increases substantially compared to x ≤ 0.37. Surprisingly, the in-plane Hc2 anisotropy disappears, indicating that the gap symmetry undergoes a transition from nematic to isotropic (possibly chiral) as carrier increases.

Conversion of a conventional superconductor into a topological superconductor by topological proximity effect

Realization of topological superconductors (TSCs) hosting Majorana fermions is a central challenge in condensed-matter physics. One approach is to use the superconducting proximity effect (SPE) in heterostructures, where a topological insulator contacted with a superconductor hosts an effective p-wave pairing by the penetration of Cooper pairs across the interface. However, this approach suffers a difficulty in accessing the topological interface buried deep beneath the surface. Here, we propose an alternative approach to realize topological superconductivity without SPE. In a Pb(111) thin film grown on TlBiSe₂, we discover that the Dirac-cone state of substrate TlBiSe₂ migrates to the top surface of Pb film and obtains an energy gap below the superconducting transition temperature of Pb. This suggests that a Bardeen-Cooper-Schrieffer superconductor is converted into a TSC by the topological proximity effect. Our discovery opens a route to manipulate topological superconducting properties of materials.

Realizing topological superconductivity is essential for applicable fault-tolerant quantum computation. Here, Trang et al. report migration of Dirac-cone from TlBiSe₂ substrate to top surface of superconducting Pb film due to topological proximity effect, suggesting realization of topological superconductivity.

Semiconductor-Ferromagnetic Insulator-Superconductor Nanowires: Stray Field and Exchange Field

Nanowires can serve as flexible substrates for hybrid epitaxial growth on selected facets, allowing for the design of heterostructures with complex material combinations and geometries. In this work we report on hybrid epitaxy of freestanding vapor-liquid-solid grown and in-plane selective area grown semiconductor-ferromagnetic insulator-superconductor (InAs/EuS/Al) nanowire heterostructures. We study the crystal growth and complex epitaxial matching of wurtzite and zinc-blende InAs/rock-salt EuS interfaces as well as rock-salt EuS/face-centered cubic Al interfaces. Because of the magnetic anisotropy originating from the nanowire shape, the magnetic structure of the EuS phase is easily tuned into single magnetic domains. This effect efficiently ejects the stray field lines along the nanowires. With tunnel spectroscopy measurements of the density of states, we show that the material has a hard induced superconducting gap, and magnetic hysteretic evolution which indicates that the magnetic exchange fields are not negligible. These hybrid nanowires fulfill key material requirements for serving as a platform for spin-based quantum applications, such as scalable topological quantum computing.

Yu-Shiba-Rusinov States in the Charge-Density Modulated Superconductor NbSe2

NbSe₂ is a remarkable superconductor in which charge-density order coexists with pairing correlations at low temperatures. Here, we study the interplay of magnetic adatoms and their Yu-Shiba-Rusinov (YSR) bound states with the charge density order. Exploiting the incommensurate nature of the charge-density wave (CDW), our measurements provide a thorough picture of how the CDW affects both the energies and the wave functions of the YSR states. Key features of the dependence of the YSR states on adsorption site relative to the CDW are explained by model calculations. Several properties make NbSe₂ a promising substrate for realizing topological nanostructures. Our results will be important in designing such systems.

Majorana fermions go for a ride

Evidence for propagating Majorana quasiparticles is found in a topological superconductor

Evidence for dispersing 1D Majorana channels in an iron-based superconductor

The possible realization of Majorana fermions as quasiparticle excitations in condensed-matter physics has created much excitement. Most studies have focused on Majorana bound states; however, propagating Majorana states with linear dispersion have also been predicted. Here, we report scanning tunneling spectroscopic measurements of crystalline domain walls (DWs) in FeSe0.45Te0.55. We located DWs across which the lattice structure shifts by half a unit cell. These DWs have a finite, flat density of states inside the superconducting gap, which is a hallmark of linearly dispersing modes in one dimension. This signature is absent in DWs in the related superconductor, FeSe, which is not in the topological phase. Our combined data are consistent with the observation of dispersing Majorana states at a π-phase shift DW in a proximitized topological material.