Optical conductivity of the Majorana mode at the s- and d-wave topological superconductor edge.

The Majorana fermion offers fascinating possibilities such as non-Abelian statistics and nonlocal robust qubits, and hunting it is one of the most important topics in current condensed matter physics. Most of the efforts have been focused on the Majorana bound state at zero energy in terms of scanning tunneling spectroscopy searching for the quantized conductance. On the other hand, a chiral Majorana edge channel appears at the surface of a three-dimensional topological insulator when engineering an interface between proximity-induced superconductivity and ferromagnetism. Recent advances in microwave spectroscopy of topological edge states open a new avenue for observing signatures of such Majorana edge states through the local optical conductivity. As a guide to future experiments, we show how the local optical conductivity and density of states present distinct qualitative features depending on the symmetry of the superconductivity, that can be tuned via the magnetization and temperature. In particular, the presence of the Majorana edge state leads to a characteristic nonmonotonic temperature dependence achieved by tuning the magnetization.

High Transition Temperature Driven by Type-II Dirac Fermions in Topological Superconductor B7Be2B7 Nanosheet.

Topological superconductors (TSCs) offer a promising avenue for delving into exotic states of matter and fundamental physics. We propose a strategy for realizing high transition temperatures (high-Tc) in TSCs by leveraging nontrivial topology alongside a high carrier density near the Fermi level in metal-doped borophenes. We identified 39 candidates with exceptional thermodynamic stability from thousands of Be-intercalated borophenes (Be1-xBx) via extensive structural searches. Seven candidates exhibit high carrier densities, with B7Be2B7 emerging as a particularly promising candidate. This nanosheet displays both type-I and type-II Dirac fermions, indicative of Z2 topological metals, thereby positioning it as an ideal platform for high-Tc TSCs. The high-density π electrons of B7Be2B7 originating from type-II Dirac fermions, coupled with the out-of-plane vibrations of B and Be atoms, significantly enhance the electron-phonon coupling (λ = 1.42), resulting in a substantially high-Tc of 31.5 K. These findings underscore the potential of metal-doped borophenes as a cutting-edge material platform for achieving high-Tc TSCs.

Competing Energy Scales in Topological Superconducting Heterostructures.

Artificially engineered topological superconductivity has emerged as a viable route to create Majorana modes. In this context, proximity-induced superconductivity in materials with a sizable spin-orbit coupling has been intensively investigated in recent years. Although there is convincing evidence that superconductivity may indeed be induced, it has been difficult to elucidate its topological nature. Here, we engineer an artificial topological superconductor by progressively introducing superconductivity (Nb), strong spin-orbital coupling (Pt), and topological states (Bi2Te3). Through spectroscopic imaging of superconducting vortices within the bare s-wave superconducting Nb and within proximitized Pt and Bi2Te3 layers, we detect the emergence of a zero-bias peak that is directly linked to the presence of topological surface states. Our results are rationalized in terms of competing energy trends which are found to impose an upper limit to the size of the minigap separating Majorana and trivial modes, its size being ultimately linked to fundamental materials properties.

Strong Correlation Between Superconductivity and Ferromagnetism in an Fe-Chalcogenide Superconductor.

The interplay among topology, superconductivity, and magnetism promises to bring a plethora of exotic and unintuitive behaviors in emergent quantum materials. The family of Fe-chalcogenide superconductors FeTexSe1-x are directly relevant in this context due to their intrinsic topological band structure, high-temperature superconductivity, and unconventional pairing symmetry. Despite enormous promise and expectation, the local magnetic properties of FeTexSe1-x remain largely unexplored, which prevents a comprehensive understanding of their underlying material properties. Exploiting nitrogen vacancy (NV) centers in diamond, here we report nanoscale quantum sensing and imaging of magnetic flux generated by exfoliated FeTexSe1-x flakes, demonstrating strong correlation between superconductivity and ferromagnetism in FeTexSe1-x. The coexistence of superconductivity and ferromagnetism in an established topological superconductor opens up new opportunities for exploring exotic spin and charge transport phenomena in quantum materials. The demonstrated coupling between NV centers and FeTexSe1-x may also find applications in developing hybrid architectures for next-generation, solid-state-based quantum information technologies.

Multifaceted properties of Andreev bound states: interplay of symmetry and topology.

Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.

Thermoelectric Efficiency of a Topological Nano-Junction.

We studied the non-equilibrium current, transport coefficients and thermoelectric performance of a nano-junction, composed by a quantum dot connected to a normal superconductor and a topological superconductor leads, respectively. We considered a one-dimensional topological superconductor, which hosts two Majorana fermion states at its edges. Our results show that the electric and thermal currents across the junction are highly mediated by multiple Andreev reflections between the quantum dot and the leads, thus leading to a strong nonlinear dependence of the current on the applied bias voltage. Remarkably, we find that our system reaches a sharp maximum of its thermoelectric efficiency at a finite bias, when an external magnetic field is imposed upon the junction. We propose that this feature can be used for accurate temperature sensing at the nanoscale.

Superconductivity in the High-Entropy Ceramics Ti0.2 Zr0.2 Nb0.2 Mo0.2 Ta0.2 Cx with Possible Nontrivial Band Topology.

Topological superconductors have drawn significant interest from the scientific community due to the accompanying Majorana fermions. Here, the discovery of electronic structure and superconductivity (SC) in high-entropy ceramics Ti0.2 Zr0.2 Nb0.2 Mo0.2 Ta0.2 Cx (x = 1 and 0.8) combined with experiments and first-principles calculations is reported. The Ti0.2 Zr0.2 Nb0.2 Mo0.2 Ta0.2 Cx high-entropy ceramics show bulk type-II SC with Tc ≈ 4.00 K (x = 1) and 2.65 K (x = 0.8), respectively. The specific heat jump (∆C/γTc ) is equal to 1.45 (x = 1) and 1.52 (x = 0.8), close to the expected value of 1.43 for the BCS superconductor in the weak coupling limit. The high-pressure resistance measurements show a robust SC against high physical pressure in Ti0.2 Zr0.2 Nb0.2 Mo0.2 Ta0.2 C, with a slight Tc variation of 0.3 K within 82.5 GPa. Furthermore, the first-principles calculations indicate that the Dirac-like point exists in the electronic band structures of Ti0.2 Zr0.2 Nb0.2 Mo0.2 Ta0.2 C, which is potentially a topological superconductor. The Dirac-like point is mainly contributed by the d orbitals of transition metals M and the p orbitals of C. The high-entropy ceramics provide an excellent platform for the fabrication of novel quantum devices, and the study may spark significant future physics investigations in this intriguing material.

Robust topological superconductivity in spin-orbit coupled systems at higher-order van Hove filling.

Van Hove singularities in proximity to the Fermi level promote electronic interactions and generate diverse competing instabilities. It is also known that a nontrivial Berry phase derived from spin-orbit coupling can introduce an intriguing decoration into the interactions and thus alter correlated phenomena. However, it is unclear how and what type of new physics can emerge in a system featured by the interplay between van Hove singularities (VHSs) and the Berry phase. Here, based on a general Rashba model on the square lattice, we comprehensively explore such an interplay and its significant influence on the competing electronic instabilities by performing a parquet renormalization group analysis. Despite the existence of a variety of comparable fluctuations in the particle-particle and particle-hole channels associated with higher-order VHSs, we find that the chiral p±ip pairings emerge as two stable fixed trajectories within the generic interaction parameter space, namely the system becomes a robust topological superconductor. The chiral pairings stem from the hopping interaction induced by the nontrivial Berry phase. The possible experimental realization and implications are discussed. Our work sheds new light on the correlated states in quantum materials with strong spin-orbit coupling (SOC) and offers fresh insights into the exploration of topological superconductivity.

Orbital-Hybridization-Driven Charge Density Wave Transition in CsV3 Sb5 Kagome Superconductor.

Owing to its inherent non-trivial geometry, the unique structural motif of the recently discovered kagome topological superconductor AV3 Sb5 (A = K, Rb, Cs) is an ideal host of diverse topologically non-trivial phenomena, including giant anomalous Hall conductivity, topological charge order, charge density wave (CDW), and unconventional superconductivity. Despite possessing a normal-state CDW order in the form of topological chiral charge order and diverse superconducting gaps structures, it remains unclear how fundamental atomic-level properties and many-body effects including Fermi surface nesting, electron-phonon coupling, and orbital hybridization contribute to these symmetry-breaking phenomena. Here, the direct participation of the V3d-Sb5p orbital hybridization in mediating the CDW phase transition in CsV3 Sb5 is reported. The combination of temperature-dependent X-ray absorption and first-principles studies clearly indicates the inverse Star-of-David structure as the preferred reconstruction in the low-temperature CDW phase. The results highlight the critical role that Sb orbitals play and establish orbital hybridization as the direct mediator of the CDW states and structural transition dynamics in kagome unconventional superconductors. This is a significant step toward the fundamental understanding and control of the emerging correlated phases from the kagome lattice through the orbital interactions and provides promising approaches to novel regimes in unconventional orders and topology.

Magnetization Control of Zero-Field Intrinsic Superconducting Diode Effect.

The superconducting diode effect (SDE), which causes a superconducting state in one direction and a normal-conducting state in another, has significant potential for developing ultralow power consumption circuits and non-volatile memory. However, the practical control of the SDE necessities the precise tuning of current, temperature, magnetic field, or magnetism. Therefore, the mechanisms of the SDE must be understood to develop novel materials and devices capable of realizing the SDE under more controlled and robust conditions. This study demonstrates an intrinsic zero-field SDE with an efficiency of up to 40% in Fe/Pt-inserted non-centrosymmetric Nb/V/Ta superconducting artificial superlattices. The polarity and magnitude of the zero-field SDE are controllable by the direction of magnetization, indicating that the effective exchange field acts on Cooper pairs. Furthermore, the first-principles calculation indicates that the SDE can be enhanced by an asymmetric configuration of proximity induced magnetic moments in superconducting layers, which induces a magnetic toroidal moment. This study has important implications regarding the development of novel materials and devices that can effectively control the SDE. Moreover, the magnetization control of the SDE is expected to aid in the designing of superconducting quantum devices and establishing a material platform for topological superconductors.

Universal characteristics of one-dimensional non-Hermitian superconductors.

We establish a non-Bloch band theory for one-dimensional(1D) non-Hermitian topological superconductors. The universal physical properties of non-Hermitian topological superconductors are revealed based on the theory. According to the particle-hole symmetry, there exist reciprocal particle and hole loops of generalized Brillouin zone. The critical point of quantum phase transition, where the energy gap closes, appears when the particle and hole loops intersect at Bloch points. If the non-Hermitian system has non-Hermitian skin effects, the non-Hermitian skin effect should be theZ2skin effect: the corresponding eigenstates of particle and hole localize at opposite ends of an open chain, respectively. The non-Bloch band theory is applied to two examples, non-Hermitianp- ands-wave topological superconductors. In terms of Majorana Pfaffian, aZ2non-Bloch topological invariant is defined to establish the non-Hermitian bulk-boundary correspondence for the non-Hermitian topological superconductors.

Probing the topological surface states in superconducting Sn4Au single crystal: a magneto transport study.

Materials exhibiting bulk superconductivity along with magnetoresistance (MR) in their normal state have emerged as suitable candidates for topological superconductivity. In this article, we report a flux free method to synthesize single crystal of topological superconductor candidate Sn4Au. The phase purity and single crystalline nature are confirmed through various characterizations viz. x-ray diffraction, field emission scanning electron microscopy, selected area electron diffraction, and transmission electron microscopy. Chemical states of the constituent element viz. Sn and Au are analysed through x-ray photoelectron spectroscopy. Superconductivity in synthesized Sn4Au single crystal is evident formρ-Tplot, for which the critical field (Hc) is determined throughρ-Hplot at 2 K i.e. just below critical temperatureTc. A positive MR is observed inρ-Hmeasurements at different temperatures aboveTc, viz. at 3 K, 5 K, 10 K and 20 K. Further, the magnetoconductivity (MC) is analysed by using Hikami-Larkin-Nagaoka formalism, which signifies the presence of weak antilocalization (WAL) effect in Sn4Au. Angle dependent magneto-transport measurement has been performed to detect the origin of observed WAL effect in Sn4Au single crystal. Normalized MC vsHcosθplot shows presence of topological surface states in the studied system. It is evident that Sn4Au is a 2.6 K topological superconductor.

Correlation of Magnetism and Disordered Shiba Bands in Fe Monolayer Islands on Nb(110).

Two-dimensional (2D) magnet-superconductor hybrid systems are intensively studied due to their potential for the realization of 2D topological superconductors with Majorana edge modes. It is theoretically predicted that this quantum state is ubiquitous in spin-orbit-coupled ferromagnetic or skyrmionic 2D spin-lattices in proximity to an s-wave superconductor. However, recent examples suggest that the requirements for topological superconductivity are complicated by the multiorbital nature of the magnetic components and disorder effects. Here, we investigate Fe monolayer islands grown on a surface of the s-wave superconductor with the largest gap of all elemental superconductors, Nb, with respect to magnetism and superconductivity using spin-resolved scanning tunneling spectroscopy. We find three types of islands which differ by their reconstruction inducing disorder, the magnetism and the subgap electronic states. All three types are ferromagnetic with different coercive fields, indicating diverse exchange and anisotropy energies. On all three islands, there is finite spectral weight throughout the substrate's energy gap at the expense of the coherence peak intensity, indicating the formation of Shiba bands overlapping with the Fermi energy. A strong lateral variation of the spectral weight of the Shiba bands signifies substantial disorder on the order of the substrate's pairing energy with a length scale of the period of the three different reconstructions. There are neither signs of topological gaps within these bands nor of any kind of edge modes. Our work illustrates that a reconstructed growth mode of magnetic layers on superconducting surfaces is detrimental for the formation of 2D topological superconductivity.

One-dimensional spin-orbit coupled Dirac system with extendeds-wave superconductivity: Majorana modes and Josephson effects.

Motivated by the spin-momentum locking of electrons at the boundaries of certain topological insulators, we study a one-dimensional system of spin-orbit coupled massless Dirac electrons withs-wave superconducting pairing. As a result of the spin-orbit coupling, our model has only two kinds of linearly dispersing modes, and we take these to be right-moving spin-up and left-moving spin-down. Both lattice and continuum models are studied. In the lattice model, we find that a single Majorana zero energy mode appears at each end of a finite system provided that thes-wave pairing has an extended form, with the nearest-neighbor pairing being larger than the on-site pairing. We confirm this both numerically and analytically by calculating the winding number. We find that the continuum model also has zero energy end modes. Next we study a lattice version of a model with both Schrödinger and Dirac-like terms and find that the model hosts a topological transition between topologically trivial and non-trivial phases depending on the relative strength of the Schrödinger and Dirac terms. We then study a continuum system consisting of twos-wave superconductors with different phases of the pairing, with aδ-function potential barrier lying at the junction of the two superconductors. Remarkably, we find that the system has asingleAndreev bound state (ABS) which is localized at the junction. When the pairing phase difference crosses a multiple of 2π, an ABS touches the top of the superconducting gap and disappears, and a different state appears from the bottom of the gap. We also study the AC Josephson effect in such a junction with a voltage bias that has both a constantV0and a term which oscillates with a frequencyω. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequencyωJ= 2eV0/ℏis a rational fraction ofω. We discuss experiments which can realize such junctions.

Fermi surfaces of the topological semimetal CaSn3probed through de Haas van Alphen oscillations.

In the search of topological superconductors, nailing down the Fermiology of the normal state is as crucial a prerequisite as unraveling the superconducting pairing symmetry. In particular, the number of time-reversal-invariant momenta (TRIM) in the Brillouin zone enclosed by Fermi surfaces is closely linked to the topological class of time-reversal-invariant systems, and can experimentally be investigated. We report here a detailed study of de Haas van Alphen quantum oscillations in single crystals of the topological semimetal CaSn3with torque magnetometry in high magnetic fields up to 35 T. In conjunction with density functional theory based calculations, the observed quantum oscillations frequencies indicate that the Fermi surfaces of CaSn3enclose an odd number of TRIM, satisfying one of the proposed criteria to realize topological superconductivity. Nonzero Berry phases extracted from the magnetic oscillations also support the nontrivial topological nature of CaSn3.

Unusual upper critical fields of the topological nodal-line semimetal candidate SnxNbSe2-δ.

We report superconductivity in SnxNbSe2-δ, a topological nodal-line semimetal candidate with a noncentrosymmetric crystal structure. The superconducting transition temperatureTcof this compound is extremely sensitive to Sn concentrationxand Se deficiencyδ, 5.0 K for Sn0.13NbSe1.70and 8.6 K for Sn0.14NbSe1.71and Sn0.15NbSe1.69. In all samples, the temperature dependence of the upper critical fieldHc2(T) differs from the prediction of the Werthamer-Helfand-Hohenberg theory. While the zero-temperature value of the in-plane upper critical field of SnxNbSe2-δwith the higherTcis lower than the BCS Pauli paramagnetic limitHP, that of the lowerTcsample exceedsHPby a factor of ∼2. Our observations suggest that a possible odd-parity contribution dominates the superconducting gap function of SnxNbSe2-δ, and it can be fine-tuned by the Sn concentration and Se deficiency.

Band Structure Extraction at Hybrid Narrow-Gap Semiconductor-Metal Interfaces.

The design of epitaxial semiconductor-superconductor and semiconductor-metal quantum devices requires a detailed understanding of the interfacial electronic band structure. However, the band alignment of buried interfaces is difficult to predict theoretically and to measure experimentally. This work presents a procedure that allows to reliably determine critical parameters for engineering quantum devices; band offset, band bending profile, and number of occupied quantum well subbands of interfacial accumulation layers at semiconductor-metal interfaces. Soft X-ray angle-resolved photoemission is used to directly measure the quantum well states as well as valence bands and core levels for the InAs(100)/Al interface, an important platform for Majorana-zero-mode based topological qubits, and demonstrate that the fabrication process strongly influences the band offset, which in turn controls the topological phase diagrams. Since the method is transferable to other narrow gap semiconductors, it can be used more generally for engineering semiconductor-metal and semiconductor-superconductor interfaces in gate-tunable superconducting devices.

Superconductivity in Cu Co-Doped SrxBi2Se3 Single Crystals.

In this study, we grew Cu co-doped single crystals of a topological superconductor candidate Sr x Bi 2 Se 3 , and studied their structural and transport properties. We reveal that the addition of even as small an amount of Cu co-dopant as 0.6 atomic %, completely suppresses superconductivity in Sr x Bi 2 Se 3 . Critical temperature (∼2.7 K) is rather robust with respect to co-doping. We show that Cu systematically increases the electron density and lattice parameters a and c. Our results demonstrate that superconductivity in Sr x Bi 2 Se 3 -based materials is induced by significantly lower Sr doping level x < 0.02 than commonly accepted x ∼ 0.06 , and it strongly depends on the specific arrangement of Sr atoms in the host matrix. The critical temperature in superconductive Sr-doped Bi 2 Se 3 is shown to be insensitive to carrier density.

A zero-dimensional topologically nontrivial state in a superconducting quantum dot.

The classification of topological states of matter in terms of unitary symmetries and dimensionality predicts the existence of nontrivial topological states even in zero-dimensional systems, i.e., systems with a discrete energy spectrum. Here, we show that a quantum dot coupled with two superconducting leads can realize a nontrivial zero-dimensional topological superconductor with broken time-reversal symmetry, which corresponds to the finite size limit of the one-dimensional topological superconductor. Topological phase transitions corresponds to a change of the fermion parity, and to the presence of zero-energy modes and discontinuities in the current-phase relation at zero temperature. These fermion parity transitions therefore can be revealed by the current discontinuities or by a measure of the critical current at low temperatures.