Phase diagram of superconductivity in the integer quantum Hall regime.

An interplay between pairing and topological orders has been predicted to give rise to superconducting states supporting exotic emergent particles, such as Majorana particles obeying non-Abelian braid statistics. We consider a system of spin polarized electrons on a Hofstadter lattice with nearest-neighbor attractive interaction and solve the mean-field Bogoliubov-de Gennes equations in a self-consistent fashion, leading to gauge-invariant observables and a rich phase diagram as a function of the chemical potential, the magnetic field, and the interaction. As the strength of the attractive interaction is increased, the system first makes a transition from a quantum Hall phase to a skyrmion lattice phase that is fully gapped in the bulk but has topological chiral edge current, characterizing a topologically nontrivial state. This is followed by a vortex phase in which the vortices carrying Majorana modes form a lattice; the spectrum contains a low-energy Majorana band arising from the coupling between neighboring vortex-core Majorana modes but does not have chiral edge currents. For some parameters, a dimer vortex lattice occurs with no Majorana band. The experimental feasibility and the observable consequences of skyrmions as well as Majorana modes are indicated.

Selective-Area-Grown PbTe-Pb Planar Josephson Junctions for Quantum Devices.

Planar Josephson junctions are predicted to host Majorana zero modes. The material platforms in previous studies are two-dimensional electron gases (InAs, InSb, InAsSb, and HgTe) coupled to a superconductor such as Al or Nb. Here, we introduce a new material platform for planar JJs, the PbTe-Pb hybrid. The semiconductor, PbTe, was grown as a thin film via selective area epitaxy. The Josephson junction was defined by a shadow wall during the deposition of superconductor Pb. Scanning transmission electron microscopy reveals a sharp semiconductor-superconductor interface. Gate-tunable supercurrents and multiple Andreev reflections are observed. A perpendicular magnetic field causes interference patterns of the switching current, exhibiting Fraunhofer-like and SQUID-like behaviors. We further demonstrate a prototype device for Majorana detection wherein phase bias and tunneling spectroscopy are applicable.

Phase-induced topological superconductivity in a planar heterostructure.

Topological superconductivity in quasi-one-dimensional systems is a novel phase of matter with possible implications for quantum computation. Despite years of effort, a definitive signature of this phase in experiments is still debated. A major cause of this ambiguity is the side effects of applying a magnetic field: induced in-gap states, vortices, and alignment issues. Here we propose a planar semiconductor-superconductor heterostructure as a platform for realizing topological superconductivity without applying a magnetic field to the two-dimensional electron gas hosting the topological state. Time-reversal symmetry is broken only by phase biasing the proximitizing superconductors, which can be achieved using extremely small fluxes or bias currents far from the quasi-one-dimensional channel. Our platform is based on interference between this phase biasing and the phase arising from strong spin-orbit coupling in closed electron trajectories. The principle is demonstrated analytically using a simple model, and then shown numerically for realistic devices. We show a robust topological phase diagram, as well as explicit wavefunctions of Majorana zero modes. We discuss experimental issues regarding the practical implementation of our proposal, establishing it as an accessible scheme with contemporary experimental techniques.

Current-Phase Relation of a WTe2 Josephson Junction.

When a topological insulator is incorporated into a Josephson junction, the system is predicted to reveal the fractional Josephson effect with a 4π-periodic current-phase relation. Here, we report the measurement of a 4π-periodic switching current through an asymmetric SQUID, formed by the higher-order topological insulator WTe2. Contrary to the established opinion, we show that a high asymmetry in critical current and negligible loop inductance are not sufficient by themselves to reliably measure the current-phase relation. Instead, we find that our measurement is heavily influenced by additional inductances originating from the self-formed PdTex inside the junction. We therefore develop a method to numerically recover the current-phase relation of the system and find the 1.5 µm long junction to be best described in the short ballistic limit. Our results highlight the complexity of subtle inductance effects that can give rise to misleading topological signatures in transport measurements.

Spectroscopic fingerprint of chiral Majorana modes at the edge of a quantum anomalous Hall insulator/superconductor heterostructure.

With the recent discovery of the quantum anomalous Hall insulator (QAHI), which exhibits the conductive quantum Hall edge states without external magnetic field, it becomes possible to create a topological superconductor (SC) by introducing superconductivity into these edge states. In this case, 2 distinct topological superconducting phases with 1 or 2 chiral Majorana edge modes were theoretically predicted, characterized by Chern numbers (N) of 1 and 2, respectively. We present spectroscopic evidence from Andreev reflection experiments for the presence of chiral Majorana modes in an Nb/(Cr0.12Bi0.26Sb0.62)2Te3 heterostructure with distinct signatures attributed to 2 different topological superconducting phases. The results are in qualitatively good agreement with the theoretical predictions.

Gate-Tunable Transport in Quasi-One-Dimensional α-Bi4I4 Field Effect Transistors.

Bi4I4 belongs to a novel family of quasi-one-dimensional (1D) topological insulators (TIs). While its ß phase was demonstrated to be a prototypical weak TI, the α phase, long thought to be a trivial insulator, was recently predicted to be a rare higher order TI. Here, we report the first gate tunable transport together with evidence for unconventional band topology in exfoliated α-Bi4I4 field effect transistors. We observe a Dirac-like longitudinal resistance peak and a sign change in the Hall resistance; their temperature dependences suggest competing transport mechanisms: a hole-doped insulating bulk and one or more gate-tunable ambipolar boundary channels. Our combined transport, photoemission, and theoretical results indicate that the gate-tunable channels likely arise from novel gapped side surface states, two-dimensional (2D) TI in the bottommost layer, and/or helical hinge states of the upper layers. Markedly, a gate-tunable supercurrent is observed in an α-Bi4I4 Josephson junction, underscoring the potential of these boundary channels to mediate topological superconductivity.

Topological Superconductivity Based on Antisymmetric Spin-Orbit Coupling.

Topological superconductivity (TSC) has drawn much attention for its fundamental interest and application in quantum computation. An outstanding challenge is the lack of intrinsic TSC materials with a p-wave pairing gap, which has led to the development of an effective p-wave theory of coupling s-wave gap with Rashba spin-orbit coupling (RSOC). However, the RSOC-strict mechanism and materials pose still both fundamental and practical limitations. Here, we generalize this theory to antisymmetric SOC (ASOC). Using k·p perturbation theory, we demonstrate that 2D crystals, with point groups of C2, C4, C6, C2v, C4v, C6v, D2, D4, D6, S4, or D2d, can all facilitate the desired ASOC. Remarkably, this enables us to discover 314 TSC candidates by screening 2D material databases, which are further confirmed by first-principles calculations of Majorana boundary modes and the topological invariant of the superconducting gap. Our work fundamentally enriches TSC theory and greatly expands the classes of TSC materials for experimental exploration.

InSbAs Two-Dimensional Electron Gases as a Platform for Topological Superconductivity.

Topological superconductivity can be engineered in semiconductors with strong spin-orbit interaction coupled to a superconductor. Experimental advances in this field have often been triggered by the development of new hybrid material systems. Among these, two-dimensional electron gases (2DEGs) are of particular interest due to their inherent design flexibility and scalability. Here, we discuss results on a 2D platform based on a ternary 2DEG (InSbAs) coupled to in situ grown aluminum. The spin-orbit coupling in these 2DEGs can be tuned with the As concentration, reaching values up to 400 meV Å, thus exceeding typical values measured in its binary constituents. In addition to a large Landé g-factor of ∼55 (comparable to that of InSb), we show that the clean superconductor-semiconductor interface leads to a hard induced superconducting gap. Using this new platform, we demonstrate the basic operation of phase-controllable Josephson junctions, superconducting islands, and quasi-1D systems, prototypical device geometries used to study Majorana zero modes.

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.

Ising pairing in atomically thin superconductors.

Ising-type pairing in atomically thin superconducting materials has emerged as a novel means of generating devices with resilience to a magnetic field applied parallel to the two-dimensional (2D) plane. In this mini-review, we canvas the state of the field by giving a historical account of 2D superconductors with strongly enhanced in-plane upper critical fields, together with the type-I and type-II Ising pairing mechanisms. We highlight the vital role of spin-orbit coupling in these superconductors and discuss other effects such as symmetry breaking, atomic thicknesses, etc. Finally, we summarize the recent theoretical proposals and highlight the open questions, such as exploring topological superconductivity in these systems and looking for more materials with Ising pairing.

Nontrivial superconductivity in topological MoTe2-x S x crystals.

Topological Weyl semimetals (TWSs) with pairs of Weyl points and topologically protected Fermi arc states have broadened the classification of topological phases and provide superior platform for study of topological superconductivity. Here we report the nontrivial superconductivity and topological features of sulfur-doped Td -phase MoTe2 with enhanced Tc compared with type-II TWS MoTe2 It is found that Td -phase S-doped MoTe2 (MoTe2-x S x , x ∼ 0.2) is a two-band s-wave bulk superconductor (∼0.13 meV and 0.26 meV), where the superconducting behavior can be explained by the s+- pairing model. Further, measurements of the quasi-particle interference (QPI) patterns and a comparison with band-structure calculations reveal the existence of Fermi arcs in MoTe2-x S x More interestingly, a relatively large superconducting gap (∼1.7 meV) is detected by scanning tunneling spectroscopy on the sample surface, showing a hint of topological nontrivial superconductivity based on the pairing of Fermi arc surface states. Our work demonstrates that the Td -phase MoTe2-x S x is not only a promising topological superconductor candidate but also a unique material for study of s+- superconductivity.

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.

Hard Superconducting Gap and Diffusion-Induced Superconductors in Ge-Si Nanowires.

We show a hard superconducting gap in a Ge-Si nanowire Josephson transistor up to in-plane magnetic fields of 250 mT, an important step toward creating and detecting Majorana zero modes in this system. A hard gap requires a highly homogeneous tunneling heterointerface between the superconducting contacts and the semiconducting nanowire. This is realized by annealing devices at 180 °C during which aluminum interdiffuses and replaces the germanium in a section of the nanowire. Next to Al, we find a superconductor with lower critical temperature (TC = 0.9 K) and a higher critical field (BC = 0.9-1.2 T). We can therefore selectively switch either superconductor to the normal state by tuning the temperature and the magnetic field and observe that the additional superconductor induces a proximity supercurrent in the semiconducting part of the nanowire even when the Al is in the normal state. In another device where the diffusion of Al rendered the nanowire completely metallic, a superconductor with a much higher critical temperature (TC = 2.9 K) and critical field (BC = 3.4 T) is found. The small size of these diffusion-induced superconductors inside nanowires may be of special interest for applications requiring high magnetic fields in arbitrary direction.

Conductance Spectroscopy of Exfoliated Thin Flakes of Nb xBi2Se3.

We study unconventional superconductivity in exfoliated single crystals of a promising three-dimensional (3D) topological superconductor candidate, Nb-doped Bi2Se3 through differential conductance spectroscopy and magneto-transport. The strong anisotropy of the critical field along the out-of-plane direction suggests that the thin exfoliated flakes are in the quasi-2D limit. Normal metal-superconductor (NS) contacts with either high or low transparencies made by depositing gold leads onto Nb-doped Bi2Se3 flakes both show significant enhancement in zero bias conductance and coherence dips at the superconducting energy gap. Such behavior is inconsistent with conventional Blonder-Tinkham-Klapwijk theory. Instead, we discuss how our results are consistent with p-wave pairing symmetry, supporting the possibility of topological superconductivity in Nb-doped Bi2Se3. Finally, we observe signatures of multiple superconducting energy gaps, which could originate from multiple Fermi surfaces reported earlier in bulk crystals.

High Mobility HgTe Microstructures for Quantum Spin Hall Studies.

The topic of two-dimensional topological insulators has blossomed after the first observation of the quantum spin Hall (QSH) effect in HgTe quantum wells. However, studies have been hindered by the relative fragility of the edge states. Their stability has been a subject of both theoretical and experimental investigation in the past decade. Here, we present a new generation of high quality (Cd,Hg)Te/HgTe-structures based on a new chemical etching method. From magnetotransport measurements on macro- and microscopic Hall bars, we extract electron mobilities µ up to about 400 × 103 cm2/(V s), and the mean free path λmfp becomes comparable to the sample dimensions. The Hall bars show quantized spin Hall conductance, which is remarkably stable up to 15 K. The clean and robust edge states allow us to fabricate high quality side-contacted Josephson junctions, which are significant in the context of topological superconductivity. Our results open up new avenues for fundamental research on QSH effect as well as potential applications in spintronics and topological quantum computation.

Hard Superconducting Gap in InSb Nanowires.

Topological superconductivity is a state of matter that can host Majorana modes, the building blocks of a topological quantum computer. Many experimental platforms predicted to show such a topological state rely on proximity-induced superconductivity. However, accessing the topological properties requires an induced hard superconducting gap, which is challenging to achieve for most material systems. We have systematically studied how the interface between an InSb semiconductor nanowire and a NbTiN superconductor affects the induced superconducting properties. Step by step, we improve the homogeneity of the interface while ensuring a barrier-free electrical contact to the superconductor and obtain a hard gap in the InSb nanowire. The magnetic field stability of NbTiN allows the InSb nanowire to maintain a hard gap and a supercurrent in the presence of magnetic fields (∼0.5 T), a requirement for topological superconductivity in one-dimensional systems. Our study provides a guideline to induce superconductivity in various experimental platforms such as semiconductor nanowires, two-dimensional electron gases, and topological insulators and holds relevance for topological superconductivity and quantum computation.

Exploring a Proximity-Coupled Co Chain on Pb(110) as a Possible Majorana Platform.

Linear chains of magnetic atoms proximity coupled to an s-wave superconductor are predicted to host Majorana zero modes at the chain ends in the presence of strong spin-orbit coupling. Specifically, iron (Fe) chains on Pb(110) have been explored as a possible system to exhibit topological superconductivity and host Majorana zero modes [ Nadj-Perge , S. et al., Science 2014 , 346 , 602 - 607 ]. Here, we study chains of the transition metal cobalt (Co) on Pb(110) and check for topological signatures. Using spin-polarized scanning tunneling spectroscopy, we resolve ferromagnetic order in the d bands of the chains. Interestingly, also the subgap Yu-Shiba-Rusinov (YSR) bands carry a spin polarization as was predicted decades ago. Superconducting tips allow us to resolve further details of the YSR bands and in particular resonances at zero energy. We map the spatial distribution of the zero-energy signal and find it delocalized along the chain. Hence, despite the ferromagnetic coupling within the chains and the strong spin-orbit coupling in the superconductor, we do not find clear evidence of Majorana modes. Simple tight-binding calculations suggest that the spin-orbit-split bands may cross the Fermi level four times which suppresses the zero-energy modes.

Magnetisation control of the nematicity direction and nodal points in a superconducting doped topological insulator.

We study the effects of magnetisation on the properties of the doped topological insulator of theBi2Se3family with nematic superconductivity. We found that the direction of the in-plane magnetisation fixes the direction of the nematicity in the system. The chiral state is more favourable than the nematic state for large values of out-of-plane magnetisation. Overall, the critical temperature of the nematic superconductivity is robust against magnetisation. We explore in detail the spectrum of the system with the pinned direction of the nematic order parameterΔy. Without magnetisation, there is a full gap in the spectrum because of finite hexagonal warping. At an out-of-planemzor orthogonal in-planemxmagnetisation that is strong enough, the spectrum is closed at the nodal points that are split by the magnetisation. Flat Majorana surface states connect such split bulk nodal points. Parallel magnetisationmylifts the nodal points and opens a full gap in the spectrum. We discuss relevant experiments and propose experimental verifications of our theory.

Topological Quantum Well States in Pb/Sb Thin-Film Heterostructures.

Composite topological heterostructures, wherein topologically protected states are electronically tuned due to their proximity to other matter, are key avenues for exploring emergent physical phenomena. Particularly, pairing a topological material with a superconductor such as Pb is a promising means for generating a topological superconducting phase with exotic Majorana quasiparticles, but oft-neglected is the emergence of bulklike spin-polarized states that are quite relevant to applications. Using high-resolution photoemission spectroscopy and first-principles calculations, we report the emergence of bulk-like spin-polarized topological quantum well states with long coherence lengths in Pb films grown on the topological semimetal Sb. The results establish Pb/Sb heterostructures as topological superconductor candidates and advance the current understanding of topological coupling effects required for realizing emergent physics and for designing advanced spintronic device architectures.

Realizing a Superconducting Square-Lattice Bismuth Monolayer.

Interplay of crystal symmetry, strong spin-orbit coupling (SOC), and many-body interactions in low-dimensional materials provides a fertile ground for the discovery of unconventional electronic and magnetic properties and versatile functionalities. Two-dimensional (2D) allotropes of group 15 elements are appealing due to their structures and controllability over symmetries and topology under strong SOC. Here, we report the heteroepitaxial growth of a proximity-induced superconducting 2D square-lattice bismuth monolayer on superconducting Pb films. The square lattice of monolayer bismuth films in a C4 symmetry together with a stripey moiré structure is clearly resolved by our scanning tunneling microscopy, and its atomic structure is revealed by density functional theory (DFT) calculations. A Rashba-type spin-split Dirac band is predicted by DFT calculations to exist at the Fermi level and becomes superconducting through the proximity effect from the Pb substrate. We suggest the possibility of a topological superconducting state in this system with magnetic dopants/field. This work introduces an intriguing material platform with 2D Dirac bands, strong SOC, topological superconductivity, and the moiré superstructure.