Topological Superconductivity in Dirac Semimetals

Pressure-Dependent Superconductivity in Topological Dirac Semimetal SrCuBi

The discovery of superconducting states in diverse topological materials generates a burgeoning interest to explore a topological superconductor and to realize a fault-tolerant topological quantum computation. A variety of routes to realize topological superconductors are proposed, and many types of topological materials are developed. However, a pristine topological material with a natural superconducting state is relatively rare as compared to topological materials with artificially induced superconductivity. Here, it is reported that the planar honeycomb structured 3D topological Dirac semimetal (TDS) SrCuBi, which is the Zintl phase, shows a natural surface superconductivity at 2.1 K under ambient pressure. It is clearly identified from theoretical calculations that a topologically nontrivial state exists on the (100) surface. Further, its superconducting transition temperature (Tc) increases by applying pressure, exhibiting a maximal Tc of 4.8 K under 6.2 GPa. It is believed that this discovery opens up a new possibility of exploring exotic Majorana fermions at the surface of 3D TDS superconductors.

Coexistence of topological node surface and Dirac fermions in phonon-mediated superconductor YB2C2

The interaction between nontrivial topology and superconductivity in condensed matter physics has attracted tremendous research interest as it could give rise to exotic phenomena. Herein, based on first-principles calculations, we investigate the electronic structures, mechanical properties, topological properties, dynamic stability, electron-phonon coupling (EPC), and superconducting properties of the synthesized real material YB2C2. It is a tetragonal structure with P4/mbm symmetry and exhibits excellent stability. The calculated electronic band structures reveal that a zero-dimension (0D) Dirac point and two-dimensional (2D) nodal surface coexist near the Fermi level. A spin-orbit coupling (SOC) Dirac point with the topological Fermi arc is observed on the (001) surface. These nodal surfaces are protected by a two-fold screw axis and time-reversal symmetry. Based on the Bardeen-Cooper-Schrieffer theory, the superconducting transition temperature (Tc) in the range 1.25-4.45 K with different Coulomb repulsion constant μ* for YB2C2 is estimated to be consistent with previous experimental results. In addition, the EPC is mainly from the coupling between the dx2-y2 and dz2 orbitals of the Y atom and low-energy phonon modes. The presence of superconductivity and nontrivial topological surface state in YB2C2 suggests that it may be a candidate material for topological superconductors.

Broad and colossal edge supercurrent in Dirac semimetal Cd3As2 Josephson junctions

Edge supercurrent has attracted great interest recently due to its crucial role in achieving and manipulating topological superconducting states. Proximity-induced superconductivity has been realized in quantum Hall and quantum spin Hall edge states, as well as in higher-order topological hinge states. Non-Hermitian skin effect, the aggregation of non-Bloch eigenstates at open boundaries, promises an abnormal edge channel. Here we report the observation of broad edge supercurrent in Dirac semimetal Cd3As2-based Josephson junctions. The as-grown Cd3As2 nanoplates are electron-doped by intrinsic defects, which enhance the non-Hermitian perturbations. The superconducting quantum interference indicates edge supercurrent with a width of ~1.6 μm and a magnitude of ~1 μA at 10 mK. The wide and large edge supercurrent is inaccessible for a conventional edge system and suggests the presence of non-Hermitian skin effect. A supercurrent nonlocality is also observed. The interplay between band topology and non-Hermiticity is beneficial for exploiting exotic topological matter.

Nodeless superconductivity in topologically nontrivial materials HfRuP and ZrRuAs

Topologically nontrivial electronic states are recently found in a family of noncentrosymmetric transition metal pnictidesTRuX(T=Zr, Hf;X=P, As), presenting a unique platform for superconductivity to interplay with topological electronic states and asymmetric spin-orbit coupling. Here, we investigate the superconducting order parameter of HfRuP and ZrRuAs by measuring the magnetic penetration depth changeΔλ(T)using a method based on the tunnel-diode oscillator. Both compounds show clear exponential temperature dependence inΔλ(T)at low temperatures, suggesting fully-gapped superconductivity. Moreover, the superfluid densities in both HfRuP and ZrRuAs can be reasonably described by ans-wave superconducting model.

Superconductivity and topological states in hexagonal TaC and NbC

Materials with superconductivity and a nontrivial band structure near the Fermi level are promising candidates in realizing topological superconductivity. Using first-principles calculations, we systematically investigated the stability, mechanical properties, superconductivity, electronic structures, and topological states of hexagonal TaC and NbC. The results show that they are stable and have excellent mechanical properties. We predicted that these two carbides are strong electron-phonon coupling superconductors with superconducting transition temperatures of 14.8 and 17.1 K, respectively. Strong coupling is mainly dominated by in-plane Ta/Nb atomic vibrations and in-plane Ta/Nb-d_xy/d_x²-y² electronic orbitals. The electronic structure calculations demonstrate that a nodal line and a triply degenerate point coexist when not including the spin-orbit coupling (SOC) effect. After including the SOC effect, the nodal line is gapped. The complicated surface states are also calculated and need further experiments to verify. The present results indicate that the hexagonal TaC and NbC are potential candidates as topological superconductors, and pave the way towards exploring the superconductivity and topological materials in condensed matter systems.

Spin-Nernst Effect in Time-Reversal-Invariant Topological Superconductors

We investigate the spin-Nernst effect in time-reversal-invariant topological superconductors, and show that it provides smoking-gun evidence for helical Cooper pairs. The spin-Nernst effect stems from asymmetric, in spin space, scattering of quasiparticles at nonmagnetic impurities, and generates a transverse spin current by the temperature gradient. Both the sign and the magnitude of the effect sensitively depend on the scattering phase shift at impurity sites. Therefore the spin-Nernst effect is uniquely suitable for identifying time-reversal-invariant topological superconducting orders.

Dirac semimetal phase and switching of band inversion in XMg2Bi2 (X = Ba and Sr)

Topological Dirac semimetals (TDSs) offer an excellent opportunity to realize outstanding physical properties distinct from those of topological insulators. Since TDSs verified so far have their own problems such as high reactivity in the atmosphere and difficulty in controlling topological phases via chemical substitution, it is highly desirable to find a new material platform of TDSs. By angle-resolved photoemission spectroscopy combined with first-principles band-structure calculations, we show that ternary compound BaMg₂Bi₂ is a TDS with a simple Dirac-band crossing around the Brillouin-zone center protected by the C₃ symmetry of crystal. We also found that isostructural SrMg₂Bi₂ is an ordinary insulator characterized by the absence of band inversion due to the reduction of spin–orbit coupling. Thus, XMg₂Bi₂ (X = Sr, Ba, etc.) serves as a useful platform to study the interplay among crystal symmetry, spin–orbit coupling, and topological phase transition around the TDS phase.

Emergence of topological superconductivity in doped topological Dirac semimetals under symmetry-lowering lattice distortions

Recently, unconventional superconductivity having a zero-bias conductance peak is reported in doped topological Dirac semimetal (DSM) with lattice distortion. Motivated by the experiments, we theoretically study the possible symmetry-lowering lattice distortions and their effects on the emergence of unconventional superconductivity in doped topological DSM. We find four types of symmetry-lowering lattice distortions that reproduce the crystal symmetries relevant to experiments from the group-theoretical analysis. Considering inter-orbital and intra-orbital electron density-density interactions, we calculate superconducting phase diagrams. We find that the lattice distortions can induce unconventional superconductivity hosting gapless surface Andreev bound states (SABS). Depending on the lattice distortions and superconducting pairing interactions, the unconventional inversion-odd-parity superconductivity can be either topological nodal superconductivity hosting a flat SABS or topological crystalline superconductivity hosting a gapless SABS. Remarkably, the lattice distortions increase the superconducting critical temperature, which is consistent with the experiments. Our work opens a pathway to explore and control pressure-induced topological superconductivity in doped topological semimetals.

Tip-induced superconductivity

It is widely believed that topological superconductivity, a hitherto elusive phase of quantum matter, can be achieved by inducing superconductivity in topological materials. In search of such topological superconductors, certain topological insulators (like, Bi₂Se₃) were successfully turned into superconductors by metal-ion (Cu, Pd, Sr, Nb etc) intercalation. Superconductivity could be induced in topological materials through applying pressure as well. For example, a pressure-induced superconducting phase was found in the topological insulator Bi₂Se₃. However, in all such cases, no conclusive signature of topological superconductivity was found. In this review, we will discuss about another novel way of inducing superconductivity in a non-superconducting topological material-by creating a mesoscopic interface on the material with a non-superconducting, normal metallic tip where the mesoscopic interface becomes superconducting. Such a phase is now known as a tip-induced superconducting (TISC) phase. This was first realized on Cd₃As₂in India. Following that, a large number of other topological materials were shown to display TISC. Since the TISC phase emerges only at a confined region under a mesoscopic point contact, traditional bulk tools for characterizing superconductivity cannot be employed to detect/confirm such a phase. On the other hand, such a point contact geometry is ideal for probing the possible existence of a temperature and magnetic field dependent superconducting energy gap and a temperature and magnetic field dependent critical current. We will review the details of the experimental signatures that can be used to prove the existence of superconductivity even when the 'text-book' tests for detecting superconductivity cannot be performed. Then, we will review various systems where a TISC phase could be realized.

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.

The Tetragonal Monoxide of Platinum: A New Platform for Investigating Nodal-Line and Nodal-Point Semimetallic Behavior

The search for new topological materials that are realistic to synthesize has attracted increasing attention. In this study, we systematically investigated the electronic, mechanical, and topological semimetallic properties, as well as the interesting surface states, of the tetragonal monoxide of platinum, which is realistic to synthesize, via a first-principles approach. Our calculated results indicate that PtO is a novel topological semimetal with double nodal lines in the k z = 0 plane and a pair of triple topological nodal points along the A'-M-A directions. Obvious surface states, including Fermi arc and drum-head-like surfaces, could be found around nodal points and nodal lines. The dynamic and mechanical stabilities of P4 2 /mmc-type PtO were examined in detail via calculation of the phonon dispersion and determination of elastic constants, respectively. Some other mechanical properties, including the bulk modulus, Young's modulus, shear modulus, Poisson's ratio, and Pugh's index, were considered in this study. P4 2 /mmc-type PtO provides a good research platform for investigation of novel behaviors that combine mechanical properties and rich topological elements.

Nodal Flexible-surface Semimetals: Case of Carbon Nanotube Networks

Nodal surface-based topological semimetals (TSMs) are drawing attention due to their unique excitation and plasmon behaviors. However, only nodal flat-surface and nodal sphere TSMs are theoretically proposed due to strict symmetry requirements. Here, we propose that a series of surface-based topological phases can be realized in a tight-binding (TB) model with sublattice symmetry. These topological phases, named as nodal flexible-surface semimetals, include not only nodal surface and nodal sphere TSMs but also novel phases, like nodal tube, nodal crossbar, and nodal hourglass-like surface TSMs. According to the TB model, a family of carbon nanotube networks are then identified as nodal flexible-surface TSMs by first-principles calculations, and the topological phase transitions between these TSMs can be induced by strains. Moreover, the nodal flexible-surface TSMs with intrinsic high density of states at the Fermi level and special drumhead surface states are promising for studying high-temperature superconductors and strong correlation effects.

Vortex End Majorana Zero Modes in Superconducting Dirac and Weyl Semimetals

Time-reversal invariant Dirac and Weyl semimetals in three dimensions (3D) can host open Fermi arcs and spin-momentum locking Fermi loops on the surfaces. We find that when they become superconducting with s-wave pairing and the doping is lower than a critical level, straight π-flux vortex lines terminating at surfaces with Fermi arcs or spin-momentum locking Fermi loops can realize 1D topological superconductivity and harbor Majorana zero modes at their ends. Remarkably, we find that the vortex-generation-associated Zeeman field can open (when the surfaces have only Fermi arcs) or enhance the topological gap protecting Majorana zero modes, which is contrary to the situation in superconducting topological insulators. By studying the tilting effect of bulk Dirac and Weyl cones, we further find that type-I Dirac and Weyl semimetals in general have a much broader topological regime than type-II ones. Our findings build up a connection between time-reversal invariant Dirac and Weyl semimetals and Majorana zero modes in vortices.

The superconductivity and topological surface state of type-II Dirac semimetal NiTe2

NiTe₂ is a type-II Dirac semimetal with the Dirac point very close to the Fermi level. In this paper, its electronic structure, phonon structure and electron-phonon interaction are studied via first-principles calculations. The noteworthy result is that the nontrival bands around the type-II Dirac point are strongly coupled with phonon modes, suggesting that they play an important role in superconductivity. Furthermore, the topological surface states on the (0 0 1) cleavage plane originated from the nontrivial Z ₂ are well separated from the bulk states and can be tuned to approach the Fermi level by adding holes or by V substitution. The possible topological superconductivity in type-II Dirac semimetal NiTe₂ is discussed.

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.

Quantum Phase Transition of Correlated Iron-Based Superconductivity in LiFe_{1-x}Co_{x}As

The interplay between unconventional Cooper pairing and quantum states associated with atomic scale defects is a frontier of research with many open questions. So far, only a few of the high-temperature superconductors allow this intricate physics to be studied in a widely tunable way. We use scanning tunneling microscopy to image the electronic impact of Co atoms on the ground state of the LiFe_{1-x}Co_{x}As system. We observe that impurities progressively suppress the global superconducting gap and introduce low energy states near the gap edge, with the superconductivity remaining in the strong-coupling limit. Unexpectedly, the fully opened gap evolves into a nodal state before the Cooper pair coherence is fully destroyed. Our systematic theoretical analysis shows that these new observations can be quantitatively understood by the nonmagnetic Born-limit scattering effect in an s±-wave superconductor, unveiling the driving force of the superconductor to metal quantum phase transition.

Majorana Multipole Response of Topological Superconductors

In contrast to elementary Majorana particles, emergent Majorana fermions (MFs) in condensed-matter systems may have electromagnetic multipoles. We developed a general theory of magnetic multipoles for helical MFs on time-reversal-invariant superconductors. The results show that the multipole response is governed by crystal symmetry, and that a one-to-one correspondence exists between the symmetry of Cooper pairs and the representation of magnetic multipoles under crystal symmetry. The latter property provides a way to identify unconventional pairing symmetry via the magnetic response of helical MFs. We also find that most helical MFs exhibit a magnetic-dipole response, but those on superconductors with spin-3/2 electrons may display a magnetic-octupole response in leading order, which uniquely characterizes high-spin superconductors. Detection of such an octupole response provides direct evidence of high-spin superconductivity, such as in half-Heusler superconductors.

Quasi-1D Topological Nodal Vortex Line Phase in Doped Superconducting 3D Dirac Semimetals

We study vortex bound states in three-dimensional (3D) superconducting Dirac semimetals with time reversal symmetry. We find that there exist robust gapless vortex bound states propagating along the vortex line in the s-wave superconducting state. We refer to this newly found phase as the quasi-1D nodal vortex line phase. According to the Altland-Zirnbauer classification, the phase is characterized by a topological index (ν;N) at k_{z}=0 and k_{z}=π, where ν is the Z_{2} topological invariant for a 0D class-D system and N is the Z topological invariant for a 0D class-A system. Furthermore, we show that the vortex end Majorana zero mode can coexist with the quasi-1D nodal phase in certain types of Dirac semimetals. The possible experimental observations and material realization of such nodal vortex line states are discussed.

Z_{4} Topological Superconductivity in UCoGe

Topological nonsymmorphic crystalline superconductivity (TNCS) is an intriguing phase of matter, offering a platform to study the interplay between topology, superconductivity, and nonsymmorphic crystalline symmetries. Interestingly, some of TNCSs are classified into Z_{4} topological phases, which have unique surface states referred to as a Möbius strip or an hourglass, and they have not been achieved in symmorphic superconductors. However, material realization of Z_{4} TNCS has never been known, to the best of our knowledge. Here, we propose that the paramagnetic superconducting phase of UCoGe under pressure is a promising candidate of Z_{4}-nontrivial TNCS enriched by glide symmetry. We evaluate Z_{4} invariants of UCoGe by deriving the formulas relating Z_{4} invariants to the topology of Fermi surfaces. Applying the formulas and previous ab initio calculations, we clarify that three odd-parity representations out of four are Z_{4}-nontrivial TNCS, whereas the other is also Z_{2}-nontrivial TNCS. We also discuss possible Z_{4} TNCS in CrAs and related materials.

Inter-node superconductivity in strained Weyl semimetals

The effects of a strain-induced pseudomagnetic field on inter-node spin-triplet superconducting states in Weyl semimetals are studied by using the quasiclassical Eilenberger formalism. It is found that the Cooper pairing with spins parallel to the pseudomagnetic field has the lowest energy among the spin-triplet states and its gap does not depend on the strength of the field. In such a state, both electric and chiral superconducting currents are absent. This is in contrast to the superconducting states with the spins of Cooper pairs normal to the field, which support a nonzero chiral current and are inhibited by the strain-induced pseudomagnetic field. The corresponding critical value of the field, which separates the normal and superconducting phases, is estimated.

Observation of an Odd-Integer Quantum Hall Effect from Topological Surface States in Cd_{3}As_{2}

The quantum Hall effect (QHE) in a 3D Dirac semimetal thin film is attributed to either the quantum confinement induced bulk subbands or the Weyl orbits that connect the opposite surfaces via bulk Weyl nodes. However, it is still unknown whether the QHE based on the Weyl orbit can survive as the bulk Weyl nodes are gapped. Moreover, there are closed Fermi loops rather than open Fermi arcs on the Dirac semimetal surface, which can also host the QHE. Here we report the QHE in the 3D Dirac semimetal Cd_{3}As_{2} nanoplate by tuning the gate voltage under a fixed 30 T magnetic field. The quantized Hall plateaus at odd filling factors are observed as a magnetic field along the [001] crystal direction, indicating a Berry's phase π from the topological surface states. Furthermore, even filling factors are observed when the magnetic field is along the [112] direction, indicating the C_{4} rotational symmetry breaking and a topological phase transition. The results shed light on the understanding of QHE in 3D Cd_{3}As_{2}.

4π-Periodic Supercurrent from Surface States in Cd_{3}As_{2} Nanowire-Based Josephson Junctions

The combination of superconductivity and surface states in Dirac semimetal can produce a 4π-periodic supercurrent in a Josephson junction configuration, which can be revealed by the missing of odd Shapiro steps (especially the n=1 step). However, the suppression of the n=1 step is also anticipated in the high-power oscillatory regime of the ordinary 2π-periodic Josephson effect, which is irrelevant to the 4π-periodic supercurrent. Here, in order to identify the origin of the suppressed n=1 step, we perform the measurements of radio frequency irradiation on Nb-Dirac semimetal Cd_{3}As_{2} nanowire-Nb junctions with continuous power dependence at various frequencies. Besides the n=1 step suppression, we uncover a residual supercurrent of first node at the n=0 step, which provides a direct and predominant signature of the 4π-periodic supercurrent. Furthermore, by tuning the gate voltage, we can modulate the surface and bulk state contribution and the visibility of the n=1 step. Our results provide deep insights to explore the topological superconductivity in Dirac semimetals.

Tip-induced or enhanced superconductivity: a way to detect topological superconductivity

Topological materials, hosting topological nontrivial electronic band, have attracted widespread attentions. As an application of topology in physics, the discovery and study of topological materials not only enrich the existing theoretical framework of physics, but also provide fertile ground for investigations on low energy excitations, such as Weyl fermions and Majorana fermions, which have not been observed yet as fundamental particles. These quasiparticles with exotic physical properties make topological materials the cutting edge of scientific research and a new favorite of high tech. As a typical example, Majorana fermions, predicted to exist in the edge state of topological superconductors, are proposed to implement topological error-tolerant quantum computers. Thus, the detection of topological superconductivity has become a frontier in condensed matter physics and materials science. Here, we review a way to detect topological superconductivity triggered by the hard point contact: tip-induced superconductivity (TISC) and tip-enhanced superconductivity (TESC). The TISC refers to the superconductivity induced by a non-superconducting tip at the point contact on non-superconducting materials. We take the elaboration of the chief experimental achievement of TISC in topological Dirac semimetal Cd₃As₂ and Weyl semimetal TaAs as key components of this article for detecting topological superconductivity. Moreover, we also briefly introduce the main results of another exotic effect, TESC, in superconducting Au₂Pb and Sr₂RuO₄ single crystals, which are respectively proposed as the candidates of helical topological superconductor and chiral topological superconductor. Related results and the potential mechanism are conducive to improving the comprehension of how to induce and enhance the topological superconductivity.

Band Structure Perfection and Superconductivity in Type-II Dirac Semimetal Ir1-x Ptx Te2

The discovery of a new type-II Dirac semimetal in Ir_1-x Pt_x Te₂ with optimized band structure is described. Pt dopants protect the crystal structure holding the Dirac cones and tune the Fermi level close to the Dirac point. The type-II Dirac dispersion in Ir_1-x Pt_x Te₂ is confirmed by angle-resolved photoemission spectroscopy and first-principles calculations. Superconductivity is also observed and persists when the Fermi level aligns with the Dirac points. Ir_1-x Pt_x Te₂ is an ideal platform for further studies on the exotic properties and potential applications of type-II DSMs, and opens up a new route for the investigation of the possible topological superconductivity and Majorana physics.

Inducing Strong Superconductivity in WTe2 by a Proximity Effect

The search for proximity-induced superconductivity in topological materials has generated widespread interest in the condensed matter physics community. The superconducting states inheriting nontrivial topology at interfaces are expected to exhibit exotic phenomena such as topological superconductivity and Majorana zero modes, which hold promise for applications in quantum computation. However, a practical realization of such hybrid structures based on topological semimetals and superconductors has hitherto been limited. Here, we report the strong proximity-induced superconductivity in type-II Weyl semimetal WTe₂, in a van der Waals hybrid structure obtained by mechanically transferring NbSe₂ onto various thicknesses of WTe₂. When the WTe₂ thickness ( t_WTe2) reaches 21 nm, the superconducting transition occurs around the critical temperature ( T_c) of NbSe₂ with a gap amplitude (Δ_p) of 0.38 meV and an unexpected ultralong proximity length ( l_p) up to 7 μm. With the thicker 42 nm WTe₂ layer, however, the proximity effect yields T_c ≈ 1.2 K, Δ_p = 0.07 meV, and a short l_p of less than 1 μm. Our theoretical calculations, based on the Bogoliubov-de Gennes equations in the clean limit, predict that the induced superconducting gap is a sizable fraction of the NbSe₂ superconducting one when t_WTe2 is less than 30 nm and then decreases quickly as t_WTe2 increases. This agrees qualitatively well with the experiments. Such observations form a basis in the search for superconducting phases in topological semimetals.

Dirac Cones and Nodal Line in Borophene

Two-dimensional single-layer boron (borophene) has emerged as a new material with several intriguing properties. Recently, the β₁₂ polymorph of borophene was grown on Ag(111), and observed to host Dirac fermions. Similar to graphene, β₁₂ borophene can be described as atom-vacancy pseudoalloy on a closed-packed triangular lattice; however, unlike graphene, the origin of its Dirac fermions  is yet unclear. Here, using first-principles calculations, we probe the origin of Dirac fermions in freestanding and Ag(111)-supported β₁₂ borophene. The freestanding β₁₂ sheet hosts two Dirac cones and a topologically nontrivial Dirac nodal line with interesting Dirac-like edge states. On Ag(111), the Dirac cones develop a gap, whereas the topologically protected nodal line remains intact, and its position in the Brillouin zone matches that of the Dirac-like electronic states seen in the experiment. The presence of nontrivial topological states near the Fermi level in borophene makes its electronic properties important for both fundamental and applied research.

Topological superconductors: a review

This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin-orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

Superconductivity Provides Access to the Chiral Magnetic Effect of an Unpaired Weyl Cone

The massless fermions of a Weyl semimetal come in two species of opposite chirality, in two cones of the band structure. As a consequence, the current j induced in one Weyl cone by a magnetic field B [the chiral magnetic effect (CME)] is canceled in equilibrium by an opposite current in the other cone. Here, we show that superconductivity offers a way to avoid this cancellation, by means of a flux bias that gaps out a Weyl cone jointly with its particle-hole conjugate. The remaining gapless Weyl cone and its particle-hole conjugate represent a single fermionic species, with renormalized charge e^{*} and a single chirality ± set by the sign of the flux bias. As a consequence, the CME is no longer canceled in equilibrium but appears as a supercurrent response ∂j/∂B=±(e^{*}e/h^{2})μ along the magnetic field at chemical potential μ.

Discovery of tip induced unconventional superconductivity on Weyl semimetal

Weyl fermion is a massless Dirac fermion with definite chirality, which has been long pursued since 1929. Though it has not been observed as a fundamental particle in nature, Weyl fermion can be realized as low-energy excitation around Weyl point in Weyl semimetal, which possesses Weyl fermion cones in the bulk and nontrivial Fermi arc states on the surface. As a firstly discovered Weyl semimetal, TaAs crystal possesses 12 pairs of Weyl points in the momentum space, which are topologically protected against small perturbations. Here, we report for the first time the tip induced superconductivity on TaAs crystal by point contact spectroscopy. The zero bias conductance peak as well as a conductance plateau with double conductance peaks and sharp double dips are observed in the point contact spectra simultaneously, indicating unconventional superconductivity. Our further theoretical study suggests that the induced superconductivity may have nontrivial topology. The present work opens a new route in investigating the novel superconducting states based on Weyl materials.

Superconductivity in the antiperovskite Dirac-metal oxide Sr3-xSnO

Investigations of perovskite oxides triggered by the discovery of high-temperature and unconventional superconductors have had crucial roles in stimulating and guiding the development of modern condensed-matter physics. Antiperovskite oxides are charge-inverted counterpart materials to perovskite oxides, with unusual negative ionic states of a constituent metal. No superconductivity was reported among the antiperovskite oxides so far. Here we present the first superconducting antiperovskite oxide Sr_3-xSnO with the transition temperature of around 5 K. Sr₃SnO possesses Dirac points in its electronic structure, and we propose from theoretical analysis a possibility of a topological odd-parity superconductivity analogous to the superfluid ³He-B in moderately hole-doped Sr_3-xSnO. We envision that this discovery of a new class of oxide superconductors will lead to a rapid progress in physics and chemistry of antiperovskite oxides consisting of unusual metallic anions.

Three-dimensional Majorana fermions in chiral superconductors

Using a systematic symmetry and topology analysis, we establish that three-dimensional chiral superconductors with strong spin-orbit coupling and odd-parity pairing generically host low-energy nodal quasiparticles that are spin-nondegenerate and realize Majorana fermions in three dimensions. By examining all types of chiral Cooper pairs with total angular momentum J formed by Bloch electrons with angular momentum j in crystals, we obtain a comprehensive classification of gapless Majorana quasiparticles in terms of energy-momentum relation and location on the Fermi surface. We show that the existence of bulk Majorana fermions in the vicinity of spin-selective point nodes is rooted in the nonunitary nature of chiral pairing in spin-orbit-coupled superconductors. We address experimental signatures of Majorana fermions and find that the nuclear magnetic resonance spin relaxation rate is significantly suppressed for nuclear spins polarized along the nodal direction as a consequence of the spin-selective Majorana nature of nodal quasiparticles. Furthermore, Majorana nodes in the bulk have nontrivial topology and imply the presence of Majorana bound states on the surface, which form arcs in momentum space. We conclude by proposing the heavy fermion superconductor PrOs4Sb12 and related materials as promising candidates for nonunitary chiral superconductors hosting three-dimensional Majorana fermions.

Substantial Band-Gap Tuning and a Strain-Controlled Semiconductor to Gapless/Band-Inverted Semimetal Transition in Rutile Lead/Stannic Dioxide

By first-principle calculations, we have systematically studied the effect of strain/pressure on the electronic structure of rutile lead/stannic dioxide (PbO₂/SnO₂). We find that pressure/strain has a significant impact on the electronic structure of PbO₂/SnO₂. Not only can the band gap be substantially tuned by pressure/strain, but also a transition between a semiconductor and a gapless/band-inverted semimetal can be manipulated. Furthermore, the semimetallic state is robust under strain, indicating a bright perspective for electronics applications. In addition, a practical approach to realizing strain in SnO₂ is then proposed by substituting tin (Sn) with lead (Pb), which also can trigger the transition from a large-band-gap to a moderate-gap semiconductor with enhanced electron mobility. This work is expected to provide guidance for full utilization of the flexible electronic properties in PbO₂ and SnO₂.

Z_{2} and Chiral Anomalies in Topological Dirac Semimetals

We demonstrate that topological Dirac semimetals, which possess two Dirac nodes, separated in momentum space along a rotation axis and protected by rotational symmetry, exhibit an additional quantum anomaly, distinct from the chiral anomaly. This anomaly, which we call the Z_{2} anomaly, is a consequence of the fact that the Dirac nodes in topological Dirac semimetals carry a Z_{2} topological charge. The Z_{2} anomaly refers to nonconservation of this charge in the presence of external fields due to quantum effects and has observable consequences due to its interplay with the chiral anomaly. We discuss possible implications of this for the interpretation of magnetotransport experiments on topological Dirac semimetals. We also provide a possible explanation for the magnetic field dependent angular narrowing of the negative longitudinal magnetoresistance, observed in a recent experiment on Na_{3}Bi.

Unconventional superconductivity at mesoscopic point contacts on the 3D Dirac semimetal Cd3As2

Three-dimensional (3D) Dirac semimetals exist close to topological phase boundaries which, in principle, should make it possible to drive them into exotic new phases, such as topological superconductivity, by breaking certain symmetries. A practical realization of this idea has, however, hitherto been lacking. Here we show that the mesoscopic point contacts between pure silver (Ag) and the 3D Dirac semimetal Cd3As2 (ref. ) exhibit unconventional superconductivity with a critical temperature (onset) greater than 6 K whereas neither Cd3As2 nor Ag are superconductors. A gap amplitude of 6.5 meV is measured spectroscopically in this phase that varies weakly with temperature and survives up to a remarkably high temperature of 13 K, indicating the presence of a robust normal-state pseudogap. The observations indicate the emergence of a new unconventional superconducting phase that exists in a quantum mechanically confined region under a point contact between a Dirac semimetal and a normal metal.