Magnetic Breakdown and Klein Tunneling in a Type-II Weyl Semimetal

Thickness-Dependent Magnetic Breakdown in ZrSiSe Nanoplates

We report a study of thickness-dependent interband and intraband magnetic breakdown by thermoelectric quantum oscillations in ZrSiSe nanoplates. Under high magnetic fields of up to 30 T, quantum oscillations arising from degenerated hole pockets were observed in thick ZrSiSe nanoplates. However, when decreasing the thickness, plentiful multifrequency quantum oscillations originating from hole and electron pockets are captured. These multiple frequencies can be explained by the emergent interband magnetic breakdown enclosing individual hole and electron pockets and intraband magnetic breakdown within spin-orbit coupling (SOC) induced saddle-shaped electron pockets, resulting in the enhanced contribution to thermal transport in thin ZrSiSe nanoplates. These experimental frequencies agree well with theoretical calculations of the intriguing tunneling processes. Our results introduce a new member of magnetic breakdown to the field and open up a dimension for modulating magnetic breakdown, which holds fundamental significance for both low-dimensional topological materials and the physics of magnetic breakdown.

Bound-Extended Mode Transition in Type-II Synthetic Photonic Weyl Heterostructures

Photonic structures with Weyl points (WPs), including type I and type II, promise nontrivial surface modes and intriguing light manipulations for their three-dimensional topological bands. While previous studies mainly focus on exploring WPs in a uniform Weyl structure, here we establish Weyl heterostructures (i.e., a nonuniform Weyl lattice) with different rotational orientations in the synthetic dimension by nanostructured photonic waveguides. In this work, we unveil a transition between bound and extended modes on the interface of type-II Weyl heterostructures by tuning their rotational phases, despite the reversed topological order across the interface. This mode transition is also manifested from the total transmission to total reflection at the interface. All of these unconventional effects are attributed to the tilted dispersion of type-II Weyl band structure that can lead to mismatched bands and gaps across the interface. As a comparison, the type-I Weyl heterostructures lack the phase transition due to the untilted band structure. This work establishes a flexible scheme of artificial Weyl heterostructures that opens a new avenue toward high-dimensional topological effects and significantly enhances our capabilities in on-chip light manipulations.

Quantum oscillations of the quasiparticle lifetime in a metal

Following nearly a century of research, it remains a puzzle that the low-lying excitations of metals are remarkably well explained by effective single-particle theories of non-interacting bands1-4. The abundance of interactions in real materials raises the question of direct spectroscopic signatures of phenomena beyond effective single-particle, single-band behaviour. Here we report the identification of quantum oscillations (QOs) in the three-dimensional topological semimetal CoSi, which defy the standard description in two fundamental aspects. First, the oscillation frequency corresponds to the difference of semiclassical quasiparticle (QP) orbits of two bands, which are forbidden as half of the trajectory would oppose the Lorentz force. Second, the oscillations exist up to above 50 K, in strong contrast to all other oscillatory components, which vanish below a few kelvin. Our findings are in excellent agreement with generic model calculations of QOs of the QP lifetime (QPL). Because the only precondition for their existence is a nonlinear coupling of at least two electronic orbits, for example, owing to QP scattering on defects or collective excitations, such QOs of the QPL are generic for any metal featuring Landau quantization with several orbits. They are consistent with certain frequencies in topological semimetals5-9, unconventional superconductors10,11, rare-earth compounds12-14 and Rashba systems15, and permit to identify and gauge correlation phenomena, for example, in two-dimensional materials16,17 and multiband metals18.

Revealing a 3D Fermi Surface Pocket and Electron-Hole Tunneling in UTe_{2} with Quantum Oscillations

Spin triplet superconductor UTe_{2} is widely believed to host a quasi-two-dimensional Fermi surface, revealed by first-principles calculations, photoemission, and quantum oscillation measurements. An outstanding question still remains as to the existence of a three-dimensional Fermi surface pocket, which is crucial for our understanding of the exotic superconducting and topological properties of UTe_{2}. This 3D Fermi surface pocket appears in various theoretical models with different physics origins, but has not been unambiguously detected in experiment. Here for the first time we provide concrete evidence for a relatively isotropic, small Fermi surface pocket of UTe_{2} via quantum oscillation measurements. In addition, we observed high frequency quantum oscillations corresponding to electron-hole tunneling between adjacent electron and hole pockets. The coexistence of 2D and 3D Fermi surface pockets, as well as the breakdown orbits, provide a test bed for theoretical models and aid the realization of a unified understanding of the superconducting state of UTe_{2} from the first-principles approach.

Observation of flat band, Dirac nodal lines and topological surface states in Kagome superconductor CsTi3Bi5

Kagome lattices of various transition metals are versatile platforms for achieving anomalous Hall effects, unconventional charge-density wave orders and quantum spin liquid phenomena due to the strong correlations, spin-orbit coupling and/or magnetic interactions involved in such a lattice. Here, we use laser-based angle-resolved photoemission spectroscopy in combination with density functional theory calculations to investigate the electronic structure of the newly discovered kagome superconductor CsTi₃Bi₅, which is isostructural to the AV₃Sb₅ (A = K, Rb or Cs) kagome superconductor family and possesses a two-dimensional kagome network of titanium. We directly observe a striking flat band derived from the local destructive interference of Bloch wave functions within the kagome lattice. In agreement with calculations, we identify type-II and type-III Dirac nodal lines and their momentum distribution in CsTi₃Bi₅ from the measured electronic structures. In addition, around the Brillouin zone centre, [Formula: see text] nontrivial topological surface states are also observed due to band inversion mediated by strong spin-orbit coupling.

Two step I to II type transitions in layered Weyl semi-metals and their impact on superconductivity

Novel "quasi two dimensional" typically layered (semi) metals offer a unique opportunity to control the density and even the topology of the electronic matter. Along with doping and gate voltage, a robust tuning is achieved by application of the hydrostatic pressure. In Weyl semi-metals the tilt of the dispersion relation cones, [Formula: see text] increases with pressure, so that one is able to reach type II ([Formula: see text]starting from the more conventional type I Weyl semi-metals [Formula: see text]. The microscopic theory of such a transition is constructed. It is found that upon increasing pressure the I to II transition occurs in two continuous steps. In the first step the cones of opposite chirality coalesce so that the chiral symmetry is restored, while the second transition to the Fermi surface extending throughout the Brillouin zone occurs at higher pressures. Flattening of the band leads to profound changes in Coulomb screening. Superconductivity observed recently in wide range of pressure and chemical composition in Weyl semi-metals of both types. The phonon theory of pairing including the Coulomb repulsion for a layered material is constructed and applied to recent extensive experiments on [Formula: see text].

Nontrivial Topological Properties and Synthesis of Sn2CoS with L21 Structure

We synthesize Sn₂CoS in experiment and study its topological properties in theory. By first-principles calculations, we study the band structure and surface state of Sn₂CoS with L2₁ structure. It is found that the material has type-II nodal line in the Brillouin zone and clear drumhead-like surface state when the spin-orbit coupling is not considered. In the case of spin-orbit coupling, the nodal line will open gap, leaving the Dirac points. To check the stability of the material in nature, we synthesize Sn₂CoS nanowires with L2₁ structure in an anodic aluminum oxide (AAO) template directly by the electrochemical deposition (ECD) method with direct current (DC). Additionally, the diameter of the typical Sn₂CoS nanowires is about 70 nm, with a length of about 70 μm. The Sn₂CoS nanowires are single crystals with an axis direction of [100], and the lattice constant determined by XRD and TEM is 6.0 Å. Overall, our work provides realistic material to study the nodal line and Dirac fermions.

Prediction of novel two-dimensional Dirac nodal line semimetals in Al2B2 and AlB4 monolayers

Topological semimetal phases in two-dimensional (2D) materials have gained widespread interest due to their potential applications in novel nanoscale devices. Despite the growing number of studies on 2D topological nodal lines (NLs), candidates with significant topological features that combine nontrivial topological semimetal phase with superconductivity are still rare. Herein, we predict Al₂B₂ and AlB₄ monolayers as new 2D nonmagnetic Dirac nodal line semimetals with several novel features. Our extensive electronic structure calculations combined with analytical studies reveal that, in addition to multiple Dirac points, these 2D configurations host various highly dispersed NLs around the Fermi level, all of which are semimetal states protected by time-reversal and in-plane mirror symmetries. The most intriguing NL in Al₂B₂ encloses the K point and crosses the Fermi level, showing a considerable dispersion and thus providing a fresh playground to explore exotic properties in dispersive Dirac nodal lines. More strikingly, for the AlB₄ monolayer, we provide the first evidence for a set of 2D nonmagnetic open type-II NLs coexisting with superconductivity at a rather high transition temperature. The coexistence of superconductivity and nontrivial band topology in AlB₄ not only makes it a promising material to exhibit novel topological superconducting phases, but also a rather large energy dispersion of type-II nodal lines in this configuration may offer a platform for the realization of novel topological features in the 2D limit.

On-Site Synthesis and Characterizations of Atomically-Thin Nickel Tellurides with Versatile Stoichiometric Phases through Self-Intercalation

Self-intercalation of native metal atoms in two-dimensional (2D) transition metal dichalcogenides has received rapidly increasing interest, due to the generation of intriguing structures and exotic physical properties, however, only reported in limited materials systems. An emerging type-II Dirac semimetal, NiTe₂, has inspired great attention at the 2D thickness region, but has been rarely achieved so far. Herein, we report the direct synthesis of mono- to few-layer Ni-tellurides including 1T-NiTe₂ and Ni-rich stoichiometric phases on graphene/SiC(0001) substrates under ultra-high-vacuum conditions. Differing from previous chemical vapor deposition growth accompanied with transmission electron microscopy imaging, this work combines precisely tailored synthesis with on-site atomic-scale scanning tunneling microscopy observations, offering us visual information about the phase modulations of Ni-tellurides from 1T-phase NiTe₂ to self-intercalated Ni₃Te₄ and Ni₅Te₆. The synthesis of Ni self-intercalated Ni_xTe_y compounds is explained to be mediated by the high metal chemical potential under Ni-rich conditions, according to density functional theory calculations. More intriguingly, the emergence of superconductivity in bilayer NiTe₂ intercalated with 50% Ni is also predicted, arising from the enhanced electron-phonon coupling strength after the self-intercalation. This work provides insight into the direct synthesis and stoichiometric phase modulation of 2D layered materials, enriching the family of self-intercalated materials and propelling their property explorations.

Various half-metallic nodal loops in organic Cr2N6C3 monolayers

Topological nodal-line semimetals, as a type of exotic quantum electronic state, have drawn considerable research interest recently. In this work, we propose a new two-dimensional covalent-organic Cr₂N₆C₃ monolayer (ML) material, which has a combined honeycomb and effective Kagome lattice and has various half-metallic nodal loops (HMNLs). First-principles calculations show that the Cr₂N₆C₃ ML is dynamically and thermally stable and has an out-of-plane ferromagnetic order. Remarkably, various nodal loops, including types I-III, are found coexisting in the material, all of which are rare half-metallic states. The obtained HMNLs, simultaneously possessing the merits of spintronics and semimetals, are robust against spin-orbit coupling and biaxial strain. A topological phase transition, characterized by loop-winding indexes, can be induced in the ML by applying uniaxial strain. Tight-binding model calculations show that the obtained HMNLs originate primarily from the band inversion between Cr d_x²-y²/xy and N p_z orbitals, accommodated on the honeycomb and Kagome sublattices, respectively. The various predicted HMNLs and topological behaviors mean that the Cr₂N₆C₃ MLs have promisingly versatile applications in future low-power-consuming spintronics and electronics.

Emergent superconductivity by Re doping in type -II Weyl semimetal NiTe2

Recently topological semimetals emerge as a new platform to realise topological superconductivity. Here we report the emergent superconductivity in single-crystals of Re doped type-II Weyl semimetal NiTe₂. The magnetic and transport measurements highlight that Re substitution in Ni-site induces superconductivity at a maximum temperature of 2.36 K. Hall effect and specific heat measurements indicate that Re substitution is doping hole and facilitates the emergence of superconductivity by phonon softening and enhancing the electron-phonon coupling.

Weyl Fermions in VI3 Monolayer

We report the presence of a Weyl fermion in VI₃ monolayer. The material shows a sandwich-like hexagonal structure and stable phonon spectrum. It has a half-metal band structure, where only the bands in one spin channel cross the Fermi level. There are three pairs of Weyl points slightly below the Fermi level in spin-up channel. The Weyl points show a clean band structure and are characterized by clear Fermi arcs edge state. The effects of spin-orbit coupling, electron correlation, and lattice strain on the electronic band structure were investigated. We find that the half-metallicity and Weyl points are robust against these perturbations. Our work suggests VI₃ monolayer is an excellent Weyl half-metal.

Evidence of higher-order topology in multilayer WTe2 from Josephson coupling through anisotropic hinge states

Td-WTe₂ (non-centrosymmetric and orthorhombic), a type-II Weyl semimetal, is expected to have higher-order topological phases with topologically protected, helical one-dimensional hinge states when its Weyl points are annihilated. However, the detection of these hinge states is difficult due to the semimetallic behaviour of the bulk. In this study, we have spatially resolved the hinge states by analysing the magnetic field interference of the supercurrent in Nb-WTe₂-Nb proximity Josephson junctions. The Josephson current along the a axis of the WTe₂ crystal, but not along the b axis, showed a sharp enhancement at the edges of the junction, and the amount of enhanced Josephson current was comparable to the upper limits of a single one-dimensional helical channel. Our experimental observations suggest a higher-order topological phase in WTe₂ and its corresponding anisotropic topological hinge states, in agreement with theoretical calculations. Our work paves the way for the study of hinge states in topological transition-metal dichalcogenides and analogous phases.

Recent advances in 2D, 3D and higher-order topological photonics

Over the past decade, topology has emerged as a major branch in broad areas of physics, from atomic lattices to condensed matter. In particular, topology has received significant attention in photonics because light waves can serve as a platform to investigate nontrivial bulk and edge physics with the aid of carefully engineered photonic crystals and metamaterials. Simultaneously, photonics provides enriched physics that arises from spin-1 vectorial electromagnetic fields. Here, we review recent progress in the growing field of topological photonics in three parts. The first part is dedicated to the basics of topological band theory and introduces various two-dimensional topological phases. The second part reviews three-dimensional topological phases and numerous approaches to achieve them in photonics. Last, we present recently emerging fields in topological photonics that have not yet been reviewed. This part includes topological degeneracies in nonzero dimensions, unidirectional Maxwellian spin waves, higher-order photonic topological phases, and stacking of photonic crystals to attain layer pseudospin. In addition to the various approaches for realizing photonic topological phases, we also discuss the interaction between light and topological matter and the efforts towards practical applications of topological photonics.

Bond-breaking induced Lifshitz transition in robust Dirac semimetal VAI3

Understanding the relation between crystal structure and electronic properties is crucial for designing new quantum materials with desired functionality. So far, controlling a chemical bond is less considered as an effective way to manipulate the topological electrons. In this paper, we show that the V–Al bond acts as a shield for protecting the topological electrons in Dirac semimetal VAl3. The Dirac electrons remain intact in the V1−xTixAl3 solid solutions, even after a substantial part of V atoms have been replaced. A Lifshitz transition from Dirac semimetal to trivial metal occurs as long as the V–Al bond is completely broken. Our finding highlights a rational approach for designing new quantum materials via controlling their chemical bond.

Topological electrons in semimetals are usually vulnerable to chemical doping and environment change, which restricts their potential application in future electronic devices. In this paper, we report that the type-II Dirac semimetal VAl3 hosts exceptional, robust topological electrons which can tolerate extreme change of chemical composition. The Dirac electrons remain intact, even after a substantial part of V atoms have been replaced in the V1−xTixAl3 solid solutions. This Dirac semimetal state ends at x=0.35, where a Lifshitz transition to p-type trivial metal occurs. The V–Al bond is completely broken in this transition as long as the bonding orbitals are fully depopulated by the holes donated from Ti substitution. In other words, the Dirac electrons in VAl3 are protected by the V–Al bond, whose molecular orbital is their bonding gravity center. Our understanding on the interrelations among electron count, chemical bond, and electronic properties in topological semimetals suggests a rational approach to search robust, chemical-bond-protected topological materials.

Theory of universal differential conductance of magnetic Weyl type-II junctions

We study the transport properties of junctions of normal and superconducting Weyl semimetal with tilted dispersion, in the presence of magnetization induced by magnetic strips. The sub gap tunnelling conductance shows robust signatures in the presence of different orientation and strength of magnetization of the magnetic strips. We obtain the analytical results for the normal-magnetic-superconducting junction in the thin barrier limit and demonstrate that these results have no analogues to their conventional counterparts and junctions with Dirac electrons in two-dimensions. We discuss possible experimental setups to test our theoretical predictions.

Ternary compound HfCuP: An excellent Weyl semimetal with the coexistence of type-I and type-II Weyl nodes

In most Weyl semimetal (WSMs), the Weyl nodes with opposite chiralities usually have the same type of band dispersions (either type-I or type-II), whereas realistic candidate materials hosting different types of Weyl nodes have not been identified to date. Here we report for the first time that, a ternary compound HfCuP, is an excellent WSM with the coexistence of type-I and type-II Weyl nodes. Our results show that, HfCuP totally contains six pairs of type-I and six pairs of type-II Weyl nodes in the Brillouin zone, all locating at the H-K path. These Weyl nodes situate slightly below the Fermi level, and do not coexist with other extraneous bands. The nontrivial band structure in HfCuP produces clear Fermi arc surface states in the (1 0 0) surface projection. Moreover, we find the Weyl nodes in HfCuP can be effectively tuned by strain engineering. These characteristics make HfCuP a potential candidate material to investigate the novel properties of type-I and type-II Weyl fermions, as well as the potential entanglements between them.

Deterministic Scheme for Two-Dimensional Type-II Dirac Points and Experimental Realization in Acoustics

Low-energy electrons near Dirac/Weyl nodal points mimic massless relativistic fermions. However, as they are not constrained by Lorentz invariance, they can exhibit tipped-over type-II Dirac/Weyl cones that provide highly anisotropic physical properties and responses, creating unique possibilities. Recently, they have been observed in several quantum and classical systems. Yet, there is still no simple and deterministic strategy to realize them since their nodal points are accidental degeneracies, unlike symmetry-guaranteed type-I counterparts. Here, we propose a band-folding scheme for constructing type-II Dirac points, and we use a tight-binding analysis to unveil its generality and deterministic nature. Through realizations in acoustics, type-II Dirac points are experimentally visualized and investigated using near-field mappings. As a direct effect of tipped-over Dirac cones, strongly tilted kink states originating from their valley-Hall properties are also observed. This deterministic scheme could serve as a platform for further investigations of intriguing physics associated with various strongly Lorentz-violating nodal points.

Electron collimator in Weyl semimetals with periodic magnetic barriers

We investigate theoretically the effect of periodic magnetic barriers on the transport for a Weyl semimetal. We find that there are momentum and spin filtering tunneling behaviors, which is controlled by the numbers of the magnetic barriers. For the tunneling through periodic square-shaped magnetic barriers, the transmission is angular φ asymmetry, and the asymmetrical transmission probability becomes more pronounced with increasing the superlattice number n. However, the transmission is symmetric with respect to angle γ, and the window of the transmission become more and more narrower with increasing the number of barriers, i.e., the collimator behavior. This feature comes from the electron Fabry-Pérot modes among the barriers. We find that the constructive interference of the backscattering amplitudes suppress transmissions, and consequently form the minigaps of the transmission. The transmission can be switched on/off by tuning the incident energies and angles, the heights and numbers of the magnetic barriers, and result in the interesting collimator behavior.

Polar and phase domain walls with conducting interfacial states in a Weyl semimetal MoTe2

Much of the dramatic growth in research on topological materials has focused on topologically protected surface states. While the domain walls of topological materials such as Weyl semimetals with broken inversion or time-reversal symmetry can provide a hunting ground for exploring topological interfacial states, such investigations have received little attention to date. Here, utilizing in-situ cryogenic transmission electron microscopy combined with first-principles calculations, we discover intriguing domain-wall structures in MoTe2, both between polar variants of the low-temperature(T) Weyl phase, and between this and the high-T higher-order topological phase. We demonstrate how polar domain walls can be manipulated with electron beams and show that phase domain walls tend to form superlattice-like structures along the c axis. Scanning tunneling microscopy indicates a possible signature of a conducting hinge state at phase domain walls. Our results open avenues for investigating topological interfacial states and unveiling multifunctional aspects of domain walls in topological materials.

Experimental Realization of Type-II Weyl Points and Fermi Arcs in Phononic Crystal

Weyl points (WPs), as the doubly degenerate points in three-dimensional momentum band structures, carry quantized topological charges and give rise to a variety of extraordinary properties, such as robust surface wave and chiral anomaly. Type-II Weyl semimetals, which have conical dispersions in Fermi surfaces and a strongly tilted dispersion with respect to type I, have recently been proposed in condensed-matter systems and photonics. Although the type-II WPs have been theoretically predicted in acoustics, the experimental realization in phononic crystals has not been reported so far. Here, we experimentally realize a type-II Weyl phononic crystal. We demonstrate the topological transitions observed at the WP frequencies and the topological surface acoustic waves between the Weyl frequencies. The experiment results are in good accordance with our theoretical analyses. Due to the violation of the Lorentz symmetry, the type-II WPs only exist in low energy systems. As the analog counterpart in classical waves, the phononic crystal brings a platform for the research of type-II WPs in macroscopic systems.

Electron-Hole Tunneling Revealed by Quantum Oscillations in the Nodal-Line Semimetal HfSiS

We report a study of quantum oscillations in the high-field magnetoresistance of the nodal-line semimetal HfSiS. In the presence of a magnetic field up to 31 T parallel to the c axis, we observe quantum oscillations originating both from orbits of individual electron and hole pockets, and from magnetic breakdown between these pockets. In particular, we reveal a breakdown orbit enclosing one electron and one hole pocket in the form of a "figure of eight," which is a manifestation of Klein tunneling in momentum space, although in a regime of partial transmission due to the finite separation between the pockets. The observed very strong dependence of the oscillation amplitude on the field angle and the cyclotron masses of the orbits are in agreement with the theoretical predictions for this novel tunneling phenomenon.

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.

Inversion symmetry breaking induced triply degenerate points in orderly arranged PtSeTe family materials

k paths exactly with [Formula: see text] symmetry allow to find triply degenerate points (TDPs) in band structures. The paths that host the type-II Dirac points in PtSe₂ family materials also have the [Formula: see text] spatial symmetry. However, due to Kramers degeneracy (the systems have both inversion symmetry and time reversal symmetry), the crossing points in them are Dirac ones. In this work, based on symmetry analysis, first-principles calculations, and [Formula: see text] method, we predict that PtSe₂ family materials should undergo topological transitions if the inversion symmetry is broken, i.e. the Dirac fermions in PtSe₂ family materials split into TDPs in PtSeTe family materials (PtSSe, PtSeTe, and PdSeTe) with orderly arranged S/Se (Se/Te). It is different from the case in high-energy physics that breaking inversion symmetry I leads to the splitting of Dirac fermion into Weyl fermions. We also address a possible method to achieve the orderly arranged in PtSeTe family materials in experiments. Our study provides a real example that Dirac points transform into TDPs, and is helpful to investigate the topological transition between Dirac fermions and TDP fermions.

Bulk Fermi Surfaces of the Dirac Type-II Semimetallic Candidates MAl_{3} (Where M=V, Nb, and Ta)

We report a de Haas-van Alphen (dHvA) effect study on the Dirac type-II semimetallic candidates MAl_{3} (where, M=V, Nb and Ta). The angular dependence of their Fermi surface (FS) cross-sectional areas reveals a remarkably good agreement with our first-principles calculations. Therefore, dHvA supports the existence of tilted Dirac cones with Dirac type-II nodes located at 100, 230 and 250 meV above the Fermi level ϵ_{F} for VAl_{3}, NbAl_{3} and TaAl_{3} respectively, in agreement with the prediction of broken Lorentz invariance in these compounds. However, for all three compounds we find that the cyclotron orbits on their FSs, including an orbit nearly enclosing the Dirac type-II node, yield trivial Berry phases. We explain this via an analysis of the Berry phase where the position of this orbit, relative to the Dirac node, is adjusted within the error implied by the small disagreement between our calculations and the experiments. We suggest that a very small amount of doping could displace ϵ_{F} to produce topologically nontrivial orbits encircling their Dirac node(s).

Rules for Phase Shifts of Quantum Oscillations in Topological Nodal-Line Semimetals

Nodal-line semimetals are topological semimetals in which band touchings form nodal lines or rings. Around a loop that encloses a nodal line, an electron can accumulate a nontrivial π Berry phase, so the phase shift in the Shubnikov-de Haas (SdH) oscillation may give a transport signature for the nodal-line semimetals. However, different experiments have reported contradictory phase shifts, in particular, in the WHM nodal-line semimetals (W=Zr/Hf, H=Si/Ge, M=S/Se/Te). For a generic model of nodal-line semimetals, we present a systematic calculation for the SdH oscillation of resistivity under a magnetic field normal to the nodal-line plane. From the analytical result of the resistivity, we extract general rules to determine the phase shifts for arbitrary cases and apply them to ZrSiS and Cu_{3}PdN systems. Depending on the magnetic field directions, carrier types, and cross sections of the Fermi surface, the phase shift shows rich results, quite different from those for normal electrons and Weyl fermions. Our results may help explore transport signatures of topological nodal-line semimetals and can be generalized to other topological phases of matter.

Shot noise and electronic properties in the inversion-symmetric Weyl semimetal resonant structure

Using the transfer matrix method, the authors combine the analytical formula with numerical calculation to explore the shot noise and conductance of massless Weyl fermions in the Weyl semimetal resonant junction. By varying the barrier strength, the structure of the junction, the Fermi energy, and the crystallographic angle, the shot noise and conductance can be tuned efficiently. For a quasiperiodic superlattice, in complete contrast to the conventional junction case, the effect of the disorder strength on the shot noise and conductance depends on the competition of classical tunneling and Klein tunneling. Moreover, the delta barrier structure is also vital in determining the shot noise and conductance. In particular, a universal Fano factor has been found in a single delta potential case, whereas the resonant structure of the Fano factor perfectly matches with the number of barriers in a delta potential superlattice. These results are crucial for engineering nanoelectronic devices based on this topological semimetal material.

Ubiquitous formation of bulk Dirac cones and topological surface states from a single orbital manifold in transition-metal dichalcogenides

Transition-metal dichalcogenides (TMDs) are renowned for their rich and varied bulk properties, while their single-layer variants have become one of the most prominent examples of two-dimensional materials beyond graphene. Their disparate ground states largely depend on transition metal d-electron-derived electronic states, on which the vast majority of attention has been concentrated to date. Here, we focus on the chalcogen-derived states. From density-functional theory calculations together with spin- and angle-resolved photoemission, we find that these generically host a co-existence of type-I and type-II three-dimensional bulk Dirac fermions as well as ladders of topological surface states and surface resonances. We demonstrate how these naturally arise within a single p-orbital manifold as a general consequence of a trigonal crystal field, and as such can be expected across a large number of compounds. Already, we demonstrate their existence in six separate TMDs, opening routes to tune, and ultimately exploit, their topological physics.

Geometric Phase and Orbital Moment in Quantization Rules for Magnetic Breakdown

The modern semiclassical theory of a Bloch electron in a magnetic field encompasses the orbital magnetization and geometric phase. Beyond this semiclassical theory lies the quantum description of field-induced tunneling between semiclassical orbits, known as magnetic breakdown. Here, we synthesize the modern semiclassical notions with quantum tunneling-into a single Bohr-Sommerfeld quantization rule that is predictive of magnetic energy levels. This rule is applicable to a host of topological solids with unremovable geometric phase, that also unavoidably undergo breakdown. A notion of topological invariants is formulated that nonperturbatively encode tunneling, and is measurable in the de Haas-van Alphen effect. Case studies are discussed for topological metals near a metal-insulator transition and overtilted Weyl fermions.

Type-II Dirac semimetal stabilized by electron-phonon coupling

There is major interest, in condensed matter physics, in understanding the role of topology: remarkable progress has been made in classifying topological properties of non-interacting electrons, and on understanding the interplay between topology and electron–electron interactions. We extend such studies to interactions with the lattice, and predict non-trivial topological effects in infinitely long-lived polaron bands. Specifically, for a two-dimensional many-band model with realistic electron–phonon coupling, we verify that sharp level crossings are possible for polaron eigenstates, and prove that they are responsible for a novel type of sharp transition in the ground state of the polaron that can occur at a fixed momentum. Furthermore, they result in the appearance of Dirac cones stabilized by electron–phonon coupling. Thus, electron–phonon coupling opens an avenue to create and control Dirac and Weyl semimetals.

How lattice degrees of freedom affect the topological properties of electrons remains rarely explored. Here, Moeller et al. predict non-trivial topological effects in long-lived polaron bands by studying a two-dimensional model stabilized by realistic electron-phonon coupling.

Evidence for topological type-II Weyl semimetal WTe2

Recently, a type-II Weyl fermion was theoretically predicted to appear at the contact of electron and hole Fermi surface pockets. A distinguishing feature of the surfaces of type-II Weyl semimetals is the existence of topological surface states, so-called Fermi arcs. Although WTe₂ was the first material suggested as a type-II Weyl semimetal, the direct observation of its tilting Weyl cone and Fermi arc has not yet been successful. Here, we show strong evidence that WTe₂ is a type-II Weyl semimetal by observing two unique transport properties simultaneously in one WTe₂ nanoribbon. The negative magnetoresistance induced by a chiral anomaly is quite anisotropic in WTe₂ nanoribbons, which is present in b-axis ribbon, but is absent in a-axis ribbon. An extra-quantum oscillation, arising from a Weyl orbit formed by the Fermi arc and bulk Landau levels, displays a two dimensional feature and decays as the thickness increases in WTe₂ nanoribbon.

Exotic transport properties of type-II Weyl semimetals have been predicted but are yet to be experimentally evidenced. Here, Li et al. report evidences of an anisotropy of negative magnetoresistance and a quantum oscillation arising from the predicted Weyl orbit in the type-II Weyl semimetal WTe₂.

From Type-II Triply Degenerate Nodal Points and Three-Band Nodal Rings to Type-II Dirac Points in Centrosymmetric Zirconium Oxide

Using first-principles calculations, we report that ZrO is a topological material with the coexistence of three pairs of type-II triply degenerate nodal points (TNPs) and three nodal rings (NRs), when spin-orbit coupling (SOC) is ignored. Noticeably, the TNPs reside around the Fermi energy with a large linear energy range along the tilt direction (>1 eV), and the NRs are formed by three strongly entangled bands. Under symmetry-preserving strain, each NR would evolve into four droplet-shaped NRs before fading away, producing distinct evolution compared with that in usual two-band NR. When SOC is included, TNPs would transform into type-II Dirac points while all of the NRs are gapped. Remarkably, the type-II Dirac points inherit the advantages of TNPs: residing around the Fermi energy and exhibiting a large linear energy range. Both features facilitate the observation of interesting phenomena induced by type-II dispersion. The symmetry protections and low-energy Hamiltonian for the nontrivial band crossings are discussed.

Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg3Bi2

Topological type-II nodal line semimetal (NLS) was proposed quite recently and exhibits distinct properties compared with conventional type-I NLS. To date, no ambient-condition stable candidate material has been reported. Here we propose that a stable Kagome compound Mg₃Bi₂ can host a type-II nodal line state with the protection of time reversal and spatial inversion symmetries. Similar to type-I NLSs, the type-II nodal line in Mg₃Bi₂ is characterized by the drumhead surface states, which has not been observed in the previous type-II NLSs. The nodal line in Mg₃Bi₂ can open a minor gap, and a pair of 3D Dirac points occurs when SOC is included. The SOC-induced gap around the nodal line is quite small, and the formation of 3D Dirac points is independent of the nodal line. Therefore, the Mg₃Bi₂ compound is expected to be a good candidate to investigate the exotic properties of both type-II NLS and 3D Dirac semimetal states.

Tuning the electrical transport of type II Weyl semimetal WTe2 nanodevices by Ga+ ion implantation

Here we introduce lattice defects in WTe₂ by Ga+ implantation (GI), and study the effects of defects on the transport properties and electronic structures of the samples. Theoretical calculation shows that Te Frenkel defects is the dominant defect type, and Raman characterization results agree with this. Electrical transport measurements show that, after GI, significant changes are observed in magnetoresistance and Hall resistance. The classical two-band model analysis shows that both electron and hole concentration are significantly reduced. According to the calculated results, ion implantation leads to significant changes in the band structure and the Fermi surface of the WTe₂. Our results indicate that defect engineering is an effective route of controlling the electronic properties of WTe₂ devices.

Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide PtTe2

Topological semimetals have recently attracted extensive research interests as host materials to condensed matter physics counterparts of Dirac and Weyl fermions originally proposed in high energy physics. Although Lorentz invariance is required in high energy physics, it is not necessarily obeyed in condensed matter physics, and thus Lorentz-violating type-II Weyl/Dirac fermions could be realized in topological semimetals. The recent realization of type-II Weyl fermions raises the question whether their spin-degenerate counterpart-type-II Dirac fermions-can be experimentally realized too. Here, we report the experimental evidence of type-II Dirac fermions in bulk stoichiometric PtTe2 single crystal. Angle-resolved photoemission spectroscopy measurements and first-principles calculations reveal a pair of strongly tilted Dirac cones along the Γ-A direction, confirming PtTe2 as a type-II Dirac semimetal. Our results provide opportunities for investigating novel quantum phenomena (e.g., anisotropic magneto-transport) and topological phase transition.Whether the spin-degenerate counterpart of Lorentz-violating Weyl fermions, the Dirac fermions, can be realized remains as an open question. Here, Yan et al. report experimental evidence of such type-II Dirac fermions in bulk PtTe2 single crystal with a pair of strongly tilted Dirac cones.

Type-II Symmetry-Protected Topological Dirac Semimetals

The recent proposal of the type-II Weyl semimetal state has attracted significant interest. In this Letter, we propose the concept of the three-dimensional type-II Dirac fermion and theoretically identify this new symmetry-protected topological state in the large family of transition-metal icosagenides, MA_{3} (M=V, Nb, Ta; A=Al, Ga, In). We show that the VAl_{3} family features a pair of strongly Lorentz-violating type-II Dirac nodes and that each Dirac node can be split into four type-II Weyl nodes with chiral charge ±1 via symmetry breaking. Furthermore, we predict that the Landau level spectrum arising from the type-II Dirac fermions in VAl_{3} is distinct from that of known Dirac or Weyl semimetals. We also demonstrate a topological phase transition from a type-II Dirac semimetal to a quadratic Weyl semimetal or a topological crystalline insulator via crystalline distortions.

Discovery of Lorentz-violating type II Weyl fermions in LaAlGe

In quantum field theory, Weyl fermions are relativistic particles that travel at the speed of light and strictly obey the celebrated Lorentz symmetry. Their low-energy condensed matter analogs are Weyl semimetals, which are conductors whose electronic excitations mimic the Weyl fermion equation of motion. Although the traditional (type I) emergent Weyl fermions observed in TaAs still approximately respect Lorentz symmetry, recently, the so-called type II Weyl semimetal has been proposed, where the emergent Weyl quasiparticles break the Lorentz symmetry so strongly that they cannot be smoothly connected to Lorentz symmetric Weyl particles. Despite some evidence of nontrivial surface states, the direct observation of the type II bulk Weyl fermions remains elusive. We present the direct observation of the type II Weyl fermions in crystalline solid lanthanum aluminum germanide (LaAlGe) based on our photoemission data alone, without reliance on band structure calculations. Moreover, our systematic data agree with the theoretical calculations, providing further support on our experimental results.

Thermoelectric quantum oscillations in ZrSiS

Topological semimetals are systems in which conduction and valence bands cross each other and the crossings are protected by topological constraints. These materials provide intriguing tests for fundamental theories, while their unique physical properties promise a wide range of possible applications in low-power spintronics, optoelectronics, quantum computing and green energy harvesting. Here we report our study of the thermoelectric power of single-crystalline ZrSiS that is believed to be a topological nodal-line semimetal. We show that the thermoelectric power is an extremely sensitive probe of multiple quantum oscillations that are visible in ZrSiS at temperatures as high as 100 K. Two of these oscillations are shown to arise from three- and two-dimensional electronic bands, each with linear dispersion and the additional Berry phase predicted theoretically for materials with non-trivial topology. Our work not only provides further information on ZrSiS but also suggests a different route for studying other topological semimetals.

Field-Selective Anomaly and Chiral Mode Reversal in Type-II Weyl Materials

Three-dimensional condensed matter incarnations of Weyl fermions generically have a tilted dispersion-in sharp contrast to their elusive high-energy relatives where a tilt is forbidden by Lorentz invariance, and with the low-energy excitations of two-dimensional graphene sheets where a tilt is forbidden by either crystalline or particle-hole symmetry. Very recently, a number of materials (MoTe_{2}, LaAlGe, and WTe_{2}) have been identified as hosts of so-called type-II Weyl fermions whose dispersion is so strongly tilted that a Fermi surface is formed, whereby the Weyl node becomes a singular point connecting electron and hole pockets. We here predict that these systems have remarkable properties in the presence of magnetic fields. Most saliently, we show that the nature of the chiral anomaly depends crucially on the relative angle between the applied field and the tilt, and that an inversion-asymmetric overtilting creates an imbalance in the number of chiral modes with positive and negative slopes. The field-selective anomaly gives a novel magneto-optical resonance, providing an experimental way to detect concealed Weyl nodes.

Predicted Unusual Magnetoresponse in Type-II Weyl Semimetals

We show several distinct signatures in the magnetoresponse of type-II Weyl semimetals. The energy tilt tends to squeeze the Landau levels (LLs), and, for a type-II Weyl node, there always exists a critical angle between the B field and the tilt, at which the LL spectrum collapses, regardless of the field strength. Before the collapse, signatures also appear in the magneto-optical spectrum, including the invariable presence of intraband peaks, the absence of absorption tails, and the special anisotropic field dependence.