Dirac Nodes Research Articles

Two-dimensional Dirac-line semimetals resistant to strong spin–orbit coupling

Several intriguing electronic phenomena and electric properties were discovered in three-dimensional Dirac nodal line semimetals (3D-DNLSM), which are, however, easy to be perturbed under strong spin-orbit coupling (SOC). While two-dimensional (2D) layers are an emerging material category with many advantages, 2D-DNLSM against SOC is yet to be uncovered. Here, we report a 2D-DNLSM in odd-atomic-layer Bi (the brick phase, another Bi allotrope), whose robustness against SOC is protected by the little co-group C_2v \times Z^T_2, the unique protecting symmetry we found in 2D.Specially, (4n+2) valence electrons fill the electronic bands in the brick phase, so that the Dirac nodal line with fourfold degeneracy locates across the Fermi level. There are almost no other low energy states close to the Fermi level; this allows to feasibly observe the neat DNLSM-induced phenomena in transport measurements without being affected by other bands. In contrast, Other VA-group elements also form the brick phases, but their DNL states are mixed with the extra states around the Fermi level. This unprecedented category of layered materials allows for exploring nearly isolated 2DDNL states in 2D.

Dirac nodal lines and nodal loops in the topological kagome superconductor CsV3Sb5

The intertwining of charge order, superconductivity and band topology has promoted the AV$_3$Sb$_5$ (A=K, Rb, Cs) family of materials to the center of attention in condensed matter physics. Underlying those mysterious macroscopic properties such as giant anomalous Hall conductivity (AHC) and chiral charge density wave is their nontrivial band topology. While there have been numerous experimental and theoretical works investigating the nontrivial band structure and especially the van Hove singularities, the exact topological phase of this family remains to be clarified. In this work, we identify CsV$_3$Sb$_5$ as a Dirac nodal line semimetal based on the observation of multiple Dirac nodal lines and loops close to the Fermi level. Combining photoemission spectroscopy and density functional theory, we identify two groups of Dirac nodal lines along $k_z$ direction and one group of Dirac nodal loops in the A-H-L plane. These nodal loops are located at the Fermi level within the instrumental resolution limit. Importantly, our first-principle analyses indicate that these nodal loops may be a crucial source of the mysterious giant AHC observed. Our results not only provide a clear picture to categorize the band structure topology of this family of materials, but also suggest the dominant role of topological nodal loops in shaping their transport behavior.

Orbital dependent band degeneracy and edge states in single layer and AA bilayer honeycomb lattice systems with p orbital degeneracy

We have investigated band structure and edge states of single and AA bilayer honeycomb lattice systems with p orbital degeneracy which would be relevant for relatively heavy metals such as Pb and Bi. The p electrons, which are triply degenerate in the atomic limit, form quadruple degeneracy at K point in a single layer honeycomb lattice. The spin-orbit interaction splits the quartet at K point into two singlets and one doublet which are mixtures of p x , p y and p z orbitals. In the AA bilayer honeycomb lattice, the Dirac node at K point is deformed into circular line node around it. With the intermediate interlayer coupling, there is a degenerate point along Γ-K with p z character. A p x /p y type degenerate point exists under the strong interlayer coupling. This degenerate point is robust against the spin-orbit interaction and the antiferromagnetic spin arrangement only along the intralayer bonds. Without SOI, localized edge states with p x /p y or p z character are created at the zigzag edges both in the single and AA bilayer systems. In the AA bilayer system with SOI, only the p x /p y edge states exhibit the Dirac-like dispersion.

Electronic Topological Transition of 2D Boron by the Ion Exchange Reaction

We systematically investigated electronic evolutions of non-symmorphic borophene with chemical environments that were realized by the ion exchange method. Electronic structures can be characterized by the topological $Z_2$ invariant. Spectroscopic experiments and DFT calculations unveiled that a sheet of hydrogenated borophene (borophane) is the Dirac nodal loop semimetal ($Z_2=-1$), while a layered crystal of YCrB$_4$ is an insulator ($Z_2=1$). The results demonstrate the electronic topological transition by replacement of the counter atoms on the non-symmorphic borophene layer.

Two-dimensional Dirac nodal line state protected against spin-orbit coupling in MoTe monolayer

We theoretically study a kind of protected double-nodal line state in two-dimensional (2D) transition metal monochalcogenide MoTe by using symmetry analysis and first-principles calculation. Among this kind of materials, there are various intriguing electronic properties, such as the long-range Coulomb interaction. The 2D layer four-fold degeneracy nodal line material is susceptible to the perturbation of strong spin-orbit coupling (SOC). We find that the Dirac degeneracy of two low-energy bands around the S point of the Brillouin zone is robust against SOC in P21/m-MoTe. The degeneracy nodal lines are protected by nonsymmorphic screw-rotational crystal symmetry and time-reversal symmetry. Due to the protection of symmetries, the Dirac double node line states under SOC are different from the accidental band touching. Furthermore, several isolog structures of P21/m-MoTe are also discussed. We find that the fourfold degeneracy nodal line state of MoSe also is robust against SOC, similar to MoTe.

Controlling and visualizing Dirac physics in topological semimetal heterostructures.

A bulk crystal of cadmium arsenide is a three-dimensional Dirac semimetal, but, in a thin film, it can behave like a three-dimensional topological insulator. This tunability provides unique opportunities to manipulate and explore a topological insulator phase. However, an obstacle to engineering such tunability is the subtlety of transport-based discriminants for topological phases. In this work, the quantum capacitance of cadmium arsenide–based heterostructures provides two direct experimental signatures of three-dimensional topological insulator physics: an insulating three-dimensional bulk and a Landau level at zero energy that does not disperse in a magnetic field. We proceed to join our ability to see these fingerprints of the topological surface states with flexibility afforded by our epitaxial heterostructures to demonstrate a route toward controlling the energy of the Dirac nodes on each surface. These results point to new avenues for engineering topological insulators based on cadmium arsenide.

Magnetic and transport properties of the topological compound DySbTe

We report the magnetic susceptibilities $\ensuremath{\chi}(T)$, magnetization $M(H)$, transport properties $\ensuremath{\rho}(T,$ $H)$, and heat capacity ${C}_{P}(T)$ of DySbTe single crystal. DySbTe is an isostructural compound with the nonmagnetic Dirac nodal line (DNL) semimetal ZrSiS, in which Dy spins show a long-range antiferromagnetic (AFM) ordering below N\'eel temperature ${T}_{N}=7\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. A steplike magnetization curve and a 1/2 magnetization plateau were observed at 2 K for a magnetic field $H$ applied parallel to the $ab$ plane. The temperature dependence of electrical resistivity of DySbTe shows a bad-metal-like state and a peak at about 7 K associated with the AFM phase transition. The transverse magnetoresistance exhibits a crossover at a critical field from the semiclassical weak-field ${H}^{2}$ dependence to the high-field linear dependence, due to the Dirac nodal line states in DySbTe. The first principles calculations of band structures illustrate that DySbTe is a DNL semimetal and turns to be a weak topological insulator if the spin-orbit coupling is considered. DySbTe will be a new topological compound exhibiting the complex interplay between magnetism, topology, and electron correlations.

Possible topological states in two dimensional Kagome ferromagnet MnGe

From first principles computation, we propose hexagonal MnGe as a ferromagnetic Kagome candidate. As for three dimensional (3D) MnGe, the nearest, next-nearest and next-next-nearest magnetic exchange interaction parameters are respectively estimated to be 0.4 meV, 7.8 meV and 1.8 meV within the Heisenberg model while the calculated Curie temperature Tc is approaching 115 K based on Monte Carlo (MC) simulation. As for two dimensional (2D) MnGe monolayer, the spin-orbit coupling (SOC) would bring in a tiny gap (~ 1.9 meV) at the Dirac nodes and quantum anomalous Hall (QAH) effect, characterized by the chiral edge states and nonzero Chern number (C = 1). Moreover, the Dirac points can be tuned to the fermi level EF by 1.5 electrons doping or 3.3% compressive biaxial strain. Besides, possible synthetic scheme as well as the mechanical, dynamical and thermal stability are confirmed for both bulk and monolayer MnGe. Thus our work provides a promising Kagome ferromagnet to study intriguing topological physics associated with Dirac fermions towards spintronics application.

Robust structure symmetry and electronic band structure of the chiral topological semimetal CoSi against high pressure

The chiral crystal symmetry in the transition metal monosilicide CoSi plays a requisite role in protecting the multifold chiral fermion and its robustness to external pressures is very crucial for the applications of this important material. By performing high-pressure X-ray diffraction and electronic transport measurements, we reveal that the chiral structure of CoSi is robust against external pressure up to ~ 39 GPa. The ab initio calculations unveil that the external pressure negligibly influences the electric band structure in which the threefold degenerate spin-1 chiral fermion at the Γ point and the fourfold degenerate charge-2 Dirac node at the R point in the Brillouin zone could be well preserved. Through studying the high pressure effects on the unconventional topological semimetal CoSi, our results have unveiled robust multifold chiral fermions under the high pressure.

Lorentz violation in Dirac and Weyl semimetals

We propose a correspondence between the description of emergent Lorentz symmetry in condensed-matter systems and the established general effective field theory for Lorentz violation in fundamental theories of spacetime and matter. This correspondence has potential implications in both directions. We illustrate the proposal by investigating its consequences for the spectral and transport properties of Dirac and Weyl semimetals. Particular realizations of this framework give rise to Dirac nodal spectra with nodal lines and rings. We demonstrate a bulk-boundary correspondence between bulk topological invariants and drumhead surface states of these Dirac nodal semimetals. We calculate their transport coefficients in leading-order perturbation theory, thereby characterizing the unconventional electromagnetic response due to small deviations from emergent Lorentz invariance. Some prospective future applications of the correspondence are outlined.

Theory of topological superconductivity in doped IV-VI semiconductors

We theoretically study potential unconventional superconductivity in doped $AB$-type IV-VI semiconductors, based on a minimal effective model with interaction up to the next-nearest neighbors. According to the experimental implications, we focus on the spin-triplet channels and obtain the superconducting phase diagram with respect to the anisotropy of the Fermi surfaces and the interaction strength. Abundant nodal and nodeless states with different symmetry breaking appear in the phase diagram, and all the states are time-reversal invariant and topologically nontrivial. Specifically, the various nodal superconducting ground states, dubbed as the topological Dirac superconductors, are featured by Dirac nodes in the bulk and Majorana arcs on the surface; among the full-gap states, there exist a mirror-symmetry-protected second-order topological superconductor state favoring helical Majorana hinge cones, and different first-order topological superconductor states supporting four surface Majorana cones. The experimental verification of the different kinds of superconducting ground states is also discussed.

Photonic Dirac nodal-line semimetals realized by a hypercrystal

Recently, the gapless Dirac/Weyl nodal semimetals with linear dispersion and topologically protected modes degeneracy are rapidly growing frontiers of topological physics. Especially, type-I, type-II, and critical type-III nodal semimetals are discovered according to the tilt angles of the Dirac/Weyl cones. Here, by introducing hyperbolic metamaterials into one-dimensional photonic crystals, we design the "hyper-crystal" and study the photonic four-fold degenerate Dirac nodal line semimetals (DNLSs) with two types of perpendicularly polarized waves. Moreover, the flexibly controlled photonic DNLSs using the phase compensation effect of hyperbolic dispersion are studied. Our results not only demonstrate a new platform to realize the various photonic DNLSs, where the optical polarization plays the role of electron spin in electronic DNLSs, but also may pave a novel way to explore the abundant Dirac/Weyl physics in the simple classical wave systems.

Scanning tunneling microscopy and spectroscopy of NiTe2

Nickel ditelluride is an unusual member of the transition metal dichalcogenide family which has garnered interest due to potential valley spin-polarized surface states near the Fermi level and the presence of Dirac nodes in its electronic band structure. In this work, exfoliation of bulk nickel ditelluride is performed under ultra-high vacuum in order to generate a clean surface for scanning tunneling microscopy and spectroscopy at 4.8 K. Multiple features in the observed electronic density of states are observed in the vicinity of the Fermi level and compared to calculated band structures to elucidate their origins. Our results are consistent with the presence of the spin-polarized surface states, yet indicate trivial states, which are close to the Fermi level, can interfere with their potential utility in spintronic applications.

Observation of a Flat and Extended Surface State in a Topological Semimetal.

A flat band structure in momentum space is considered key for the realization of novel phenomena. A topological flat band, also known as a drumhead state, is an ideal platform to drive new exotic topological quantum phases. Using angle-resolved photoemission spectroscopy experiments, we reveal the emergence of a highly localized surface state in a topological semimetal BaAl and provide its full energy and momentum space topology. We find that the observed surface state is localized in momentum, inside a square-shaped bulk Dirac nodal loop, and in energy, leading to a flat band and a peak in the density of state. These results imply this class of materials as an experimental realization of drumhead surface states and provide an important reference for future studies of the fundamental physics of correlated quantum effects in topological materials.

Nonsymmorphic symmetry-protected band crossings in a square-net metal PtPb4

Topological semimetals with symmetry-protected band crossings have emerged as a rich landscape to explore intriguing electronic phenomena. Nonsymmorphic symmetries in particular have been shown to play an important role in protecting the crossings along a line (rather than a point) in momentum space. Here we report experimental and theoretical evidence for Dirac nodal line crossings along the Brillouin zone boundaries in PtPb4, arising from the nonsymmorphic symmetry of its crystal structure. Interestingly, while the nodal lines would remain gapless in the absence of spin–orbit coupling (SOC), the SOC, in this case, plays a detrimental role to topology by lifting the band degeneracy everywhere except at a set of isolated points. Nevertheless, the nodal line is observed to have a bandwidth much smaller than that found in density functional theory (DFT). Our findings reveal PtPb4 to be a material system with narrow crossings approximately protected by nonsymmorphic crystalline symmetries.

Dirac nodal-line semimetal zinc polynitride at high pressure

Topological semimetals exhibit novel quantum states and electronic structures with interesting Fermi surfaces. In contrast to Weyl and Dirac nodal points, nodal lines can have multiple topological configurations in momentum space that lead to new properties, such as long-range Coulomb interaction and flat Landau levels. Herein, we report the discovery of a Dirac nodal-line semimetal polynitride ${\mathrm{ZnN}}_{4}$, synthesized from the elements under high pressure via laser-heated diamond anvil cell technique. The crystal structure of ${\mathrm{ZnN}}_{4}$ has an $Ibam$ space group, and features infinite one-dimensional nitrogen chains in the pressure range of 14--130 GPa. The Dirac cone is observed midway between $S$ and $X$ points in the Brillouin zone and the Dirac nodal-line is found near the $T$ point. The measured electrical resistance of the ${\mathrm{ZnN}}_{4}$ sample shows an anomalous increase with pressure, which is attributed to the localization of valence electrons. The case of ${\mathrm{ZnN}}_{4}$ provides a unique example of Dirac nodal-line semimetal in nitrides.

Observation of multiple nodal lines in SmSbTe

Having been a ground for various topological fermionic phases, the family of ZrSiS-type 111 materials has been under experimental and theoretical investigations. Within this family of materials, the subfamily $\mathit{Ln}\mathrm{SbTe}$ $(Ln=\text{lanthanide}\phantom{\rule{4.pt}{0ex}}\text{elements})$ is gaining interest in recent times as the strong correlation effects and magnetism arising from the $4f$ electrons of the lanthanides can provide an important platform to study the link between topology, magnetism, and correlation. In this Letter, we report the systematic study of the electronic structure of $\mathrm{SmSbTe}$---a member of the $\mathit{Ln}\mathrm{SbTe}$ subfamily---by utilizing angle-resolved photoemission spectroscopy in conjunction with first-principles calculations, transport, and magnetic measurements. Our experimental results identify multiple Dirac nodes forming the nodal lines along the $\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}X$ and $Z\text{\ensuremath{-}}R$ directions in the bulk Brillouin zone (BZ) as predicted by our theoretical calculations. A surface Dirac-like state is also observed at the $\overline{X}$ point of the surface BZ. Our study highlights $\mathrm{SmSbTe}$ as a promising candidate to understand the topological electronic structure of $\mathit{Ln}\mathrm{SbTe}$ materials.

Half-negative Poisson’s ratio in graphene+ with intrinsic Dirac nodal loop

Carbon materials have attracted extensive attention in the past decades due to the rich configurations and outstanding properties. Numerous efforts have been dedicated to obtaining two-dimensional (2D) materials inheriting the properties of graphene. Herein, from first-principles calculations, we report a novel 2D carbon allotrope with a hybridized sp2-sp3 network, i.e., “grapheneplus (graphene+)”. Both the Dirac properties of graphene and the negative Poisson’s ratio (NPR) of penta-graphene are inherited in graphene+. The NPR behavior manifests itself along the out-of-plane direction only when it is stretched, namely, out-of-plane half-auxeticity, which is different from the already known NPR behavior. Besides, the Dirac nodes form a nodal loop that remains robust regardless of the application of uniaxial/biaxial strain and electric field. In addition, graphene+ is found to be energetically more stable than penta-graphene, which promises a potentially easier synthesis. In short, graphene+ could act as a strong competitor to graphene.

Two-Dimensional Dirac Nodal Line Carbon Nitride to Anchor Single-Atom Catalyst for Oxygen Reduction Reaction.

Two-dimensional carbon nitride (2DCN) materials have emerged as an important class of 2D materials beyond graphene. However, 2DCN materials with nodal-line semimetal characteristic are rarely reported. In this work, a new nodal-line semimetal 2DCN with the stoichiometry C4 N4 is designed by using density functional theory (DFT) calculations and its application to anchor single-atom catalysts (SACs) for the oxygen reduction reaction (ORR) is investigated. C4 N4 is a planar covalent network (sp2 hybridization) with regular holes formed by the four N atoms, which is dynamically, thermodynamically, and mechanically stable. The nodal line is contributed by the pz orbitals of C and px/y orbitals of N atoms. C4 N4 shows an anisotropic Fermi velocity and high electron mobility. Because of its porous structure, C4 N4 can anchor heteroatoms as SACs for electrocatalysis. C4 N4 anchored with Fe or Co is shown to be highly active for the ORR with a rather high half-wave potential of around 0.90 V, which is higher than those of SACs on other carbon nitrides. These findings may provide a new strategy to design novel substrates for SACs.

Thickness-dependent topological phase transition and Rashba-like preformed topological surface states of α-Sn(001) thin films on InSb(001)

Topological materials, possessing spin-momentum locked topological surface states (TSS), have attracted much interest due to their potential applications in spintronics. \ensuremath{\alpha}-phase Sn (\ensuremath{\alpha}-Sn), being one of them, displays enriched topological phases via band-gap engineering through a strain or confinement effect. In this work, we investigated the band evolution of in-plane compressively strained \ensuremath{\alpha}-Sn(001) thin films on InSb(001) in a wide range of thickness from 3 bilayers (BL) to 370 BL by combining angle-resolved photoemission spectra and first-principles calculations. Gapped surface states evolved to a linearly dispersive TSS at a critical thickness of 6 BL, indicating that the system undergoes a phase transition from topologically trivial to nontrivial. For films thicker than 30 BL, additional Rashba-like surface states (RSS) were identified. These RSS served as preformed TSS in another strain-induced topological phase transition. In thick films, 370-BL \ensuremath{\alpha}-Sn(001), so as to preclude the confinement effect in thin films, our results were consistent with a Dirac semimetal phase with Dirac nodes located along $\mathrm{\ensuremath{\Gamma}}--Z$. This thickness-dependent band-structure study deepens our understanding of topological phase transitions and the evolution of Dirac states. Furthermore, the coexistence of TSS and RSS in a Dirac semimetal \ensuremath{\alpha}-Sn might significantly enhance the potential for spintronic applications.