Topological Nodal-line States Research Articles

Coexistence of topological semimetal states in holography

We introduce a holographic model that exhibits a coexistence state of the Weyl semimetal and the topological nodal line state, providing us with a valuable tool to investigate the system’s behavior in the strong coupling regime. Nine types of bulk solutions exhibiting different IR behaviors have been identified, corresponding to nine different types of boundary states. These nine states include four distinct phases, namely the Weyl-nodal phase, the gap-nodal phase, the Weyl gap phase and the gap-gap phase, four phase boundaries, which are the Weyl-Dirac phase, the gap-Dirac phase, the Dirac-gap phase and the Dirac-nodal phase, and finally a double critical point. A phase diagram is plotted that exhibits qualitative similarity to the one obtained in the weak coupling limit. The anomalous Hall conductivity, which serves as an order parameter, and the free energy are calculated, with the latter showing the continuity of the topological phase transitions within the system. Our study highlights the similarities and differences in such a topological system between the weak and strong coupling regimes, paving the way for further experimental observations.

Ni-Intercalation-Induced Topological Nodal Line States and Superconductivity in NiTe

Based on symmetry analyses and first-principles calculations, we report the predictions of topological fermions and superconductivity in Ni-intercalated transition metal chalcogenide NiTe. The self-intercalation of Ni introduces nonsymmorphic symmetry operations, protecting a pair of Dirac points and three intersecting Dirac nodal lines near Fermi energy level (EF), which generate spin-textured surface states across EF. Moreover, the Ni-intercalation strongly enhances electron–phonon coupling in NiTe and makes it an anisotropic Bardeen–Cooper–Schrieffer superconductor with a full superconducting gap and a critical temperature Tc ∼1.5 K. The coexistence of topological surface states and superconductivity implies that NiTe is a potential material platform for exploring topological superconductivity.

An ideal Weyl nodal ring with a large drumhead surface state in the orthorhombic compound TiS2.

Topological metals or semimetals have attracted great research attention and interest in condensed matter physics and chemistry due to their exotic properties. Different from the conventional topological insulators, topological metals or semimetals are characterized by distinct topological surface states, such as a Fermi arc or a drumhead surface state, which are often used in experiments to verify the corresponding topological properties. However, the current study in this field is strongly limited in the experimental characterization because of the extreme lack of perfect material candidates with a clean band structure and clear surface states. In this work, based on theoretical calculations, we propose a new topological semimetal TiS2, which has an orthorhombic structure and exhibits excellent stability. Calculated electronic band structures reveal that there is a single Weyl nodal ring in the ky = 0 plane. A detailed symmetry analysis is provided and the corresponding surface state is calculated, which exhibits both a large energy variation of 1.5 eV and wide space distribution without and with the spin orbit coupling effect. Besides, the surface states are well separated from the bulk state. These ideal features together make TiS2 a promising nodal line semimetal for experimental investigation. In combination with the other two isostructural compounds TiSe2 and TiTe2 with similar properties, their further experimental synthesis and characterization can be highly expected and the corresponding study for the topological nodal line state can thus be greatly facilitated.

Anisotropic Nodal‐Line‐Derived Large Anomalous Hall Conductivity in ZrMnP and HfMnP (Adv. Mater. 48/2021)

Topological Nodal Line States A mirror helps light rays to make an image of an object; the mirror planes in a material's crystal structure similarly guide electrons and enhance the properties. In article number 2104126, Chandra Shekhar and co-workers demonstrate that the abundance of such mirror planes is primarily responsible for topological nodal line states at the Fermi energy in some ferromagnetic metals, leading to a large anomalous Hall conductivity.

Topological nodal line and superconductivity of highly thermally stable two-dimensional TiB4

By means of first-principles calculations, we study the electronic structures, lattice dynamics, electron-phonon coupling (EPC), and superconductivity of ${\mathrm{TiB}}_{4}$ monolayer (TBML) and ${\mathrm{TiB}}_{4}$ bilayer (TBBL). We find that both TBML and TBBL are nodal line semimetals, and the occurrences of their nodal lines are mainly due to the band inversions between B-${p}_{x}+{p}_{y}$ and B-${p}_{z}$ for TBML and between Ti-${d}_{xz}+{d}_{yz}$ and Ti-${d}_{{z}^{2}}$ for TBBL. The distortion of Ti atoms in the TBBL induces a horizontal glide mirror plane, which protects its nodal line against the spin-orbit coupling. The computed EPC constant $\ensuremath{\lambda}$ of TBML is 0.65, higher than that of the TBBL with $\ensuremath{\lambda}=0.35$. Both TBML and TBBL are identified to be phonon-mediated two-dimensional (2D) superconductors with the calculated ${T}_{c}=1.66\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and 0.82 K, respectively. The ${T}_{c}$ of the TBBL can be further enhanced to 6.43 K by applying a tensile strain of 11%. Moreover, they exhibit excellent thermal stability. The coexistence of the topological nodal-line states around the Fermi level and superconductivity in the square-lattice ${\mathrm{TiB}}_{4}$ monolayer may show more potential for realization of exotic physics.

Observation of Nodal-Line Plasmons in ZrSiS.

Nodal-line semimetals (NLSMs), a large family of new topological phases of matter with continuous linear band crossing points in the momentum space, attract considerable attention. Here, we report the direct observation of plasmons originating from topological nodal-line states in a prototypical NLSM ZrSiS by high-resolution electron energy loss spectroscopy. There exist three temperature-independent plasmons with energies ranging from the near- to the mid-infrared frequencies. With first-principles calculations of a slab model, these plasmons can be ascribed to the correlations of electrons in the bulk nodal lines and their projected surface states, dubbed nodal-line plasmons. An anomalous surface plasmon has higher excitation energy than the bulk plasmon due to the larger contribution from the nodal-line projected surface states. This work reveals the novel plasmons related to the unique nodal-line states in a NLSM.

Evidence of topological nodal lines and surface states in the centrosymmetric superconductor SnTaS2

The discovery of signatures of topological superconductivity in superconducting bulk materials with topological surface states has attracted intensive research interest recently. Utilizing angle-resolved photoemission spectroscopy and first-principles calculations, here, we demonstrate the existence of topological nodal-line states and drumheadlike surface states in centrosymmetric superconductor $\mathrm{Sn}\mathrm{Ta}{\mathrm{S}}_{2}$, which is a type-II superconductor with a critical transition temperature of about 3 K. The valence bands from Ta $5d$ orbitals and the conduction bands from Sn $5p$ orbitals cross each other, forming two nodal lines in the vicinity of the Fermi energy without the inclusion of spin-orbit coupling (SOC), protected by the spatial-inversion symmetry and time-reversal symmetry. The nodal lines are gapped out by SOC. The drumheadlike surface states, the typical characteristics in nodal-line semimetals, are quite visible near the Fermi level. Our findings indicate that $\mathrm{Sn}\mathrm{Ta}{\mathrm{S}}_{2}$ offers a promising platform for exploring the exotic properties of the topological nodal-line fermions and helps in the study of topological superconductivity.

Pnma metal hydride system LiBH: a superior topological semimetal with the coexistence of twofold and quadruple degenerate topological nodal lines

To date, a handful of topological semimetals (TMs) with multiple types of topological nodal line (TNL) states have been theoretically predicted in novel materials. However, their TNLs are often affected by many factors, such as spin–orbit coupling (SOC) effect, strain, and the extraneous bands near the band crossing points, and therefore, the TNL states cannot be easily verified by experiments. Here, by using first-principles calculations, we report that the Pnma LiBH is a potential TM with twofold and quadruple degenerate topological nodal lines. These TNLs situate very close to the Fermi level, and do not coexist with other extraneous bands. More importantly, the TNLs of this material are very robust to the effect of SOC, uniform strain, and biaxial strain. The nontrivial band structure in LiBH produces drum-head-like surface states in the (001) surface projection. Our result reveals that LiBH material is an excellent candidate to study the multiple kinds of TNLs.

Structural transition, metallization, and superconductivity in quasi-two-dimensional layered PdS2 under compression

Based on first-principles simulations and calculations, we explore the evolution of crystal structure, electronic structure and transport properties of quasi 2D layered PdS2 under uniaxial stress and hydrostatic pressure. The coordination of the Pd ions plays crucial roles in the structural transition, electronic structure and transport properties of PdS2. An interesting ferroelastic phase transition with lattice reorientation is revealed under uniaxial compressive stress, which originates from the bond reconstructions of the unusual PdS4 square-planar coordination. By contrast, the layered structure transforms to 3D cubic pyrite-type structure under hydrostatic pressure. In contrast to the experimental proposed coexistence of layered PdS2-type structure with cubic pyrite-type structure at intermediate pressure range, we predict that the compression-induced intermediate phase showing the same structural symmetry with the ambient phase, except of sharply contracted interlayer-distances. The coordination environments of the Pd ions have changed from square-planar to distorted octahedra in the intermediate phase, which results in the bandwidth broaden and orbital-selective metallization. In addition, the superconductivity comes from the cubic pyrite-type structure protected topological nodal-line states. The strong correlations between structural transition, electronic structure and transport properties in PdS2 provide a platform to study the fundamental physics of the interplay between crystal structure and transport behavior, and the competition between diverse phases.

Superconducting and Topological Properties in Centrosymmetric PbTaS2 Single Crystals

We report the superconductivity of PbTaS2 single crystals with the centrosymmetric structure. The systematic measurements of magnetization, electric transport and specific heat indicate that PbTaS2 is a weakly coupled type-II superconductor with transition temperature Tc = 2.6 K. Furthermore, the band structure calculations predicted four nodal lines near the Fermi energy with drumhead-like surface states, suggesting centrosymmetric PbTaS2 is a candidate of topological nodal line semimetals. These results demonstrate that PbTaS2 may open up another avenue for further exploring the properties of superconductivity and topological nodal-line states.

Rich novel zero-dimensional (0D), 1D, and 2D topological elements predicted in the P63/m type ternary boride HfIr3B4.

Topological semimetals, including topological nodal point semimetals (TNPSs), topological nodal line state semimetals (TNLSs), and topological nodal surface semimetals (TNSSs), featuring zero-dimensional (0D), one-dimensional (1D), and two-dimensional (2D) topological elements (TEs), respectively, have attracted widespread attention in recent years. In this work, based on first-principles calculations, we propose for the first time that three different (0D, 1D, and 2D) TEs are simultaneously present in a synthetic compound, HfIr3B4, with a P63/m type structure. In detail, HfIr3B4 hosts a Dirac point (DP) state at the K point, a TNL state in the kz = 0 plane, and a 2D TNS state in the kz = π plane, respectively. All sorts of topological elements, 0D, 1D, and 2D TEs, coexisting in the P63/m type HfIr3B4, provide an ideal platform to study the rich fermionic states and their related physical properties in this type of compound. In addition, because the 0D, 1D, and 2D TEs of HfIr3B4 are equally distributed in different energy ranges relative to the Fermi level, an approach is proposed to utilize individual TEs to build on-demand devices.

Unique topological nodal line states and associated exceptional thermoelectric power factor platform in Nb3GeTe6 monolayer and bulk.

To date, ideal topological nodal line semimetal (TNLS) candidates in high dynamically stable and high thermally stable two-dimensional (2D) materials are still extremely scarce. Herein, by performing first-principles calculations, on the one hand, we found that three-dimensional Nb3GeTe6 bulk possesses a single closed TNL in the kx = 0 plane and a fourfold TNL in the S-R direction without considering spin-orbit coupling (SOC). Under the SOC effect, a new topological signature, i.e., hourglass-like Dirac nodal line, occurs in Nb3GeTe6 bulk. On the other hand, we found that the 2D Nb3GeTe6 monolayer features a doubly degenerate TNL along surface X-S paths. Importantly, this monolayer enjoys the following advantages: (i) it has high thermal stability at room temperature and above; (ii) its TNL is nearly flat in energy and is very close to the Fermi level (EF), which provides a fantastic maximum value platform of the thermoelectric power factor around the EF; and (iii) no extraneous bands are close to the TNL, near the Fermi level. Moreover, we explore the entanglement between the topological states and thermolectric properties for the 2D Nb3GeTe6 monolayer. Our work not only reports the discovery of a novel TNL material, but also builds the link between the TNL and thermoelectric properties.

Topological nodal line states in three-dimensional ball-and-stick sonic crystals

Nodal line states in electronic systems are the extended band crossings in three-dimensional (3D) momentum space, which recently has been widely explored in classical systems in analogs. With the Dirac cones in two-dimensional (2D) hexagonal lattices, the linear degeneracy points in the stacking 2D hexagonal lattices (3D lattices) are elongated into degeneracy lines in the momentum space. In this work, we show that by coupling the stacked hexagonal lattices with the time-reversal symmetry and inversion symmetry protected, degeneracy points will form a closed nodal ring in the momentum space in the strong coupling regime. We observe flat drumhead dispersion surfaces in the band gaps, which verifies the existence of the intriguing nodal line states. Based on full-wave simulations, we show the field confinement at the truncated surface and the field enhancement due to the large density of states in flat bands. Furthermore, topological robustness of the drumhead surface states is investigated against various randomly distributed defects, such as site disorders and hopping disorders. Our work may serve as the platform of the sonic-crystal based semimetal for versatile applications like sound trapping, vibration isolation, and absorption.

First-principles study of electronic structure and superconductivity of PbTa2Se

The existences of topological nodal-line states and topological Dirac surface states in the non-centrosymmetric superconductor PbTaSe2 have been demonstrated by some recent experiments combining with first-principles simulations. Here, we predict by means of the first-principles calculations that PbTa2Se is also a phonon-mediated superconductor. The superconducting temperature Tc is estimated to be ∼7 K, greatly larger than that of PbTaSe2 (Tc ∼ 3.8 K). The main contribution to the electron-phonon coupling and superconductivity is from the in-plane vibrations of Ta atoms. After including the spin–orbit coupling, a continuous energy gap in the whole Brillouin-zone is opened.

Topological nodal line state in superconducting NaAlSi compound

NaAlSi manifests both topological band structures and superconductivity, which is promising to realize a topological superconducting state.

Alkali-metal-induced topological nodal line semimetal in layered XN2 (X = Cr, Mo, W)

Based on first principles calculations and the K·p effective model, we propose that alkali metal deposition on the surface of hexagonal XN2 (X = Cr, Mo, W) nanosheets induces topologically nontrivial phases in these systems. When spin orbit coupling (SOC) is disregarded, the electron-like conduction band from N-pz orbitals can be considered to cross the hole-like valence band from X-dz2 orbitals, thereby giving rise to a topological nodal line state in lithium-functionalized XN2 sheets (Li2MoN2 and Li2WN2). Such band crossing is protected by the existence of mirror reflection and time reversal symmetry. More interestingly, the bands cross exactly at the Fermi level, and the linear dispersion regions of such band crossings extend to as high as 0.9 eV above the crossing. For Li2CrN2, the results reveal the emergence of a Dirac cone at the Fermi level. Our calculations show that lattice compression decreases the thickness of a Li2CrN2 nanosheet, leading to phase transition to a nodal line semimetal. The evolution of the band gap of Li2XN2 at the Γ point indicates that the nontrivial topological character of Li2XN2 nanolayers is stable over a large strain range. When SOC is included, the band crossing point is gapped out giving rise to quantum spin Hall states in Li2CrN2 nanosheets, while for Li2MoN2, the SOC-induced gap at the crossing points is negligible.

Topological Nodal States in Circuit Lattice

The search for artificial structure with tunable topological properties is an interesting research direction of today's topological physics. Here, we introduce a scheme to realize topological nodal states with a three-dimensional periodic inductor-capacitor (LC) circuit lattice, where the topological nodal line state and Weyl state can be achieved by tuning the parameters of inductors and capacitors. A tight-binding-like model is derived to analyze the topological properties of the LC circuit lattice. The key characters of the topological states, such as the drumhead-like surface bands for nodal line state and the Fermi arc-like surface bands for Weyl state, are found in these systems. We also show that the Weyl points are stable with the fabrication errors of electric devices.

Dependence of topological and optical properties on surface-terminated groups in two-dimensional molybdenum dinitride and tungsten dinitride nanosheets.

Using ab initio methods, the topological and optical properties of surface-functionalized XN2 sheets (X = Mo, W) were investigated. Based on first principles calculations and the K·p effective model, the existence of topological nodal-line states in potassium-functionalized XN2 sheets (K2MoN2 and K2WN2) is reported. This study shows that a nodal line ring exists near the Fermi level in the absence of spin-orbit coupling (SOC). When SOC is included, the band-crossing points are gapped, giving rise to a new nodal ring along Γ-K. This band-crossing is protected due to the existence of mirror reflection and time-reversal symmetry. These calculations demonstrate the inclusion of electron-hole (e-h) interactions, which were further confirmed through the optical absorption of functionalized MoN2, revealing the presence of strongly bound excitons below the absorption onset where they depend strongly on the terminated surface groups. Moreover, the surface terminated groups change the energy distribution range of the exciton, which can be used to tune the absorption of infrared (IR) and visible light. Interestingly, F2MoN2 has several strongly bound excitons, with the first exciton having a binding energy of 1.35 eV, larger than the corresponding one in the transition metal dichalcogenide MoS2.

Topological nodal-line fermions in spin-orbit metal PbTaSe2.

Topological semimetals can support one-dimensional Fermi lines or zero-dimensional Weyl points in momentum space, where the valence and conduction bands touch. While the degeneracy points in Weyl semimetals are robust against any perturbation that preserves translational symmetry, nodal lines require protection by additional crystalline symmetries such as mirror reflection. Here we report, based on a systematic theoretical study and a detailed experimental characterization, the existence of topological nodal-line states in the non-centrosymmetric compound PbTaSe2 with strong spin-orbit coupling. Remarkably, the spin-orbit nodal lines in PbTaSe2 are not only protected by the reflection symmetry but also characterized by an integer topological invariant. Our detailed angle-resolved photoemission measurements, first-principles simulations and theoretical topological analysis illustrate the physical mechanism underlying the formation of the topological nodal-line states and associated surface states for the first time, thus paving the way towards exploring the exotic properties of the topological nodal-line fermions in condensed matter systems.