The prediction of a family group of two-dimensional node-line semimetals

C-Me-graphene: an ideal two-dimensional nodal line semimetal with ultrahigh Young's modulus

Nodal line semimetal (NLSM) has become a captivating medium for studying varieties of novel quantum phenomena. Here, based on first-principles calculations, we identify a square compound lattice (SCL) structure, namely C-Me-graphene, featuring a NLSM, wherein the nodal line of this configuration resides precisely at the Fermi energy without any extraneous bands in the vicinity, manifesting the quintessential characteristics of an ideal NLSM. As a corollary, utilizing symmetry analysis, we propose that nodal lines can be generated by exploiting the two-dimensional (2D) SCL of carbon. This is because the SCL not only satisfies time-reversal symmetry and inversion symmetry but also conforms to glide mirror symmetry. Additionally, this structure reveals remarkable mechanical attributes, exemplifying the highest Young's modulus within the realm of 2D materials, second only to graphene. Our work not only identifies an ideal carbon-based NLSM but also advances a scheme for crafting NLSMs, which would greatly enrich topological materials with exotic properties.

Dual Dirac Nodal Line in Nearly Freestanding Electronic Structure of β-Sn Monolayer

Two-dimensional topological insulators (2D TIs) have distinct electronic properties that make them attractive for various applications, especially in spintronics. The conductive edge states in 2D TIs are protected from disorder and perturbations and are spin-polarized, which restrict current flow to a single spin orientation. In contrast, topological nodal line semimetals (TNLSM) are distinct from TIs because of the presence of a 1D ring of degeneracy formed from two bands that cross each other along a line in the Brillouin zone. These nodal lines are protected by topology and can be destroyed only by breaking certain symmetry conditions, making them highly resilient to disorder and defects. However, 2D TNLSMs do not possess protected boundary modes, which makes their investigation challenging. There have been several theoretical predictions of 2D TNLSMs, however, experimental realizations are rare. β-Sn, a metallic allotrope of tin with a superconducting temperature of 3.72 K, may be a candidate for a topological superconductor that can host Majorana Fermions for quantum computing. In this work, single layers of α-Sn and β-Sn on a Cu(111) substrate are successfully prepared and studied using scanning tunneling microscopy, angle-resolved photoemission spectroscopy, and density functional theory calculations. The lattice and electronic structure undergo a topological transition from 2D topological insulator α-Sn to 2D TNLSM β-Sn, with two types of nodal lines coexisting in monolayer β-Sn. Such a realization of two types of nodal lines in one 2D material has not been reported to date. Moreover, we also observed an unexpected phenomenon of freestanding-like electronic structures of β-Sn/Cu(111), highlighting the potential of ultrathin β-Sn films as a platform for exploring the electronic properties of 2D TNLSM and topological superconductors, such as few-layer superconducting β-Sn in lateral contact with topological nodal line single-layer β-Sn.

Two -dimensional semimetal AlSb monolayer with multiple nodal-loops and extraordinary transport properties under uniaxial strain

Two-dimensional (2D) nodal-loop semimetal (NLSM) materials have attracted much attention for their high-speed and low-consumption transporting properties as well as their fantastic symmetry protection mechanisms. In this paper, using systematic first-principles calculations, we present an excellent NLSM candidate, a 2D AlSb monolayer, in which the conduction and valence bands cross with each other forming fascinating multiple nodal-loop (NL) states. The NLSM properties of the AlSb monolayer are protected by its glide mirror symmetry, which was confirmed using a symmetry-constrained six-band tight-binding model. The transport properties of the AlSb monolayer under in-plane uniaxial strains are also studied, based on a non-equilibrium Green's function method. It is found that both compressive and tensile strains from -10% to 10% improve the transporting properties of AlSb, and it is interesting to see that flexure configurations are energetically favored when compressive uniaxial strains are applied. Our studies not only provide a novel 2D NLSM candidate with a new symmetry protection mechanism, but also raise the novel possibility for the detection of out-of-plane flexure in 2D semimetal materials.

Two-dimensional antiferromagnetic nodal-line semimetal and quantum anomalous Hall state in the van der Waals heterostructure germanene/Mn2S2

Materials with interactions between the topology and magnetism are triggering increasing interest. We constructed a two-dimensional (2D) van der Waals heterostructure germanene/Mn₂S₂, where the germanene is a quantum spin Hall insulator and Mn₂S₂provides antiferromagnetic (AFM) interactions. In this structure, a 2D AFM nodal-line semimetal (NLSM) phase is expected without the spin-orbit coupling (SOC), which is of a high density of states around the Fermi level. The band touching rings originate from the intersection between different spin components ofporbitals of germanene. This result provides a possible 2D realization of NLSMs, which are usually realized in three-dimensional systems. When the SOC is present, a quantum anomalous Hall (QAH) state emerges with the annihilation of the band-touching rings. The nontrivial topology is determined by calculating the Chern number and Wannier charge centers. This provides an alternative platform to realize QAH states. These results could also provide the possibility of further understanding the topological states in NLSM and electronic applications.

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.

Tunable topological electronic states in the honeycomb-kagome lattices of nitrogen/oxygen-doped graphene nanomeshes

The rich and exotic electronic properties of graphene nanomeshes (GNMs) have been attracting interest due to their superiority to pristine graphene. Using first-principles calculations, we considered three graphene meshes doped with nitrogen and oxygen atoms (C10N3, C9N4 and C10O3). The electronic band structures of these GNMs in terms of the proximity of the Fermi level featured a two-dimensional (2D) honeycomb-kagome lattice with concurrent kagome and Dirac bands. The position of the Fermi level can be regulated by the doping ratio, resulting in different topological quantum states, namely topological Dirac semimetals and Dirac nodal line (DNL) semimetals. More interestingly, the adsorption of rhenium (Re) atoms in the voids of the C10N3 (Re@ C10N3) GNMs induced quantum anomalous Hall (QAH) states, as verified by the nonzero Chern numbers and chiral edge states. These GNMs offer a promising platform superior to pristine graphene for regulating multiple topological states.

Anisotropic nodal loop in NiB2 monolayer with nonsymmorphic configuration

Two-dimensional (2D) materials featuring a nodal-loop (NL) state have been drawing considerable attention in condensed matter physics and materials science. Owing to their structural polymorphism, recent high-profile metal-boride films have great advantages and the potential to realize a NL. Herein, a 2D NiB₂ monolayer with an anisotropic NL nature is proposed and investigated using first-principles calculations. We show that the NiB₂ monolayer has excellent thermal dynamics stability, suggesting the possibility of its synthesis in experiments. Remarkably, the NL with a considerable Fermi velocity is demonstrated to be protected by nonsymmorphic glide mirror symmetry, instead of the widely known horizontal mirror symmetry. Accompanied by the proper preservation of the NL, strain engineering can not only regulate the anisotropy of the NL but also give rise to a self-doping phenomenon characterized by effective modulation of the carrier type and concentration. Moreover, this NL state is robust against the correlation effect. These findings pave the way for exploring nonsymmorphic-symmetry-enabled NL nature in 2D metal-borides.

Two-dimensional Weyl points and nodal lines in pentagonal materials and their optical response

Two-dimensional pentagonal structures based on the Cairo tiling are the basis of a family of layered materials with appealing physical properties. In this work we present a theoretical study of the symmetry-based electronic and optical properties of these pentagonal materials. We provide a complete classification of the space groups that support pentagonal structures for binary and ternary systems. By means of first-principles calculations, the electronic band structures and the local spin textures in momentum space are analyzed for four examples of these materials, namely, PdSeTe, PdSeS, InP₅ and GeBi₂, all of which are dynamically stable. Our results show that pentagonal structures can be realized in chiral and achiral lattices with Weyl nodes pinned at high-symmetry points and nodal lines along the Brillouin zone boundary; these degeneracies are protected by the combined action of crystalline and time-reversal symmetries. Additionally, we computed the linear and nonlinear optical features of the proposed pentagonal materials and discuss some particular features such as the shift current, which shows an enhancement due to the presence of nodal lines and points, and their possible applications.

A novel spin-valley-coupled nodal-ring semimetal in single-layer Ta2C3

Nodal-ring semimetals with band crossing are the new type of quantum materials that have attracted considerable interest from scholars for research. In general, the spin-orbit coupling (SOC) effect opens a band gap at the Dirac point. Therefore, finding 2D nodal-ring semimetals with resistance to SOC has more challenges. Based on first-principles calculations, we propose here that the two-dimensional (2D) Ta2C3 monolayer is a novel nodal-ring semimetal. In particular, Ta2C3 forms six closed rings in the Brillouin zone (BZ) with SOC, which originates from the dxy,x2-y2 orbitals of Ta and the pz orbitals of C. The nodal-ring bands at the K point in Ta2C3 exhibits characteristics of valley splitting and spin polarization due to the breaking of inversion symmetry and SOC. The masximal spin-splitting at the K point is as large as 268.87 meV and 61.90 meV for the conduction band minimum (CBM) and valence band maximum (VBM), respectively. The massless Dirac fermions in the non-equivalent valley have the opposite Berry curvature and spin moment. Therefore, 2D Ta2C3 is novel spin-valley-coupled nodal-ring semimetal. In addition, we found interesting negative differential resistance effects when calculating its transport properties. Our results not only provide an ideal platform for studying the combination of new physical properties, spintronics and valleytronics, but also open the way for designing low-power and fast-transport electronic devices.

Pure Zirconium: Type II Nodal Line and Nodal Surface States

Type II nodal line states have novel properties, such as direction-reliant chiral anomalies and high anisotropic negative magneto-resistance. These type II nodal line states have been widely investigated. Compared to nodal line materials, there are far fewer proposed nodal surface materials, and furthermore, a very recent challenge is to find a realistic material that co-exhibits both nodal line and nodal surface states. In this manuscript, we present the study of the electronic and topological states of pure zirconium within the density functional theory. We found that pure Zr is an interesting material that rarely exhibits both the type II nodal line state (in kz = 0 plane) and nodal surface state (in kz = π plane). The nontrivial topological states are explained based on the orbital-resolved band structures. Our study shows that pure Zr can serve as a new platform to investigate the interplay between the nodal line state and the nodal surface state.

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 k_x = 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 (E_F), which provides a fantastic maximum value platform of the thermoelectric power factor around the E_F; 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.

Strain tuning of closed topological nodal lines and opposite pockets in quasi-two-dimensional α-phase FeSi2

Following topological nodal point semimetals, topological nodal line semimetals with one dimensional (1D) topological elements have recently aroused great interest worldwide in the fields of quantum chemistry and condensed matter physics. In this study, by means of first-principles, we predict that quasi-two-dimensional (2D) α-FeSi2 with a P4/mmm space group is a topological nodal line semimetal with two nodal lines close to the Fermi level, in the kz = 0 and kz = π planes. Usually, topological nodal line semimetals can be classified into type I, type II, and hybrid-type categories, each type with different physical properties. Importantly, for the first time, we find that type I, type II, and hybrid-type nodal lines can be realized in a realistic material, i.e., quasi-2D α-FeSi2, by strain switching. The realization of tunable nodal line types occurs because quasi-2D α-FeSi2 has special opposite-pocket-behaving bands around the Fermi level. The results presented herein reflect that α-FeSi2 is a valuable candidate for spintronics application by utilization of type I, type II, and hybrid-type topological nodal line semimetals in a single material tuned by mechanical strain.

Fully Spin-Polarized Nodal Loop Semimetals in Alkaline Metal Monochalcogenide Monolayers

Topological semimetals in ferromagnetic materials have attracted an enormous amount of attention due to potential applications in spintronics. Using first-principles density functional theory together with an effective lattice model, here we present a new family of topological semimetals with a fully spin-polarized nodal loop in alkaline metal monochalcogenide MX (M = Li, Na, K, Rb, or Cs; X = S, Se, or Te) monolayers. The half-metallic ferromagnetism can be established in MX monolayers, in which one nodal loop formed by two crossing bands with the same spin components is found at the Fermi energy. This nodal loop half-metal survives even when considering the spin-orbit coupling owing to the symmetry protection provided by the Mz mirror plane. The quantum anomalous Hall state and Weyl-like semimetal in this system can be also achieved by rotating the spin from the out-of-plane to the in-plane direction. The MX monolayers hosting rich topological phases thus offer an excellent platform for realizing advanced spintronic concepts.

Two-dimensional honeycomb borophene oxide: strong anisotropy and nodal loop transformation

The search for topological semimetals is mainly focused on heavy-element compounds by following the footsteps of previous research on topological insulators, with less attention on light-element materials. However, the negligible spin orbit coupling with light elements may turn out to be beneficial for realizing topological band features. Here, using first-principles calculations, we propose a new two-dimensional light-element material-the honeycomb borophene oxide (h-B2O), which has nontrivial topological properties. The proposed structure is based on the recently synthesized honeycomb borophene on an Al (111) substrate [W. Li, L. Kong, C. Chen, J. Gou, S. Sheng, W. Zhang, H. Li, L. Chen, P. Cheng and K. Wu, Sci. Bull., 2018, 63, 282-286]. The h-B2O monolayer is completely flat, unlike the oxides of graphene or silicene. We systematically investigate the structural properties of h-B2O, and find that it has very good stability and exhibits significant mechanical anisotropy. Interestingly, the electronic band structure of h-B2O hosts a nodal loop centered around the Y point in the Brillouin zone, protected by the mirror symmetry. Furthermore, under moderate lattice strain, the single nodal loop can be transformed into two loops, each penetrating through the Brillouin zone. The loops before and after the transition are characterized by different [Doublestruck Z] × [Doublestruck Z] topological indices. Our work not only predicts a new two-dimensional material with interesting physical properties, but also offers an alternative approach to search for new topological phases in 2D light-element systems.

Discovery of a novel spin-polarized nodal ring in a two-dimensional HK lattice

Nodal-ring materials with a spin-polarized feature have attracted intensive interest recently due to their exotic properties and potential applications in spintronics. However, such a type of two-dimensional (2D) lattice is rather rare and difficult to realize experimentally. Here, we identify the first 2D Honeycomb-Kagome (HK) lattice, Mn-Cyanogen, as a new single-spin nodal-ring material by using first-principles calculations. Mn-Cyanogen shows gapless and semiconducting properties in spin-up and spin-down orientations, respectively, indicating a spin-gapless semiconductor nature. Remarkably, a spin-polarized nodal ring induced by px,y/pz band inversion is captured from the 3D band structure, which is irrelevant to spin-orbit coupling. The origin of the single-spin nodal-ring can be further clarified by the effective tight-binding (TB) model. These results open a new avenue to achieving spin-polarized nodal-ring materials with promising applications in spintronic devices.

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.

Hourglass Dirac chain metal in rhenium dioxide

Nonsymmorphic symmetries, which involve fractional lattice translations, can generate exotic types of fermionic excitations in crystalline materials. Here we propose a topological phase arising from nonsymmorphic symmetries-the hourglass Dirac chain metal, and predict its realization in the rhenium dioxide. We show that ReO2 features hourglass-type dispersion in the bulk electronic structure dictated by its nonsymmorphic space group. Due to time reversal and inversion symmetries, each band has an additional two-fold degeneracy, making the neck crossing-point of the hourglass four-fold degenerate. Remarkably, close to the Fermi level, the neck crossing-point traces out a Dirac chain-a chain of connected four-fold-degenerate Dirac loops-in the momentum space. The symmetry protection, the transformation under symmetry-breaking, and the associated topological surface states of the Dirac chain are revealed. Our results open the door to an unknown class of topological matters, and provide a platform to explore their intriguing physics.

Experimental realization of two-dimensional Dirac nodal line fermions in monolayer Cu2Si

Topological nodal line semimetals, a novel quantum state of materials, possess topologically nontrivial valence and conduction bands that touch at a line near the Fermi level. The exotic band structure can lead to various novel properties, such as long-range Coulomb interaction and flat Landau levels. Recently, topological nodal lines have been observed in several bulk materials, such as PtSn₄, ZrSiS, TlTaSe₂ and PbTaSe₂. However, in two-dimensional materials, experimental research on nodal line fermions is still lacking. Here, we report the discovery of two-dimensional Dirac nodal line fermions in monolayer Cu₂Si based on combined theoretical calculations and angle-resolved photoemission spectroscopy measurements. The Dirac nodal lines in Cu₂Si form two concentric loops centred around the Γ point and are protected by mirror reflection symmetry. Our results establish Cu₂Si as a platform to study the novel physical properties in two-dimensional Dirac materials and provide opportunities to realize high-speed low-dissipation devices.