Two-dimensional Topological Materials Research Articles

From topological nodal-line semimetals to quantum spin Hall insulators in tetragonal SnX monolayers (X=F, Cl, Br, I)

Abstract Two-dimensional (2D) topological materials have recently garnered significant interest due to their profound physical properties and promising applications for future quantum nanoelectronics. Achieving various topological states within one type of materials is, however, seldom reported. Based on first-principles calculations and tight-binding models, we investigate topological electronic states in a novel family of 2D halogenated tetragonal stanene (T-SnX, X = F, Cl, Br, I). All the four monolayers are found being unusual topological nodal-line semimetals (NLSs), protected by a glide mirror symmetry. When spin-orbit coupling (SOC) is turned on, T-SnF and TSnCl are still ascertained as topological NLSs due to the remaining band inversion, primarily composed of Sn pxy orbitals, while T-SnBr and T-SnI become quantum spin Hall insulators. The phase transition is ascribed to moving up in energy of Sn s orbitals and increasing of SOC strengths. The topology origin in the materials is uniformly rationalized through elementary band representations. The robust and diverse topological states found in the 2D T-SnX monolayers position them as an excellent material platform for the development of innovative topological electronics.

Two-dimensional quadratic Weyl points, nodal loops, and spin-orbit Dirac points in PtS, PtSe, and PtTe monolayers.

Topological quasiparticles have garnered significant research attention in condensed matter physics. However, they are exceedingly rare in two-dimensional systems, particularly those hosting unconventional topological quasiparticles. In this work, employing first-principles calculations and symmetry analysis, we demonstrate that PtS, PtSe, and PtTe monolayers serve as high-quality two-dimensional topological semimetal materials. These materials exhibit multiple types of topological quasiparticles around the Fermi level in the absence of spin-orbit coupling, such as conventional linear Weyl points and unconventional quadratic Weyl points in the PtS monolayer, as well as nodal loops in PtSe and PtTe monolayers. When spin-orbit coupling (SOC) is introduced, a tiny gap opens, transforming the systems into quantum spin hall insulators. Simultaneously, three spin-orbit Dirac points, robust against SOC, appear at the X, Y, and M points. We illustrate the symmetry protection, low-energy effective model, and edge states of these topological states. Our work provides an excellent material platform for studying novel two-dimensional topological quasiparticles and topological insulators.

Growth and characterization of Bi(110)/CrTe2 heterostructures: Exploring interplay between magnetism and topology

The interplay between topology and magnetism is vital for realizing exotic quantum phenomena, significant examples including quantum anomalous Hall effect, axion insulators, and high-order topological states. These states host great potential for future applications in high-speed and low-consumption electronic devices. Despite being extensively investigated, practical platforms are still scarce. In this work, with molecular beam epitaxy (MBE), we provide the first experimental report on high-quality Bi(110)/CrTe2 magnetic heterostructure. By employing in-situ high-resolution scanning tunneling microscopy, we are able to examine the interaction between magnetism and topology. There is a potential edge state at an energy level above the Fermi level, but no edge states observed near the Fermi level The absence of high-order topological corner states near E F highlights the importance of lattice matching and interface engineering in designing high-order topological states. Our study provides key insights into the interplay between two-dimensional magnetic and topological materials and offers an important dimension for engineering magnetic topological states.

Two-dimensional antiferromagnetic type-II Weyl fermions in monolayer ZnSb and type-I Weyl fermions and topological insulator in bilayer ZnSb

Two-dimensional (2D) topological materials have attracted extensive attention due to the discovery of graphene. In addition, their counterparts in magnetic materials are also a fascinating subject of research. However, it remains challenging to find 2D materials that have the desired topological phases. Here, based on first-principles calculations, we investigate the properties of a newly synthesized 2D material, monolayer/bilayer (ML/BL) ZnSb. We find that the ML ZnSb is antiferromagnetic (AFM) and that its magnetism can be changed by an external strain, from AFM to ferromagnetic. Without any external strain, the ML ZnSb can be considered as an almost ideal 2D type-II Weyl material with a pair of type-II Weyl fermions exactly located at the Fermi level. When taking into account spin–orbit coupling (SOC), a gap can be introduced at the Weyl point. We further show that by applying a biaxial strain to the BL ZnSb material, it is possible to open different gaps at the Weyl points, leading to a topological phase transition without breaking symmetry. When SOC is included, these Weyl points become topological insulators with a large bandgap, where the helical edge states can be identified. Overall, this study provides an example of a material that can be used to explore the physical consequences of 2D topological phases and provide a promising candidate for future research in this field.

Modulating intrinsic anomalous Hall effect in Fe3GeTe2 monolayer via strain engineering

The Fe3GeTe2 monolayer, a ferromagnetic topological candidate with a high Curie temperature of 130 K, has recently garnered considerable interest. We investigated the impact of strain on the electronic, magnetic, and topological properties of the Fe3GeTe2 monolayer using density functional theory calculations. Our results showed that the Fe3GeTe2 monolayer was an itinerant ferromagnet with a high spin polarization of 69.93% and out-of-plane easy magnetization. Tensile strain had no profound impact on the anomalous Hall conductivity (AHC). However, when applying a compressive biaxial strain of −3%, the AHC at the Fermi level was dramatically enhanced to 1.62 e2/h, which is much larger than that of the bulk (0.7 e2/h) and bilayer (1.5 e2/h) material. The large AHC nearly coincided with the singularity of the energy bands near the M point. Our results highlight the potential of using strain engineering to control and optimize the properties of two-dimensional topological materials.

Quantum computation at the edge of a disordered Kitaev honeycomb lattice

We analyze propagation of quantum information along chiral Majorana edge states in two-dimensional topological materials. The use of edge states may facilitate the braiding operation, an important ingredient in topological quantum computations. For the edge of the Kitaev honeycomb model in a topological phase, we discuss how the edge states can participate in quantum-information processing, and consider a two-qubit logic gate between distant external qubits coupled to the edge. Here we analyze the influence of disorder and noise on properties of the edge states and quantum-gate fidelity. We find that realistically weak disorder does not prevent one from implementation of a high-fidelity operation via the edge.

Giant room-temperature nonlinearities in a monolayer Janus topological semiconductor

Nonlinear optical materials possess wide applications, ranging from terahertz and mid-infrared detection to energy harvesting. Recently, the correlations between nonlinear optical responses and certain topological properties, such as the Berry curvature and the quantum metric tensor, have attracted considerable interest. Here, we report giant room-temperature nonlinearities in non-centrosymmetric two-dimensional topological materials—the Janus transition metal dichalcogenides in the 1 T’ phase, synthesized by an advanced atomic-layer substitution method. High harmonic generation, terahertz emission spectroscopy, and second harmonic generation measurements consistently show orders-of-the-magnitude enhancement in terahertz-frequency nonlinearities in 1 T’ MoSSe (e.g., > 50 times higher than 2H MoS2 for 18th order harmonic generation; > 20 times higher than 2H MoS2 for terahertz emission). We link this giant nonlinear optical response to topological band mixing and strong inversion symmetry breaking due to the Janus structure. Our work defines general protocols for designing materials with large nonlinearities and heralds the applications of topological materials in optoelectronics down to the monolayer limit.

High Chern number van der Waals magnetic topological multilayers MnBi2Te4/hBN

Chern insulators are two-dimensional magnetic topological materials that conduct electricity along their edges via the one-dimensional chiral modes. The number of these modes is a topological invariant called the first Chern number C that defines the quantized Hall conductance as Sxy = Ce2/h. Increasing C is pivotal for the realization of low-power-consumption topological electronics, but there has been no clear-cut solution to this problem so far, with the majority of existing Chern insulators showing C = 1. Here, by using state-of-the-art theoretical methods, we propose an efficient approach for the realization of the high-C state in MnBi2Te4/hBN van der Waals multilayer heterostructures. We show that a stack of n MnBi2Te4 films with C = 1 intercalated by hBN monolayers gives rise to a high Chern number state with C = n, characterized by n chiral edge modes. This state can be achieved both under the external magnetic field and without it, both cases leading to the quantized Hall conductance Sxy = Ce2/h. Our results, therefore, pave the way to practical high-C quantized Hall systems.

Schoch Effect in Topological Phononic Crystals

This article investigates the Schoch negative displacement phenomenon at the interface between two-dimensional topological acoustic materials and traditional materials. The results show that a negative Schoch displacement occurs at the frequency of the Dirac point in the phononic crystal. At this point, the reciprocal of the effective bulk modulus of the phononic crystal tends to zero, making it an acoustic metamaterial with a refractive index close to zero. At the same time, the maximum value of the effective impedance of the phononic crystal and the real part of the reflection coefficient undergo a 2π phase transition at the corresponding frequency. When the phononic crystal is spliced into a topological supercell structure, a larger negative Schoch displacement is observed. This study demonstrates that topological phononic crystals can achieve negative Schoch displacement. The negative Schoch displacement realized in this article provides a new theoretical reference for designing acoustic devices based on interface waves.

Topological tailoring-induced Dirac cone in ultrathin niobium diboride nanosheets for electrocatalytic sulfur reduction reaction

Two-dimensional topological Dirac materials with novel electronic structures have attracted increasing attention in materials science, where the Weyl fermions can emerge in topological insulating multilayers with unusual quantum transport properties. However, their practical catalytic performance is seldom reported. Here, we design ultrathin niobium diboride (NbB2) nanosheets by utilizing topological electronic states to power catalytic activity. In the lithium-sulfur batteries, the stable topological surface states of NbB2 with high carrier mobility are a recipe for high-activity catalysts to spur the sluggish electrocatalytic sulfur reduction reaction, which originates from the Dirac cone electronic structure. Most remarkably, it delivers stable capacities with 1500 cycles under the 3.6 mA/cm2 (3 C), which is far superior to most electrode catalysts. The proposed functional Weyl and/or Dirac materials would have broad applications in other related fields such as the hydrogen evolution reaction, nitrogen reduction reaction, and energy conversion.

Amorphous Kane-Mele model in disordered hyperuniform two-dimensional networks

As a prototype system of the quantum spin Hall effect, the Kane-Mele model which was proposed initially in graphene promotes the search for two-dimensional topological materials of hexagonal lattices. Here we generalize the Kane-Mele model to exotic amorphous systems which possess a remarkable structural property called hyperuniformity. We show that, in general, the Quantum spin Hall state still survives in disordered hyperuniform lattices that are constructed by structural transformation involving Stone-Wales defects. However, compared to that in the honeycomb lattice, the gapped topological region in the phase diagram shrinks and the size of the corresponding topological gap decreases remarkably in disordered hyperuniform lattices. By introducing random vacancies in either perfectly ordered or disordered hyperuniform lattices, we further show that the degradation or destruction of hyperuniformity is detrimental to the existence of gapped topological states. We therefore propose that the hyperuniform metric of an amorphous lattice, which quantifies the extent of disordered hyperuniformity, reflects its ability to preserve topological states. Our finding not only establishes the possible underlying link between electronic topology and disordered hyperuniformity but also provides useful guidance for the seeking of topological materials in amorphous states of matter.

Two-dimensional anisotropic Dirac materials PtN4C2 and Pt2N8C6 with quantum spin and valley Hall effects

We propose two different two-dimensional topological Dirac materials, planar ${\mathrm{PtN}}_{4}{\mathrm{C}}_{2}$ and ${\mathrm{Pt}}_{2}{\mathrm{N}}_{8}{\mathrm{C}}_{6}$, which exhibit graphenelike electronic structures with linearly dispersive Dirac-cone states exactly at the Fermi level. Moreover, the Dirac cone is anisotropic, resulting in anisotropic Fermi velocities and making it possible to realize orientation-dependent quantum devices. Using first-principles electronic structure calculations, we have systemically studied the structural, electronic, and topological properties. We find that spin-orbit coupling opens a sizable topological band gap so that the materials can be classified as quantum spin Hall insulators as well as quantum valley Hall insulators. Helical edge states that reside in the insulating band gap connecting the bulk conduction and valence bands are observed. Our work not only expands the Dirac-cone material family, but also provides another avenue to search for more two-dimensional topological quantum spin and valley Hall insulators.

Optical N -insulators: Topological obstructions to optical Wannier functions in the atomistic susceptibility tensor

A powerful result of topological band theory is that nontrivial phases manifest obstructions to constructing localized Wannier functions. In Chern insulators, it is impossible to construct Wannier functions that respect translational symmetry in both directions. Similarly, Wannier functions that respect time-reversal symmetry cannot be formed in quantum spin Hall insulators. This molecular orbital interpretation of topology has been enlightening and was recently extended to topological crystalline insulators which include obstructions tied to space group symmetries. In this article, we introduce a new class of two-dimensional topological materials known as optical $N$-insulators that possess obstructions to constructing localized molecular polarizabilities. The optical $N$-invariant $N\in\mathbb{Z}$ is the winding number of the atomistic susceptibility tensor $\chi$ and counts the number of singularities in the electromagnetic linear response theory. We decipher these singularities by analyzing the optical band structure of the material -- the eigenvectors of the susceptibility tensor -- which constitutes the collection of optical Bloch functions. The localized basis of these eigenvectors are optical Wannier functions which represent the molecular polarizabilities at different lattice sites. We prove that in a nontrivial optical phase $N\neq 0$, such a localized polarization basis is impossible to construct. Utilizing the mathematical machinery of $K$-theory, these optical $N$-phases are refined further to account for the underlying crystalline symmetries of the material, generating a classification of the topological electromagnetic phase of matter.

Machine learning for materials discovery: Two-dimensional topological insulators

One of the main goals and challenges of materials discovery is to find the best candidates for each interest property or application. Machine learning rises in this context to efficiently optimize this search, exploring the immense materials space, consisting of simultaneously the atomic, compositional, and structural spaces. Topological insulators, presenting symmetry-protected metallic edge states, are a promising class of materials for different applications. However, further development is limited by the scarcity of viable candidates. Here we present and discuss machine learning–accelerated strategies for searching the materials space for two-dimensional topological materials. We show the importance of detailed investigations of each machine learning component, leading to different results. Using recently created databases containing thousands of ab initio calculations of 2D materials, we train machine learning models capable of determining the electronic topology of materials, with an accuracy of over 90%. We can then generate and screen thousands of novel materials, efficiently predicting their topological character without the need for a priori structural knowledge. We discover 56 non-trivial materials, of which 17 are novel insulating candidates for further investigation, for which we corroborate their topological properties with density functional theory calculations. This strategy is 10× more efficient than the trial-and-error approach while a few orders of magnitude faster and is a proof of concept for guiding improved materials discovery search strategies.

Tuning topological phases and electronic properties of monolayer ternary transition metal chalcogenides (ABX4, A/B = Zr, Hf, or Ti; X = S, Se, or Te)

Topological materials are very promising materials for technological applications ranging from spintronics to quantum computation. Here, based on first-principles calculations, we predict a family of two-dimensional (2D) topological materials in nine ternary transition metal chalcogenides (TTMCs) ABX4, where A/B = Zr, Hf, or Ti and X = S, Se, or Te. A total of three compounds (ZrTiTe4, HfZrTe4, and HfTiTe4) are identified to be nontrivial within a hybrid functional calculation. The nontrivial phase originated from the p-d band inversion at the Г point with spin–orbit coupling. The structural stability of these monolayers is confirmed by phonon spectrum analysis, showing no negative phonon frequencies. The diversity of TTMCs will open a wide possibility for tuning the bandgap and will provide a variety of opportunities for 2D and topological materials research.

Observation of Photonic Antichiral Edge States.

Chiral edge states are a hallmark feature of two-dimensional topological materials. Such states must propagate along the edges of the bulk either clockwise or counterclockwise, and thus produce oppositely propagating edge states along the two parallel edges of a strip sample. However, recent theories have predicted a counterintuitive picture, where the two edge states at the two parallel strip edges can propagate in the same direction; these anomalous topological edge states are named as antichiral edge states. Here, we report the experimental observation of antichiral edge states in a gyromagnetic photonic crystal. The crystal consists of gyromagnetic cylinders in a honeycomb lattice, with the two triangular sublattices magnetically biased in opposite directions. With microwave measurement, unique properties of antichiral edge states have been observed directly, which include tilted dispersion, chiral-like robust propagation in samples with certain shapes, and 100% scattering into backward bulk states at certain terminations. These results extend and supplement the current understanding of chiral edge states.

Quantum-confinement-induced periodic surface states in two-dimensional metal-organic frameworks

Recently, a series of single-layer metal–organic frameworks (MOFs) was theoretically predicted to be two-dimensional organic topological materials. However, the experimental evidence of their nontrivial topological states has not been found. Here, combining the use of angle-resolved photoemission spectroscopy and scanning tunneling microscopy, we report the electronic structure studies on a single-layer Cu-coordinated 2,4,6-tri(4-pyridyl)-1,3,5-triazine (Cu-T4PT) MOF supported by a Cu(111) substrate and identify periodic surface states with the period of the Cu-T4PT reciprocal lattice. These periodic surface states, which have identical features to the Cu(111) Shockley surface states, can be attributed to the quantum confinement of the surface states of the underlying Cu(111) substrate by the network lattices of the Cu-T4PT MOF. Our work indicates that the surface states of the metal substrate can be tailored in a controlled manner by the network structures of MOFs with different periodic lattices. The lack of intrinsic bands and the possible topological properties of the single-layer Cu-T4PT MOF may be attributed to the strong electronic coupling between the Cu-T4PT MOF and the Cu(111) substrates. In order to exploit organic topological materials predicted in MOFs, it is necessary to grow them on weak van der Waals interaction substrates in the future.

Computational search for magnetic and non-magnetic 2D topological materials using unified spin\u2013orbit spillage screening

Two-dimensional topological materials (2D TMs) have a variety of properties that make them attractive for applications including spintronics and quantum computation. However, there are only a few such experimentally known materials. To help discover new 2D TMs, we develop a unified and computationally inexpensive approach to identify magnetic and non-magnetic 2D TMs, including gapped and semi-metallic topological classifications, in a high-throughput way using density functional theory-based spin–orbit spillage, Wannier-interpolation, and related techniques. We first compute the spin–orbit spillage for the ~1000 2D materials in the JARVIS-DFT dataset, resulting in 122 materials with high-spillage values. Then, we use Wannier-interpolation to carry-out Z2, Chern-number, anomalous Hall conductivity, Curie temperature, and edge state calculations to further support the predictions. We identify various topologically non-trivial classes such as quantum spin-Hall insulators, quantum anomalous-Hall insulators, and semimetals. For a few predicted materials, we run G0W0+SOC and DFT+U calculations. We find that as we introduce many-body effects, only a few materials retain non-trivial band-topology, suggesting the importance of high-level density functional theory (DFT) methods in predicting 2D topological materials. However, as an initial step, the automated spillage screening and Wannier-approach provide useful predictions for finding new topological materials and to narrow down candidates for experimental synthesis and characterization.

Emergence of d -orbital magnetic Dirac fermions in a MoS2 monolayer with squared pentagon structure

We find that an atomically thin $\mathrm{Mo}{\mathrm{S}}_{2}$ monolayer with squared pentagon structure motifs, distinct from all the conventional $\mathrm{Mo}{\mathrm{S}}_{2}$ monolayers, has unique electronic band structures with Dirac cones comprising $d$ orbitals. The squared pentagon $\mathrm{Mo}{\mathrm{S}}_{2}$ monolayer is an intrinsic ferromagnet with a high Curie temperature and behaves as a half semimetal, which possesses spin-polarized Dirac fermions around the Fermi level. Moreover, when spin-orbit couplings are included, the magnetic Dirac band is gapped, hosting quantized conductance channels. This work provides insights in achieving two-dimensional topological materials with unique atomistic patterns.

Metal-organic frameworks: possible new two-dimensional magnetic and topological materials.

Finding new two-dimensional (2D) materials with novel quantum properties is highly desirable for technological innovations. In this work, we studied a series of metal-organic frameworks (MOFs) with different metal cores and discovered various attractive properties, such as room-temperature magnetic ordering, strong perpendicular magnetic anisotropy, huge topological band gap (>200 meV), and excellent spin-filtering performance. As many MOFs have been successfully synthesized in experiments, our results suggest realistic new 2D functional materials for the design of spintronic nanodevices.